hypothesis contrary to fact fallacy examples

Think Critically, Live Honestly

Hypothesis Contrary To Fact

Imagine arguing about a reality that never happened, asserting cause and effect from a non-existent event, and presenting it as a fact - that's the intriguing world of this logical fallacy. It's like building castles in the air and then claiming they can actually house people, a captivating yet deceptive illusion that can mislead by creating a false sense of understanding or control over a situation.

Cause and Effect
Misrepresentation

Definition of Hypothesis Contrary To Fact

Hypothesis Contrary To Fact, also known as "counterfactual fallacy" or "speculative fallacy," is a type of logical fallacy where a statement or argument is made based on a hypothetical situation that is presented as fact, but is actually contrary to what is known or proven to be true. This fallacy involves making a claim about a past event that didn't occur, and then asserting a cause and effect relationship based on that non-existent event. It is fallacious because it's impossible to definitively know the outcome of an event that did not happen. This fallacy can mislead or manipulate by creating an illusion of understanding or control over a situation, when in fact the hypothetical scenario and its supposed consequences are purely speculative. It's important to note that while hypothetical scenarios can be useful for exploring possibilities, they become fallacious when presented as factual or inevitable outcomes.

In Depth Explanation

The Hypothesis Contrary to Fact, also known as the Counterfactual Fallacy, is a logical error that occurs when an argument is built on a premise that is not true, but is presented as if it were. This fallacy involves making a claim about what would have happened in the past if a certain event had or hadn't occurred, even though there's no way to verify this claim because it's based on a hypothetical situation, not a factual one. Let's imagine a simple scenario to illustrate this fallacy. Suppose you're playing a game of chess and you lose. You then say, "If I had moved my queen instead of my pawn, I would have won the game." This statement is a hypothesis contrary to fact. You're making a claim about an alternate reality that didn't happen, and there's no way to prove whether your claim is true or false because we can't go back in time to see what would have happened if you had made a different move. The logical structure of this fallacy typically involves two statements: one that sets up a hypothetical situation ("If I had moved my queen...") and one that makes a claim about what would have happened in this situation ("...I would have won the game"). The problem is that the first statement is not true—you didn't move your queen—so any claim based on this statement is inherently flawed. This fallacy can be particularly misleading in abstract reasoning because it often sounds plausible. After all, it's easy to imagine how things might have turned out differently if we had made different choices. However, this kind of reasoning is purely speculative and doesn't provide a solid basis for an argument. The Hypothesis Contrary to Fact can have a significant impact on rational discourse because it can be used to deflect responsibility, justify poor decisions, or manipulate others. For example, a person might use this fallacy to argue that they would have succeeded if not for some external factor, thereby shifting the blame for their failure onto something beyond their control. Alternatively, a person might use this fallacy to convince others to take a certain course of action based on what they claim would have happened in a hypothetical situation. In conclusion, while it's natural to speculate about what might have been, it's important to recognize that these speculations are not facts and should not be treated as such in logical arguments. The Hypothesis Contrary to Fact is a fallacy that can lead us astray in our thinking and decision-making, so it's crucial to be aware of it and to challenge it when we encounter it.

Real World Examples

1. Sports Scenario: Imagine a basketball fan saying, "If Michael Jordan had not retired in 1993, the Chicago Bulls would have won eight consecutive NBA championships instead of six." This statement is an example of a hypothesis contrary to fact. It assumes a hypothetical scenario where Jordan didn't retire and then predicts an outcome based on that assumption. However, there's no way to prove this hypothesis because it's impossible to know how the Bulls would have performed had Jordan not retired. 2. Historical Event: A common example is the assertion, "If the United States had not entered World War II, the Allies would have lost." This is a hypothesis contrary to fact because it's based on a hypothetical scenario that didn't occur. While it's possible to speculate, there's no way to definitively know what would have happened had the U.S. not entered the war. 3. Everyday Scenario: Suppose a student who failed an exam says, "If I had just studied one more hour, I would have passed the test." This is an example of a hypothesis contrary to fact. The student is assuming that an extra hour of study would have made the difference between passing and failing, but there's no way to prove this. It's possible that the student might still have failed even with an additional hour of study, or they might have passed even without it. This statement is based on a hypothetical scenario, not on what actually happened.

Countermeasures

Addressing the logical fallacy of Hypothesis Contrary To Fact can be achieved through a few clear and concise steps. Firstly, it's important to encourage critical thinking. This involves questioning the basis of the hypothesis and examining the evidence that supports it. If the hypothesis is based on an event or circumstance that did not actually occur, it's crucial to point this out and discuss the implications of this. Secondly, promoting evidence-based reasoning is key. This means focusing on what we know to be true and what can be proven, rather than what might have been. If a hypothesis is based on a counterfactual, it's essential to redirect the conversation towards the facts at hand. Thirdly, fostering open-mindedness can help counteract this fallacy. This involves being open to alternative hypotheses and not being wedded to a particular outcome. It's important to be willing to change one's mind in the face of new evidence. Lastly, it's beneficial to cultivate a culture of intellectual humility. This means acknowledging the limits of our knowledge and being open to the possibility that we might be wrong. If a hypothesis is based on a counterfactual, it's important to acknowledge this and be willing to revise our views accordingly. In conclusion, countering the Hypothesis Contrary To Fact fallacy involves promoting critical thinking, evidence-based reasoning, open-mindedness, and intellectual humility. By fostering these qualities, we can help ensure that our hypotheses are grounded in fact, rather than in what might have been.

Thought Provoking Questions

1. Can you identify a time when you made a claim about a past event that didn't occur and asserted a cause and effect relationship based on that non-existent event? How did this impact your understanding or control over the situation? 2. Have you ever presented a hypothetical scenario as a factual or inevitable outcome? How did this affect your decision-making process and the decisions of those around you? 3. Can you recall a situation where you were misled by a 'Hypothesis Contrary To Fact' fallacy? How did this influence your perception of the situation and the actions you took? 4. How do you differentiate between useful hypothetical scenarios for exploring possibilities and those that are fallacious because they are presented as factual or inevitable outcomes? How has this skill affected your critical thinking and decision-making abilities?

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Logical Fallacies | Definition, Types, List & Examples

Logical Fallacies | Definition, Types, List & Examples

Published on April 20, 2023 by Kassiani Nikolopoulou . Revised on October 9, 2023.

A logical fallacy is an argument that may sound convincing or true but is actually flawed. Logical fallacies are leaps of logic that lead us to an unsupported conclusion. People may commit a logical fallacy unintentionally, due to poor reasoning, or intentionally, in order to manipulate others.

Because logical fallacies can be deceptive, it is important to be able to spot them in your own argumentation and that of others.

Logical fallacy list (free download), what is a logical fallacy, types of logical fallacies, what are common logical fallacies, logical fallacy examples, other interesting articles, frequently asked questions about logical fallacies.

There are many logical fallacies. You can download an overview of the most common logical fallacies by clicking the blue button.

Logical fallacy list (Google Docs)

A logical fallacy is an error in reasoning that occurs when invalid arguments or irrelevant points are introduced without any evidence to support them. People often resort to logical fallacies when their goal is to persuade others. Because fallacies appear to be correct even though they are not, people can be tricked into accepting them.

The majority of logical fallacies involve arguments—in other words, one or more statements (called the premise ) and a conclusion . The premise is offered in support of the claim being made, which is the conclusion.

There are two types of mistakes that can occur in arguments:

A factual error in the premises . Here, the mistake is not one of logic. A premise can be proven or disproven with facts. For example, If you counted 13 people in the room when there were 14, then you made a factual mistake.
The premises fail to logically support the conclusion . A logical fallacy is usually a mistake of this type. In the example above, the students never proved that English 101 was itself a useless course—they merely “begged the question” and moved on to the next part of their argument, skipping the most important part.

In other words, a logical fallacy violates the principles of critical thinking because the premises do not sufficiently support the conclusion, while a factual error involves being wrong about the facts.

There are several ways to label and classify fallacies, such as according to the psychological reasons that lead people to use them or according to similarity in their form. Broadly speaking, there are two main types of logical fallacy, depending on what kind of reasoning error the argument contains:

Informal logical fallacies

Formal logical fallacies.

An informal logical fallacy occurs when there is an error in the content of an argument (i.e., it is based on irrelevant or false premises).

Informal fallacies can be further subdivided into groups according to similarity, such as relevance (informal fallacies that raise an irrelevant point) or ambiguity (informal fallacies that use ambiguous words or phrases, the meanings of which change in the course of discussion).

“ Some philosophers argue that all acts are selfish . Even if you strive to serve others, you are still acting selfishly because your act is just to satisfy your desire to serve others.”

A formal logical fallacy occurs when there is an error in the logical structure of an argument.

Premise 2: The citizens of New York know that Spider-Man saved their city.

Conclusion: The citizens of New York know that Peter Parker saved their city.

This argument is invalid, because even though Spider-Man is in fact Peter Parker, the citizens of New York don’t necessarily know Spider-Man’s true identity and therefore don’t necessarily know that Peter Parker saved their city.

A logical fallacy may arise in any form of communication, ranging from debates to writing, but it may also crop up in our own internal reasoning. Here are some examples of common fallacies that you may encounter in the media, in essays, and in everyday discussions.

Red herring logical fallacy

The red herring fallacy is the deliberate attempt to mislead and distract an audience by bringing up an unrelated issue to falsely oppose the issue at hand. Essentially, it is an attempt to change the subject and divert attention elsewhere.

Bandwagon logical fallacy

The bandwagon logical fallacy (or ad populum fallacy ) occurs when we base the validity of our argument on how many people believe or do the same thing as we do. In other words, we claim that something must be true simply because it is popular.

This fallacy can easily go unnoticed in everyday conversations because the argument may sound reasonable at first. However, it doesn’t factor in whether or not “everyone” who claims x is in fact qualified to do so.

Straw man logical fallacy

The straw man logical fallacy is the distortion of an opponent’s argument to make it easier to refute. By exaggerating or simplifying someone’s position, one can easily attack a weak version of it and ignore their real argument.

Person 2: “So you are fine with children taking ecstasy and LSD?”

Slippery slope logical fallacy

The slippery slope logical fallacy occurs when someone asserts that a relatively small step or initial action will lead to a chain of events resulting in a drastic change or undesirable outcome. However, no evidence is offered to prove that this chain reaction will indeed happen.

Hasty generalization logical fallacy

The hasty generalization fallacy (or jumping to conclusions ) occurs when we use a small sample or exceptional cases to draw a conclusion or generalize a rule.

A false dilemma (or either/or fallacy ) is a common persuasion technique in advertising. It presents us with only two possible options without considering the broad range of possible alternatives.

In other words, the campaign suggests that animal testing and child mortality are the only two options available. One has to save either animal lives or children’s lives.

People often confuse correlation (i.e., the fact that two things happen one after the other or at the same time) with causation (the fact that one thing causes the other to happen).

It’s possible, for example, that people with MS have lower vitamin D levels because of their decreased mobility and sun exposure, rather than the other way around.

It’s important to carefully account for other factors that may be involved in any observed relationship. The fact that two events or variables are associated in some way does not necessarily imply that there is a cause-and-effect relationship between them and cannot tell us the direction of any cause-and-effect relationship that does exist.

If you want to know more about fallacies , research bias , or AI tools , make sure to check out some of our other articles with explanations and examples.

ChatGPT vs human editor
ChatGPT citations
Is ChatGPT trustworthy?
Using ChatGPT for your studies
Sunk cost fallacy
Straw man fallacy
Slippery slope fallacy
Either or fallacy
Appeal to emotion fallacy
Non sequitur fallacy

Research bias

Implicit bias
Framing bias
Cognitive bias
Optimism bias
Hawthorne effect
Affect heuristic

An ad hominem (Latin for “to the person”) is a type of informal logical fallacy . Instead of arguing against a person’s position, an ad hominem argument attacks the person’s character or actions in an effort to discredit them.

This rhetorical strategy is fallacious because a person’s character, motive, education, or other personal trait is logically irrelevant to whether their argument is true or false.

Name-calling is common in ad hominem fallacy (e.g., “environmental activists are ineffective because they’re all lazy tree-huggers”).

An appeal to ignorance (ignorance here meaning lack of evidence) is a type of informal logical fallacy .

It asserts that something must be true because it hasn’t been proven false—or that something must be false because it has not yet been proven true.

For example, “unicorns exist because there is no evidence that they don’t.” The appeal to ignorance is also called the burden of proof fallacy .

People sometimes confuse cognitive bias and logical fallacies because they both relate to flawed thinking. However, they are not the same:

Cognitive bias is the tendency to make decisions or take action in an illogical way because of our values, memory, socialization, and other personal attributes. In other words, it refers to a fixed pattern of thinking rooted in the way our brain works.
Logical fallacies relate to how we make claims and construct our arguments in the moment. They are statements that sound convincing at first but can be disproven through logical reasoning.

In other words, cognitive bias refers to an ongoing predisposition, while logical fallacy refers to mistakes of reasoning that occur in the moment.

Sources in this article

We strongly encourage students to use sources in their work. You can cite our article (APA Style) or take a deep dive into the articles below.

Nikolopoulou, K. (2023, October 09). Logical Fallacies | Definition, Types, List & Examples. Scribbr. Retrieved June 18, 2024, from https://www.scribbr.com/fallacies/logical-fallacy/

Jin, Z., Lalwani, A., Vaidhya, T., Shen, X., Ding, Y., Lyu, Z., Sachan, M., Mihalcea, R., & Schölkopf, B. (2022). Logical Fallacy Detection. arXiv (Cornell University) . https://doi.org/10.48550/arxiv.2202.13758

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Structure & Outlining

Logical fallacies handlist.

Logical Fallacies Handlist: Fallacies are statements that might sound reasonable or superficially true but are actually flawed or dishonest. When readers detect them, these logical fallacies backfire by making the audience think the writer is (a) unintelligent or (b) deceptive. It is important to avoid them in your own arguments, and it is also important to be able to spot them in others’ arguments so a false line of reasoning won’t fool you. Think of this as intellectual kung-fu : the vital art of self-defense in a debate. For extra impact, learn both the Latin terms and the English equivalents. You can click here to download a PDF version of this material . In general, one useful way to organize fallacies is by category. We have below fallacies of relevance , component fallacies , fallacies of ambiguity , and fallacies of omission . We will discuss each type in turn. The last point to discuss is Occam’s Razor . FALLACIES OF RELEVANCE : These fallacies appeal to evidence or examples that are not relevant to the argument at hand. Appeal to Force ( Argumentum Ad Baculum or the “Might-Makes-Right” Fallacy): This argument uses force, the threat of force, or some other unpleasant backlash to make the audience accept a conclusion. It commonly appears as a last resort when evidence or rational arguments fail to convince a reader. If the debate is about whether or not 2+2=4, an opponent’s argument that he will smash your nose in if you don’t agree with his claim doesn’t change the truth of an issue. Logically, this consideration has nothing to do with the points under consideration. The fallacy is not limited to threats of violence, however. The fallacy includes threats of any unpleasant backlash–financial, professional, and so on. Example: “Superintendent, you should cut the school budget by $16,000. I need not remind you that past school boards have fired superintendents who cannot keep down costs.” While intimidation may force the superintendent to conform, it does not convince him that the choice to cut the budget was the most beneficial for the school or community. Lobbyists use this method when they remind legislators that they represent so many thousand votes in the legislators’ constituencies and threaten to throw the politician out of office if he doesn’t vote the way they want. Teachers use this method if they state that students should hold the same political or philosophical position as the teachers or risk failing the class. Note that it is isn’t a logical fallacy, however, to assert that students must fulfill certain requirements in the course or risk failing the class! Genetic Fallacy : The genetic fallacy is the claim that an idea, product, or person must be untrustworthy because of its racial, geographic, or ethnic origin. “That car can’t possibly be any good! It was made in Japan!” Or, “Why should I listen to her argument? She comes from California, and we all know those people are flakes.” Or, “Ha! I’m not reading that book. It was published in Tennessee, and we know all Tennessee folk are hillbillies and rednecks!” This type of fallacy is closely related to the fallacy of argumentum ad hominem or personal attack , appearing immediately below. Personal Attack ( Argumentum Ad Hominem , literally, “argument toward the man.” Also called “Poisoning the Well”): Attacking or praising the people who make an argument, rather than discussing the argument itself. This practice is fallacious because the personal character of an individual is logically irrelevant to the truth or falseness of the argument itself. The statement “2+2=4” is true regardless if it is stated by criminals, congressmen, or pastors. There are two subcategories: (1) Abusive : To argue that proposals, assertions, or arguments must be false or dangerous because they originate with atheists, Christians, Muslims, communists, capitalists, the John Birch Society, Catholics, anti-Catholics, racists, anti-racists, feminists, misogynists (or any other group) is fallacious. This persuasion comes from irrational psychological transference rather than from an appeal to evidence or logic concerning the issue at hand. This is similar to the genetic fallacy , and only an anti-intellectual would argue otherwise.

(2) Circumstantial : To argue that an opponent should accept or reject an argument because of circumstances in his or her life. If one’s adversary is a clergyman, suggesting that he should accept a particular argument because not to do so would be incompatible with the scriptures is such a fallacy. To argue that, because the reader is a Republican or Democrat, she must vote for a specific measure is likewise a circumstantial fallacy. The opponent’s special circumstances have no control over the truth or untruth of a specific contention. The speaker or writer must find additional evidence beyond that to make a strong case. This is also similar to the genetic fallacy in some ways. If you are a college student who wants to learn rational thought, you simply must avoid circumstantial fallacies.

Argumentum ad Populum (Literally “Argument to the People”): Using an appeal to popular assent, often by arousing the feelings and enthusiasm of the multitude rather than building an argument. It is a favorite device with the propagandist, the demagogue, and the advertiser. An example of this type of argument is Shakespeare’s version of Mark Antony’s funeral oration for Julius Caesar. There are three basic approaches:

(1) Bandwagon Approach : “Everybody is doing it.” This argumentum ad populum asserts that, since the majority of people believes an argument or chooses a particular course of action, the argument must be true, or the course of action must be followed, or the decision must be the best choice. For instance, “85% of consumers purchase IBM computers rather than Macintosh; all those people can’t be wrong. IBM must make the best computers.” Popular acceptance of any argument does not prove it to be valid, nor does popular use of any product necessarily prove it is the best one. After all, 85% of people may once have thought planet earth was flat, but that majority’s belief didn’t mean the earth really was flat when they believed it! Keep this in mind, and remember that everybody should avoid this type of logical fallacy.

(2) Patriotic Approach : “Draping oneself in the flag.” This argument asserts that a certain stance is true or correct because it is somehow patriotic, and that those who disagree are unpatriotic. It overlaps with pathos and argumentum ad hominem to a certain extent. The best way to spot it is to look for emotionally charged terms like Americanism, rugged individualism, motherhood, patriotism, godless communism, etc. A true American would never use this approach. And a truly free man will exercise his American right to drink beer, since beer belongs in this great country of ours. This approach is unworthy of a good citizen. (3) Snob Approach : This type of argumentum ad populum doesn’t assert “everybody is doing it,” but rather that “all the best people are doing it.” For instance, “Any true intellectual would recognize the necessity for studying logical fallacies.” The implication is that anyone who fails to recognize the truth of the author’s assertion is not an intellectual, and thus the reader had best recognize that necessity.

In all three of these examples, the rhetorician does not supply evidence that an argument is true; he merely makes assertions about people who agree or disagree with the argument. For Christian students in religious schools like Carson-Newman, we might add a fourth category, “ Covering Oneself in the Cross .” This argument asserts that a certain political or denominational stance is true or correct because it is somehow “Christian,” and that anyone who disagrees is behaving in an “un-Christian” or “godless” manner. (It is similar to the patriotic approach except it substitutes a gloss of piety instead of patriotism.) Examples include the various “Christian Voting Guides” that appear near election time, many of them published by non-Church related organizations with hidden financial/political agendas, or the stereotypical crooked used-car salesman who keeps a pair of bibles on his dashboard in order to win the trust of those he would fleece. Keep in mind Moliere’s question in Tartuffe : “Is not a face quite different than a mask?” Is not the appearance of Christianity quite different than actual Christianity? Christians should beware of such manipulation since they are especially vulnerable to it.

Appeal to Tradition ( Argumentum Ad Traditionem; aka Argumentum Ad Antiquitatem ): This line of thought asserts that a premise must be true because people have always believed it or done it. For example, “We know the earth is flat because generations have thought that for centuries!” Alternatively, the appeal to tradition might conclude that the premise has always worked in the past and will thus always work in the future: “Jefferson City has kept its urban growth boundary at six miles for the past thirty years. That has been good enough for thirty years, so why should we change it now? If it ain’t broke, don’t fix it.” Such an argument is appealing in that it seems to be common sense, but it ignores important questions. Might an alternative policy work even better than the old one? Are there drawbacks to that long-standing policy? Are circumstances changing from the way they were thirty years ago? Has new evidence emerged that might throw that long-standing policy into doubt?

Appeal to Improper Authority ( Argumentum Ad Verecundium, literally “argument from that which is improper”): An appeal to an improper authority, such as a famous person or a source that may not be reliable or who might not know anything about the topic. This fallacy attempts to capitalize upon feelings of respect or familiarity with a famous individual. It is not fallacious to refer to an admitted authority if the individual’s expertise is within a strict field of knowledge. On the other hand, to cite Einstein to settle an argument about education or economics is fallacious. To cite Darwin, an authority on biology, on religious matters is fallacious. To cite Cardinal Spellman on legal problems is fallacious. The worst offenders usually involve movie stars and psychic hotlines. A subcategory is the Appeal to Biased Authority . In this sort of appeal, the authority is one who actually is knowledgeable on the matter, but one who may have professional or personal motivations that render his professional judgment suspect: for instance, “To determine whether fraternities are beneficial to this campus, we interviewed all the frat presidents.” Or again, “To find out whether or not sludge-mining really is endangering the Tuskogee salamander’s breeding grounds, we interviewed the owners of the sludge-mines, who declared there is no problem.” Indeed, it is important to get “both viewpoints” on an argument, but basing a substantial part of your argument on a source that has personal, professional, or financial interests at stake may lead to biased arguments. As Upton Sinclair once stated, “It’s difficult to get a man to understand something when his salary depends upon his not understanding it.” Sinclair is pointing out that even a knowledgeable authority might not be entirely rational on a topic when he has economic incentives that bias his thinking.

Appeal to Emotion (Argumentum Ad Misericordiam , literally, “argument from pity”): An emotional appeal concerning what should be a logical issue during a debate. While pathos generally works to reinforce a reader’s sense of duty or outrage at some abuse, if a writer tries to use emotion merely for the sake of getting the reader to accept what should be a logical conclusion, the argument is a fallacy. For example, in the 1880s, prosecutors in a Virginia court presented overwhelming proof that a boy was guilty of murdering his parents with an ax. The defense presented a “not-guilty” plea for on the grounds that the boy was now an orphan, with no one to look after his interests if the court was not lenient. This appeal to emotion obviously seems misplaced, and the argument is irrelevant to the question of whether or not he did the crime.

Argument from Adverse Consequences: Asserting that an argument must be false because the implications of it being true would create negative results. For instance, “The medical tests show that Grandma has advanced cancer. However, that can’t be true because then she would die! I refuse to believe it!” The argument is illogical because truth and falsity are not contingent based upon how much we like or dislike the consequences of that truth. Grandma, indeed, might have cancer, in spite of how negative that fact may be or how cruelly it may affect us.

Argument from Personal Incredulity : Asserting that opponent’s argument must be false because you personally don’t understand it or can’t follow its technicalities. For instance, one person might assert, “I don’t understand that engineer’s argument about how airplanes can fly. Therefore, I cannot believe that airplanes are able to fly.” Au contraire , that speaker’s own mental limitations do not limit the physical world—so airplanes may very well be able to fly in spite of a person’s inability to understand how they work. One person’s comprehension is not relevant to the truth of a matter.

Begging the Question (also called Petitio Principii , this term is sometimes used interchangeably with Circular Reasoning ): If writers assume as evidence for their argument the very conclusion they are attempting to prove, they engage in the fallacy of begging the question. The most common form of this fallacy is when the first claim is initially loaded with the very conclusion one has yet to prove. For instance, suppose a particular student group states, “Useless courses like English 101 should be dropped from the college’s curriculum.” The members of the student group then immediately move on in the argument, illustrating that spending money on a useless course is something nobody wants. Yes, we all agree that spending money on useless courses is a bad thing. However, those students never did prove that English 101 was itself a useless course–they merely “begged the question” and moved on to the next “safe” part of the argument, skipping over the part that’s the real controversy, the heart of the matter, the most important component. Begging the question is often hidden in the form of a complex question (see below).

Circular Reasoning is closely related to begging the question . Often the writers using this fallacy word take one idea and phrase it in two statements. The assertions differ sufficiently to obscure the fact that that the same proposition occurs as both a premise and a conclusion. The speaker or author then tries to “prove” his or her assertion by merely repeating it in different words. Richard Whately wrote in Elements of Logic (London 1826): “To allow every man unbounded freedom of speech must always be on the whole, advantageous to the state; for it is highly conducive to the interest of the community that each individual should enjoy a liberty perfectly unlimited of expressing his sentiments.” Obviously the premise is not logically irrelevant to the conclusion, for if the premise is true the conclusion must also be true. It is, however, logically irrelevant in proving the conclusion. In the example, the author is repeating the same point in different words, and then attempting to “prove” the first assertion with the second one. A more complex but equally fallacious type of circular reasoning is to create a circular chain of reasoning like this one: “God exists.” “How do you know that God exists?” “The Bible says so.” “Why should I believe the Bible?” “Because it’s the inspired word of God.” If we draw this out as a chart, it looks like this:

The so-called “final proof” relies on unproven evidence set forth initially as the subject of debate. Basically, the argument goes in an endless circle, with each step of the argument relying on a previous one, which in turn relies on the first argument yet to be proven. Surely God deserves a more intelligible argument than the circular reasoning proposed in this example!

Hasty Generalization ( Dicto Simpliciter , also called “Jumping to Conclusions,” “Converse Accident”): Mistaken use of inductive reasoning when there are too few samples to prove a point. Example: “Susan failed Biology 101. Herman failed Biology 101. Egbert failed Biology 101. I therefore conclude that most students who take Biology 101 will fail it.” In understanding and characterizing general situations, a logician cannot normally examine every single example. However, the examples used in inductive reasoning should be typical of the problem or situation at hand. Maybe Susan, Herman, and Egbert are exceptionally poor students. Maybe they were sick and missed too many lectures that term to pass. If a logician wants to make the case that most students will fail Biology 101, she should (a) get a very large sample–at least one larger than three–or (b) if that isn’t possible, she will need to go out of his way to prove to the reader that her three samples are somehow representative of the norm. If a logician considers only exceptional or dramatic cases and generalizes a rule that fits these alone, the author commits the fallacy of hasty generalization.

One common type of hasty generalization is the Fallacy of Accident . This error occurs when one applies a general rule to a particular case when accidental circumstances render the general rule inapplicable. For example, in Plato’s Republic , Plato finds an exception to the general rule that one should return what one has borrowed: “Suppose that a friend when in his right mind has deposited arms with me and asks for them when he is not in his right mind. Ought I to give the weapons back to him? No one would say that I ought or that I should be right in doing so. . . .” What is true in general may not be true universally and without qualification. So remember, generalizations are bad. All of them. Every single last one. Except, of course, for those that are not.

Another common example of this fallacy is the misleading statistic . Suppose an individual argues that women must be incompetent drivers, and he points out that last Tuesday at the Department of Motor Vehicles, 50% of the women who took the driving test failed. That would seem to be compelling evidence from the way the statistic is set forth. However, if only two women took the test that day, the results would be far less clear-cut. Incidentally, the cartoon Dilbert makes much of an incompetent manager who cannot perceive misleading statistics. He does a statistical study of when employees call in sick and cannot come to work during the five-day work week. He becomes furious to learn that 40% of office “sick-days” occur on Mondays (20%) and Fridays (20%)–just in time to create a three-day weekend. Suspecting fraud, he decides to punish his workers. The irony, of course, is that these two days compose 40% of a five day work week, so the numbers are completely average. Similar nonsense emerges when parents or teachers complain that “50% of students perform at or below the national average on standardized tests in mathematics and verbal aptitude.” Of course they do! The very nature of an average implies that!

False Cause : This fallacy establishes a cause/effect relationship that does not exist. There are various Latin names for various analyses of the fallacy. The two most common include these types:

(1) Non Causa Pro Causa (Literally, “Not the cause for a cause”): A general, catch-all category for mistaking a false cause of an event for the real cause. (2) Post Hoc, Ergo Propter Hoc (Literally: “After this, therefore because of this”): This type of false cause occurs when the writer mistakenly assumes that, because the first event preceded the second event, it must mean the first event caused the later one. Sometimes it does, but sometimes it doesn’t. It is the honest writer’s job to establish clearly that connection rather than merely assert it exists. Example: “A black cat crossed my path at noon. An hour later, my mother had a heart-attack. Because the first event occurred earlier, it must have caused the bad luck later.” This is how superstitions begin. The most common examples are arguments that viewing a particular movie or show, or listening to a particular type of music “caused” the listener to perform an antisocial act–to snort coke, shoot classmates, or take up a life of crime. These may be potential suspects for the cause, but the mere fact that an individual did these acts and subsequently behaved in a certain way does not yet conclusively rule out other causes. Perhaps the listener had an abusive home-life or school-life, suffered from a chemical imbalance leading to depression and paranoia, or made a bad choice in his companions. Other potential causes must be examined before asserting that only one event or circumstance alone earlier in time caused a event or behavior later. For more information, see correlation and causation .

Irrelevant Conclusion ( Ignorantio Elenchi ): This fallacy occurs when a rhetorician adapts an argument purporting to establish a particular conclusion and directs it to prove a different conclusion. For example, when a particular proposal for housing legislation is under consideration, a legislator may argue that decent housing for all people is desirable. Everyone, presumably, will agree. However, the question at hand concerns a particular measure. The question really isn’t, “Is it good to have decent housing?” The question really is, “Will this particular measure actually provide it or is there a better alternative?” This type of fallacy is a common one in student papers when students use a shared assumption–such as the fact that decent housing is a desirable thing to have–and then spend the bulk of their essays focused on that fact rather than the real question at issue. It’s similar to begging the question , above.

One of the most common forms of Ignorantio Elenchi is the “ Red Herring .” A red herring is a deliberate attempt to change the subject or divert the argument from the real question at issue to some side-point; for instance, “Senator Jones should not be held accountable for cheating on his income tax. After all, there are other senators who have done far worse things.” Another example: “I should not pay a fine for reckless driving. There are many other people on the street who are dangerous criminals and rapists, and the police should be chasing them, not harassing a decent tax-paying citizen like me.” Certainly, worse criminals do exist, but that it is another issue! The questions at hand are (1) did the speaker drive recklessly, and (2) should he pay a fine for it?

Another similar example of the red herring is the fallacy known as Tu Quoque (Latin for “And you too!”), which asserts that the advice or argument must be false simply because the person presenting the advice doesn’t consistently follow it herself. For instance, “Susan the yoga instructor claims that a low-fat diet and exercise are good for you–but I saw her last week pigging out on oreos, so her argument must be a load of hogwash.” Or, “Reverend Jeremias claims that theft is wrong, but how can theft be wrong if Jeremias himself admits he stole objects when he was a child?” Or “Thomas Jefferson made many arguments about equality and liberty for all Americans, but he himself kept slaves, so we can dismiss any thoughts he had on those topics.”

Straw Man Argument : A subtype of the red herring , this fallacy includes any lame attempt to “prove” an argument by overstating, exaggerating, or over-simplifying the arguments of the opposing side. Such an approach is building a straw man argument. The name comes from the idea of a boxer or fighter who meticulously fashions a false opponent out of straw, like a scarecrow, and then easily knocks it over in the ring before his admiring audience. His “victory” is a hollow mockery, of course, because the straw-stuffed opponent is incapable of fighting back. When a writer makes a cartoon-like caricature of the opposing argument, ignoring the real or subtle points of contention, and then proceeds to knock down each “fake” point one-by-one, he has created a straw man argument.

For instance, one speaker might be engaged in a debate concerning welfare. The opponent argues, “Tennessee should increase funding to unemployed single mothers during the first year after childbirth because they need sufficient money to provide medical care for their newborn children.” The second speaker retorts, “My opponent believes that some parasites who don’t work should get a free ride from the tax money of hard-working honest citizens. I’ll show you why he’s wrong . . .” In this example, the second speaker is engaging in a straw man strategy, distorting the opposition’s statement about medical care for newborn children into an oversimplified form so he can more easily appear to “win.” However, the second speaker is only defeating a dummy-argument rather than honestly engaging in the real nuances of the debate.

Non Sequitur (literally, “It does not follow”): A non sequitur is any argument that does not follow from the previous statements. Usually what happened is that the writer leaped from A to B and then jumped to D, leaving out step C of an argument she thought through in her head, but did not put down on paper. The phrase is applicable in general to any type of logical fallacy, but logicians use the term particularly in reference to syllogistic errors such as the undistributed middle term , non causa pro causa , and ignorantio elenchi . A common example would be an argument along these lines: “Giving up our nuclear arsenal in the 1980’s weakened the United States’ military. Giving up nuclear weaponry also weakened China in the 1990s. For this reason, it is wrong to try to outlaw pistols and rifles in the United States today.” There’s obviously a step or two missing here.

The “Slippery Slope” Fallacy (also called “The Camel’s Nose Fallacy”) is a non sequitur in which the speaker argues that, once the first step is undertaken, a second or third step will inevitably follow, much like the way one step on a slippery incline will cause a person to fall and slide all the way to the bottom. It is also called “the Camel’s Nose Fallacy” because of the image of a sheik who let his camel stick its nose into his tent on a cold night. The idea is that the sheik is afraid to let the camel stick its nose into the tent because once the beast sticks in its nose, it will inevitably stick in its head, and then its neck, and eventually its whole body. However, this sort of thinking does not allow for any possibility of stopping the process. It simply assumes that, once the nose is in, the rest must follow–that the sheik can’t stop the progression once it has begun–and thus the argument is a logical fallacy. For instance, if one were to argue, “If we allow the government to infringe upon our right to privacy on the Internet, it will then feel free to infringe upon our privacy on the telephone. After that, FBI agents will be reading our mail. Then they will be placing cameras in our houses. We must not let any governmental agency interfere with our Internet communications, or privacy will completely vanish in the United States.” Such thinking is fallacious; no logical proof has been provided yet that infringement in one area will necessarily lead to infringement in another, no more than a person buying a single can of Coca-Cola in a grocery store would indicate the person will inevitably go on to buy every item available in the store, helpless to stop herself. So remember to avoid the slippery slope fallacy; once you use one, you may find yourself using more and more logical fallacies.

Either/Or Fallacy (also called “the Black-and-White Fallacy,” “Excluded Middle,” “False Dilemma,” or “False Dichotomy”): This fallacy occurs when a writer builds an argument upon the assumption that there are only two choices or possible outcomes when actually there are several. Outcomes are seldom so simple. This fallacy most frequently appears in connection to sweeping generalizations: “Either we must ban X or the American way of life will collapse.” “We go to war with Canada, or else Canada will eventually grow in population and overwhelm the United States.” “Either you drink Burpsy Cola, or you will have no friends and no social life.” Either you must avoid either/or fallacies, or everyone will think you are foolish.

Faulty Analogy : Relying only on comparisons to prove a point rather than arguing deductively and inductively. For example, “education is like cake; a small amount tastes sweet, but eat too much and your teeth will rot out. Likewise, more than two years of education is bad for a student.” The analogy is only acceptable to the degree a reader thinks that education is similar to cake. As you can see, faulty analogies are like flimsy wood, and just as no carpenter would build a house out of flimsy wood, no writer should ever construct an argument out of flimsy material.

Undistributed Middle Term : A specific type of error in deductive reasoning in which the minor premise and the major premise of a syllogism might or might not overlap. Consider these two examples: (1) “All reptiles are cold-blooded. All snakes are reptiles. All snakes are cold-blooded.” In the first example, the middle term “snakes” fits in the categories of both “reptile” and “things-that-are-cold-blooded.” (2) “All snails are cold-blooded. All snakes are cold-blooded. All snails are snakes.” In the second example, the middle term of “snakes” does not fit into the categories of both “things-that-are-cold-blooded” and “snails.” Sometimes, equivocation (see below) leads to an undistributed middle term.

Contradictory Premises (also known as a logical paradox): Establishing a premise in such a way that it contradicts another, earlier premise. For instance, “If God can do anything, he can make a stone so heavy that he can’t lift it.” The first premise establishes a deity that has the irresistible capacity to move other objects. The second premise establishes an immovable object impervious to any movement. If the first object capable of moving anything exists, by definition, the immovable object cannot exist, and vice-versa .

Closely related is the fallacy of Special Pleading , in which the writer creates a universal principle, then insists that principle does not for some reason apply to the issue at hand. For instance, “Everything must have a source or creator. Therefore God must exist and he must have created the world. What? Who created God? Well, God is eternal and unchanging–He has no source or creator.” In such an assertion, either God must have His own source or creator, or else the universal principle of everything having a source or creator must be set aside—the person making the argument can’t have it both ways.

FALLACIES OF AMBIGUITY : These errors occur with ambiguous words or phrases, the meanings of which shift and change in the course of discussion. Such more or less subtle changes can render arguments fallacious.

Equivocation : Using a word in a different way than the author used it in the original premise, or changing definitions halfway through a discussion. When we use the same word or phrase in different senses within one line of argument, we commit the fallacy of equivocation. Consider this example: “Plato says the end of a thing is its perfection; I say that death is the end of life; hence, death is the perfection of life.” Here the word end means “goal” in Plato’s usage, but it means “last event” or “termination” in the author’s second usage. Clearly, the speaker is twisting Plato’s meaning of the word to draw a very different conclusion. Compare with amphiboly , below.

Amphiboly (from the Greek word “indeterminate”): This fallacy is similar to equivocation. Here, the ambiguity results from grammatical construction. A statement may be true according to one interpretation of how each word functions in a sentence and false according to another. When a premise works with an interpretation that is true, but the conclusion uses the secondary “false” interpretation, we have the fallacy of amphiboly on our hands. In the command, “Save soap and waste paper,” the amphibolous use of “waste” results in the problem of determining whether “waste” functions as a verb or as an adjective.

Composition : This fallacy is a result of reasoning from the properties of the parts of the whole to the properties of the whole itself–it is an inductive error. Such an argument might hold that, because every individual part of a large tractor is lightweight, the entire machine also must be lightweight. This fallacy is similar to Hasty Generalization (see above), but it focuses on parts of a single whole rather than using too few examples to create a categorical generalization. Also compare it with Division (see below).

Division : This fallacy is the reverse of composition . It is the misapplication of deductive reasoning. One fallacy of division argues falsely that what is true of the whole must be true of individual parts. Such an argument notes that, “Microtech is a company with great influence in the California legislature. Egbert Smith works at Microtech. He must have great influence in the California legislature.” This is not necessarily true. Egbert might work as a graveyard shift security guard or as the copy-machine repairman at Microtech–positions requiring little interaction with the California legislature. Another fallacy of division attributes the properties of the whole to the individual member of the whole: “Sunsurf is a company that sells environmentally safe products. Susan Jones is a worker at Sunsurf. She must be an environmentally minded individual.” (Perhaps she is motivated by money alone?)

Fallacy of Reification (Also called “ Fallacy of Misplaced Concreteness ” by Alfred North Whitehead): The fallacy of treating a word or an idea as equivalent to the actual thing represented by that word or idea, or the fallacy of treating an abstraction or process as equivalent to a concrete object or thing. In the first case, we might imagine a reformer trying to eliminate illicit lust by banning all mention of extra-marital affairs or certain sexual acts in publications. The problem is that eliminating the words for these deeds is not the same as eliminating the deeds themselves. In the second case, we might imagine a person or declaring “a war on poverty.” In this case, the fallacy comes from the fact that “war” implies a concrete struggle with another concrete entity which can surrender or be exterminated. “Poverty,” however is an abstraction that cannot surrender or sign peace treaties, cannot be shot or bombed, etc. Reification of the concept merely muddles the issue of what policies to follow and leads to sloppy thinking about the best way to handle a problem. It is closely related to and overlaps with faulty analogy and equivocation .

FALLACIES O F OMISSION : These errors occur because the logician leaves out necessary material in an argument or misdirects others from missing information.

Stacking the Deck : In this fallacy, the speaker “stacks the deck” in her favor by ignoring examples that disprove the point and listing only those examples that support her case. This fallacy is closely related to hasty generalization, but the term usually implies deliberate deception rather than an accidental logical error. Contrast it with the straw man argument .

‘No True Scotsman’ Fallacy : Attempting to stack the deck specifically by defining terms in such a narrow or unrealistic manner as to exclude or omit relevant examples from a sample. For instance, suppose speaker #1 asserts, “The Scottish national character is brave and patriotic. No Scottish soldier has ever fled the field of battle in the face of the enemy.” Speaker #2 objects, “Ah, but what about Lucas MacDurgan? He fled from German troops in World War I.” Speaker #1 retorts, “Well, obviously he doesn’t count as a true Scotsman because he did not live up to Scottish ideals, thus he forfeited his Scottish identity.” By this fallacious reasoning, any individual who would serve as evidence contradicting the first speaker’s assertion is conveniently and automatically dismissed from consideration. We commonly see this fallacy when a company asserts that it cannot be blamed for one of its particularly unsafe or shoddy products because that particular one doesn’t live up to its normally high standards, and thus shouldn’t “count” against its fine reputation. Likewise, defenders of Christianity as a positive historical influence in their zeal might argue the atrocities of the eight Crusades do not “count” in an argument because the Crusaders weren’t living up to Christian ideals, and thus aren’t really Christians, etc. So, remember this fallacy. Philosophers and logicians never use it, and anyone who does use it by definition is not really a philosopher or logician.

Argument from the Negative : Arguing from the negative asserts that, since one position is untenable, the opposite stance must be true. This fallacy is often used interchangeably with Argumentum Ad Ignorantium (listed below) and the either/or fallacy (listed above). For instance, one might mistakenly argue that, since the Newtonian theory of mathematics is not one hundred percent accurate, Einstein’s theory of relativity must be true. Perhaps not. Perhaps the theories of quantum mechanics are more accurate, and Einstein’s theory is flawed. Perhaps they are all wrong. Disproving an opponent’s argument does not necessarily mean your own argument must be true automatically, no more than disproving your opponent’s assertion that 2+2=5 would automatically mean your argument that 2+2=7 must be the correct one. Keeping this mind, students should remember that arguments from the negative are bad, arguments from the positive must automatically be good.

Appeal to a Lack of Evidence ( Argumentum Ad Ignorantium , literally “Argument from Ignorance”): Appealing to a lack of information to prove a point, or arguing that, since the opposition cannot disprove a claim, the opposite stance must be true. An example of such an argument is the assertion that ghosts must exist because no one has been able to prove that they do not exist. Logicians know this is a logical fallacy because no competing argument has yet revealed itself.

Hypothesis Contrary to Fact ( Argumentum Ad Speculum ): Trying to prove something in the real world by using imaginary examples alone, or asserting that, if hypothetically X had occurred, Y would have been the result. For instance, suppose an individual asserts that if Einstein had been aborted in utero , the world would never have learned about relativity, or that if Monet had been trained as a butcher rather than going to college, the impressionistic movement would have never influenced modern art. Such hypotheses are misleading lines of argument because it is often possible that some other individual would have solved the relativistic equations or introduced an impressionistic art style. The speculation might make an interesting thought-experiment, but it is simply useless when it comes to actually proving anything about the real world. A common example is the idea that one “owes” her success to another individual who taught her. For instance, “You owe me part of your increased salary. If I hadn’t taught you how to recognize logical fallacies, you would be flipping hamburgers at McDonald’s for minimum wages right now instead of taking in hundreds of thousands of dollars as a lawyer.” Perhaps. But perhaps the audience would have learned about logical fallacies elsewhere, so the hypothetical situation described is meaningless.

Complex Question (Also called the “Loaded Question”): Phrasing a question or statement in such as way as to imply another unproven statement is true without evidence or discussion. This fallacy often overlaps with begging the question (above), since it also presupposes a definite answer to a previous, unstated question. For instance, if I were to ask you “Have you stopped taking drugs yet?” my hidden supposition is that you have been taking drugs. Such a question cannot be answered with a simple yes or no answer. It is not a simple question but consists of several questions rolled into one. In this case the unstated question is, “Have you taken drugs in the past?” followed by, “If you have taken drugs in the past, have you stopped taking them now?” In cross-examination, a lawyer might ask a flustered witness, “Where did you hide the evidence?” or “when did you stop beating your wife?” The intelligent procedure when faced with such a question is to analyze its component parts. If one answers or discusses the prior, implicit question first, the explicit question may dissolve.

Complex questions appear in written argument frequently. A student might write, “Why is private development of resources so much more efficient than any public control?” The rhetorical question leads directly into his next argument. However, an observant reader may disagree, recognizing the prior, implicit question remains unaddressed. That question is, of course, whether private development of resources really is more efficient in all cases, a point which the author is skipping entirely and merely assuming to be true without discussion.

To master logic more fully, become familiar with the tool of Occam’s Razor .

Logical Fallacies Handlist. Authored by : Dr. Kip Wheeler. Provided by : Carson Newman University. Located at : https://web.cn.edu/kwheeler/fallacies_list.html . License : CC BY-SA: Attribution-ShareAlike

Primary links

Avoiding logical fallacies.

Logic can go wrong in many ways. We’ve talked about building logical arguments. Now let’s consider how to avoid building illogical ones. The logical fallacies below can slip into your own and others’ arguments. Learn to identify them.

Distortions in Logic

(Instead of engaging the claim, the response dismisses its importance.)

(Does this mean we need to minimize or maximize the amount of water?)

(This claim dismisses opposition by saying poverty is just a fact of life.)

(This claim generalizes from some spousal abuse to all domestic violence.)

(In place of an argument, the same assertion is made three times.)

(To answer the question would be to admit to destroying the country.)

(This statement incorrectly assumes that the president’s location caused the 9/11 attacks.)

(This analogy does not accurately represent the process, in which the winner becomes arguably the most powerful person in the world.)

(This claim swaps the cause and the effect. The worsening economy causes the Fed to lower interest rates, not the other way around.)

(This claim ignores the difference between free speech and treason.)

(The agency can function on a reduced budget without shutting down.)

(This statement ignores the long evolution of both parties.)

(Both candidates won’t back down, so both should get the same praise or blame.)

(This statement does not follow. A person’s body fat percentage does not relate to his or her ability to balance governmental budgets.)

Your Turn Find a political debate online and listen to it. Write down as many examples as you can of the fallacies on these two pages.

(Instead of changing the false assumption that all Scotsmen are brave, the person discounts the counterexample of cowardly Andrew.)

(This statement means “We should get rid of harmful influences,” an idea so obvious that it really doesn’t need to be stated.)

(This oversimplification ignores the fact that such an act would catastrophically devalue the dollar.)

(This statement applies a reasonable principle to absurd specificity.)

(The language in this statement allows for no reasonable discussion.)

(Immigration reform does not require pardoning all illegal activity.)

(That a politician wants to destroy the country is a dummy argument.)

Your Turn Pick four of the fallacies on this page and write your own examples. Share your answers with a partner and discuss the faulty logic in each.

Misusing Evidence

(Someone else’s bad behavior doesn’t justify one’s own bad behavior.)

(Absence of evidence is not evidence of absence.)

(A proposal should be accepted on its own merits, not due to hard work.)

(A sentimental name doesn’t make peanut butter worth buying.)

(An actor who plays a doctor is not a medical authority.)

(The horror of the idea does not preclude its possibility.)

(Actually, quantum physicists have proven this idea.)

(This ad hominem attack diverts attention from the real issue: taxes.)

(A stronger argument would focus on the value of the paper.)

(The thousands who are injured are a tiny fraction of the millions who use power tools safely and who rely on them to make a living.)

(There is no way to prove or disprove what would have happened if the other candidate had won, so the argument is meaningless.)

(The use of numbers baffles the audience into acceptance.)

(World hunger is a serious problem that shouldn’t be dismissed with a joke.)

(The Ebola virus, a separate problem, should not be used to distract from the abhorrent use of child soldiers.)

(Threats are never an acceptable form of persuasion.)

Your Turn Watch commercials on television or on the Internet and write down two examples of the misuse of evidence on pages 111–112 .

Additional Resources

Web Site: Fallacies

Web Page: Logical Fallacies

Web Page: Common Fallacies in Reasoning

Web Page: Fallacies of Ambiguity

Web Page: Marketing Plots

Web Page: Hasty Generalization

Video: Correlation and Causation

Web Page: False Cause

Web Page: False Dichotomy

Web Page: Non Sequitur

Blog Post: The New Illiteracy--Obfuscation

Web Page: Reductio ad Absurdum

Web Page: Slippery Slope

Web Page: Argumentum Ad Hominem

Web Page: Bandwagon Appeal

Web Page: Red Herring

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Friday, March 18, 2016

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Counterfactuals

Modal discourse concerns alternative ways things can be, e.g., what might be true, what isn’t true but could have been, what should be done. This entry focuses on counterfactual modality which concerns what is not, but could or would have been. What if Martin Luther King had died when he was stabbed in 1958 (Byrne 2005: 1) ? What if the Americas had never been colonized? What if I were to put that box over here and this one over there? These modes of thought and speech have been the subject of extensive study in philosophy, linguistics, psychology, artificial intelligence, history, and many other allied fields. These diverse investigations are united by the fact that counterfactual modality crops up at the center of foundational questions in these fields.

In philosophy, counterfactual modality has given rise to difficult semantic, epistemological, and metaphysical questions:

Semantic How do we communicate and reason about possibilities which are remote from the way things actually are?
Epistemic How can our experience in the actual world justify thought and talk about remote possibilities?
Metaphysical Do these remote possibilities exist independently from the actual world, or are they grounded in things that actually exist?

These questions have attracted significant attention in recent decades, revealing a wealth of puzzles and insights. While other entries address the epistemic— the epistemology of modality —and metaphysical questions— possible worlds and the possibilism-actualism debate —this entry focuses on the semantic question. It will aim to refine this question, explain its central role in certain philosophical debates, and outline the main semantic analyses of counterfactuals.

Section 1 begins with a working definition of counterfactual conditionals ( §1.1 ), and then surveys how counterfactuals feature in theories of agency, mental representation, and rationality ( §1.2 ), and how they are used in metaphysical analysis and scientific explanation ( §1.3 ). Section 1.4 then details several ways in which the logic and truth-conditions of counterfactuals are puzzling. This sets the stage for the sections 2 and 3 , which survey semantic analyses of counterfactuals that attempt to explain this puzzling behavior.

Section 2 focuses on two related analyses that were primarily developed to study the logic of counterfactuals: strict conditional analyses and similarity analyses. These analyses were not originally concerned with saying what the truth-conditions of particular counterfactuals are. Attempts to extend them to that domain, however, have attracted intense criticism. Section 3 surveys more recent analyses that offer more explicit models of when counterfactuals are true. These analyses include premise semantics ( §3.1 ), conditional probability analyses ( §3.2 ) and structural equations/causal models ( §3.3 ). They are more closely connected to work on counterfactuals in psychology, artificial intelligence, and the philosophy of science.

Sections 2 and 3 of this entry employ some basic tools from set theory and logical semantics. But these sections also provide intuitive characterizations alongside formal definitions, so familiarity with these tools is not a pre-requisite. Readers interested in more familiarity with these tools will find basic set theory , as well as Gamut (1991) and Sider (2010) useful.

1.1 What are Counterfactuals?

1.2.1 agency, choice, and free will, 1.2.2 rationality, 1.2.3 mental representation, content, and knowledge, 1.3 metaphysical analysis and scientific explanation, 1.4 semantic puzzles, 2.1 introducing strict and similarity analyses, 2.2.1 second wave strict conditional analyses, 2.3 similarity semantics, 2.4 comparing the logics, 2.5.1 truth-conditions and similarity, 2.5.2 truth-conditions and the strict analysis, 2.6 philosophical objections, 2.7 summary, 3.1 premise semantics, 3.2 conditional probability analyses, 3.3 bayesian networks, structural equations, and causal models, 3.4 summary, 4. conclusion, other internet resources, related entries, 1. counterfactuals and philosophy.

This section begins with some terminological issues ( §1.1 ). It then provides two broad surveys of research that places counterfactuals at the center of key philosophical issues. Section 1.2 covers the role of counterfactuals in theories of rational agency, mental representation, and knowledge. Section 1.3 focuses on the central role of counterfactuals in metaphysics and the philosophy of science. Section 1.4 will then bring a bit of drama to the narrative by explaining how counterfactuals are deeply puzzling from the perspective of classical and modal logics alike.

In philosophy and related fields, counterfactuals are taken to be sentences like:

This entry will follow this widely used terminology to avoid confusion. However, this usage also promotes a confusion worth dispelling. Counterfactuals are not really conditionals with contrary-to-fact antecedents. For example ( 2 ) can be used as part of an argument that the antecedent is true (Anderson 1951) :

On these grounds, it might be better to speak instead of subjunctive conditionals , and reserve the term counterfactual for subjunctive conditionals whose antecedent is assumed to be false in the discourse. [ 1 ] While slightly more enlightened, this use of the term does not match the use of counterfactuals in the sprawling philosophical and interdisciplinary literature surveyed here, and has its own drawbacks that will be discussed shortly. This entry will use counterfactual conditional and subjunctive conditional interchangeably, hoping to now have dispelled the suggestion that all counterfactuals, in that sense, have contrary-to-fact antecedents.

The terminology of indicative and subjunctive conditionals is also vexed, but it aims to get at a basic contrast which begins between two different forms of conditionals that can differ in truth value. (3) and (4) can differ in truth-value while holding fixed the world they are being evaluated in. [ 2 ]

It is easy to imagine a world where (3) is true, and (4) false. Consider a world like ours where Kennedy was assassinated. Further suppose Oswald didn’t do it, but some lone fanatic did for deeply idiosyncratic reasons. Then (3) is true and (4) false. Another aspect of the contrast between indicative and subjunctive conditionals is illustrated in (5) and (6) .

Indicatives like (5) are infelicitous when their antecedent has been denied, unlike the subjunctives like (6) and (7) ( Stalnaker 1975; Veltman 1986 ).

The indicative and subjunctive conditionals above differ from each other only in particular details of their linguistic form. It is therefore plausible to explain their contrasting semantic behavior in terms of the semantics of those linguistic differences. Indicatives, like (3) and (5) , feature verbs in the simple past tense form, and no modal auxiliary in the consequent. Subjunctives, like (4) and (6) , feature verbs in the past perfect (or “pluperfect”) with a modal would in the consequent. Something in the neighborhood of these linguistic and semantic differences constitutes the distinction between indicative and subjunctive conditionals —summarized in Figure 1 . [ 3 ]


	,	, …	, …	Not felicitous
	,	, , , …	, , , …	Can be felicitous

Figure 1: Rough Guide to Indicative and Subjunctive Conditionals

As with most neighborhoods, there are heated debates about the exact boundaries and the names—especially when future-oriented conditionals are included. These debates are surveyed in the supplement Indicative and Subjunctive Conditionals . The main entry will rely only on the agreed-upon paradigm examples like (3) and (4) . The labels indicative and subjunctive are also flawed since these two kinds of conditionals are not really distinguished on the basis of whether they have indicative or subjunctive mood in the antecedent or consequent. [ 4 ] But the terminology is sufficiently entrenched to permit this distortion of linguistic reality.

Much recent work has been devoted to explaining how the semantic differences between indicative and subjunctive conditionals can be derived from their linguistic differences—rather than treating them as semantically unrelated. Much of this work has been done in light of Kratzer’s ( 1986, 2012 ) general approach to modality according to which all conditionals are treated as two-place modal operators. This approach is also discussed in the supplement Indicative and Subjunctive Conditionals . [ 5 ] This entry will focus on the basic logic and truth-conditions of subjunctive conditionals as a whole, and will use the following notation for them (following Stalnaker 1968 ). [ 6 ]

Subjunctive Conditionals (Notation) $\phi>\psi$ symbolizes if it had been the case that $\phi$ then it would have been the case that $\psi$

This project and notation has an important limitation that should be highlighted: it combines the meaning of the modal would and if…then… into a single connective “$>$”. This makes it difficult to adequately represent subjunctive conditionals like:

Conditionals like (8a) have figured in debates about the semantics of counterfactuals and have been modeled either as a related connective (D. Lewis 1973b: §1.5) or a normal would -subjunctive conditional embedded under might ( Stalnaker 1980, 1984: Ch.7 ). But the more complex examples (8b) – (8d) highlight the need for a more refined compositional analysis, like those surveyed in Indicative and Subjunctive Conditionals . So, while this notation will be used in §1.4 and throughout §2 and §3 , it should be regarded as an analytic convenience rather than a defensible assumption.

1.2 Agency, Mind, and Rationality

Counterfactuals have played prominent and interconnected roles in theories of rational agency. They have figured prominently in views of what agency and free will amount to, and played important roles in particular theories of mental representation, rational decision making, and knowledge. This section will outline these uses of counterfactuals and begin to paint a broader picture of how counterfactuals connect to central philosophical questions.

A defining feature of agents is that they make choices. Suppose a citizen votes, and in doing so chooses to vote for X rather than Y . It is hard to see how this act can be a choice without a corresponding counterfactual being true:

The idea that choice entails the ability to do otherwise has been taken by many philosophers to underwrite our practice of holding agents responsible for their choices. But understanding the precise meaning of the counterfactual could have claim in (9) requires navigating the classic problem of free will: if we live in a universe where the current state of the universe is determined (or near enough) by the prior state of the universe and the physical laws, then it seems like every action of every agent, including their “choices”, are predetermined. So interpreting this intuitively plausible counterfactual (9) leads quite quickly to a deep philosophical dilemma. One can maintain, with some Incompatibilists, that (9) is a false claim about what’s physically possible, and revisit the understanding of agency, choice, and responsibility above—the entry incompatibilist theories of free will explores this further. [ 7 ] Alternatively, one can maintain that (9) is a true claim about some non-physical sense of possibility, and explain how that is appropriate to our understanding of choice and responsibility—the entry compatibilism explores this further. It is wrong to construe debates about free will as just debates about the meaning of counterfactuals. But, the semantics of counterfactuals can have a substantive impact on delimiting the space of possible solutions, and perhaps even deciding between them. The same is true for research on counterfactual thinking in psychology.

Experiments in social psychology suggest that belief in free will is linked to increased counterfactual thinking (Alquist et al. 2015) . Further, they have shown that counterfactually reflecting on past events and choices is one significant way humans imbue life experiences with meaning and create a sense of self (Galinsky et al. 2005; Heintzelman et al. 2013; Kray et al. 2010) . Incompatibilists might be able to cite this result as an explanation for why so many people believe they have free will. It is a specific form of wishful thinking: it is interwoven with the practices of counterfactual reflection that give our lives meaning. Seto et al. (2015) support this idea by showing that variation in subjects’ belief in free will predicts how much meaning they derive from relevant instances of counterfactual reflection. This might even be used as part of a pragmatic argument for believing in free will: roughly, belief in free will is so practically important, and our knowledge of the world so incomplete, that it is rational to believe that it exists. [ 8 ]

Counterfactual reflection is not just used for the “sentimental” purposes discussed above, but as part of what Byrne (2005) calls rational imagination . This capacity is implicated in many philosophical definitions of rational agency. According to the standard model, agency involves intentional action—see entries agency and action . While choices are intentional actions, intentional actions are a more general class of actions which, on most views, are in part caused by intentions—see entry intention . One prominent understanding of intentions is that they are prospective (forward looking) mental states that play a crucial role in planning actions. Byrne (2005, 2016: 138) details psychological evidence showing that counterfactual thinking is central to forming rational intentions. People use counterfactual thinking after particular events to formulate plans that will improve the outcome of their actions in related scenarios. Examples include aviation pilots thinking after a near-accident “if I had understood the controller’s words accurately, I wouldn’t have initiated the inappropriate landing attempt”, and blackjack players thinking “If I’d gotten the 2, I would have beaten the dealer”. People who reason in this way show more persistence and improved performance in related tasks, while those who dwell on how things could have been worse, or do not counterfactually reflect at all, show less persistence and no improvement in performance. Finally, human rationality can become disordered when counterfactual thinking goes astray, e.g., in depression, anxiety, and schizophrenia (Byrne 2016: 140–143) .

This psychological research shows that rational human agents do learn from the past and plan for the future engaging in counterfactual thinking. Many researchers in artificial intelligence have voiced similar ideas (Ginsberg 1985; Pearl 1995; Costello & Mccarthy 1999) . But, this view is distinct from a stronger philosophical claim: that the nature of rational agency consists, in part, in the ability to perform counterfactual thinking. Some versions of causal decision theory make precisely this claim, and do so to capture similar patterns of rational behavior. Newcomb’s Problem (Nozick 1969) consists of a decision problem which challenges the standard way of articulating the idea that rational agents maximize expected utility, and, according to some philosophers (Stalnaker 1972 [1980]; Gibbard & Harper 1978) , shows that causal or counterfactual reasoning must be included in rational decision procedures—see the entry causal decision theory for further details. In a similar vein, work on belief revision theory explores how a rational agent should revise their beliefs when they are inconsistent with something they have just learned—much like a counterfactual antecedent demands—and uses structures that formally parallel those used in the semantics of counterfactuals (Harper 1975; Gärdenfors 1978, 1982; Levi 1988) . See formal representations of belief for further discussion of this literature.

The idea that counterfactual reasoning is central to rational agency has surfaced in another way in cognitive science and artificial intelligence, where encoding counterfactual-supporting relationships has emerged as a major theory of mental representation (Chater et al. 2010) . These disciplines also study how states of mind like belief, desire, and intention explain rational agency. But they are not satisfied with just showing that certain states of mind can explain certain choices and actions. They aim to explain how those particular states of mind lead to those choices and actions. They do so by characterizing those states of mind in terms of representations, and formulating particular algorithms for using those representations to learn, make choices and perform actions. [ 9 ] Many recent advances in cognitive science and artificial intelligence share a starting point with Bayesian epistemology: agents must learn and decide what to do despite being uncertain what exactly the world is like, and these processes can be modeled in the probability calculus. On a simple Bayesian approach, an agent represents the world with a probability distribution over binary facts or variables that represent what the world is like. But even for very simple domains the probability calculus does not provide computationally tractable representations and algorithms for implementing Bayesian intelligence. The tools of Bayesian networks , structural equations and causal models , developed by Spirtes, Glymour, and Scheines (1993, 2000) and Pearl (2000, 2009) address this limitation, and also afford simple algorithms for causal and counterfactual reasoning, among other cognitive processes. This framework represents an agent’s knowledge in a way that puts counterfactuals and causal connections at the center, and the tools it provides have been influential beyond cognitive science and AI. It has also been applied to topics covered later in this entry: the semantic analyses of counterfactuals ( §3.2 ) and metaphysical dependence, causation and scientific explanation ( §1.3 ). For this reason, it will be useful to describe its basics now, though still focusing on its applications to mental representation. What follows is a simplified version of the accessible introduction in Sloman (2005: Ch.4) . For a more thorough introduction, see Pearl (2009: Ch.1) .

In a Bayesian framework, probabilities are real numbers between 0 and 1 assigned to propositional variables A , B , C ,…. These probabilities reflect an agent’s subjective credence, e.g., $P(A)=0.6$ reflects that they think A is slightly more likely than not to be true. [ 10 ] At the heart of Bayesian Networks are the concepts of conditional probability and two variables being probabilistically independent . $P(B \mid A)$ is the credence in B conditional on A being true and is defined as follows:

Definition 1 (Conditional Probability) $\displaystyle P(B\mid A)\dequal \frac{P(A\land B)}{P(A)}$

Conditional probabilities allow one to say when B is probabilistically independent of A : when an agent’s credence in B is the same as their credence in B conditional on A and conditional on $\neg A$.

Definition 2 (Probabilistic Independence) B is probabilistically independent of A just in case $P(B)=P(B\mid A)=P(B\mid\neg A)$.

Bayesian networks represent relations of probabilistic dependence. For example, an agent’s knowledge about a system containing eight variables could be represented by the directed acyclic graph and system of structural equations between those variables in Figure 2 .

Figure 2: Bayesian Network and Structural Equations. [An extended description of figure 2 is in the supplement.]

While the arrows mark relations of probabilistic dependence, the equations characterize the nature of the dependence, e.g., “$H\dequal F\lor G$” means that the value of H is determined by the value of $F\lor G$ (but not vice versa). [ 11 ] This significantly reduces the number of values that must be stored. [ 12 ] But it also stores information that is useful to agents. It facilitates counterfactual reasoning—e.g., if C had been true then G would have been true—reasoning about actions—e.g., if we do A then C will be true—and explanatory reasoning—e.g., H is true in part because C is true (Pearl 2002) .

The usefulness of Bayesian networks is evidenced by their many applications in psychology (e.g., Glymour 2001; Sloman 2005 ) and artificial intelligence (e.g., Pearl 2009, 2002) ). They are among the key representations employed in autonomous vehicles (Thrun et al. 2006; Parisien & Thagard 2008) , and have been applied to a wide range of cognitive phenomena:

Causal learning and reasoning in AI (Pearl 2009: Chs 1–4) and humans (Glymour 2001; Gopnik et al. 2004; Sloman 2005: Chs.6–12)
Counterfactual reasoning in AI (Pearl 2009: Ch.7) and humans (Sloman & Lagnado 2005; Sloman 2005; Rips 2010; Lucas & Kemp:2015)
Conceptual categorization and action planning (Sloman 2005: Chs.9,10)
Learning and cognitive development (Gopnik & Tenenbaum 2007)

As Sloman (2005: 177) highlights, this form of representation fits well with a guiding idea of embodied cognition: mental representations in biological agents are constrained by the fact that their primary function is to facilitate successful action despite uncertain information and bounded computational resources. Bayesian networks have also been claimed to address a deep and central issue in artificial intelligence called the frame problem (e.g., Glymour 2001: Ch. 3 ). For the purposes of this entry, it is striking how fruitful this approach to mental representation has been, since counterfactual dependence is at its core.

Counterfactual dependence has also featured prominently in theories of mental content, which explain how a mental representation like the concept dog comes to represent dogs. Informational theories take their inspiration from natural representations like tree rings, which represent, in some sense, how old the tree is (Dretske 2011) . While some accounts in this family are called “causal theories of mental content”, it is somewhat limiting to formulate the view as: X represents Y just in case Y causes X . Even for the tree rings, it is metaphysically controversial to claim that the tree rings are caused by the age of the tree, rather than thinking they have a common cause or are merely causally related via a number of laws and factors, e.g., rainfall, seasons, growth periods. For this and other reasons, Dretske (1981, 1988, 2002) formulates the relationship in terms of conditional probabilities:

Definition 3 (Dretske’s Probabilistic Theory of Information) State s carries the information that a is F , given background conditions g , just in case $P(a\text{ is }F\mid s, g)=1$.

On this view, the state of the tree rings carries the information that the tree is a certain age, since given the background conditions in our world the relevant conditional probability is 1. As argued by Loewer (1983: 76) and Cohen and Meskin (2006) , this formulation introduces problematic issues in how to interpret the probabilities involved and these problems are avoided by a counterfactual formulation:

Definition 4 (Loewer’s Counterfactual Theory of Information) State s carries the information that a is F , given background conditions g , just in case, given g , if s were to obtain, a would have to have been F .

Even this theory of information requires several elaborations to furnish a plausible account of mental content. For example, Dretske (1988, 2002) holds that a mental representation r represents that a is F just in case r has the function of indicating that a is F . The teleological (“function”) component is added to explain how a deer on a dark night can cause tokens of the concept dog without being part of the information carried by thoughts that token dog . Fodor (1987, 1990) pursues another, non-teleological solution, the asymmetric dependence theory. Counterfactuals feature here in another way:

“ a being F causes r ” is a law.
For any other cause c of r , c would not have caused r if a being F had not caused r . ( c ’s causing r asymmetrically depends on a being F causing r .)

This approach also appeals to laws, which are another key philosophical concept connected to counterfactuals—see §1.3 below.

Counterfactuals are not just used to analyze how a given mental state represents reality, but also when a mental state counts as knowledge. Numerous counterexamples, like Gettier cases, make the identification of knowledge with justified true belief problematic—for further details see the analysis of knowledge . But some build on this analysis by proposing further conditions to address these counterexamples. Two counterfactual conditions are prominent in this literature:

Sensitivity If p were false, S would not believe that p .
Safety If S were to believe that p , p would not be false.

Both concepts are ways of articulating the idea that S ’s beliefs must be formed in a way that is responsive to p being true. The semantics of counterfactuals have interacted with this project in a number of ways: in establishing their non-equivalence, refining them, and adjudicating putative counterexamples.

Counterfactuals have played an equally central role in metaphysics and the philosophy of science. They have featured in metaphysical theories of causation, supervenience, grounding, ontological dependence, and dispositions. They have also featured in issues at the intersection of metaphysics and philosophy of science like laws of nature and scientific explanation. This section will briefly overview these applications, largely linking to related entries that cover these applications in more depth. But, this overview is more than just a list of how counterfactuals have been applied in these areas. It helps identify a cluster of inter-related concepts (and/or properties) that are fruitfully studied together rather than in isolation.

Many philosophers have proposed to analyze causal concepts in terms of counterfactuals (e.g., D. Lewis 1973a , Mackie 1974 ). The basic idea is that (10) can be understood in terms of something like (11) (see counterfactual theories of causation for further discussion).

This basic idea has been elaborated and developed in several ways. D. Lewis (1973a, c) refines it using his similarity semantics for counterfactuals—see §2.3 . The resulting counterfactual analysis of causation faces a number of challenges—see counterfactual theories of causation for discussion and references. But this has simply inspired a new wave of counterfactual analyses that use different tools.

Hitchcock (2001, 2007) and Woodward (2003: Ch.5) develop counterfactual analyses of causation using the tools of Bayesian networks (or “causal models”) and structural equations described back in §1.2.3 . The rough idea of the analysis is as follows. Given a graph like the one in Figure 2 , X can be said to be a cause of Y just in case there is a path from X to Y and changing just the value of X changes the value of Y . According to Hitchcock (2001) and Woodward (2002, 2003) , this analysis of causation counts as a counterfactual analysis because the basic structural equations, e.g., $C\dequal A\land B$, are best understood as primitive counterfactual claims, e.g., if A and B had been true, C would have been true. While not all theories of causation that employ structural equations are counterfactual theories, structural equations are central to many of the contemporary counterfactual theories of causation. [ 13 ] See counterfactual theories of causation for further developments and critical reactions to this account of causation.

Recently, Schaffer (2016) and Wilson (2018) have also used structural equations to articulate a counterfactual theory of metaphysical grounding. [ 14 ] Metaphysical grounding is a concept widely employed in metaphysics throughout its history, but has been the focus of intense attention only recently—see entry metaphysical grounding for further details. As Schaffer (2016) puts it, the fact that Koko the gorilla lives in California is not a fundamental fact because it is grounded in more basic facts about the physical world, perhaps facts about spacetime and certain physical fields. Statements articulating these grounding facts constitute distinct metaphysical explanations. So conceived, metaphysical grounding is among the most central concepts in metaphysics. The key proposals in Schaffer (2016) and Wilson (2018) are to use structural equations to model grounding relations, and not just causal relations, and in doing so capture parallels between causation and grounding. Indeed, they define grounding in terms of structural equations in the same way as the authors above defined causation in terms of structural equations. The key difference is that the equations articulate what grounds what. While this approach to grounding has its critics (e.g., Koslicki 2016 ), it is worth noting here since it places counterfactuals at the center of metaphysical explanations. [ 15 ] Counterfactuals have been implicated in other key metaphysical debates. Work on dispositions is a prominent example. A glass’s fragility is a curious property: the glass has it in virtue of possibly shattering in certain conditions, even if those conditions are never manifested in the actual world, unlike say, the glass’s shape. This dispositional property is quite naturally understood in terms of a counterfactual claim:

Early analyses of this form were pursued by Ryle (1949) , Quine (1960) , and Goodman (1955) , and have remained a major position in the literature on dispositions. See dispositions for further discussion and references.

It is not just metaphysical explanation where counterfactuals have been central. They also feature prominently in accounts of scientific explanation and laws of nature. Strict empiricists have attempted to characterize scientific explanation without reliance on counterfactuals, despite the fact that they tend to creep in—for further background on this see scientific explanation . Scientific explanations appeal to laws of nature, and laws of nature are difficult to separate from counterfactuals. Laws of nature are crucially different from accidental generalizations, but how? One prominent idea is that they “support counterfactuals”. As Chisholm (1955: 97) observed, the counterfactual (14) follows from the corresponding law (13) but the counterfactual (16) does not follow from the corresponding accidental generalization (15) .

A number of prominent views have emerged from pursuing this connection. Woodward (2003) argues that the key feature of an explanation is that it answers what-if-things-had-been-different questions, and integrates this proposal with a structural equations approach to causation and counterfactuals. [ 16 ] Lange (1999, 2000, 2009) proposes an anti-reductionist account of laws according to which they are identified by their invariance under certain counterfactuals. Maudlin (2007: Ch.1) also proposes an anti-reductionist account of laws, but instead uses laws to define the truth-conditions of counterfactuals relevant to physical explanations. For more on these views see laws of nature .

It should now be clear that a wide variety of central philosophical topics rely crucially on counterfactuals. This highlights the need to understand their semantics: how can we systematically specify what the world must be like if a given counterfactual is true and capture patterns of valid inference involving them? It turns out to be rather difficult to answer this question using the tools of classical logic, or even modal logic. This section will explain why.

Logical semantics (Frege 1893; Tarski 1936; Carnap 1948) provided many useful analyses of English connectives like and and not using Boolean truth-functional connectives like $\land$ and $\neg$. Unfortunately, such an analysis is not possible for counterfactuals. In truth-functional semantics, the truth of a complex sentence is determined by the truth of its parts because a connective’s meaning is modeled as a truth-function—a function from one or more truth-values to another. Many counterfactuals have false antecedents and consequents, but some are true and others false. (17a) is false—given Joplin’s critiques of consumerism—and (17b) is true.

It may be useful to state the issue a bit more precisely.

In truth-functional semantics, the truth-value (True/False: 1/0) of a complex sentence is determined by the truth-values of its parts and particular truth-function expressed by the connective. This is illustrated by the truth-tables for negation $\neg$, conjunction $\land$, and the material conditional $\supset$ in Figure 3 .

$\phi$	$\neg\phi$
1	0
0	1

$\phi$	$\psi$	$\phi\land\psi$
1	1	1
1	0	0
0	1	0
0	0	0

$\phi$	$\psi$	$\phi\supset\psi$
1	1	1
1	0	0
0	1	1
0	0	1

Figure 3: Negation ($\neg$), Conjunction ($\land$), Material Conditional ($\supset$)

Truth-functional logic is inadequate for counterfactuals not just because the material conditional $\supset$ does not capture the fact that some counterfactuals with false antecedents like (17a) are false. It is inadequate because there is, by definition, no truth-functional connective whatsoever that simultaneously combines two false sentences to make a true one like (17b) and combines two false ones to make a false one like (17a) . In contemporary philosophy, this is overwhelmingly seen as a failing of classical logic. But there was a time at which it fueled skepticism about whether counterfactuals really make true or false claims about the world at all. Quine ( 1960: §46, 1982: Ch.3 ) voices this skepticism and supports it by highlighting puzzling pairs like (18) and (19) :

Quine (1982: Ch.3) suggests that no state of the world could settle whether (19a) or (19b) is true. Similarly he contends that it is not the world, but sympathetically discerning the speaker’s imagination and purpose in speaking that matters for the truth of (18b) versus (18a) (Quine 1960: §46) . Rather than promoting skepticism about a semantic analysis of counterfactuals, Lewis (1973b: 67) took these examples as evidence that their truth-conditions are context-sensitive : the possibilities that are considered when evaluating the antecedent are constrained by the context in which the counterfactual is asserted, including the intentions and practical ends of the speaker. All contemporary accounts of counterfactuals incorporate some version of this idea. [ 17 ]

Perhaps the most influential semantic puzzle about counterfactuals was highlighted by Goodman (1947) , who noticed that adding more information to the antecedent can actually turn a true counterfactual into a false one. For example, (20a) could be true, while (20b) is false.

Lewis ( 1973c: 419; 1973b: 10 ) dramatized the problem by considering sequences such as (21) , where adding more information to the antecedent repeatedly flips the truth-value of the counterfactual.

The English discourse (21) is clearly consistent: it is nothing like saying I shirked my duty and I did not shirk my duty . This property of counterfactual antecedents is known by a technical name, non-monotonicity , and is one of the features all contemporary accounts are designed to capture. As will be discussed in §2.2 , even modal logic does not have the resources to capture semantically non-monotonic operators.

Goodman (1947) posed another influential problem. Examples (20a) and (20b) show that the truth-conditions of counterfactuals depend on assumed background facts like the presence of oxygen. However, a moment’s reflection reveals that specifying all of these background facts is quite difficult. The match must be dry, oxygen must be present, wind must be below a certain threshold, the friction between the striking surface and the match must be sufficient to produce heat, that heat must be sufficient to activate the chemical energy stored in the match head, etc. Further, counterfactuals like (20a) also rely for their truth on physical laws specific to our world, e.g., the conservation of energy. Goodman’s problem is this: it is difficult to adequately specify these background conditions and laws without further appealing to counterfactuals. This is clearest for laws. As discussed in §1.3 , some have aimed to distinguish laws from accidental generalizations by noting that only the former support counterfactuals. But if this is a defining feature of laws, and laws are part of the definition of when a counterfactual is true, circularity becomes a concern. Explicit analyses of laws in terms of counterfactuals, like Lange (2009) , would make an analysis of counterfactuals in terms of laws circular.

The potential circularity for background conditions takes a bit more explanation. Suppose one claims to have specified all of the background conditions relevant to the truth of (20a) , as in (22a) . Then it is tempting to say that (20a) is true because (22c) follows from (22a) , (22c) , and the physical laws.

But now suppose there is an agent seeing to it that a fire is not started, and will only strike the match if it is wet. In this case the counterfactual (20a) is intuitively false. However, unless one adds the counterfactual, if the match were struck, it would have to be wet , to the background conditions, (22c) still follows from (22a) , (22c) , and the physical laws. That would incorrectly predict the counterfactual to be true. In short, it seems that the background conditions must themselves consist of counterfactuals. Any analysis of counterfactuals that captures their sensitivity to background facts must either eliminate these appeals to counterfactuals, or show how this appeal is non-circular, e.g., part of a recursive, non-reductive analysis.

To summarize, this section has identified three key theses about the semantics of counterfactuals and a central problem:

Counterfactuals are not truth-functional.
Counterfactuals have context-sensitive truth-conditions.
Counterfactual antecedents are interpreted non-monotonically.
Goodman’s Problem The truth-conditions of counterfactuals depend on background facts and laws. It is challenging to specify these facts and laws in general, but particularly difficult to specify them in non-counterfactual terms.

These theses, along with Goodman’s Problem, were once grounds for skepticism about the coherence of counterfactual discourse. But with advances in semantics and pragmatics, they have instead become the central features of counterfactuals that contemporary analyses aim to capture.

2. The Logic of Counterfactuals

This section will survey two semantic analyses of counterfactuals: the strict conditional analysis and the similarity analysis. These conceptually related analyses also have a shared explanatory goal: to capture logically valid inferences involving counterfactuals, while treating them non-truth-functionally, leaving room for their context dependence, and addressing the non-monotonic interpretation of counterfactual antecedents. Crucially, these analyses abstract away Goodman’s Problem because they are not primarily concerned with the truth-conditions of particular counterfactuals—just as classical logic does not take a stand on which atomic sentences are actually true. Instead, they say only enough about truth-conditions to settle matters of logic, e.g., if $\phi$ and $\phi>\psi$ are true, then $\psi$ is true. Sections 2.5 and 2.6 will revisit questions about the truth-conditions of particular counterfactuals, Goodman’s Problem and the philosophical projects surveyed in §1 .

The following subsections will detail strict conditional and similarity analyses. But it is useful at the outset to consider simplified versions of these two analyses alongside each other. This will clarify their key differences and similarities. Both analyses are also stated in the framework of possible world semantics developed in Kripke (1963) for modal logics. The following subsection provides this background and an overview of the two analyses.

The two key concepts in possible worlds semantics are possible worlds and accessibility spheres (or relations). Intuitively, a possible world w is simply a way the world could be or could have been. Formally, they are treated as primitive points in the set of all possible worlds W . But their crucial role comes in assigning truth-conditions to sentences: a sentence $\phi$ can only said to be true given a possible world w , but since w is genuinely possible, it cannot be the case that both $\phi$ and $\neg\phi$ are true at w . Accessibility spheres provide additional structure for reasoning about what’s possible: for each world w , $R(w)$ is the set of worlds accessible from w . [ 18 ] This captures the intuitive idea that given a possible world w , a certain range of other worlds $R(w)$ are possible, in a variety of senses. $R_1(w)$ might specify what’s nomologically possible in w by including only worlds where w ’s natural laws hold, while $R_2(w)$ specifies what’s metaphysically possible in w .

These tools furnish truth-conditions for a formal language including non-truth-functional necessity (${{\medsquare}}$) and possibility (${{\meddiamond}}$) operators: [ 19 ]

Definition 6 (Kripkean Semantics) \[ \begin{align} {\llbracket A\rrbracket}^{R}_v & =\set{w\mid v(w,A)=1} & \tag*{1.}\\ {\llbracket\neg\phi\rrbracket}^{R}_v & = W-{\llbracket\phi\rrbracket}^{R}_v & \tag*{2.}\\ {\llbracket\phi\land\psi\rrbracket}^{R}_v & ={\llbracket\phi\rrbracket}^{R}_v\cap{\llbracket\psi\rrbracket}^{R}_v & \tag*{3.}\\ {\llbracket\phi\lor\psi\rrbracket}^{R}_v & ={\llbracket\phi\rrbracket}^{R}_v\cup{\llbracket\psi\rrbracket}^{R}_v & \tag*{4.}\\ {\llbracket\phi\supset\psi\rrbracket}^{R}_v & =(W-{\llbracket\phi\rrbracket}^{R}_v)\cup{\llbracket\psi\rrbracket}^{R}_v & \tag*{5.}\\ {\llbracket\medsquare\phi\rrbracket}^{R}_v & =\set{w\mid R(w)\subseteq{\llbracket\phi\rrbracket}^{R}_v} & \tag*{6.}\\ {\llbracket\meddiamond\phi\rrbracket}^{R}_v & =\set{w\mid R(w)\cap{\llbracket\phi\rrbracket}^{R}_v\neq\emptyset} & \tag*{7.} \end{align} \]

In classical logic, the meaning of $\phi$ is simply its truth-value. But in modal logic, it is the set of possible worlds where $\phi$ is true: ${\llbracket}\phi{\rrbracket}$. So $\phi$ is true in w , relative to v and R , just in case $w\in{\llbracket}\phi{\rrbracket}^R_v$:

Definition 7 (Truth) $w,v,R\vDash \phi \iff w\in{\llbracket}\phi{\rrbracket}^R_v$

Only clauses 6 and 7 rely crucially on this richer notion of meaning. ${{\medsquare}}\phi$ says that in all accessible worlds $R(w)$, $\phi$ is true. ${{\meddiamond}}\phi$ says that there are some accessible worlds where $\phi$ is true. Logical concepts like consequence are also defined in terms of relations between sets of possible worlds. The intersection of the premises must be a subset of the conclusion (i.e., every world where the premises are true, the conclusion is true):

Definition 8 (Logical Consequence) $\phi_1,{\ldots},\phi_n\vDash\psi \iff \forall R,v{{:\thinspace}}({\llbracket}\phi_1{\rrbracket}^R_v\cap\cdots\cap{\llbracket}\phi_n{\rrbracket}^R_v)\subseteq{\llbracket}\psi{\rrbracket}^R_v$

Given this framework, the strict analysis can be formulated very simply: $\phi > \psi$ should be analyzed as ${{\medsquare}}(\phi\supset\psi)$. This says that all accessible $\phi$-worlds are $\psi$-worlds. This analysis can be depicted as in Figure 4 . [ 20 ]

Figure 4: Truth in $w_0$ relative to R . [An extended description of figure 4 is in the supplement.]

The red circle delimits the worlds accessible from $w_0$, the x -axis divides $\phi$ and $\neg\phi$-worlds, and the y -axis $\psi$ and $\neg\psi$-worlds. ${{\medsquare}}(\phi\supset\psi)$ says that there are no worlds in the blue shaded region.

It is crucial to highlight that this semantics does not capture the non-monotonic interpretation of counterfactual antecedents. For example, ${\llbracket}\mathsf{A}\land \mathsf{B}{\rrbracket}^R_v$ is a subset of ${\llbracket}\mathsf{A}{\rrbracket}$, and this means that any time ${{\medsquare}}(\mathsf{A\supset C})$ is true, so is ${{\medsquare}}(\mathsf{(A\land B)\supset C})$. After all, if all $\mathsf{A}$-worlds are in the red quadrant of Figure 4 , so are all of the $\mathsf{A\land B}$-worlds, since the $\mathsf{A\land B}$-worlds are just a subset of the $\mathsf{A}$-worlds. A crucial point here is that on this semantics the domain of worlds quantified over by a counterfactual is constant across counterfactuals with different antecedents. As will be discussed in §2.2 , advocates of strict conditional analyses aim to instead capture the non-monotonic behavior of antecedents pragmatically by incorporating it into a model of their context-sensitivity. The most important difference between strict analyses and similarity analyses is that similarity analyses capture this non-monotonicity semantically.

On the similarity analysis, $\phi >\psi$ is true in $w_0$, roughly, just in case all the $\phi$-worlds most similar to $w_0$ are $\psi$-worlds. To model this notion of similarity, one needs more than a simple accessibility sphere. One way to capture it is with with a nested system of spheres $\mathcal{R}$ around a possible world $w_0$ (D. Lewis 1973b: §1.3) —this is just a particular kind of set of accessibility spheres. As one goes out in the system, one gets to less and less similar worlds. This analysis can be depicted as in Figure 5 . [ 21 ]

Figure 5: Truth in $w_0$ relative to $\mathcal{R}$. [An extended description of figure 5 is in the supplement.]

The most similar $\phi$-worlds are in the innermost gray region. So, this analysis excludes any worlds from being in the shaded innermost blue region. Comparing Figures 4 and 5 , one difference stands out: the similarity analyses does not require that there be no $\phi\land\neg\psi$-worlds in any sphere, just in the innermost sphere. For example, world $w_1$ does not prevent the counterfactual $\phi >\psi$ from being true. It is not in the $\phi$-sphere most similar to w . This is the key to semantically capturing the non-monotonic interpretation of antecedents. The truth of $\mathsf{A > C}$ does not guarantee the truth of $\mathsf{(A\land B)> C}$ precisely because the most similar $\mathsf{A}$-worlds may be in the innermost sphere, and the most similar $\mathsf{A\land B}$ may be in an intermediate sphere, and include worlds like $w_1$ where the consequent is false. In this sense, the domain of worlds quantified over by a similarity-based counterfactual varies across counterfactuals with different antecedents, though it does express a strict conditional over this varying domain. For this reason, D. Lewis (1973b) and many others call the similarity analysis a variably-strict analysis.

Since antecedent monotonicity is the key division between strict and similarity analyses, it is worthwhile being a bit more precise about what it is, and what its associated inference patterns are.

Definition 9 (Antecedent Monotonicity) If $\phi_1>\psi$ is true at some $w,R,v$ and ${\llbracket}\phi_2{\rrbracket}^R_v\subseteq{\llbracket}\phi_1{\rrbracket}^R_v$, then $\phi_2 >\psi$ is true at $w,R,v$.

The crucial patterns associated with antecedent monotonicity are:

Antecedent Strengthening (AS) $\phi_1>\psi\vDash (\phi_1\land\phi_2)>\psi$
Simplification of Disjunctive Antecedents (SDA) $(\phi_1\lor\phi_2)>\psi\vDash (\phi_1>\psi)\land(\phi_2>\psi)$
Transitivity $\phi_2>\phi_1,\phi_1 > \psi\vDash \phi_2>\psi$
Contraposition $\phi>\psi\,\leftmodels\vDash \neg\psi>\neg\phi$

AS and SDA clearly follow from antecedent monotonicity. By contrast, Transitivity and a plausible auxiliary assumption entail antecedent monotonicity, [ 22 ] and the same is true for Contraposition. [ 23 ] With these basics in place, it is possible to focus in on each of these analyses in more detail. In doing so, it will become clear that there are important differences even among variants of the similarity analysis and variants of the strict analysis. This entry will focus on what these analyses predict about valid inferences involving counterfactuals.

2.2 Strict Conditional Analyses

The strict conditional analysis has a long history, but its contemporary form was first articulated by Peirce: [ 24 ]

“If A is true then B is true”… is expressed by saying, “In any possible state of things, [ w ], either $[A]$ is not true [in w ], or $[B]$ is true [in w ]”. (Peirce 1896: 33)

C.I. Lewis (1912, 1914) defended the strict conditional analysis of subjunctives and developed an axiomatic system for studying their logic, but offered no semantics. A precise model-theoretic semantics for the strict conditional was first presented in Carnap (1956: Ch. 5) . However, that account did not appeal to accessibility relations, and ranged only over logically possible worlds. Since counterfactuals are often non-logical, it it was only after Kripke (1963) introduced a semantics for modal logic featuring an accessibility relation, that the modern form of the strict analysis was precisely formulated: [ 25 ]

${\llbracket\phi>\psi\rrbracket}^R_v= {\llbracket\medsquare(\phi\supset\psi)\rrbracket}^R_v$
I.e., all $\phi$-worlds in $R(w)$ are $\psi$-worlds
$ \begin{align*} \llbracket\medsquare(\phi\supset\psi)\rrbracket^{R} & =\{w\mid R(w)\subseteq\llbracket\phi\supset\psi\rrbracket^{R}_v\} & \\ & =\{w\mid (R(w)\cap\llbracket\phi\rrbracket^{R}_v)\subseteq\llbracket\psi\rrbracket^{R}_v\}& \\[-18px] \end{align*} $
$\phi\strictif\psi\mathbin{:=}\medsquare(\phi\supset\psi)$

Just as the logic of ${{\medsquare}}$ will vary with constraints that can be placed on R , so too will the logic of strict conditionals. [ 26 ] For example, if one does not assume that $w\in R(w)$ then modus ponens will not hold for the strict conditional: $\psi$ will not follow from $\phi$ and ${{\medsquare}}(\phi\supset\psi)$. But even without settling these constraints, some basic logical properties of the analysis can be established. The discussion to follow is by no means exhaustive. [ 27 ] Instead, it will highlight the logical patterns which are central to the debates between competing analyses.

The core idea of the basic strict analysis leads to the following validities.

$\vDash(\phi\land\neg\phi)\strictif\psi$
$\vDash\psi\strictif(\phi\lor\neg\phi)$
$\neg{{\meddiamond}}\phi\vDash\phi\strictif\psi$
${{\medsquare}}\psi\vDash\phi\strictif\psi$

In these validities, some see a plausible and attractive logic (C.I. Lewis 1912, 1914) . Others see them as “so utterly devoid of rationality [as to be] a reductio ad absurdum of any view which involves them” ( Nelson 1933: 271) , earning them the title paradoxes of strict implication . Patterns 3 and 4 are more central to debates about counterfactuals, so they will be the focus here. Pattern 3 clearly follows from the core idea of the basic strict analysis : the premise guarantees that there are no accessible $\phi$-worlds, from which it vacuously follows that all accessible $\phi$-worlds are $\psi$-worlds. Much the same is true of pattern 4: if all the accessible worlds are $\psi$-worlds then all the accessible $\phi$-worlds are $\psi$-worlds. Both 3 and 4 are seem incorrect for English counterfactuals.

Contrary to pattern 3, the false (23b) does not intuitively follow from the true (23a) . Similarly, for pattern 4. Suppose one’s origin from a particular sperm and egg is an essential feature of oneself. Then (24a) is true.

And, yet, many would hesitate to infer (24b) on the basis of (24a) . Each of these patterns follow from the core idea of the strict analysis. While these counterexamples may not constitute a conclusive objection, they do present a problem for the basic strict analysis. The second wave strict analyses surveyed in §2.2.1 are designed to solve it, however. They are also designed to address another suite of validities that are even more problematic.

The strict analysis is widely criticized for validating antecedent monotonic patterns. It is worth saying a bit more precisely, using Definition 9 and Figure 6 , why antecedent monotonicity holds for the strict conditional.

Figure 6: Strict Conditionals are Antecedent Monotonic. [An extended description of figure 6 is in the supplement.]

If $\phi_1\strictif\psi$ is true, then the shaded blue region is empty, and the position of $\phi_2$ reflects the fact that ${\llbracket}\phi_2{\rrbracket}^R_v\subseteq{\llbracket}\phi_1{\rrbracket}^R_v$—recall that all worlds above the x -axis are $\phi_1$-worlds. Since the shaded blue region within $\phi_2$ is also empty, all $\phi_2$ worlds in $R(w)$ are $\psi$-worlds. That is, $\phi_2\strictif \psi$ is true.

Recall that Transititivity and Contraposition entail antecedent monotonicity , so it remains to show that both hold for the strict conditional. To see why Contraposition holds for the strict conditional, note again that if $\phi\strictif\psi$ is true in w , then all $\phi$-worlds in $R(w)$ are $\psi$-worlds, as depicted in the left Venn diagram in Figure 7 . Now suppose w is a $\neg\psi$-world in $R(w)$. As the diagram makes clear, w has to be a $\neg\phi$-world, and so $\neg\psi\strictif\neg\phi$ must be true in w . Similarly, if $\neg\psi\strictif\neg\phi$ is true in w , then all $\neg\psi$-worlds in $R(w)$ are $\neg\phi$-worlds, as depicted in the right Venn diagram in Figure 7 . Now suppose w is a $\phi$-world in $R(w)$. As depicted, w has to be a $\psi$-world, and so $\phi\strictif\psi$ must be true in w .

Figure 7: $w\in {\llbracket\phi\strictif\psi\rrbracket}^R_v \iff w\in {\llbracket\neg\psi\strictif\neg\phi\rrbracket}^R_v$ (Contraposition). [An extended description of figure 7 is in the supplement.]

The validity of Transititivity for the strict conditional is also easy to see with a Venn diagram.

Figure 8: $w\in{\llbracket\phi_2\strictif\phi_1\rrbracket}^R_v\cap{ \llbracket\phi_1\strictif\psi\rrbracket}^R_v\Leftrightarrow w\in{\llbracket\phi_2\strictif\psi\rrbracket}^R_v$ (Transitivity). [An extended description of figure 8 is in the supplement.]

The premises guarantee that all $\phi_2$-worlds in $R(w)$ are $\phi_1$-worlds, and that all $\phi_1$-worlds in $R(w)$ are $\psi$-worlds. That gives one the relationships depicted in Figure 8 . To show that $\phi_2\strictif\psi$ follows, suppose that w is a $\phi_2$-world in $R(w)$. As Figure 8 makes evident, w must then be a $\psi$-world.

Antecedent monotonic patterns are an ineliminable part of a strict conditional logic. Examples of them often sound compelling. For example, the transitive inference (25) sounds perfectly reasonable, as does the antecedent strengthening inference (26) .

Similar examples for SDA are easy to find. However, counterexamples to each of the four patterns have been offered.

Counterexamples to Antecedent Strengthening were already discussed back in §1.4 . Against Transititivity , Stalnaker (1968: 48) points out that (27c) does not intuitively follow from (27a) and (27b) .

Contra Contraposition , D. Lewis (1973b: 35) presents (28) .

Suppose Boris wanted to go, but stayed away to avoid Olga. Then (28b) is false. Further suppose that Olga would have been even more excited to attend if Boris had. In that case (28a) is true. Against SDA , Mckay & van Inwagen (1977: 354) offer:

(29b) does not intuitively follow from (29a) .

These counterexamples have been widely taken to be conclusive evidence against the strict analysis (e.g., D. Lewis 1973b; Stalnaker 1968 ), since they follow from the core assumptions of that analysis. As a result, D. Lewis (1973b) and Stalnaker (1968) developed similarity analyses which build the non-monotonicity of antecendents into the semantics of counterfactuals—see §2.3 . However, there was a subsequent wave of strict analyses designed to systematically address these counterexamples. In fact, they do so by unifying two features of counterfactuals: the non-monotonic interpretation of their antecedents and their context-sensitivity.

Beginning with Daniels and Freeman (1980) and Warmbrōd (1981a,b) , there was a second wave of strict analyses developed explicitly to address the non-monotonic interpretation of counterfactual antecedents. Warmbrōd (1981a,b) , Lowe (1983, 1990) , and Lycan (2001) account for the counterexamples to antecedent monotonic patterns within a systematic theory of how counterfactuals are context-sensitive. More recently, Gillies (2007) has argued that a strict analysis along those lines is actually preferable to an account that builds the non-monotonicity of counterfactual antecedents into their semantics, i.e., similarity analyses. This section will outline the basic features of these second wave strict conditional analyses.

The key idea in Warmbrōd (1981a,b) is that the accessibility sphere in the basic strict analysis should be viewed as a parameter of the context. Roughly, the idea is that $R(w)$ corresponds to background facts assumed by the participants of a discourse context. For example, if they are assuming propositions (modeled as sets of possible worlds) A , B , and C then $R(w)=A\cap B\cap C$. The other key idea is that trivial strict conditionals are not pragmatically useful in conversation. If a strict conditional $\mathsf{A\strictif C}$ is asserted in a context with background facts $R(w)$ and $\mathsf{A}$ is inconsistent with $R(w)$—${\llbracket}\mathsf{A}{\rrbracket}^R_v\cap R(w)={\emptyset}$, then asserting $\mathsf{A\strictif C}$ does not provide any information. If there are no $\mathsf{A}$-worlds in $R(w)$, then, trivially, all $\mathsf{A}$-worlds in $R(w)$ are $\mathsf{C}$-worlds. Warmbrōd (1981a,b) proposes that conversationalists adapt a pragmatic rule of charitable interpretation to avoid trivialization:

On this view, $R(w)$ may very well change over the course of a discourse as a result of conversationalists adhering to (P) . This part of the view is central to explaining away counterexamples to antecedent monotonic validities.

Consider again the example from Goodman 1947 that appeared to be a counterexample to Antecedent Strengthening .

Now note that if (30a) is going to come out true, the proposition that there is oxygen in the room O must be true in all worlds in the initial accessibility sphere $R_0(w)$. However, if (30b) is interpreted against $R_0(w)$, the antecedent will be inconsistent with $R_0(w)$ and so express a trivial, uninformative proposition. Warmbrōd (1981a,b) proposes that in interpreting (30b) we are forced by to adopt a new, modified accessibility sphere $R_1(w)$ where O is no longer assumed. But if this is right, (30a) and (30b) don’t constitute a counterexample to Antecedent Strengthening because they are interpreted against different accessibility spheres. It’s like saying All current U.S. presidents are intelligent doesn’t entail All current U.S. presidents are unintelligent because this sentence before Donald Trump was sworn in was true, but uttering it afterwards was false. There is an equivocation of context, or so Warmbrōd (1981a,b) contends.

Warmbrōd (1981a,b) outlines parallel explanations of the counterexamples presented to SDA , Contraposition , and Transititivity . This significantly complicates the issue of whether antecedent monotonicity is the key issue in understanding the semantics of counterfactuals. It appears that the non-monotonic interpretation of counterfactual antecedents can either be captured pragmatically in the way that accessibility spheres change in context (Warmbrōd 1981a,b) , or it can be captured semantically as we will see from similarity analyses in §2.3 . There are significant limitations to Warmbrōd’s ( (1981a,b) ) analysis: it does not capture nested conditionals, and does not actually predict how $R(w)$ evolves to satisfy (P) . Fintel (2001) and Gillies (2007) offer accounts that remove these limitations, and pose a challenge for traditional similarity analyses.

Fintel (2001) and Gillies (2007) propose analyses where counterfactuals have strict truth-conditions, but they also have a dynamic meaning which effectively changes $R(w)$ non-monotonically. They argue that such a theory can better explain particular phenomena. Chief among them is reverse Sobel sequences. Recall the sequence of counterfactuals (21) presented by Lewis ( 1973b, 1973c: 419 ), and attributed to Howard Sobel. Reversing these sequences is not felicitous:

Fintel (2001) and Gillies (2007) observe that similarity analyses render sequences like (31) semantically consistent. Their theories predict this infelicity by providing a theory of how counterfactuals in context can change $R(w)$. Unlike Fintel (2001) , Gillies (2007) does not rely essentially on a similarity ordering over possible worlds to compute these changes to $R(w)$, and so clearly counts as a second wave strict analysis. [ 28 ] The debate over whether counterfactuals are best given a strict or similarity analysis is very much ongoing. Moss (2012) , Starr (2014) , and K. Lewis (2018) have proposed three different ways of explaining reverse Sobel sequences within a similarity analysis. But Willer (2015, 2017, 2018) has argued on the basis of other data that a dynamic second wave strict analysis is preferable. This argument takes one into a logical comparison of strict and similarity analyses, which will be taken up in §2.4 after the similarity analysis has been presented in more detail.

Recall the rough idea of the similarity analysis sketched in §2.1 : worlds can be ordered by their similarity to the actual world, and counterfactuals say that the most similar—or least different—worlds where the antecedent is true are worlds where the consequent is also true. This idea is commonly attributed to David Lewis and Robert Stalnaker, but the actual history is a bit more nuanced. Although publication dates do not tell the full story, the approach was developed roughly contemporaneously by Stalnaker (1968) , Stalnaker and Thomason (1970) , D. Lewis (1973b) , Nute (1975b) , and Sprigge (1970) . [ 29 ] And, there is an even earlier statement of the view:

When we allow for the possibility of the antecedent’s being true in the case of a counterfactual, we are hypothetically substituting a different world for the actual one. It has to be supposed that this hypothetical world is as much like the actual one as possible so that we will have grounds for saying that the consequent would be realized in such a world. (Todd 1964: 107)

Recall the major difference between this proposal and the basic strict analysis : the similarity analysis uses a graded notion of similarity instead of an absolute notion of accessibility. It also allows most similar worlds to vary between counterfactuals with different antecedents. These differences invalidate antecedent monotonic inference patterns. This section will introduce similarity analyses in a bit more formal detail and describe the differences between analyses within this family.

The similarity analysis has come in many varieties and formulations, including the system of spheres approach informally described in §2.1 . That formulation is easiest for comparison to strict analyses. But there is a different formulation that is more intuitive and better facilitates comparison among different similarity analyses. This formulation appeals to a (set) selection function f , which takes a world w , a proposition p , and returns the set of p -worlds most similar to w : $f(w,p)$. [ 30 ] $\phi>\psi$ is then said to be true when the most f -similar $\phi$-worlds to w are $\psi$-worlds, i.e., every world in $f(w,{\llbracket}\phi{\rrbracket}^f_v)$ is in ${\llbracket}\psi{\rrbracket}^f_v$. The basics of this approach can be summed up thus.

Most similar according to the selection function f
f takes a proposition p and a world w and returns the p -worlds most similar to w
$\llbracket\phi > \psi\rrbracket^{f}_v=\{w\mid f(w,\llbracket\phi\rrbracket^{f}_v)\subseteq\llbracket\psi\rrbracket^{f}_v\}$
Making “limit assumption”: $\phi$-worlds do not get indefinitely more and more similar to w

As noted, this formulation makes the limit assumption: $\phi$-worlds do not get indefinitely more and more similar to w . While D. Lewis (1973b) rejected this assumption, adopting it will serve exposition. It is discussed at length in the supplement Formal Constraints on Similarity . The logic of counterfactuals generated by a similarity analysis will depend on the constraints imposed on f . Different theorists have defended different constraints. Table 1 lists them, where $p,q\subseteq W$ and $w\in W$:

(a)	$f(w,p)\subseteq p$
(b)	$f(w,p)=\{w\}$, if $w\in p$
(c)	$f(w,p)\subseteq q$ & $f(w,q)\subseteq p \; {\Longrightarrow}\; f(w,p)=f(w,q)$
(d)	$f(w,p)$ contains one world

Table 1: Candidate Constraints on Selection Functions

Modulo the limit assumption, Table 2 provides an overview of which analyses have adopted which constraints.


Pollock (1976)	success, strong centering
D. Lewis (1973b), Nute (1975a)	success, strong centering, uniformity
Stalnaker (1968)	success, strong centering, uniformity, uniqueness

Table 2: Similarity Analyses, modulo Limit Assumption

simply enforces that $f(w,p)$ is indeed a set of p -worlds. Recall that $f(w,p)$ is supposed to be the set of most similar p -worlds to w . The other constraints correspond to certain logical validities, as detailed in the supplement Formal Constraints on Similarity . This means that Pollock (1976) endorses the weakest logic for counterfactuals and Stalnaker (1968) the strongest. It is worth seeing how, independently of constraints (b)–(d), this semantics invalidates an antecedent monotonicity pattern like Antecedent Strengthening .

Consider an instance of Antecedent Strengthening involving $\mathsf{A > C}$ and $\mathsf{(A\land B)>C}$, and where the space of worlds is that given in Table 3 .

World	$\mathsf{A}$	$\mathsf{B}$	$\mathsf{C}$
$w_{0}$	1	1	1
$w_{1}$	1	1	0
$w_{2}$	1	0	1
$w_{3}$	1	0	0
$w_{4}$	0	1	1
$w_{5}$	0	1	0
$w_{6}$	0	0	1
$w_{7}$	0	0	0

Table 3: A space of worlds W , and truth-values at each world

Now evaluate $\mathsf{A > C}$ and $\mathsf{(A\land B)>C}$ in $w_{5}$ using a selection function $f_1$ with the following features:

$f_1(w_{5},{\llbracket}\mathsf{A}{\rrbracket}^{f_1}_v)=\{w_{2}\}$

$f_1(w_{5},{\llbracket}\mathsf{A}\land\mathsf{B}{\rrbracket}^{f_1}_v)=\{w_{1}\}$

Since $\mathsf{C}$ is true in $w_{2}$, $\mathsf{A > C}$ is true in $w_{5}$ according to $f_1$. But, since $\mathsf{C}$ is false in $w_{1}$, $\mathsf{(A\land B) > C}$ is false in $w_{5}$ according to $f_1$. No constraints are needed here other than success . While $f_1$ satisfies uniqueness , the counterexample works just as well if, say, $f_1(w_{5},{\llbracket}\mathsf{A}{\rrbracket}^{f_1}_v)=\{w_{2},w_0\}$. Accordingly, all similarity analyses allow for the non-monotonic interpretation of counterfactual antecedents.

While Stalnaker (1968) and D. Lewis (1973b) remain the most popular similarity analyses, there are substantial logical issues which separate similarity analyses. These issues, and the constraints underlying them, are detailed in the supplement Formal Constraints on Similarity . Table 4 summarizes which validities go with which constraints.


Strong Centering	$\phi>\psi, \phi\vDash \psi$ $\phi\land\psi \vDash \phi>\psi$
Uniformity	$\phi_1>\phi_2,\phi_2>\phi_1,\phi_1>\psi\vDash \phi_2>\psi$ $\phi_1>\phi_2,(\phi_1\land\phi_2)>\psi\vDash \phi_1>\psi$ $\phi_1>\phi_2,\neg(\phi_1>\neg\psi)\vDash(\phi_1\land\phi_2)>\psi$
Uniqueness	$\vDash (\phi>\psi)\lor(\phi>\neg\psi)$ ${\Diamond}\phi\land\neg(\phi>\psi)\,\leftmodels\vDash {\Diamond}\phi\land\phi>\neg\psi$ $\phi>(\psi_1\lor\psi_2)\vDash(\phi>\psi_1)\lor(\phi>\psi_2)$
Limit Assumption	If $\Gamma={\{\phi_2,\phi_3,{\ldots}\}}$, $\phi_1>\phi_2,\phi_1>\phi_3,{\ldots}$ are true and $\Gamma\vDash\psi$ then $\phi_1>\psi$

Table 4: Selection Constraints & Associated Validities

A few comments are in order here, though. Strong centering is sufficient but not necessary for Modus Ponens, weak centering would do: $w\in f(w,p)$ if $w\in p$. LT and LAS follow from SSE, and allow similarity theorists to say why some instances of Transititivity and Antecedent Strengthening are intuitively compelling.

The issue of whether a second wave strict analysis ( §2.2.1 ) or a similarity analysis provides a better logic of counterfactuals is very much an open and subtle issue. As sections 2.2.1 and 2.3 detailed, both analyses have their own way of capturing the non-monotonic interpretation of antecedents. Both analyses also have their own way of capturing instances of monotonic inferences that do sound good. Perhaps this issue is destined for a stalemate. [ 31 ] But before declaring it such, it is important to investigate two patterns that are potentially more decisive: Simplification of Disjunctive Antecedents , and a pattern not yet discussed called Import-Export .

Both SDA and Import-Export are valid in a strict analyses and invalid on standard similarity analyses. Crucially, the counterexamples to them that have been offered by similarity theorists are significantly less compelling than those offered to patterns like Antecedent Strengthening . Import-Export relates counterfactuals like (33a) and (33b) .

It is hard to imagine one being true without the other. The basic strict analysis agrees: it renders them equivalent.

Import-Export $(\phi_1\land\phi_2)>\psi\,\leftmodels\vDash \phi_1>(\phi_2>\psi)$

But it is not valid on a similarity analysis . [ 32 ] While Import-Export is generally regarded as a plausible principle, some have challenged it. Kaufmann (2005: 213) presents an example involving indicative conditionals which can be adapted to subjunctives. Consider a case where there is a wet match which will light if tossed in the campfire, but not if it is struck. It has not been lit. Consider now:

One might then deny (34a) . This match would not have lit if it had been struck, and if it had lit it would have to have been thrown into the campfire. (34b) , on the other hand, seems like a straightforward logical truth. However, it is worth noting that this intuition about (34a) is very fragile. The slight variation of (34a) in (35) is easy to hear as true.

This subtle issue may be moot, however. Starr (2014) shows that a dynamic semantic implementation of the similarity analysis can validate Import-Export , so it may not be important for settling between strict and similarity analyses.

As for the Simplification of Disjunctive Antecedents (SDA) , Fine (1975) , Nute (1975b) , Loewer (1976) , and Warmbrōd (1981) each object to the similarity analysis predicting that this pattern is invalid. Counterexamples like (29) from Mckay & van Inwagen 1977: 354) have a suspicious feature.

Starr (2014: 1049) and Warmbrōd (1981a: 284) observe that (29a) seems to be another way of saying that Spain would never have fought for the Allies. While Warmbrōd (1981a: 284) uses this to pragmatically explain-away this counterexample to his strict analysis, Starr (2014: 1049) makes a further critical point: it sounds inconsistent to say (29a) after asserting that Spain could have fought for the Allies.

Starr (2014: 1049) argues that this makes it inconsistent for a similarity theorist to regard this as a counterexample to SDA . On a similarity analysis of the could claim, it follows that there are no worlds in which Spain fought for the Allies most similar to the actual world: $f(w_@,{\llbracket}\mathsf{Allies}{\rrbracket})={\emptyset}$. But if that’s the case, then (29b) is vacuously true on a similarity analysis, and so a similarity theorist cannot consistently claim that this is a case where the premise is true and conclusion false. It is, however, too soon for the strict theorist to declare victory. Nute (1980a) , Alonso-Ovalle (2009) , and Starr (2014: 1049) each develop similarity analyses where disjunction is given a non-Boolean interpretation to validate SDA without validating the other antecedent monotonic patterns. But even this is not the end of the SDA debate.

Nute (1980b: 33) considers a similar antecedent simplification pattern involving negated conjunctions:

Simplification of Negated Conjunctive Antecedents (SNCA) $\neg(\phi_1\land \phi_2)>\neg \psi\vDash (\neg\phi_1>\psi)\land(\neg\phi_2>\psi)$

Nute (1980b: 33) presents (37) in favor of SNCA.

Note that $\mathsf{\neg(N\land A)}$ and $\mathsf{\neg N\lor\neg A}$ are Boolean equivalents. However, non-Boolean analyses like Nute (1980a) , Alonso-Ovalle (2009) , and Starr (2014: 1049) designed to capture SDA break this equivalence, and so fail to predict that SNCA is valid. Willer (2015, 2017) develops a dynamic strict analysis which validates both SDA and SNCA. Fine (2012a,b) advocates for a departure from possible worlds semantics altogether in order to capture both SDA and SNCA. However, these accounts also face counterexamples. Fine (2012a,b) and Willer (2015, 2017) render $(\neg\phi_1\lor\neg\phi_2)>\psi$ and $\neg(\phi_1\land\phi_2)>\psi$ equivalent, while Champollion, Ciardelli, and Zhang (2016) present a powerful counterexample to this equivalence.

Champollion, Ciardelli, and Zhang (2016) consider a light which is on when switches A and B are both up, or both down. Currently, both switches are up, and the light is on. Consider (38a) and (38b) whose antecedents are Boolean equivalents:

While (38a) is intuitively true, (38b) is not. [ 33 ] This is not a counterexample to SNCA , since the premise of that pattern is false. But such a counterexample is not hard to think up. [ 34 ]

Suppose the baker’s apprentice completely failed at baking our cake. It was burnt to a crisp, and the thin, lumpy frosting came out puke green. The baker planned to redecorate it to make it at least look delicious, but did not have time. We may explain our extreme dissatisfaction by asserting (39a) . But the baker should not infer (39b) and assume that his redecoration plan would have worked.

Willer (2017: §4.2) suggests that such a counterexample trades on interpreting $\mathsf{\neg(B\land U)>H}$ as $\mathsf{\neg B\land\neg U)>H}$, and provides an independent explanation of this on the basis of how negation and conjunction interact. If this is right, then an analysis which validates SDA and SNCA without rendering $\neg(\phi_1\land\phi_2)>\psi$ and $\neg\phi_1\lor\neg\phi_2>\psi$ equivalent is what’s needed. Ciardelli, Zhang, and Champollion (forthcoming) develop just such an analysis. As Ciardelli, Zhang, and Champollion (forthcoming: §6.4) explain, SDA and SNCA turn out to be valid for very different reasons. Champollion, Ciardelli, and Zhang (2016) and Ciardelli, Zhang, and Champollion (forthcoming) also argue that the falsity of (38b) cannot be predicted on a similarity analysis. This example must be added to a long list of examples which have been presented not as counterexamples to the logic of the similarity analysis, but to what it predicts (or fails to predict) about the truth of particular counterfactuals in particular contexts. This will be the topic of §2.5 , where it will also be explained why the strict analysis faces similar challenges.

Where does this leave us in logical the debate between strict and similarity analyses of counterfactuals? Even Import-Export and SDA fail to clearly identify one analysis as superior. It is possible to capture SDA on either analysis. Existing similarity analyses that validate SDA , however, also invalidate SNCA (Alonso-Ovalle 2009; Starr 2014) . By contrast existing strict analyses that validate SDA also validate SNCA (Willer 2015, 2017) . However, this is far from decisive. The validity of SNCA is still being investigated, and it is far from clear that it is impossible to have a similarity analysis that validates both SDA and SNCA, or a strict analysis that validates only SDA (perhaps using a non-Boolean semantics for disjunction). So even SNCA may fail to be the conclusive pattern needed to separate these analyses.

2.5 Truth-Conditions Revisited

In their own ways, Stalnaker (1968, 1984) and D. lewis (1973b) are candid that the similarity analysis is not a complete analysis of counterfactuals. As should be clear from §2.3 , the formal constraints they place on similarity are quite minimal and only serve to settle matters of logic. There are, in general, very many possible selection functions—and corresponding conceptions of similarity—for any given counterfactual. To explain how a given counterfactual like (40) expresses a true proposition, a similarity analysis must specify which particular conception of similarity informs it.

Of course, the strict analysis is in the same position. It cannot predict the truth of (40) without specifying a particular accessibility relation. In turn, the same question arises: on what basis do ordinary speakers determine some worlds to be accessible and others not? This section will overview attempts to answer these questions, and the many counterexamples those attempts have invited. These counterexamples have been a central motivation for pursuing alternative semantic analyses, which will be covered in §3 . While this section follows the focus of the literature on the similarity analysis ( §2.5.1 ), §2.5.2 will briefly detail how parallel criticisms apply to strict analyses.

What determines which worlds are counted as most similar when evaluating a counterfactual? Stalnaker (1968) explicitly sets this issue aside, but D. Lewis (1973b: 92) makes a clear proposal:

Lewis’ (1973b: 92) Proposal Our familiar, intuitive concept of comparative overall similarity, just applied to possible worlds, is employed in assessing counterfactuals.

Just as counterfactuals are context-dependent and vague, so is our intuitive notion of overall similarity. In comparing cost of living, New York and San Francisco may count as similar, but not in comparing topography. And yet, Lewis’ (1973b: 92) Proposal has faced a barrage of counterexamples. Lewis and Stalnaker parted ways in their responses to these counterexamples, though both grant that Lewis’ (1973b: 92) Proposal was not viable. Stalnaker (1984: Ch.7) proposes the projection strategy : similarity is determined by the way we “project our epistemic policies onto the world”. D. Lewis 1979) proposes a new system of weights that amounts to a kind of curve-fitting: we must first look to which counterfactuals are intuitively true, and then find ways of weighting respects of similarity—however complex—that support the truth of counterfactuals. Since Lewis’ (1973b: 92) Proposal and Lewis’ ( 1979 ) system of weights are more developed, and have received extensive critical attention, they will be the focus of this section. [ 35 ] It will begin with the objections to Lewis’ (1973b: 92) Proposal that motivated Lewis’ ( 1979 ) system of weights, and then some objections to that approach.

Fine (1975: 452) presents the future similarity objection to Lewis’ (1973b: 92) Proposal . (41) is plausibly a true statement about world history.

Suppose, optimistically, that there never will be a nuclear holocaust. Then, for every $\mathsf{B\land H}$-world, there will be a more similar $\mathsf{B\land\neg H}$-world, one where a small difference prevents the holocaust, such as a malfunction in the electrical detonation system. In short, a world where Nixon presses the button and a malfunction prevents a nuclear holocaust is more like our own than one where there is a nuclear holocaust that changes the face of the planet. But then Lewis’ (1973b: 92) Proposal incorrectly predicts that (41) is false.

Tichý (1976: 271) offers a similar counterexample. Given (42a) – (42c) , (42d) sounds false.

Lewis’ (1973b: 92) Proposal does not seem to predict the falsity of (42d) . After all, Jones is wearing his hat in the actual world, so isn’t a world where it’s not raining and he’s wearing his hat more similar to the actual one than one where it’s not raining and he isn’t wearing his hat?

(1979: 472) responds to these examples by proposing a ranked system of weights that give what he calls the standard resolution of similarity , which may be further modulated in context:

Avoid big, widespread, diverse violations of law. (“big miracles”)
Maximize the time period over which the worlds match exactly in matters of fact
Avoid even small, localized, simple violations of law. (“little miracles”)
It is of little or no importance to secure approximate similarity of particular fact, even in matters that concern us greatly.

While weight 2 gives high importance to keeping particular facts fixed up to the change required by the counterfactual, weight 4 makes clear that particular facts after that point need not be kept fixed. In the case of (42d) the fact that Jones is wearing his hat need not be kept fixed. It was a post-rain fact, so when one counterfactually supposes that it had not been raining, there is no reason to assume that Jones is still wearing his hat. Similarly, with example (41) . A world where Nixon pushes the button, a small miracle occurs to short-circuit the equipment and the nuclear holocaust is prevented will count as less similar than one where there is no small miracle and a nuclear holocaust results. A small-miracle and no-holocaust world is similar to our own only in one insignificant respect (particular matters of fact) and dissimilar in one important respect (the small miracle).

It is clear, however, that Lewis’ (1979) System of Weights is insufficiently general. Particular matters of fact often are held fixed.

Example (43) crucially holds fixed the outcome of a highly contingent particular fact: the coin outcome. Cases of this kind are discussed extensively by Edgington (2004) . Example (44) shows that a chancy outcome is not an essential feature of these cases. Noting the existence of recalcitrant cases, (1979: 472) simply says he wishes he knew why they came out differently. Additional counterexamples to the Lewis’ (1979) System of Weights have been proposed by Bowie (1979) , Kment (2006) , and Wasserman (2006) . [ 36 ] Kment (2006: 458) proposes a new similarity metric to handle this example which is sensitive to the way particular facts are explained, and is integrated into a general account of metaphysical modality in Kment (2014) . Ippolito (2016) proposes a new theory of how context determines similarity for counterfactuals which aims to make the correct predictions about many of the above cases.

Another response to these counterexamples has been to develop alternative semantic analyses of counterfactuals such as premise semantics (Kratzer 1989, 2012; Veltman 2005) and causal models (Schulz 2007 2011; Briggs 2012; Kaufmann 2013) . These accounts start from the observation that the counterexamples can be easily explained in a model where matters of fact depend on each other. In (42) , when we counterfactually retract the fact that it rained, we don’t keep the fact that the man was wearing his hat because that fact depended on it raining. Hence, (42d) is false. In (43) , when we counterfactually retract that you didn’t bet on heads, we keep the fact that the coin came up heads because it is independent of the fact that you didn’t bet on heads. These accounts offer models of how laws, and law-like generalizations, make facts dependent on each other, and argue that once this is done, there is no work left for similarity to do in the semantics of counterfactuals. While these accounts are the focus of §3 , it is worth presenting one of the additional counterexamples to the similarity analysis that has emerged from this literature.

Recall (38) from §2.4 . Champollion, Ciardelli, and Zhang (2016) and Ciardelli, Zhang, and Champollion (forthcoming) argue on the basis of this example that any similarity analysis will make incorrect predictions about the truth-conditions of counterfactuals. In this example a light is on either when Switch A and B are both up, or they are both down. Otherwise the light is off. Suppose both switches are up and the light is on.

Intuitively, (38a) is true, as are $\mathsf{\neg A >\neg L}$ and $\mathsf{\neg B >\neg L}$, but (38b) is false. Champollion, Ciardelli, and Zhang (2016: 321) argue that a similarity analysis cannot predict $\mathsf{\neg A >\neg L}$ and $\mathsf{\neg B >\neg L}$ to be true, while (38b) is false. In order for $\mathsf{\neg A >\neg L}$ to be true, the particular fact that Switch B is up must count towards similarity. Similarly, for $\mathsf{\neg B >\neg L}$ to be true, the particular fact that Switch A is up must count towards similarity. But then it follows that (38b) is true on a similarity analysis: the most similar worlds where A and B are not both up have to either be worlds where Switch B is down but Switch A is still up, or Switch A is down and Switch B is still up. In those worlds, the light would be off, so the similarity analysis incorrectly predicts (38b) to be true. Champollion, Ciardelli, and Zhang (2016) instead pursue a semantics in terms of causal models where counterfactually making $\neg \mathsf{(A\land B)}$ true and making $\mathsf{\neg A\lor\neg B}$ true come apart.

Do strict analyses avoid the troubles faced by similarity analyses when it comes to truth-conditions? This question is difficult to answer, and has not been explicitly discussed in the literature. Other than the theory of Warmbrōd (1981a,b) , strict theorists have not made proposals for the accessibility relation analogous to Lewis’ (1973b: 92) Proposal for similarity. And, Warmbrōd’s proposal about the pragmatics of the accessibility relation is this:

Warmbrōd’s (1981b: 280) Proposal In the normal case of interpreting a conditional with a nonabsurd antecedent p , the worlds accessible from w will be those that are as similar to w as the most similar p -worlds.

All subsequent second wave strict analyses have ended up in similar territory. The dynamic analyses developed by Fintel (2001) , Gillies (2007) , and Willer (2015, 2017, 2018) assign strict truth-conditions to counterfactuals, but have them induce changes in an evolving space of possible worlds. These changes must render the antecedent consistent with an evolving body of discourse. While Fintel (2001) and Willer (2018) explicitly appeal to a similarity ordering for this purpose, Gillies (2007) and Willer (2017) do not. Nevertheless, the formal structures used by Gillies (2007) and Willer (2017) for this purpose give rise to the same question: which facts stay and which facts go when rendering the counterfactual antecedent consistent? Accordingly, at present, it does not appear that the strict analysis avoids the kinds of concerns raised for the similarity analysis in §2.5.1 .

Recall Goodman’s Problem from §1.4 : the truth-conditions of counterfactuals intuitively depend on background facts and laws, but it is difficult to specify these facts and laws in a way that does not itself appeal to counterfactuals. Strict and similarity analyses make progress on the logic of conditionals without directly confronting this problem. But the discussion of § 2.5 makes salient a related problem. Lewis’ (1979) System of Weights amounts to reverse-engineering a similarity relation to fit the intuitive truth-conditions of counterfactuals. While Lewis’ ( 1979 ) approach avoids characterizing laws and facts in counterfactual terms, Bowie (1979: 496–497) argues that it does not explain why certain counterfactuals are true without appealing to counterfactuals. Suppose one asks why certain counterfactuals are true and the similarity theorist replies with Lewis’ ( 1979 ) recipe for similarity. If one asks why those facts about similarity make counterfactuals true, the similarity theorist cannot reply that they are basic self-evident truths about the similarity of worlds. Instead, they must say that those similarity facts make those counterfactuals true. Bowie’s ( 1979: 496–497 ) criticism is that this is at best uninformative, and at worst circular.

A related concern is voiced by Horwich (1987: 172) who asks “why we should have evolved such a baroque notion of counterfactual dependence”, namely that captured by Lewis’ (1979) System of Weights . The concern has two components: why would humans find it useful, and why would human psychology ground counterfactuals in this concept of similarity rather than our ready-at-hand intuitive concept of overall similarity? These questions are given more weight given the centrality of counterfactuals to human rationality and scientific explanation outlined in §1 . Psychological theories of counterfactual reasoning and representation have found tools other than similarity more fruitful ( §1.2 ). Similarly, work on scientific explanation has not assigned any central role for similarity ( 1.3 ), and as Hájek (2014: 250) puts it:

Science has no truck with a notion of similarity; nor does Lewis’ ( 1979 ) ordering of what matters to similarity have a basis in science.

Morreau (2010) has recently argued on formal grounds that similarity is poorly suited to the task assigned to it by the similarity analysis. The similarity analysis, especially as elaborated by D. Lewis (1979) , tries to weigh some similarities between worlds against their differences to arrive at a notion of overall comparative similarity between those worlds. Morreau (2010: 471) argues that:

[w]e cannot add up similarities or weigh them against differences. Nor can we combine them in any other way… No useful comparisons of overall similarity result. (Morreau 2010: §4)

articulates this argument formally via a reinterpretation of Arrow’s Theorem in social choice theory. Arrow’s Theorem shows that it is not possible to aggregate individuals’ preferences regarding some alternative outcomes into a coherent “collective preference” ordering over those outcomes, given minimal assumptions about their rationality and autonomy. As summarized in §6.3 of Arrow’s theorem , Morreau (2010) argues that the same applies to aggregating respects of similarity and difference: there is no way to add them up into a coherent notion of overall similarity.

Strict and similarity analyses of counterfactuals showed that it was possible to address the semantic puzzles described in §1.4 with formally explicit logical models. This dispelled widespread skepticism of counterfactuals and established a major area of interdisciplinary research. Strict analyses have been revealed to provide a stronger, more classical, logic, but must be integrated with a pragmatic explanation of how counterfactual antecedents are interpreted non-monotonically. Similarity analyses provide a much weaker, more non-classical, logic, but capture the non-monotonic interpretation of counterfactual antecedents within their core semantic model. It is now a highly subtle and intensely debated question which analysis provides a better logic for counterfactuals, and which version of each kind of analysis is best. This intense scrutiny and development has also generated a wave of criticism focused on their treatment of truth-conditions, Goodman’s Problem , and integration with thinking about counterfactuals in psychology and the philosophy of science ( §2.5 , §2.6 ). None of these criticisms are absolutely conclusive, and these two analyses, particularly the similarity analysis, remain standard in philosophy and linguistics. However, the criticisms are serious enough to merit exploring alternative analyses. These alternative accounts take inspiration from a particular diagnosis of the counterexamples discussed in §2.5 : facts depend on each other, so counterfactually assuming p involves not just giving up not-p , but any facts which depended on not-p . The next section will examine analyses of this kind.

3. Semantic Theories of Counterfactual Dependence

Similarity and strict analyses nowhere refer to facts, or propositions, depending on each other. Indeed, 1979 was primarily concerned with explaining which true counterfactuals, given a similarity analysis, manifest a relation of counterfactual dependence. Other analyses have instead started with the idea that facts depend on each other, and then explain how these relations of dependence make counterfactuals true. As will become clear, none of these analyses endorse the naive idea that $\mathsf{A > B}$ is true only when B counterfactually depends on A . The dependence can be more complex, indirect, or B could just be true and independent of A . Theories in this family differ crucially in how they model counterfactual dependence. In premise semantics ( §3.1 ) dependence is modeled in terms of how facts, which are modeled as parts of worlds, are distributed across a space of worlds that has been constrained by laws, or law-like generalizations. In probabilistic semantics ( §3.2 ), this dependence is modeled as some form of conditional probability. In Bayesian networks, structural equations, and causal models ( §3.3 ), it is modeled in terms of the Bayesian networks discussed at the beginning of §1.2.3 . Because theories of these three kinds are very much still in development and often involve even more sophisticated formal models than those covered in §2 , this section will have to be more cursory than §2 to ensure breadth and accessibility.

Veltman (1976) and Kratzer (1981b) approached counterfactuals from a perspective closer to Goodman (1947) : counterfactuals involve explicitly adjusting a body of premises, facts or propositions to be consistent with the counterfactual’s antecedent, and checking to see if the consequent follows from the revised premise set—in a sense of “follow” to be articulated carefully. Since facts or premises hang together, changing one requires changing others that depend on it. The function of counterfactuals is to allow us to probe these connections between facts. While D. Lewis (1981) proved that the Kratzer (1981b) analysis was a special case of similarity semantics, subsequent refinements of premise semantics in Kratzer (1989, 1990, 2002, 2012) and Veltman (2005) evidenced important differences. Kratzer (1989: 626) nicely captures the key difference:

[I]t is not that the similarity theory says anything false about [particular] examples… It just doesn’t say enough. It stays vague where our intuitions are relatively sharp. I think we should aim for a theory of counterfactuals that is able to make more concrete predictions with respect to particular examples.

From a logical point of view, premise semantics and similarity semantics do not diverge. They diverge in the concrete predictions made about the truth-conditions of counterfactuals in particular contexts without adding additional constraints to the theory like Lewis’ (1979) System of Weights .

How does premise semantics aim to improve on the predictions of similarity semantics? It re-divides the labor between context and the semantics of counterfactuals to more accurately capture the intuitive truth-conditions of counterfactuals, and intuitive characterizations of how context influences counterfactuals. In premise semantics, context provides facts and law-like relations among them, and the counterfactual semantics exploits this information. By contrast, the similarity analysis assumes that context somehow makes a similarity relation salient, and has to make further stipulations like Lewis’ (1979) System of Weights about how facts and laws enter into the truth-conditions of counterfactuals in particular contexts. This can be illustrated by considering how Tichý’s ( 1976 ) example (42) is analyzed in premise semantics. This illustration will use the Veltman (2005) analysis because it is simpler than Kratzer (1989, 2012) —that is not to say it is preferable. The added complexity in Kratzer (1989, 2012) provides more flexibility and a broader empirical range including quantification and modal expressions other than would -counterfactuals.

Recall Tichý’s ( 1976 ) example, with the intuitively false counterfactual (42d) :

Veltman (2005) models how the sentences leading up to the counterfactual (42d) determine the facts and laws relevant to its interpretation. The law-like generalization in (42a) is treated as a strict conditional which places a hard constraint on the space of worlds relevant to evaluating the counterfactual. [ 37 ] The particular facts introduced by (42c) provide a soft constraint on the worlds relevant to interpreting the counterfactual. Figure 9 illustrates this model of the context and its evolution, including a third atomic sentence $\mathsf{H}$ for reasons that will become clear shortly.

$C_0$	$\mathsf{R}$	$\mathsf{W}$	$\mathsf{H}$
$\boldsymbol{w_0}$
$\boldsymbol{w_1}$
$\boldsymbol{w_2}$
$\boldsymbol{w_3}$
$\boldsymbol{w_4}$
$\boldsymbol{w_5}$
$\boldsymbol{w_6}$
$\boldsymbol{w_7}$

$\hspace{15px}\underrightarrow{\medsquare(\mathsf{R\supset W})}$

$C_1$	$\mathsf{R}$	$\mathsf{W}$	$\mathsf{H}$
$\boldsymbol{w_0}$
$\boldsymbol{w_1}$
$\boldsymbol{w_2}$
$\boldsymbol{w_3}$
$\xcancel{w_4}$	$\xcancel{1}$	$\xcancel{0}$	$\xcancel{0}$
$\xcancel{w_5}$	$\xcancel{1}$	$\xcancel{0}$	$\xcancel{1}$
$\boldsymbol{w_6}$
$\boldsymbol{w_7}$

$\quad\underrightarrow{\mathsf{R\land W}}$

$C_2$	$\mathsf{R}$	$\mathsf{W}$	$\mathsf{H}$
$w_0$	0	0	0
$w_1$	0	0	1
$w_2$	0	1	0
$w_3$	0	1	1
$\xcancel{w_4}$	$\xcancel{1}$	$\xcancel{0}$	$\xcancel{0}$
$\xcancel{w_5}$	$\xcancel{1}$	$\xcancel{0}$	$\xcancel{1}$
$\boldsymbol{w_6}$
$\boldsymbol{w_7}$

Figure 9: Context for (42) , Facts in Bold, Laws Crossing out Worlds

On this model a context provides a set of worlds compatible with the facts, in $C_2$ $\textit{Facts}_{C_2}={\{w_6,w_7\}}$, and the set of worlds compatible with the laws, in $C_2$ $\textit{Universe}_{C_2}={\{w_0,w_1,w_2,w_3,w_6,w_7\}}$. This model of context is one essential component of the analysis, but so too is the way Veltman (2005) models worlds, situations, and dependencies between facts. These further components allow Veltman (2005) to offer a procedure for “retracting” the fact that $\mathsf{R}$ holds from a world.

Veltman’s ( 2005 ) analysis of counterfactuals identifies possible worlds with atomic valuations (functions from atomic sentences to truth-values) like those depicted in Figure 9 . So $w_6={\{{\langle \mathsf{R},1\rangle},{\langle \mathsf{W},1\rangle},{\langle \mathsf{H},0\rangle}\}}$. This makes it possible to offer a simple model of situations , which are parts of worlds: any subset of a world. [ 38 ] It is now easy to think about one fact (sentence having a truth-value) as determining another fact (sentence having a truth value). In context $C_3$, $\mathsf{R}$ being 1 determines that $\mathsf{W}$ will be 1. Once you know that $\mathsf{R}$ is assigned to 1, you know that $\mathsf{W}$ is too. Veltman’s ( 2005 ) proposal is that speakers evaluate a counterfactual by retracting the fact that the antecedent is false from the worlds in the context, which gives you some situations, and then consider all those worlds that contain those situations, are compatible with the laws, and make the antecedent true. If the consequent is true in all of those worlds, then we can say that the counterfactual is true in (or supported by) the context. So, to evaluate $\neg \mathsf{R>W}$, one first retracts the fact that $\mathsf{R}$ is true, i.e., that $\mathsf{R}$ is assigned to 1, then one finds all the worlds consistent with the laws that contain those situations and assign $\mathsf{R}$ to 0. If all of those worlds are also $\mathsf{W}$ worlds, then the counterfactual is true in (or supported by) the context. For Veltman (2005) , the characterization of this retraction process relies essentially on the idea of facts determining other facts.

According to Veltman (2005) , when you are “retracting” a fact from the facts in the context, you begin by considering each $w\in \textit{Facts}_C$ and find the smallest situations in w which contain only undetermined facts—he calls such a situation a basis for w . This is a minimal situation which, given the laws constraining $\textit{Universe}_C$, determines all the other facts about that world. For example, $w_6$ has only one basis, namely $s_0={\{{\langle \mathsf{R},1\rangle},{\langle \mathsf{H},0\rangle}\}}$, and $w_7$ has only one basis, namely $s_1={\{{\langle \mathsf{R},1\rangle},{\langle \mathsf{H},1\rangle}\}}$. Once you have the bases for a world, you can retract a fact by finding the smallest change to the basis that no longer forces that fact to be true. So retracting the fact that $\mathsf{R}$ is true from $s_0$ produces $s'_0={\{{\langle \mathsf{H},0\rangle}\}}$, and retracting it from $s_1$ produces $s'_1={\{{\langle \mathsf{H},1\rangle}\}}$. The set consisting of these two situations is the premise set .

To evaluate $\mathsf{\neg R>W}$, one finds the set of worlds from $\textit{Universe}_{C_3}$ that contains some member of the premise set $s'_0$ or $s'_1$: ${\{w_0,w_1,w_2,w_3\}}$—these are the worlds consistent with the premise set and the laws. Are all of the $\neg \mathsf{R}$-worlds in ${\{w_0,w_1,w_2,w_3\}}$ also $\mathsf{W}$-worlds? No, $w_2$ and $w_3$ are not. Thus, $\neg \mathsf{R>W}$ is not true in (or supported by) the context $C_3$. This was the intuitively correct prediction about example (42) . Of course, the similarity analysis supplemented with Lewis’ (1979) System of Weights also makes this prediction. But consider again example (43) , which is not predicted:

This example relies seamlessly on three pieces of background knowledge about how betting works:

If you don’t bet, you don’t win: $\mathsf{\medsquare(\neg B\supset\neg W)}$

If you bet and it comes up heads, you win: $\mathsf{\medsquare((B\land H)\supset W)}$

If you bet and it doesn’t come up heads, you don’t win: $\mathsf{\medsquare((B\land\neg H)\supset\neg W)}$

And it specifies facts: $\mathsf{\neg B\land H}$. The resulting context is detailed in Figure 10 :

	$\mathsf{B}$	$\mathsf{H}$	$\mathsf{W}$
$w_0$	0	0	0
$\xcancel{w_1}$	$\xcancel{0}$	$\xcancel{0}$	$\xcancel{1}$
$\boldsymbol{w_2}$
$\xcancel{w_3}$	$\xcancel{0}$	$\xcancel{1}$	$\xcancel{1}$
$w_4$	1	0	0
$\xcancel{w_5}$	$\xcancel{1}$	$\xcancel{0}$	$\xcancel{1}$
$\xcancel{w_6}$	$\xcancel{1}$	$\xcancel{1}$	$\xcancel{0}$
$w_7$	1	1	1

Figure 10: Context for (43)

Now, consider the counterfactual $\mathsf{B>W}$. The first step is to retract the fact that $\mathsf{B}$ is false from each world in $\textit{Facts}_{C_{(43)}}$. That’s just $w_2$. This world has two bases—minimal situations consisting of undetermined facts—$s_0={\{{\langle \mathsf{B},0\rangle},{\langle \mathsf{H},1\rangle}\}}$ and $s_1={\{{\langle \mathsf{H},1\rangle},{\langle \mathsf{W},0\rangle}\}}$. [ 39 ] The next step is to retract the fact that $\mathsf{B}$ is false from both bases. For $s_0$ this yields $s'_0={\{{\langle \mathsf{H},1\rangle}\}}$ and for $s_1$ this also yields $s'_0$—since the fact that you didn’t win together with the fact that the coin came up heads, forces it to be false that you bet. Given this situation, the premise set consists of the two worlds in Universe (43) that contain $s'_0$: ${\{w_2,w_7\}}$. Now, are all of the $\mathsf{B}$-worlds in this set also $\mathsf{W}$-worlds? Yes, $w_7$ is the only $\mathsf{B}$-world, and it is also a $\mathsf{W}$-world. So Veltman (2005) correctly predicts that (43) is true in (supported by) its natural context.

It should now be more clear how premise semantics delivers on its promise to be more predictive than similarity semantics when it comes to counterfactuals in context, and affords a more natural characterization of how a context informs the interpretation of counterfactuals. This analysis was crucially based on the idea that some facts determine other facts, and that the process of retracting a fact is constrained by these relations. However, even premise semantics has encountered counterexamples.

Schulz (2007: 101) poses the following counterexample to Veltman (2005) .

Intuitively, (45d) is true in the context. Figure 11 details the context predicted for it by Veltman (2005) .

$C_2$	$\mathsf{A}$	$\mathsf{B}$	$\mathsf{L}$
$w_0$	0	0	0
$\xcancel{w_1}$	$\xcancel{0}$	$\xcancel{0}$	$\xcancel{1}$
$w_2$	0	1	0
$\xcancel{w_3}$	$\xcancel{0}$	$\xcancel{1}$	$\xcancel{1}$
$\boldsymbol{w_4}$
$\xcancel{w_5}$	$\xcancel{1}$	$\xcancel{0}$	$\xcancel{1}$
$\xcancel{w_6}$	$\xcancel{1}$	$\xcancel{1}$	$\xcancel{0}$
$w_7$	1	1	1

Figure 11: Context for (45d)

There are two bases for $w_4$: $s_0={\{{\langle \mathsf{A},1\rangle},{\langle \mathsf{L},0\rangle}\}}$—the fact that Switch A is up and the light is off determines that Switch B is down—and $s_1={\{{\langle \mathsf{A},1\rangle},{\langle \mathsf{B},0\rangle}\}}$—the fact that Switch A is up and the fact that B is down determines that the light is off. (No smaller situation would determine the facts of $w_4$.) Retracting $\mathsf{B}$’s falsity from $s_0$ leads to trouble. $s_0$ forces $\mathsf{B}$ to be false, but there are two ways of changing this. First, one can remove the fact that the light is on, yielding $s'_0={\{{\langle \mathsf{A},1\rangle}\}}$. Second, one can eliminate the fact that Switch A is up, yielding $s''_0={\{{\langle \mathsf{L},0\rangle}\}}$. Because of $s''_0$, the premise set will contain $w_2$, meaning it allows that in retracting the fact that Switch B is down one can give up the fact that Switch A is up. But then there is a $\mathsf{B}$-world where $\mathsf{L}$ is false, and $\mathsf{B>L}$ is incorrectly predicted to be false.

Intuitively, the analysis went wrong in allowing the removal of the fact that Switch A is up when retracting the fact that Switch B is down. Schulz (2007: §5.5) provides a more sophisticated version of this diagnosis: although the fact that Switch A is up and the fact that the light is off together determine that Switch B is down, only the fact that the light is off depends on the fact that Switch B is down. If one could articulate this intuitive concept of dependence, and instead only retract facts that depend on the fact you are retracting (in this case the fact that B is down), then the error could be avoided. It is unclear how to implement this kind of dependence in Veltman’s ( 2005 ) framework. Schulz (2007: §5.5) goes on show that structural equations and causal models provide the necessary concept of dependence—for more on this approach see §3.3 below. After all, it seems plausible that the light being off causally depends on Switch B being down, but Switch A being up does not causally depend on Switch B being down. It remains to be seen whether the more powerful framework developed by Kratzer (1989, 2012) can predict (45) .

While premise semantics has been prominent among linguists, probabilistic theories have been very prominent among philosophers thinking about knowledge and scientific explanation. [ 40 ] Adams (1965, 1975) made a seminal proposal in this literature:

Adams’ Thesis The assertability of q if p is proportional to $P(q\mid p)$, where P is a probability function representing the agent’s subjective credences—see Definition 1 .

However, Adams (1970) was also aware that indicative/subjunctive pairs like (3) / (4) differ in their assertability. To explain this, he proposed the prior probability analysis of counterfactuals (Adams 1976) :

Adams’ Prior Probability Analysis The assertability of $\phi>\psi$ is proportional to $P_0(\psi\mid\phi)$, where $P_0$ is the agent’s credence prior to learning that $\phi$ was false.

It would seem that this analysis accurately predicts our intuitions in (45) about $\mathsf{B>L}$. Let $P_0$ be an agent’s credence before learning that Switch B is down. (45a) requires that $P_0(\mathsf{L}\mid \mathsf{A\land B})$ is (or is close to) 1, (45b) requires that $P_0(\mathsf{\neg L}\mid \mathsf{\neg A\lor \neg B})$ is (or is close to) 1. The agent also learns that Switch A is up, so $P_0(\mathsf{A})$ is (or is close to) 1. All of this together seems to guarantee that $P_0(\mathsf{B\mid L})$ is also very high. However, this is due to an inessential artifact of the example: the agent learned that Switch B was down after learning that Switch A is up. This detail does not matter to the intuition. As was seen with example (43) , we often hold fixed facts that happen after the antecedent turns out false. Indeed, Adams’ Prior Probability Analysis makes the incorrect prediction that (43) is unassertible in its natural context.

This problem for Adams’ Prior Probability Analysis is addressed in Edgington (2003, 2004: 21) who amends the analysis: $P_0$ may also reflect any facts the agent learns after they learn that the antecedent is false, provides that those facts are causally independent of the antecedent. This parallels the idea pursued by Schulz (2007: Ch.5) to integrate causal dependence into the analysis of counterfactuals. This idea was also pursued in a probabilistic framework by Kvart (1986, 1992) . Kvart (1986, 1992) , however, does not propose a prior probability analysis and does not regard the probabilities as subjective credences: they are instead objective probabilities (propensity or objective chance). Skyrms (1981) also proposes a propensity account, but pursues a prior propensity account analogous to the subjective one proposed by Adams (1976) .

Objective probability analyses have been popular among philosophers trying to capture the way that counterfactuals feature in physical explanations, and why they are so useful to agents like us in worlds like ours. Loewer (2007) is a good example of such an account, who grounds the truth of certain counterfactuals regarding our decisions like (46) in statistical mechanical probabilities.

Loewer (2007) proposes that (46) is true just in case (where $P_{\textit{SM}}$ is the statistical mechanical probability distribution and $M(t)$ is a description of the macro-state of the universe at t ):

Loewer (2007) acknowledges that this analysis is limited to counterfactuals like (46) . He argues that it can address the philosophical objections to the similarity analysis discussed in §2.6 , namely why counterfactuals are useful in scientific explanations, and for agents like us in a world like our own.

Conditional probability analyses do not proceed by assigning truth-conditions to (all) counterfactuals. They instead associate them with certain conditional probabilities. [ 41 ] This makes it difficult to integrate the theory into a comprehensive compositional semantics and logic for a natural language. Kaufmann (2005 2008) makes important advances here, but it remains an open issue for conditional probability analyses. Leitgeb (2012a,b) thoroughly develops a new conditional probability analysis which regards $\mathsf{\phi>\psi}$ as true when the relevant conditional probability is sufficiently high. [ 42 ] But conditional probability analyses have other limitations. Without further development, these analyses are limited in their ability to explain how humans judge particular counterfactuals to be true. There is a large literature in psychology, beginning with Kahneman, Slovic, and Tversky 1982 , showing that human reasoning diverges in predictable way from precise probabilistic reasoning. Even if these performance differences didn’t turn up in counterfactuals and conditional probabilities, there is an implementation issue. As discussed in §1.2.3 , directly implementing probabilistic knowledge makes unreasonable demands on memory. Bayesian Networks are one proposed solution to this implementation issue. They are also used in the analysis of causal dependence ( §1.3 ), which conditional probability analyses must appeal to anyway. Since Bayesian Networks can also be used to directly formulate a semantics of counterfactuals, they provide an worthwhile alternative to conditional probability analyses despite proceeding from similar assumptions.

Recall from §1.2.3 the basic idea of a Bayesian Network: rather than storing probability values for all possible combinations of some set of variables, a Bayesian Network represents only the conditional probabilities of variables whose values depend on each other. This can be illustrated for (45) .

Sentences (45a) - (45c) can be encoded by the Bayesian Network and structural equations in Figure 12 .

Figure 12: Bayesian Network and Structural Equations for (45)

Recall that $L\dequal A\land B$ means that the value of L equals the value of $A\land B$, but also asymmetrically depends on the value of $A\land B$: the value of $A\land B$ determines the value of L , and not vice-versa. How, given the network in Figure 12 , does one evaluate the counterfactual $\mathsf{B>N}$? Several different answers have been given to this question.

Pearl (1995, 2000, 2009, 2013: Ch.7) proposes:

Interventionism Evaluate $\mathsf{B > L}$ relative to a Bayesian Network by removing any incoming arrows to B , setting its value to 1, and projecting this change forward through the remaining network. If L is 1 in the resulting network, $\mathsf{B > L}$ is true; otherwise it’s false.

On this approach, one simply deletes the assignment $B=0$, replaces it with $B=1$, and solves for L using the equation $L\dequal A\land B$. Since the deletion of $B=0$ does not effect the assignment $A=1$, it follows that $L=1$ and that the counterfactual is true. This simple recipe yields the right result. Pearl nicely sums up the difference between this kind of analysis and a similarity analysis:

In contrast with Lewis’s theory, counterfactuals are not based on an abstract notion of similarity among hypothetical worlds; instead, they rest directly on the mechanisms (or “laws,” to be fancy) that produce those worlds and on the invariant properties of those mechanisms. Lewis’s elusive “miracles” are replaced by principled [interventions] which represent the minimal change (to a model) necessary for establishing the antecedent… Thus, similarities and priorities—if they are ever needed—may be read into the [interventions] as an afterthought… but they are not basic to the analysis. (Pearl 2009: 239–240)

As interventionism is stated above, it does not apply to conditionals with logically complex antecedents or consequents. This limitation is addressed by Briggs (2012) , who also axiomatizes and compares the resultant logic to D. Lewis (1973b) and Stalnaker (1968) —significantly extending the analysis and results in Pearl (2009: Ch.7) . Integrations of causal models with premise semantics (Schulz 2007, 2011; Kaufmann 2013; Santorio 2014; Champollion, Ciardelli, & Zhang 2016; Ciardelli, Zhang, & Champollion forthcoming) provide another way of incorporating an interventionist analysis into a fully compositional semantics. However, interventionism does face other limitations.

Hiddleston (2005) presents the following example.

(48c) is intuitively true in this context. The network for (48) is given in Figure 13 .

Figure 13: Bayesian Network and Structural Equations for (48)

Hiddleston (2005) observes that interventionism does not predict $\mathsf{F>B}$ to be true. It tells one to delete the arrow going in to F , set its value to 1, and project the consequences of doing so. However, none of the other values depend on F so they keep their actual values: $L=0$ and $B=0$. Accordingly, $\mathsf{F>B}$ is false, contrary to intuition. Further, because the intervention on F has destroyed its connection to B , it’s not even possible to tweak interventionism to allow values to flow backwards (to the left) through the network. [ 43 ] Hiddleston’s ( 2005 ) counterexample highlights the possibility of another kind of counterexample featuring embedded conditionals. Consider again the network in Figure 12 . The following counterfactual seems true ( Starr 2012: 13 ).

And, considering a simple match, Fisher (2017b: §1) observes that (50b) is intuitively false.

In both cases, interventionism is destined to make the wrong prediction. With (49) , the intervention in the first antecedent removes the connection between Switch A and the light, so when the antecedent of the consequent is made true by intervention, it does not result in L ’s value becoming 0. And so the whole counterfactual comes out false. Similarly with (50b) , when the first antecedent is made true by intervention, it stays true even after the second antecedent is evaluated. Hence the whole conditional is predicted to be true. Fisher (2017a) also observes that interventionism also has no way of treating counterlegal counterfactuals like if Switch A had alone controlled the light, the light would be on .

These counterexamples to interventionism have stimulated alternative accounts like Hiddleston’s ( 2005 ) minimal network analysis and further developments of that analysis (Rips 2010; Rips & Edwards 2013; Fisher 2017b) . Instead of modifying an existing network to make the antecedent true, this analysis considers alternate networks where only the parent nodes of the antecedent which directly influence it are changed to make the antecedent come true. However, Pearl’s ( 2009 ) interventionist analysis has also been incorporated into the extended structural models analysis (Lucas & Kemp 2015) . This analysis aims to capture interventions as a special case of a more general proposal about how antecedents are made true. One important aspect of this proposal is that interventions often involve inserting a hidden node that amounts to an unknown cause of the antecedent. The analysis of Snider and Bjorndahl (2015) pursues a third idea: counterfactuals are not interpreted by manipulating a background network, but instead serve to constrain the class of possible networks compatible with the information shared in a conversation, as in Stalnaker’s ( 1978 ) theory of assertion. [ 44 ] Among these relations can be cause-to-effect networks as in (45d) , but also networks that involve the antecedent and consequent having a common cause, as in (48c) . As should be clear, this is a rapidly developing area of research where it is not possible to identify one analysis as standard or representative. It does bear emphasizing that this literature is driven not only by precise formal models, but also by experimental data which is brought to bear on the predictions of these analyses.

A few final philosophical remarks are in order about the kinds of analyses discussed here. If one follows Woodward (2002) and Hitchcock (2001) in their interpretation of these networks, a structural equation should be viewed as a primitive counterfactual. It follows that this is a non-reductive analysis of counterfactual dependence: it only explains how the truth of arbitrarily complex counterfactual sentences are grounded in basic relations of counterfactual dependence. However, note in the earlier quotation above from Pearl (2009: 239–240) that he interprets structural equations as basic mechanisms or laws, and so arguably counts as an analysis of counterfactuals in terms of laws. These amount to two very different philosophical positions that interact with the philosophical debates surveyed in §1.3 .

It is also worth noting that while many working in this framework apply these networks to causal relations, there is no reason to assume that the analysis would not apply to other kinds of dependence relations. For example, constitutional dependence is at the heart of counterfactuals like:

From a Bayesian Network approach to mental representation ( §1.2.3 ), this makes perfect sense: the networks encode probabilistic dependence which can come from causal or constitutional facts.

Finally, it is worth highlighting that the philosophical objections directed at the similarity analysis in §2.6 are addressed, at least to some degree, by structural equation analyses. Because the central constructs of this analysis—structural equations and Bayesian Networks—are also employed in models of mental representation, causation, and scientific explanation, it grounds counterfactuals in a construct already taken to explain how creatures like us cope with a world like the one we live in.

Premise semantics ( 3.1 ), conditional probability analyses ( §3.2 ) and structural equation analyses ( §3.3 ) all aim to analyze counterfactuals by focusing on certain relations between facts, rather than similarities between worlds. These accounts make clearer and more accurate predictions about particular counterfactuals in context than similarity analyses. But, ultimately, both premise semantics and conditional probability analyses had to incorporate causal dependence into their theories. Structural equation analyses do this from the start, and improve further on the predictions of premise semantics and conditional probability analyses. Another strength of this analysis is that it integrates elegantly into the broader applications of counterfactuals in theories of rationality, mental representation, causation, and scientific explanation surveyed in §1.1 . There is still rapid development of structural equation analyses, though, so it is too early to say where the analysis will stabilize, or how it will fair under thorough critical examination.

Philosophers, linguists, and psychologists remain fiercely divided on how to best understand counterfactuals. Rightly so. They are at the center of questions of deep human interest ( §1 ). The renaissance on this topic in the 1970s and 1980s focused on addressing certain semantic puzzles and capturing the logic of counterfactuals ( §2 ). From this seminal literature, similarity analyses (D. Lewis 1973b; Stalnaker 1968) have enjoyed the most widespread popularity in philosophy ( §2.3 ). But the logical debate between similarity and strict analyses is still raging, and strict analyses provide a viable logical alternative ( §2.4 ). Criticisms of these logical analyses have focused recent debates on our intuitions about particular utterances of counterfactuals in particular contexts. Structural equation analyses ( §3.3 ) have emerged as a particularly prominent alternative to similarity and strict analyses, which claims to improve on both in significant respects. These analyses are now being actively developed by philosophers, linguists, psychologists, and computer scientists.

Adams, Ernest W., 1965, “The Logic of Conditionals”, Inquiry , 8: 166–197. doi:10.1080/00201746508601430
–––, 1970, “Subjunctive and Indicative Conditionals”, Foundations of Language , 6(1): 89–94.
–––, 1975, The Logic of Conditionals , Dordrecht: D. Reidel.
–––, 1976, “Prior Probabilities and Counterfactual Conditionals”, in Foundations of Probability Theory, Statistical Inference, and Statistical Theories of Science , William L. Harper and Clifford Alan Hooker (eds.) (The University of Western Ontario Series in Philosophy of Science), Springer Netherlands, 6a:1–21. doi:10.1007/978-94-010-1853-1_1
Alonso-Ovalle, Luis, 2009, “Counterfactuals, Correlatives, and Disjunction”, Linguistics and Philosophy , 32(2): 207–244. doi:10.1177/0146167214563673
Alquist, Jessica L., Sarah E. Ainsworth, Roy F. Baumeister, Michael Daly, and Tyler F. Stillman, 2015, “The Making of Might-Have-Beens: Effects of Free Will Belief on Counterfactual Thinking”, Personality and Social Psychology Bulletin , 41(2): 268–283. doi:10.1177/0146167214563673
Anderson, Alan Ross, 1951, “A Note on Subjunctive and Counterfactual Conditionals”, Analysis , 12(2): 35–38. doi:10.2307/3327037
Arregui, Ana, 2007, “When Aspect Matters: The Case of Would-Conditionals”, Natural Language Semantics , 15(3): 221–264. doi:10.1007/s11050-007-9019-6
–––, 2009, “On Similarity in Counterfactuals”, Linguistics and Philosophy , 32(3): 245–278. doi:10.1007/s10988-009-9060-7
Barker, Stephen J., 1998, “Predetermination and Tense Probabilism”, Analysis , 58(4): 290–296. doi:10.1093/analys/58.4.290
Bennett, Jonathan, 1974, “Counterfactuals and Possible Worlds”, Canadian Journal of Philosophy , 4(2): 381–402. doi:10.1080/00455091.1974.10716947
–––, 2003, A Philosophical Guide to Conditionals , Oxford: Oxford University Press.
Bennett, Karen, 2017, Making Things Up , New York: Oxford University Press.
Bittner, Maria, 2011, “Time and Modality without Tenses or Modals”, in Tense Across Languages , Renate Musan and Monika Rathers (eds.), Tübingen: Niemeyer, 147–188. [ Bittner 2011 available online ]
Bobzien, Susanne, 2011, “Dialectical School”, in The Stanford Encyclopedia of Philosophy , Edward N. Zalta (ed.), Fall 2011, URL = < http://plato.stanford.edu/archives/fall2011/entries/dialectical-school/ >
Bowie, G. Lee, 1979, “The Similarity Approach to Counterfactuals: Some Problems”, Noûs , 13(4): 477–498. doi:10.2307/2215340
Brée, D.S., 1982, “Counterfactuals and Causality”, Journal of Semantics , 1(2): 147–185. doi:10.1093/jos/1.2.147
Briggs, R.A., 2012, “Interventionist Counterfactuals”, Philosophical Studies , 160(1): 139–166. doi:10.1007/s11098-012-9908-5
Byrne, Ruth M. J., 2005, The Rational Imagination: How People Create Alternatives to Reality , Cambridge, MA: MIT Press.
–––, 2016, “Counterfactual Thought”, Annual Review of Psychology , 67(1): 135–157. doi:10.1146/annurev-psych-122414-033249
Carnap, Rudolf, 1948, Introduction to Semantics , Cambridge, MA: Harvard University Press.
–––, 1956, Meaning and Necessity , second edition, Chicago: Chicago University Press.
Champollion, Lucas, Ivano Ciardelli, and Linmin Zhang, 2016, “Breaking de Morgan’s Law in Counterfactual Antecedents”, in Proceedings from Semantics and Linguistic Theory (SALT) 26 , Mary Moroney, Carol-Rose Little, Jacob Collard, and Dan Burgdorf (eds.), Ithaca, NY: CLC Publications, 304–324. doi:10.3765/salt.v26i0.3800
Chater, Nick, Mike Oaksford, Ulrike Hahn, and Evan Heit, 2010, “Bayesian Models of Cognition”, Wiley Interdisciplinary Reviews: Cognitive Science , 1(6): 811–823. doi:10.1002/wcs.79
Chisholm, Roderick M., 1955, “Law Statements and Counterfactual Inference”, Analysis , 15(5): 97–105. doi:10.1093/analys/15.5.97
Ciardelli, Ivano, Linmin Zhang, and Lucas Champollion, forthcoming, “Two Switches in the Theory of Counterfactuals”, Linguistics and Philosophy , first online: 15 June 2018. doi:10.1007/s10988-018-9232-4
Cohen, Jonathan and Aaron Meskin, 2006, “An Objective Counterfactual Theory of Information”, Australasian Journal of Philosophy , 84(3): 333–352. doi:10.1080/00048400600895821
Copeland, B. Jack, 2002, “The Genesis of Possible Worlds Semantics”, Journal of Philosophical Logic , 31(2): 99–137. doi:10.1023/A:1015273407895
Costello, Tom and John McCarthy, 1999, “Useful Counterfactuals”, Linköping Electronic Articles in Computer and Information Science , 4(12): 1–24.
Cresswell, Max J. and G.E. Hughes, 1996, A New Introduction to Modal Logic , London: Routledge.
Daniels, Charles B. and James B. Freeman, 1980, “An Analysis of the Subjunctive Conditional”, Notre Dame Journal of Formal Logic , 21(4): 639–655. doi:10.1305/ndjfl/1093883247
Declerck, Renaat and Susan Reed, 2001, Conditionals: A Comprehensive Emprical Analysis , (Topics in English Linguistics, 37), New York: De Gruyter Mouton.
Dretske, Fred I., 1981, Knowledge and the Flow of Information , Cambridge, MA: The MIT Press.
–––, 1988, Explaining Behavior: Reasons in a World of Causes , Cambridge, MA: MIT Press.
–––, 2002, “A Recipe for Thought”, in Philosophy of Mind: Contemporary and Classical Readings , David J. Chalmers (ed.), New York: Oxford University Press, 491–499.
–––, 2011, “Information-Theoretic Semantics”, in The Oxford Handbook of Philosophy of Mind , Brian McLaughlin, Angsar Beckermann, and Sven Walter (eds.), New York: Oxford University Press, 381–393.
Dudman, Victor Howard, 1984a, “Conditional Interpretations of ‘If’ Sentences”, Australian Journal of Linguistics , 4(2): 143–204. doi:10.1080/07268608408599325
–––, 1984b, “Parsing ‘If’-Sentences”, Analysis , 44(4): 145–153. doi:10.1093/analys/44.4.145
–––, 1988, “Indicative and Subjunctive”, Analysis , 48(3): 113–122. doi:10.1093/analys/48.3.113a
Edgington, Dorothy, 2003, “What If? Questions About Conditionals”, Mind & Language , 18(4): 380–401. doi:10.1111/1468-0017.00233
–––, 2004, “Counterfactuals and the Benefit of Hindsight”, in Cause and Chance: Causation in an Indeterministic World , Phil Dowe & Paul Noordhof (ed.), New York: Routledge, 12–27.
Fine, Kit, 1975, “Review of Lewis’ Counterfactuals”, Mind , 84: 451–458. doi:10.1093/mind/LXXXIV.1.451
–––, 2012a, “Counterfactuals Without Possible Worlds”, Journal of Philosophy , 109(3): 221–246. doi:10.5840/jphil201210938
–––, 2012b, “A Difficulty for the Possible Worlds Analysis of Counterfactuals”, Synthese , 189(1): 29–57. doi:10.1007/s11229-012-0094-y
Fintel, Kai von, 1999, “The Presupposition of Subjunctive Conditionals”, in The Interpretive Tract , Uli Sauerland and Orin Percus (eds.), Cambridge, MA: MITWPL, MIT Working Papers in Linguistics 25: 29–44. [ Fintel 1999 available online ]
–––, 2001, “Counterfactuals in a Dynamic Context”, in Ken Hale, A Life in Language , Michael Kenstowicz (ed.), Cambridge, MA: The MIT Press, 123–152. [ Fintel 2001 available online ]
–––, 2012, “Subjunctive Conditionals”, in The Routledge Companion to Philosophy of Language , Gillian Russell and Delia Graff Fara (eds.), New York: Routledge, 466–477. [ Fintel 2012 available online ]
Fintel, Kai von and Sabine Iatridou, 2002, “If and When If -Clauses Can Restrict Quantifiers”, Paper for the Workshop in Philosophy and Linguistics, University of Michigan, November 8–10, 2002. [ Fintel & Iatridou 2002 available online ]
Fisher, Tyrus, 2017a, “Causal Counterfactuals Are Not Interventionist Counterfactuals”, Synthese , 194(12): 4935–4957. doi:10.1007/s11229-016-1183-0
–––, 2017b, “Counterlegal Dependence and Causation’s Arrows: Causal Models for Backtrackers and Counterlegals”, Synthese , 194(12): 4983–5003. doi:10.1007/s11229-016-1189-7
Fodor, Jerry A., 1987, Psychosemantics: The Problem of Meaning in the Philosophy of Mind , Cambridge, MA: The MIT Press.
–––, (ed.), 1990, A Theory of Content and Other Essays , Cambridge, MA: The MIT Press.
Fraassen, Bas C. Van, 1966, “Singular Terms, Truth-Value Gaps and Free Logic”, Journal of Philosophy, 3: 481–495. doi:10.2307/2024549
Frege, Gottlob, 1893, Grundgesetze Der Arithmetik , Begriffsschriftlich Abgeleitet, Vol. 1, 1st ed, Jena: H. Pohle.
Galinsky, Adam D., Katie A. Liljenquist, Laura J. Kray, and Neil J. Roes, 2005, “Finding Meaning from Mutability: Making Sense and Deriving Significance through Counterfactual Thinking”, in The Psychology of Counterfactual Thinking , David R. Mandel, Denis J. Hilton, and Patrizia Catellani (eds), New York: Routledge, 110–127.
Gamut, L.T.F., 1991, Logic, Language and Meaning: Intensional Logic and Logical Grammar , Vol. 2, Chicago: The University of Chicago Press.
Gärdenfors, Peter, 1978, “Conditionals and Changes of Belief”, in The Logic and Epistemology of Scientific Belief , Ilkka Niiniluoto and Raimo Tuomela (eds.), Amsterdam: North-Holland.
–––, 1982, “Imaging and Conditionalization”, Journal of Philosophy , 79(12): 747–760. doi:10.2307/2026039
Gibbard, Allan F., 1980, “Two Recent Theories of Conditionals”, in Harper, Stalnaker, and Pearce 1980: 211–247. doi:10.1007/978-94-009-9117-0_10
Gibbard, Allan F. and William L. Harper, 1978, “Counterfactuals and Two Kinds of Expected Utility”, in Foundations and Applications of Decision Theory , Clifford Hooker, James J. Leach, and Edward McClennen (eds.), Dordrecht: D. Reidel, 125–162. doi:10.1007/978-94-009-9789-9_5
Gillies, Anthony, 2007, “Counterfactual Scorekeeping”, Linguistics and Philosophy , 30(3): 329–360. doi:10.1007/s10988-007-9018-6
–––, 2012, “Indicative Conditionals”, in The Routledge Companion to Philosophy of Language , Gillian Russell and Delia Graff Fara (eds.), New York: Routledge, 449–465.
Ginsburg, Matthew L., 1985, “Counterfactuals”, in Proceedings of the Ninth International Joint Conference on Artificial Intelligence , Aravind Joshi (ed.), Los Altos, CA: Morgan Kaufmann, 80–86. [ Ginsburg 1985 available online ]
Glymour, Clark, 2001, The Mind’s Arrows: Bayes Nets and Graphical Causal Models in Psychology , Cambridge, MA: MIT Press.
Goodman, Nelson, 1947, “The Problem of Counterfactual Conditionals”, The Journal of Philosophy , 44(5): 113–118. doi:10.2307/2019988
–––, 1954, Fact, Fiction and Forecast , Cambridge, MA: Harvard University Press.
Gopnik, Alison, Clark Glymour, David M. Sobel, Laura E. Schulz, Tamar Kushnir, and David Danks, 2004, “A Theory of Causal Learning in Children: Causal Maps and Bayes Nets”, Psychological Review, 111(1): 3–32. doi:10.1037/0033-295X.111.1.3
Gopnik, Alison and Joshua B. Tenenbaum, 2007, “Bayesian Networks, Bayesian Learning and Cognitive Development”, Developmental Science, 10(3): 281–287. doi:10.1111/j.1467-7687.2007.00584.x
Hájek, Alan, 2014, “Probabilities of Counterfactuals and Counterfactual Probabilities”, Journal of Applied Logic , 12(3): 235–251. doi:10.1016/j.jal.2013.11.001
Halpern, Joseph and Judea Pearl, 2005a, “Causes and Explanations: A Structural-Model Approach. Part I: Causes”, British Journal for Philosophy of Science , 56(4): 843–887. doi:10.1093/bjps/axi147
–––, 2005b, “Causes and Explanations: A Structural-Model Approach. Part II: Explanations”, British Journal for Philosophy of Science , 56(4): 889–911. doi:10.1093/bjps/axi148
Hansson, Sven Ove, 1989, “New Operators for Theory Change”, Theoria , 55(2): 114–132. doi:10.1111/j.1755-2567.1989.tb00725.x
Harper, William L., 1975, “Rational Belief Change, Popper Functions and the Counterfactuals”, Synthese , 30(1–2): 221–262. doi:10.1007/BF00485309
Harper, William L., Robert Stalnaker, and Glenn Pearce (eds.), 1980, Ifs: Conditionals, Belief, Decision, Chance and Time , Dordrecht: Springer Netherlands. doi:10.1007/978-94-009-9117-0
Hawthorne, John, 2005, “Chance and Counterfactuals”, Philosophy and Phenomenological Research , 70(2): 396–405. doi:10.1111/j.1933-1592.2005.tb00534.x
Heintzelman, Samantha J., Justin Christopher, Jason Trent, and Laura A. King, 2013, “Counterfactual Thinking about One’s Birth Enhances Well-Being Judgments”, The Journal of Positive Psychology , 8(1): 44–49. doi:10.1080/17439760.2012.754925
Herzberger, Hans, 1979, “Counterfactuals and Consistency”, The Journal of Philosophy , 76(2): 83–88. doi:10.2307/2025977
Hiddleston, Eric, 2005, “A Causal Theory of Counterfactuals”, Noûs , 39(4): 632–657. doi:10.1111/j.0029-4624.2005.00542.x
Hitchcock, Christopher, 2001, “The Intransitivity of Causation Revealed by Equations and Graphs”, The Journal of Philosophy , 98(6): 273–299. doi:10.2307/2678432
–––, 2007, “Prevention, Preemption, and the Principle of Sufficient Reason”, Philosophical Review , 116(4): 495–532. doi:10.1215/00318108-2007-012
Horwich, Paul, 1987, Asymmetries in Time , Cambridge, MA: MIT Press.
Iatridou, Sabine, 2000, “The Grammatical Ingredients of Counterfactuality”, Linguistic Inquiry , 31(2): 231–270. doi:10.1162/002438900554352
Ichikawa, Jonathan, 2011, “Quantifiers, Knowledge, and Counterfactuals”, Philosophy and Phenomenological Research , 82(2): 287–313. doi:10.1111/j.1933-1592.2010.00427.x
Ippolito, Michela, 2006, “Semantic Composition and Presupposition Projection in Subjunctive Conditionals”, Linguistics and Philosophy , 29(6): 631–672. doi:10.1007/s10988-006-9006-2
–––, 2008, “Subjunctive Conditionals”, in Proceedings of Sinn Und Bedeutung 12 , Atle Grønn (ed.), Oslo: Department of Literature, Area Studies and European Languages, University of Oslo, 256–270.
–––, 2013, Subjunctive Conditionals: A Linguistic Analysis , (Linguistic Inquiry Monograph Series 65), Cambridge, MA: MIT Press.
–––, 2016, “How Similar Is Similar Enough?”, Semantics and Pragmatics , 9(6): 1–60. doi:10.3765/sp.9.6
Isard, S.D., 1974, “What Would You Have Done If…”, Theoretical Linguistics , 1(1–3): 233–255. doi:10.1515/thli.1974.1.1-3.233
Jackson, Frank, 1987, Conditionals , Oxford: Basil Blackwell.
Kahneman, Daniel, Paul Slovic, and Amos Tversky (eds.), 1982, Judgement under Uncertainty: Heuristics and Biases , Cambridge: Cambridge University Press.
Kant, Immanuel, 1781, Critique of Pure Reason , Paul Guyer and Allen Wood (trans.), Cambridge: Cambridge University Press, 1998.
Kaufmann, Stefan, 2005, “Conditional Predictions”, Linguistics and Philosophy , 28(2): 181–231. doi:10.1007/s10988-005-3731-9
–––, 2008, “Conditionals Right and Left: Probabilities for the Whole Family”, Journal of Philosophical Logic , 38(1): 1–53. doi:10.1007/s10992-008-9088-0
–––, 2013, “Causal Premise Semantics”, Cognitive Science , 37(6): 1136–1170. doi:10.1111/cogs.12063
–––, 2017, “The Limit Assumption”, Semantics and Pragmatics , 10(18). doi:10.3765/sp.10.18
Khoo, Justin, 2015, “On Indicative and Subjunctive Conditionals”, Philosophers’ Imprint , 15(32): 1–40. [ Khoo 2015 available online ]
Kment, Boris, 2006, “Counterfactuals and Explanation”, Mind , 115(458): 261–310. doi:10.1093/mind/fzl261
–––, 2014, Modality and Explanatory Reasoning , New York: Oxford University Press. doi:10.1093/acprof:oso/9780199604685.001.0001
Koslicki, Kathrin, 2016, “Where Grounding and Causation Part Ways: Comments on Schaffer”, Philosophical Studies , 173(1): 101–112. doi:10.1007/s11098-014-0436-3
Kratzer, Angelika, 1981a, “The Notional Category of Modality”, in Words, Worlds and Contexts , Hans-Jürgen Eikmeyer and Hannes Rieser (eds.), Berlin: Walter de Gruyter, 38–74.
–––, 1981b, “Partition and Revision: The Semantics of Counterfactuals”, Journal of Philosophical Logic , 10(2): 201–216. doi:10.1007/BF00248849
–––, 1986, “Conditionals”, in Proceedings from the 22nd Regional Meeting of the Chicago Linguistic Society , Chicago: University of Chicago, 1–15. [ Kratzer 1986 available online ]
–––, 1989, “An Investigation of the Lumps of Thought”, Linguistics and Philosophy , 12(5): 607–653. doi:10.1007/BF00627775
–––, 1990, “How Specific Is a Fact?”, in Proceedings of the 1990 Conference on Theories of Partial In- Formation , Center for Cognitive Science, University of Texas at Austin.
–––, 1991, “Modality”, in Semantics: An International Handbook of Contemporary Research , A. von Stechow and D. Wunderlich (eds.), Berlin: De Gruyter Mouton, 639–650.
–––, 2002, “Facts: Particulars or Information Units?”, Linguistics and Philosophy , 25(5–6): 655–670. doi:10.1023/A:1020807615085
–––, 2012, Modals and Conditionals: New and Revised Perspectives , New York: Oxford University Press. doi:10.1093/acprof:oso/9780199234684.001.0001
Kray, Laura J., Linda G. George, Katie A. Liljenquist, Adam D. Galinsky, Philip E. Tetlock, and Neal J. Roese, 2010, “From What Might Have Been to What Must Have Been: Counterfactual Thinking Creates Meaning”, Journal of Personality and Social Psychology , 98(1): 106–118. doi:10.1037/a0017905
Kripke, Saul A., 1963, “Semantical Analysis of Modal Logic I: Normal Modal Propositional Calculi”, Zeitschrift Für Mathematische Logik Und Grundlagen Der Mathematik , 9(5–6): 67–96. doi:10.1002/malq.19630090502
Kvart, Igal, 1986, A Theory of Counterfactuals , Indianapolis, IN: Hackett.
–––, 1992, “Counterfactuals”, Erkenntnis , 36(2): 139–179. doi:10.1007/BF00217472
Lange, Marc, 1999, “Laws, Counterfactuals, Stability, and Degrees of Lawhood”, Philosophy of Science , 66(2): 243–267. doi:10.1086/392686
–––, 2000, Natural Laws in Scientific Practice , New York: Oxford University Press.
–––, 2009, Laws and Lawmakers: Science, Metaphysics, and the Laws of Nature , Oxford: Oxford University Press. doi:10.1093/acprof:oso/9780195328134.001.0001
Leitgeb, Hannes, 2012a, “A Probabilistic Semantics for Counterfactuals: Part A”, The Review of Symbolic Logic , 5(1): 26–84. doi:10.1017/S1755020311000153
–––, 2012b, “A Probabilistic Semantics for Counterfactuals: Part B”, The Review of Symbolic Logic , 5(1): 85–121. doi:10.1017/S1755020311000165
Levi, Isaac, 1988, “The Iteration of Conditionals and the Ramsey Test”, Synthese , 76(1): 49–81. doi:10.1007/BF00869641
Lewis, C. I., 1912, “Implication and the Algebra of Logic”, Mind , New Series, 21(84): 522–531. doi:10.1093/mind/XXI.84.522
–––, 1914, “The Calculus of Strict Implication”, Mind , New Series, 23(90): 240–247. doi:10.1093/mind/XXIII.1.240
Lewis, David K., 1973a, “Causation”, Journal of Philosophy , 70(17): 556–567. doi:10.2307/2025310
–––, 1973b, Counterfactuals , Cambridge, MA: Harvard University Press.
–––, 1973c, “Counterfactuals and Comparative Possibility”, Journal of Philosophical Logic , 2(4): 418–446. doi:10.2307/2215339
–––, 1979, “Counterfactual Dependence and Time’s Arrow”, Noûs , 13(4): 455–476. doi:10.2307/2215339
–––, 1981, “Ordering Semantics and Premise Semantics for Counterfactuals”, Journal of Philosophical Logic , 10(2): 217–234. doi:10.1007/BF00248850
Lewis, Karen S., 2016, “Elusive Counterfactuals”, Noûs , 50(2): 286–313. doi:10.1111/nous.12085
–––, 2017, “Counterfactuals and Knowledge”, in The Routledge Handbook of Epistemic Contextualism , Jonathan Jenkins Ichikawa (ed.), New York: Routledge, 411–424.
–––, 2018, “Counterfactual Discourse in Context”, Noûs , 52(3): 481–507. doi:10.1111/nous.12194
Loewer, Barry, 1976, “Counterfactuals with Disjunctive Antecedents”, The Journal of Philosophy , 73(16): 531–537. doi:10.2307/2025717
–––, 1983, “Information and Belief”, Behavioral and Brain Sciences , 6(1): 75–76. doi:10.1017/S0140525X00014783
–––, 2007, “Counterfactuals and the Second Law”, in Causation, Physics, and the Constitution of Reality: Russell’s Republic Revisited , Huw Price and Richard Corry (eds.), New York: Oxford University Press, 293–326.
Lowe, E. J., 1983, “A Simplification of the Logic of Conditionals”, Notre Dame Journal of Formal Logic , 24(3): 357–366. doi:10.1305/ndjfl/1093870380
–––, 1990, “Conditionals, Context, and Transitivity”, Analysis , 50(2): 80–87. doi:10.1093/analys/50.2.80
Lucas, Christopher G. and Charles Kemp, 2015, “An Improved Probabilistic Account of Counterfactual Reasoning”, Psychological Review, 122(4): 700–734. doi:10.1037/a0039655
Lycan, William G., 2001, Real Conditionals , Oxford: Oxford University Press.
Lyons, John, 1977, Semantics , Vol. 2, Cambridge: Cambridge University Press. doi:10.1017/CBO9780511620614
Mackie, John L., 1974, The Cement of the Universe: A Study in Causation , Oxford: Oxford University Press.
Marr, David, 1982, Vision: A Computational Investigation into the Human Representation and Processing of Visual Information , San Francisco: W.H. Freeman.
Maudlin, Tim, 2007, Metaphysics Within Physics , New York: Oxford University Press. doi:10.1093/acprof:oso/9780199218219.001.0001
McKay, Thomas J. and Peter van Inwagen, 1977, “Counterfactuals with Disjunctive Antecedents”, Philosophical Studies , 31(5): 353–356. doi:10.1007/BF01873862
Morreau, Michael, 2009, “The Hypothetical Syllogism”, Journal of Philosophical Logic , 38(4): 447–464. doi:10.1007/s10992-008-9098-y
–––, 2010, “It Simply Does Not Add Up: Trouble with Overall Similarity”, The Journal of Philosophy , 107(9): 469–490. doi:10.5840/jphil2010107931
Moss, Sarah, 2012, “On the Pragmatics of Counterfactuals”, Noûs , 46(3): 561–586. doi:10.1111/j.1468-0068.2010.00798.x
Nelson, Everett J., 1933, “On Three Logical Principles in Intension”, The Monist , 43(2): 268–284. doi:10.5840/monist19334327
Nozick, Robert, 1969, “Newcomb’s Problem and Two Principles of Choice”, in Essays in Honor of Carl G. Hempel , Nicholas Rescher (ed.), Dordrecht: D. Reidel, 111–5.
Nute, Donald, 1975a, “Counterfactuals”, Notre Dame Journal of Formal Logic , 16(4): 476–482. doi:10.1305/ndjfl/1093891882
–––, 1975b, “Counterfactuals and the Similarity of Words”, The Journal of Philosophy , 72(21): 773–778. doi:10.2307/2025340
–––, 1980a, “Conversational Scorekeeping and Conditionals”, Journal of Philosophical Logic , 9(2): 153–166. doi:10.1007/BF00247746
–––, (ed.), 1980b, Topics in Conditional Logic , Dordrecht: Reidel. doi:10.1007/978-94-009-8966-5
Palmer, Frank Robert, 1986, Mood and Modality , Cambridge: Cambridge University Press.
Parisien, Christopher and Paul Thagard, 2008, “Robosemantics: How Stanley the Volkswagen Represents the World”, Minds and Machines , 18(2): 169–178. doi:10.1007/s11023-008-9098-2
Pearl, Judea, 1995, “Causation, Action, and Counterfactuals”, in Computational Learning and Probabilistic Reasoning , A. Gammerman (ed.), New York: John Wiley and Sons, 235–255.
–––, 2000, Causality: Models, Reasoning, and Inference , Cambridge: Cambridge University Press.
–––, 2002, “Reasoning with Cause and Effect”, AI Magazine , 23(1): 95–112. [ Pearl 2002 available online ]
–––, 2009, Causality: Models, Reasoning, and Inference , second edition, Cambridge: Cambridge University Press.
–––, 2013, “Structural Counterfactuals: A Brief Introduction”, Cognitive Science , 37(6): 977–985. doi:10.1111/cogs.12065
Peirce, Charles S., 1896, “The Regenerated Logic”, The Monist , 7(1): 19–40. doi:10.5840/monist18967121
Pendlebury, Michael, 1989, “The Projection Strategy and the Truth: Conditions of Conditional Statements”, Mind , 98(390): 179–205. doi:10.1093/mind/XCVIII.390.179
Pereboom, Derk, 2014, Free Will, Agency, and Meaning in Life , New York: Oxford University Press. doi:10.1093/acprof:oso/9780199685516.001.0001
Pollock, John L., 1976, Subjunctive Reasoning , Dordrecht: D. Reidel Publishing Co., doi:10.1007/978-94-010-1500-4
–––, 1981, “A Refined Theory of Counterfactuals”, Journal of Philosophical Logic , 10(2): 239–266. doi:10.1007/BF00248852
Quine, Willard Van Orman, 1960, Word and Object , Cambridge, MA: MIT Press.
–––, 1982, Methods of Logic , fourth edition, Cambridge, MA: Harvard University Press.
Rips, Lance J., 2010, “Two Causal Theories of Counterfactual Conditionals”, Cognitive Science , 34(2): 175–221. doi:10.1111/j.1551-6709.2009.01080.x
Rips, Lance J. and Brian J. Edwards, 2013, “Inference and Explanation in Counterfactual Reasoning”, Cognitive Science , 37(6): 1107–1135. doi:10.1111/cogs.12024
Ryle, Gilbert, 1949, The Concept of Mind , London: Hutchinson.
Sanford, David H., 1989, If P Then Q: Conditionals and the Foundations of Reasoning , London: Routledge.
Santorio, Paolo, 2014, “Filtering Semantics for Counterfactuals: Bridging Causal Models and Premise Semantics”, in the Proceedings of Semantics and Linguistic Theory (SALT) 24 , 494–513.
Schaffer, Jonathan, 2016, “Grounding in the Image of Causation”, Philosophical Studies , 173(1): 49–100. doi:10.1007/s11098-014-0438-1
Schulz, Katrin, 2007, “Minimal Models in Semantics and Pragmatics: Free Choice, Exhaustivity, and Conditionals”, PhD Thesis, Amsterdam: University of Amsterdam: Institute for Logic, Language and Computation. [ Schultz 2007 available online ]
–––, 2011, “If You’d Wiggled A, Then B Would’ve Changed”, Synthese , 179(2): 239–251. doi:10.1007/s11229-010-9780-9
–––, 2014, “Fake Tense in Conditional Sentences: A Modal Approach”, Natural Language Semantics , 22(2): 117–144. doi:10.1007/s11050-013-9102-0
Seto, Elizabeth, Joshua A. Hicks, William E. Davis, and Rachel Smallman, 2015, “Free Will, Counterfactual Reflection, and the Meaningfulness of Life Events”, Social Psychological and Personality Science , 6(3): 243–250. doi:10.1177/1948550614559603
Sider, Theodore, 2010, Logic for Philosophy , New York: Oxford University Press.
Skyrms, Brian, 1980, “The Prior Propensity Account of Subjunctive Conditionals”, in Harper, Stalnaker, and Pearce 1980: 259–265. doi:10.1007/978-94-009-9117-0_13
Sloman, Steven, 2005, Causal Models: How People Think About the World and Its Alternatives , New York: Oxford University Press. doi:10.1093/acprof:oso/9780195183115.001.0001
Sloman, Steven A. and David A. Lagnado, 2005, “Do We ‘Do’?”, Cognitive Science , 29(1): 5–39. doi:10.1207/s15516709cog2901_2
Slote, Michael, 1978, “Time in Counterfactuals”, Philosophical Review , 87(1): 3–27. doi:10.2307/2184345
Smilansky, Saul, 2000, Free Will and Illusion , New York: Oxford University Press.
Snider, Todd and Adam Bjorndahl, 2015, “Informative Counterfactuals”, Semantics and Linguistic Theory , 25: 1–17. doi:10.3765/salt.v25i0.3077
Spirtes, Peter, Clark Glymour, and Richard Scheines, 1993, Causation, Prediction, and Search , Berlin: Springer-Verlag.
–––, 2000, Causation, Prediction, and Search , second edition, Cambridge, MA: The MIT Press.
Sprigge, Timothy L.S., 1970, Facts, Worlds and Beliefs , London: Routledge & K. Paul.
–––, 2006, “My Philosophy and Some Defence of It”, in Consciousness, Reality and Value: Essays in Honour of T. L. S. Sprigge , Pierfrancesco Basile and Leemon B. McHenry (eds.), Heusenstamm: Ontos Verlag, 299–321.
Stalnaker, Robert C., 1968, “A Theory of Conditionals”, in Studies in Logical Theory , Nicholas Rescher (ed.), Oxford: Basil Blackwell, 98–112.
–––, 1972 [1980], “Letter to David Lewis”. Reprinted in Harper, Stalnaker, and Pearce 1980: 151–152. doi:10.1007/978-94-009-9117-0_7
–––, 1975, “Indicative Conditionals”, Philosophia , 5(3): 269–286. doi:10.1007/BF02379021
–––, 1978, “Assertion”, in Syntax and Semantics 9: Pragmatics , Peter Cole (ed.), New York: Academic Press, 315–332.
–––, 1980, “A Defense of Conditional Excluded Middle”, in Harper, Stalnaker, and Pearce 1980: 87–104. doi:10.1007/978-94-009-9117-0_4
–––, 1984, Inquiry , Cambridge, MA: MIT Press.
–––, 1999, Context and Content: Essays on Intentionality in Speech and Thought , Oxford: Oxford University Press.
Stalnaker, Robert C. and Richmond H. Thomason, 1970, “A Semantic Analysis of Conditional Logic”, Theoria , 36(1): 23–42. doi:10.1111/j.1755-2567.1970.tb00408.x
Starr, William B., 2012, The Structure of Possible Worlds , UCLA Workshop. [ Starr 2012 available online ]
–––, 2014, “A Uniform Theory of Conditionals”, Journal of Philosophical Logic , 43(6): 1019–1064. doi:10.1007/s10992-013-9300-8
Stone, Matthew, 1997, “The Anaphoric Parallel between Modality and Tense”. Philadelphia, PA: University of Pennsylvania Institute for Research in Cognitive Science, Technical Report No. MS-CIS-97-06. [ Stone 1997 available online ]
Swanson, Eric, 2012, “Conditional Excluded Middle without the Limit Assumption”, Philosophy and Phenomenological Research , 85(2): 301–321. doi:10.1111/j.1933-1592.2011.00507.x
Tadeschi, Philip, 1981, “Some Evidence for a Branching-Futures Semantic Model”, in Tense and Aspect , (Syntax and Semantics, 14), Philip Tedeschi and Annie Zaenen (eds.), New York: Academic Press, 239–269.
Tarski, Alfred, 1936, “Der Wahrheitsbegriff in Den Formalizierten Sprachen”, Studia Philosophica , 1: 261–405.
Thrun, Sebastian, Mike Montemerlo, Hendrik Dahlkamp, David Stavens, Andrei Aron, James Diebel, Philip Fong, et al., 2006, “Stanley: The Robot That Won the DARPA Grand Challenge”, Journal of Field Robotics , 23(9): 661–692. doi:10.1002/rob.20147
Tichý, Pavel, 1976, “A Counterexample to the Stalnaker-Lewis Analysis of Counterfactuals”, Philosophical Studies , 29(4): 271–273. doi:10.1007/BF00411887
Todd, William, 1964, “Counterfactual Conditionals and the Presuppositions of Induction”, Philosophy of Science , 31(2): 101–110. doi:10.1086/287987
Veltman, Frank, 1976, “Prejudices, Presuppositions and the Theory of Counterfactuals”, in Amsterdam Papers in Formal Grammar , J. Groenendijck and M. Stokhof (eds.) (Proceedings of the 1st Amsterdam Colloquium), University of Amsterdam, 248–281.
–––, 1985, “Logics for Conditionals”, Ph.D. Dissertation, Amsterdam: University of Amsterdam.
–––, 1986, “Data Semantics and the Pragmatics of Indicative Conditionals”, in On Conditionals , Elizabeth C. Traugott, Alice ter Meulen, Judy S. Reilly, and Charles A. Ferguson (eds.), Cambridge: Cambridge University Press.
–––, 2005, “Making Counterfactual Assumptions”, Journal of Semantics , 22(2): 159–180. doi:10.1093/jos/ffh022
Walters, Lee, 2014, “Against Hypothetical Syllogism”, Journal of Philosophical Logic , 43(5): 979–997. doi:10.1007/s10992-013-9305-3
Walters, Lee and J. Robert G. Williams, 2013, “An Argument for Conjunction Conditionalization”, The Review of Symbolic Logic , 6(04): 573–588. doi:10.1017/S1755020313000191
Warmbrōd, Ken, 1981a, “Counterfactuals and Substitution of Equivalent Antecedents”, Journal of Philosophical Logic , 10(2): 267–289. doi:10.1007/BF00248853
–––, 1981b, “An Indexical Theory of Conditionals”, Dialogue, Canadian Philosophical Review , 20(4): 644–664. doi:10.1017/S0012217300021399
Wasserman, Ryan, 2006, “The Future Similarity Objection Revisited”, Synthese , 150(1): 57–67. doi:10.1007/s11229-004-6256-9
Weatherson, Brian, 2001, “Indicative and Subjunctive Conditionals”, The Philosophical Quarterly , 51(203): 200–216. doi:10.1111/j.0031-8094.2001.00224.x
Willer, Malte, 2015, “Simplifying Counterfactuals”, in 20th Amsterdam Colloquium , Thomas Brochhagen, Floris Roelofsen, and Nadine Theiler (eds.), Amsterdam: ILLC, 428–437. [ Willer 2015 available online ]
–––, 2017, “Lessons from Sobel Sequences”, Semantics and Pragmatics , 10(4). doi:10.3765/sp.10.4
–––, 2018, “Simplifying with Free Choice”, Topoi , 37(3): 379–392. doi:10.1007/s11245-016-9437-5
Williams, J. Robert G., 2010, “Defending Conditional Excluded Middle”, Noûs , 44(4): 650–668. doi:10.1111/j.1468-0068.2010.00766.x
Williamson, Timothy, 2005, “Armchair Philosophy, Metaphysical Modality and Counterfactual Thinking”, Proceedings of the Aristotelian Society , 105(1): 1–23. doi:10.1111/j.0066-7373.2004.00100.x
–––, 2007, The Philosophy of Philosophy , Malden, MA: Blackwell.
Wilson, Alastair, 2018, “Metaphysical Causation”, Noûs , 52(4): 723–751. doi:10.1111/nous.12190
Woodward, Jim, 2002, “What Is a Mechanism? A Counterfactual Account”, Philosophy of Science , 69(S3): S366–S377. doi:10.1086/341859
–––, 2003, Making Things Happen: A Theory of Causal Explanation , Oxford: Oxford University Press. doi:10.1093/0195155270.001.0001
Zeman, Jay, 1997, “Peirce and Philo”, in Studies in the Logic of Charles Sanders Peirce , Nathan Houser, Don Roberts, and James Van Evra (eds.), Indianapolis: Indiana University Press, 402–417.

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Exploring Hypothesis Contrary To Fact: A Hidden Trap in Our ...
Hypothesis Contrary To Fact, also known as "counterfactual fallacy" or "speculative fallacy," is a type of logical fallacy where a statement or argument is made based on a hypothetical situation that is presented as fact, but is actually contrary to what is known or proven to be true.
Hypothesis Contrary to Fact - Palomar College
Hypothesis Contrary to Fact. Description: From a statement of fact, the argument draws a counterfactual claim (i.e. a claim about what would have been true if the stated fact were not true). The argument falsely assumes that any state of affairs can have only one possible cause. Examples:
26 Common Logical Fallacies To Avoid When Making an Argument
Hypothesis contrary to fact This attempts to prove a point by referring to hypothetical situations instead of concrete evidence. It’s a fallacy because imaginary events or actions have no impact on real-life events or the truth.
Logical Fallacies | Definition, Types, List & Examples - Scribbr
Logical fallacies are leaps of logic that lead us to an unsupported conclusion. People may commit a logical fallacy unintentionally, due to poor reasoning, or intentionally, in order to manipulate others. Logical fallacy example.
Logical Fallacies Handlist | Arguing Through Writing
Hypothesis Contrary to Fact (Argumentum Ad Speculum): Trying to prove something in the real world by using imaginary examples alone, or asserting that, if hypothetically X had occurred, Y would have been the result.
Avoiding Logical Fallacies | Thoughtful Learning: Curriculum ...
Hypothesis contrary to fact forms an argument on the basis of something that didn’t happen. This fallacy is also called “if only” thinking. If only my candidate had won, the economy would be fixed by now. (There is no way to prove or disprove what would have happened if the other candidate had won, so the argument is meaningless.)
Today's Logical Fallacy is...Hypothesis Contrary to Fact!
Today's Logical Fallacy is...Hypothesis Contrary to Fact! This fallacy occurs when someone argues that their specific prediction about the present would be true or accurate if a past event had happened differently. It’s fallacious because the premises are based on speculation, not fact or evidence, essentially drawing conclusions from a ...
Logical Fallacies - Stanford University
Accent: the emphasis on a word or phrase suggests a meaning contrary to what the sentence actually says; Category Errors . Composition: because the attributes of the parts of a whole have a certain property, it is argued that the whole has that property; Division: because the whole has a certain property, it is argued that the parts have that ...
Counterfactuals - Stanford Encyclopedia of Philosophy
1. Counterfactuals and Philosophy. 1.1 What are Counterfactuals? 1.2 Agency, Mind, and Rationality. 1.2.1 Agency, Choice, and Free Will.
Logically Fallacious: The Ultimate Collection of Over 300 ...
Logically Fallacious is one of the most comprehensive collections of logical fallacies with all original examples and easy to understand descriptions, perfect for educators, debaters, or...

World	\(\mathsf{A}\)	\(\mathsf{B}\)	\(\mathsf{C}\)
\(w_{0}\)	1	1	1
\(w_{1}\)	1	1	0
\(w_{2}\)	1	0	1
\(w_{3}\)	1	0	0
\(w_{4}\)	0	1	1
\(w_{5}\)	0	1	0
\(w_{6}\)	0	0	1
\(w_{7}\)	0	0	0

\(C_0\)	\(\mathsf{R}\)	\(\mathsf{W}\)	\(\mathsf{H}\)
\(\boldsymbol{w_0}\)
\(\boldsymbol{w_1}\)
\(\boldsymbol{w_2}\)
\(\boldsymbol{w_3}\)
\(\boldsymbol{w_4}\)
\(\boldsymbol{w_5}\)
\(\boldsymbol{w_6}\)
\(\boldsymbol{w_7}\)

\(C_1\)	\(\mathsf{R}\)	\(\mathsf{W}\)	\(\mathsf{H}\)
\(\boldsymbol{w_0}\)
\(\boldsymbol{w_1}\)
\(\boldsymbol{w_2}\)
\(\boldsymbol{w_3}\)
\(\xcancel{w_4}\)	\(\xcancel{1}\)	\(\xcancel{0}\)	\(\xcancel{0}\)
\(\xcancel{w_5}\)	\(\xcancel{1}\)	\(\xcancel{0}\)	\(\xcancel{1}\)
\(\boldsymbol{w_6}\)
\(\boldsymbol{w_7}\)

\(C_2\)	\(\mathsf{R}\)	\(\mathsf{W}\)	\(\mathsf{H}\)
\(w_0\)	0	0	0
\(w_1\)	0	0	1
\(w_2\)	0	1	0
\(w_3\)	0	1	1
\(\xcancel{w_4}\)	\(\xcancel{1}\)	\(\xcancel{0}\)	\(\xcancel{0}\)
\(\xcancel{w_5}\)	\(\xcancel{1}\)	\(\xcancel{0}\)	\(\xcancel{1}\)
\(\boldsymbol{w_6}\)
\(\boldsymbol{w_7}\)

	\(\mathsf{B}\)	\(\mathsf{H}\)	\(\mathsf{W}\)
\(w_0\)	0	0	0
\(\xcancel{w_1}\)	\(\xcancel{0}\)	\(\xcancel{0}\)	\(\xcancel{1}\)
\(\boldsymbol{w_2}\)
\(\xcancel{w_3}\)	\(\xcancel{0}\)	\(\xcancel{1}\)	\(\xcancel{1}\)
\(w_4\)	1	0	0
\(\xcancel{w_5}\)	\(\xcancel{1}\)	\(\xcancel{0}\)	\(\xcancel{1}\)
\(\xcancel{w_6}\)	\(\xcancel{1}\)	\(\xcancel{1}\)	\(\xcancel{0}\)
\(w_7\)	1	1	1

\(\phi\)	\(\neg\phi\)
1	0
0	1

\(\phi\)	\(\psi\)	\(\phi\land\psi\)
1	1	1
1	0	0
0	1	0
0	0	0

\(\phi\)	\(\psi\)	\(\phi\supset\psi\)
1	1	1
1	0	0
0	1	1
0	0	1

(a)	\(f(w,p)\subseteq p\)
(b)	\(f(w,p)=\{w\}\), if \(w\in p\)
(c)	\(f(w,p)\subseteq q\) & \(f(w,q)\subseteq p \; {\Longrightarrow}\; f(w,p)=f(w,q)\)
(d)	\(f(w,p)\) contains one world