Wolfgang Schwarz

Lewis 1969 on the probability of conditionals

I finally got around to adding the papers from Janssen-Lauret and Macbride 2023 to the search corpus at https://www.david-lewis.org. It's a wonderful collection with lots of treasures. I want to comment on an intriguing passage on pp.71f., from an abandoned 1969 textbook project on confirmation theory.

First, some context. At this point in the manuscript, Lewis has introduced \(\mathcal{M}\) as a probability measure on the propositions expressible in a language \(\mathcal{L}\) with classical boolean connectives; \(\mathcal{C}\) is the associated conditional probability measure, defined by the ratio formula. Lewis notes that conditional probabilities are often read as "the probability of C if A", which suggests that \(\mathcal{C}(C/A)\) might equal \(\mathcal{M}(C\textit{ if }A)\), where '\(C\textit{ if }A\)' is the material conditional. But that's obviously false. Lewis continues:

If then else

Bare indicative conditionals are bewildering, but they become surprisingly well-behaved if we add an 'else' clause.

Intuitively, 'if A then B' doesn't make an outright claim about the world. It says that B is the case if A is the case – but what if A isn't the case?

An 'else' clause resolves this question. 'If A then B else C' makes an outright claim. It says that either B or C is the case, depending on whether A is the case. That is: the world is either an A-world, in which case it is also a B-world, or it is a ¬A-world, in which case it is a C-world. For short: (A∧B)∨(¬A∧C).

A fork time puzzle

According to a popular view about counterfactuals, a counterfactual hypothesis 'if A had happened…' shifts the world of evaluation to worlds that are much like the actual world until shortly before the time of A, at which point they start to deviate from the actual world in a minimal way that allows A to happen. 'If A had happened, C would have happened' is true iff all such worlds are C worlds. The time "shortly before A" when the worlds start to deviate is the fork time.

Now remember the case of Pollock's coat (introduced in Nute (1980)). John Pollock considered 'if my coat had been stolen last night…'. He stipulates that there were two occasions on which the coat could have been stolen. By the standards of Lewis (1979), worlds where it was stolen on the second occasion are more similar to the actual world than worlds where it was stolen on the first occasion. Lewis's similarity semantics therefore predicts that if the coat had been stolen, it would have been stolen on the second occasion. This doesn't seem right.

An RSA model of SDA

In this post, I'll develop an RSA model that explains why 'if A or B then C' is usually taken to imply 'if A then C' and 'if B then C', even if the conditional has a Lewis/Stalnaker ("similarity") semantics, where the inference is invalid.

I'll write 'A>C' for the conditional 'if A then C'. For the purposes of this post, we assume that 'A>C' is true at a world w iff all the closest A worlds to w are C worlds, by some contextually fixed measure of closeness.

It has often been observed that the simplification effect resembles the "Free Choice" effect, i.e., the apparent entailment of '◇A' and '◇B' by '◇(A∨B)', where the diamond is a possibility modal (permission, in the standard example). But there are also important differences.

McGee conditionals vs SDA

Has it been noted that McGee conditionals seem to clash with the Simplification of Disjunctive Antecedents (SDA)?

Consider the following conditional, inspired by McGee (1985).

(1) If a Republican had won then if it hadn't been Reagan then it would have been Andersen.

For context, imagine a scenario in which there were exactly two Republican candidates for the office in question, called Reagan and Andersen. Neither won. In this kind of context, (1) seems fine. So does (2).

(2) If Reagan or Andersen had won then if Reagan hadn't won then Andersen would have won.

Now, SDA (in its strong form) is the hypothesis that a conditional of the form 'if A or B then C' is equivalent to the conjunction of 'if A then C' and 'if B then C'. Applying this to (2), we would predict that (2) is equivalent to the conjunction of (3) and (4).

Champollion, Ciardelli, and Zhang on de Morgan's law

Champollion, Ciardelli, and Zhang (2016) argue that truth-conditionally equivalent sentences can make different contributions to the truth-conditions of larger sentences in which they embed. This seems obviously true. 'There are infinitely many primes' and Fermat's Last Theorem are truth-conditionally equivalent, but 'I can prove that there are infinitely many primes' is true, while 'I can prove that there are no integers a, b, c, and n > 2 for which aⁿ + bⁿ = cⁿ' is false. Champollion, Ciardelli, and Zhang (henceforth, CCZ) have a more interesting case in mind. They argue that substituting logically equivalent sentences in the antecedent of a subjunctive conditional can make a difference to the conditional's truth-value.

Gallow on causal counterfactuals without miracles and backtracking

Gallow (2023) spells out an interventionist theory of counterfactuals that promises to preserve two apparently incompatible intuitions.

Suppose the laws of nature are deterministic. What would have happened if you had chosen some act that you didn't actually choose? The two apparently incompatible intuitions are:

(A1) Had you chosen differently, no law of nature would have been violated.

(A2) Had you chosen differently, the initial conditions of the universe would not have been changed.

Rejecting one of these intuitions is widely thought to spell trouble for Causal Decision Theory. Gallow argues that they can both be respected. I'll explain how. Then I'll explain why I'm not convinced.

Kocurek on chance and would

A lot of rather technical papers on conditionals have come out in recent years. Let's have a look at one of them: Kocurek (2022).

The paper investigates Al Hajek's argument (e.g. in Hájek (2021)) that "chance undermines would". It begins with a neat observation.

Reversible Sobel Sequences

A Sobel sequence is a sequence of conditionals with increasingly strong antecedent. Lewis used Sobel sequences to motivate his "variably strict" analysis of counterfactuals.

For example, intuitively (1) and (2) might both be true, which seems to contradict a simple strict analysis:

(1) If the US had destroyed its nuclear weapons in 1965, there would have been war.
(2) If every country destroyed its nuclear weapons in 1965, there would have been peace.

Gibbard and Jackson on the probability of conditionals

Gibbard's 1981 paper "Two recent theories of conditionals" contains a famous passage about a poker game on a riverboat.

Sly Pete and Mr. Stone are playing poker on a Mississippi riverboat. It is now up to Pete to call or fold. My henchman Zack sees Stone's hand, which is quite good, and signals its content to Pete. My henchman Jack sees both hands, and sees that Pete's hand is rather low, so that Stone's is the winning hand. At this point, the room is cleared. A few minutes later, Zack slips me a note which says "If Pete called, he won," and Jack slips me a note which says "If Pete called, he lost." I know that these notes both come from my trusted henchmen, but do not know which of them sent which note. I conclude that Pete folded.

One puzzle raised by this scenario is that it seems perfectly appropriate for Zack and Jack to assert the relevant conditionals, and neither Zack nor Jack has any false information. So it seems that the conditionals should both be true. But then we'd have to deny that 'if p then q' and 'if p then not-q' are contrary.

The probability that if A then B

It has often been pointed out that the probability of an indicative conditional 'if A then B' seems to equal the corresponding conditional probability P(B/A). Similarly, the probability of a subjunctive conditional 'if A were the case then B would be the case' seems to equal the corresponding subjunctive conditional probability P(B//A). Trying to come up with a semantics of conditionals that validates these equalities proves tricky. Nonetheless, people keep trying, buying into all sorts of crazy ideas to make the equalities come out true.

Might counterfactuals

A might counterfactual is a statement of the form 'if so-and-so were the case then such-and-such might be the case'. I used to think that there are different kinds of might counterfactuals: that sometimes the 'might' takes scope over the entire conditional, and other times it does not.

For example, suppose we have an indeterministic coin that we don't toss. In this context, I'd say (1) is true and (2) is false.

(1) If I had tossed the coin it might have landed heads.

(2) If I had tossed the coin it would have landed heads.

These intuitions are controversial. But if they are correct, then the might counterfactual (1) can't express that the corresponding would counterfactual is epistemically possible. For we know that the would counterfactual is false. That is, the 'might' here doesn't scope over the conditional. Rather, the might counterfactual (1) seems to express the dual of the would counterfactual (2), as Lewis suggested in Counterfactuals: 'if A then might B' seems to be equivalent to 'not: if A then would not-B'.

Ifs and cans

Is 'can' information-sensitive in an interesting way, like 'ought'?

An example of uninteresting information-sensitivity is (1):

(1) If you can lift this backpack, then you can also lift that bag.

Informally speaking, the if-clause takes wide scope in (1). The truth-value of the consequent 'you can lift that bag' varies from world to world, and the if-clause directs us to evaluate the statement at worlds where the antecedent is true.

Reduction and coordination

The following principles have something in common.

Conditional Coordination Principle.
A rational person's credence in a conditional A->B should equal the ratio of her credence in the corresponding propositions B and A&B; that is, Cr(A->B) = Cr(B/A) = Cr(B)/Cr(A&B).

Normative Coordination Principle.
On the supposition that A is what should be done, a rational agent should be motivated to do A; that is, very roughly, Des(A/Ought(A)) > 0.5.

Probability Coordination Principle.
On the supposition that the chance of A is x, a rational agent should assign credence x to A; that is, roughly, Cr(A/Ch(A)=x) = x.

Nomic Coordination Principle.
On the supposition that it is a law of nature that A, a rational agent should assign credence 1 to A; that is, Cr(A/L(A)) = 1.

All these principles claim that an agent's attitudes towards a certain kind of proposition rationally constrain their attitudes towards other propositions.

Conditional expressions

Most programming languages have conditional operators that combine a (boolean) condition and two singular terms into a singular term. For example, in Python the expression

'hi' if 2 < 7 else 'hello'

is a singular term whose value is the string 'hi' (because 2 < 7). In general, the expression

x if p else y

denotes x in case p is true and otherwise y. So, for example,

Subjunctive credence and statistical chance

In her 2012 paper "Subjunctive Credences and Semantic Humility" (2012), Sarah Moss presents an interesting case due to John Hawthorne.

Suppose that it is unlikely that you perform a certain physical movement M tomorrow, though in the unlikely event that you contract a rare disease D, the chance of your performing M is high. Suppose also that the combination of contracting D and performing M causes death. Then many judge that the objective chance of 'if you were to perform M tomorrow, you would die' is low, but the conditional objective chance of this subjunctive given that you perform M is high.

The intuitive judgments Moss reports are

Supposing the truth

Here is a coin. What would have happened if I had just tossed it? It might have landed heads, and it might have landed tails. If the coin is biased towards tails, it is more likely that it would have landed heads. If it's a fair coin, both outcomes are equally likely. That is, they are equally likely on the supposition that the coin had been tossed. Let's write this as P(Heads // Toss) = 1/2, where the double slash indicates that the supposition in question is "subjunctive" rather than "indicative".

Counterexamples to Stalnaker's Thesis

I like a broadly Kratzerian account of conditionals. On this account, the function of if-clauses is to restrict the space of possibilities on which the rest of the sentence is evaluated. For example, in a sentence of the form 'the probability that if A then B is x', the if-clause restricts the space of possibilities to those where A is true; the probability of B relative to this restricted space is x iff the unrestricted conditional probability of B given A is x. This account therefore valides something that sounds exactly like "Stalnaker's Thesis" for indicative conditionals:

Imaging, counterfactuals, and expected conditional chance

In today's installment we take a look at the "imaging analysis" of subjunctive conditional probability. We will find that the analysis is fairly empty, and therefore fairly safe. In particular, it seems invulnerable to a worry that Robbie Williams recently raised in a comment on his blog. Let's begin with an example.

What if the government hadn't bailed out the banks? Some of them would almost certainly have gone bankrupt, and other companies would probably have followed.

Here we have some sort of conditional probabilities: "if A, then probably/almost certainly C". But they aren't ordinary conditional probabilities of the kind that go in the ratio formula, P(A/B) = P(AB)/P(B). I do not believe that if the government actually didn't bail out the banks (but only made everyone believe it did), then some of the banks went bankrupt. That is, my ordinary conditional probability in the bankruptcies given that there was no bailout is fairly low. Nevertheless, I believe that if the government hadn't bailed out the banks, some of them would probably have gone bankrupt. My subjunctive conditional probability in the bankruptcies given no-bailout is high.

An argument against some causal decision theories

Here is an attempt at an argument against formulating causal decision theory in terms of counterfactuals (loosely following up on the discussion in the previous post). The point seems rather obvious, so it is probably old. Does anyone know?

Suppose you would like to go for a walk, but only if it's not raining. Unfortunately, it is raining heavily, so you have almost decided to stay inside. Then you remember Gibbard and Harper's paper "Counterfactuals and two kinds of expected utility".

Lewis on Counterfactuals, Similarity, and Morgenbesser's Coin

There is a mistake on page 49 of Lewis's "Counterfactual dependence and time's arrow" (1979). Since the mistake seems to be repeated all the time, it might be worth pointing it out.

Page 49 is where Lewis lists similarity standards for his analysis of counterfactuals. The analysis, recall, says that "if A were the case, then C" is true iff the closest A-worlds are C-worlds (or, more precisely, iff either there are no A-worlds or some A&C-worlds are closer to the actual world than any A&~C world). Closeness is a matter of similarity, and Lewis indicates what the relevant respects of similarity might be for certain ordinary counterfactuals in section 3.3 of his 1973 book, and again in the 1979 article on counterfactual dependence. Roughly, the closest A-worlds are those that perfectly match the actual world across as much of spacetime as possible without diverse and widespread violations of the actual laws. This won't do for indeterministic worlds, where generally no laws need to be violated at all in order to ensure perfect match of futures even after earlier divergence. So Lewis restricts his standards to deterministic worlds, returning to the indeterministic case in the 1986 postscript to the 1979 paper.

Quantified counterfactuals, strict conditionals and escaping animals

Speaking of chapter six, Williamson here argues that the sentence

1) if an animal escaped from the zoo, it would be a monkey

is not adequately formalized as

1')

on the grounds that according to (1'), even the elephants are such that they would be monkeys if they escaped from the zoo. Williamson suggests that an adequate formalization might rather go like this:

Inadmissible games and counterfactuals

A time traveler offers you a game. You can toss a fair coin. If it lands heads, you win $2; if it lands tails, you lose $1. The time traveler informs you that all fair coins tossed today will land tails. (He knows, because he's seen all the results before traveling back in time.) Do you play?

Suppose you decide to toss. Trusting the time traveler, you can then be confident that you will lose $1. You would not have lost anything if you hadn't tossed, so the alternative option would have been better. It seems that you've made the wrong decision.

Modal knowledge, counterfactuals and counterpossibles

Carrie, Joe and Brit have recently commented on Williamson's proposal that modal knowledge is based on counterfactual knowledge. I share their suspicion, partly for the reasons Carrie mentions: the mere fact that statements about necessity and possibility are equivalent to counterfactuals doesn't tell us that the route to knowing the former proceeds via the latter. In fact, the assumption that we have a special cognitive faculty for knowing counterfactuals already seems odd to me. After all, we don't have special faculties for knowing indicatives or negations or conjunctions.

Might

Lewis once proposed that a 'might' counterfactual ("if A had been the case, C might have been the case") is true iff is true. This is sometimes used in defense of controversial philosophical claims, like in Al Hájek's "Most Counterfactuals are False" and in Boris Kment's "Counterfactuals and Explanation". But at least in some cases, the analysis doesn't seem right.

If it rains

This appears to be a problem for pure epistemic accounts of indicative conditionals (a la Weatherson and Chalmers), on which "if A then B" is true iff the [epistemically] closest worlds verifying A also verify B.

The match cannot be played if it rains; either it has to be postponed or canceled. Which of these will happen is regulated by the rule book, but nobody has looked up the relevant passages so far. All we know is that exactly one of these two conditionals is in the rule book, and therefore true, and the other false:

A Quiz (about "or" in English and in philosophers' English)

Nomic Facts and the Future

Suppose some thing x turns F, and a little later some other thing y turns G. x is the only F throughout history, so on a Humean account of laws of nature, it may well be just a coincidence that y's being G followed x's being F. Suppose it is.

But now consider another world just like this one except that in the far future, lots of G-turnings follow lots of F-turnings so that in this world, it is a law that whenever something turns F and another thing is suitably related, then that other thing turns G. In such a world, x's turning F caused y's turning G.

Some tricky counterfactuals

Sometimes, a counterfactual is true even though the consequent is false in the closest world where the antecedent is true:

1) If Hurricane Katrina hadn't hit the town with 200 km/h, completely destroying our house, we would be at home now, watching TV.

Presumably, at the closest worlds where Hurricane Katrina doesn't hit the town with 200 km/h and completely destroys the house, it hits the town a little faster or slower, still completely destroying the house. Even at the closest worlds where the hurricane doesn't completely destroy the house, it destroys it almost completely, still preventing the TV event.

Non-Backtracking Backward Counterfactuals

If Tina is a time traveler who is free to change the past, it must be true that

1) if Tina had chosen 1928, a time traveler would have appeared in 1928.

Moreover, this must be true on a "non-back-tracking" interpretation. A back-tracking interpretation is one on which we consider how past events would have had to be in order to cause some later event. Let's see how (1) fares on Lewis' conditions for non-back-tracking counterfactuals (in "Counterfactual Dependence and Time's Arrow").

Counterfactuals and Games of Make-Believe

In chapter 10 of The Varieties of Reference, Gareth Evans endorses a counterfactual analysis of truth in games of make-believe: When children play the mud pie game, an utterance of "Harry placed the pie in the oven" is true (in the game) iff (roughly) it would be true given that these globs of mud were pies and this metal object were an oven.

He then notices that this is a problem for the possible worlds analysis of counterfactuals because the relevant counterfactuals seem to have impossible antecedents: "there simply are no possible worlds in which these mud pats are pies" (p.355).

Counterfactuals and Counterexamples

It is controversial whether indicative conditionals with false antecedents are generally true. As far as I know, which really is not very far at all, it is equally controversial whether counterfactual conditionals with necessarily false antecedents are generelly true. What's interesting is the different kinds of counterexamples that are brought forward against these views. For indicatives, the counterexamples are indicative conditionals with false antecedents that nevertheless appear to be false, e.g. "if I put diesel in my coffee, the coffee tastes fine." For counterfactuals however, the alleged counterexamples (brought forward e.g. by Field in §7.2 of Realism, Mathematics & Modality, Katz in §5 of "What mathematical knowledge could be", and Rosen in §1 of "Modal fictionalism fixed") are counterfactual conditionals with necessarily false antecedents that appear to be true, e.g. "if the axiom of choice were false, the cardinals wouldn't be linearly ordered". Isn't this quite puzzling? How can the fact that some instances are true be a problem for a theory that claims that all instances are true?

What does the Wason Selection Task test?

I'm doing a visual memory test. On the table in front of me are twelve green and fourteen red apples, and an empty basket. The lights go out, and the instructor says to me:

"Put all the green apples into the basket". (1)

I try to do what he says. When the lights go on, you, the instructor's assistant, are given a form on which you are to tick whether I've correctly or incorrectly fulfilled the task. You see twelve green and two red apples in the basket. What do you tick?