Counterfactuals

Natural language

• As well as blocking true consequent, false antecedent, augmentation, and contraposition, we have:
  • No importation φ → (ψ → χ) ⊭ (φ ∧ ψ) → χ. (Or exportation, for that matter)
    • Counterexample: if Bill had married Laura, then if he had married Hillary, he would have been happy. But it’s false that if he had married Laura and Hillary, he’d have been happy (instead, he would’ve been imprisoned for polygamy).
  • No transitivity: φ → ψ, ψ → χ ⊭ φ → χ
    • Counterexample: if I hadn’t been born, Mike would’ve been my parent’s oldest child; if my parents never met, I wouldn’t have been born. But it’s false that if my parents had never met, Mike would have been my parent’s oldest child.
  • No transposition (material conditional only) φ → (ψ → χ) ⊭ ψ → (φ → χ)
    • Counterexample: if Bill had married Laura, then if Bill had married Hillary, then his wife’s name would be Hillary. But it’s false that if Bill had married Hillary, then if Bill had married Laura, his wife’s name would be Hillary.
• Disjunctive antecedents seem like they’re handled poorly in both systems, but it’s actually OK.
  • Consider: “if we’d surrendered or tried to run, we’d have been shot” seems to entail “if we’d surrendered, we’d have been shot”.
  • But on either theory, we do not have (P ∨ Q) □→ R ⊨ P □→ R.
  • Resolution: there’s a better way to formalise the English sentence than the naïve one – really it’s saying (P □→ R) ∧ (Q □→ R).
    • Compare: even though in natural English “there ain’t no cake” doesn’t imply that there is some cake, we’re perfectly happy to have the law of double negation elimination in logic.
• Stalnaker shows that any plausible theory of counterfactuals must invalidate transitivity and augmentation.
  • In particular, any theory such that: (i) some conditionals are contingent; and (ii) a conditional A □→ B, where A is possible and B logically incompatible with A, is false.
• Intuitively, we want our counterfactual conditional □→ to be stronger than the material conditional, but weaker than the strict conditional.
  • Since the material conditional → obeys modus ponens and modus tollens, so too will □→.
• You could use strict conditionals rather than Lewis/Stalnaker, if you allow for context-shifts.
  • E.g. consider again the Mike example above. When evaluating P1, we let □ range over a specific set of worlds S₁, where I wasn’t born but my parents met. And when evaluating P2, □ ranges over a different set of worlds S₂, where my parents didn’t meet.
    • So the conditional theorist invalidates the argument because P1 and P2 are evaluated relative to different sets of worlds; syllogisms hold only when both premisses are evaluated in a fixed set of worlds S.
  • Stalnaker’s reply is that although this might be extensionally adequate, it’s not very attractive as a theory, because it complicates things.
    • If context can shift between every conditional utterance, then semantic validity becomes extremely weak – you could dismiss any argument involving conditionals as an equivocation. So we lose the ability to explain semantically why some arguments are persuasive, and the work has to be done by pragmatics.
    • Rather than having that work done by the pragmatics, it seems more parsimonious to put it into the semantics (like Stalnaker’s selection function), where it’s transparent how it interacts with the logic.

Conditional excluded middle

• ⊧ (A □→ C) ∨ (A □→ ¬C)
  • Under Stalnaker, CEM holds. There’s unique maximally close A-world to each w; call it u_w. And in u_w, either C holds or it doesn’t. So the disjunction holds at every world of every model.
  • Doesn’t hold under Lewis.
    • There may be multiple equally-close A worlds. But if some of those make C true and others make C false, both disjuncts fail.
• Lewis concedes that CEM is an attractive feature of Stalnaker’s theory, and it’s a real cost to give it up.
  • “The principal virtue and the principal vice of Stalnaker’s theory is that it makes valid the law of conditional excluded middle”
• Arguments for CEM:
  • Linguistic. We don’t distinguish ¬(A □→ C) from (A □→ ¬C).
    • “It’s not true that if she’d played she’d have won” ≈ “If she’d played she wouldn’t have won”.
    • The backwards direction (A □→ ¬C) ⊧ ¬(A □→ C) is uncontroversial: if ¬C holds at the closest A-worlds, then C is false at all of them, so A □→ C is false.
    • CEM gives us forward direction. Under Lewis, it doesn’t work because (as usual) we can have multiple closest A-worlds, where C is false at some but is true at others.
  • Distribution. (A □→ B ∨ C) ⊧ (A □→ B) ∨ (A □→ C)
    • If someone says “if I’d been a baseball player, I’d have been third-baseman or a shortstop” invites the seemingly-valid question “well, which would you have been?” – and that presupposes that one or the other would hold.
    • Lewis objects, saying that they really might have been either – and it makes sense to ask the question (to probe for whether there is additional info) even if you’re anticipating the response might be “I would have been one of the two for sure, but it could really be either one”.
      • Compare: if someone says “the coin will land heads or tails”, you might ask “well, which?”, even if you accept they could say “there’s no fact of the matter yet” in response.
• Arguments against CEM
  • Bizet-Verdi.
    • Neither “if Bizet & Verdi were compatriots, Bizet would be Italian” nor “if Bizet & Verdi were compatriots, Bizet would not be Italian” seems true.
    • Lewis says both are false. Stalnaker, on his preferred approach using supervaluations, says both are indeterminate.
    • Lewis does concede that indeterminacy is more natural.
  • Might counterfactuals (see below).
    • With CEM, “might” – on Lewis’s definition – collapses into “would”. But these two connectives clearly differ in truth-value sometimes.

Might counterfactuals

“If A, it might be that B”.

• Lewis symbolises as A ◇→ B := ¬(A □→ ¬B).
  • Why? By ruling out other candidate symbolisations as implausible.
  • Suppose I didn’t look in my pocket and there is no penny there. Then “If I had looked, I might have found a penny” is clearly false. But the following four Stalnaker-friendly formalisations make it come out true:
    • ◇(φ ∧ ψ)
    • ◇(φ □→ ψ)
    • φ □→ ◇ψ
    • φ □→ ◇ (φ ∧ ψ)
• Stalnaker’s response: symbolise using as ◇(A □→ B), where ◇ is for epistemic modality.
  • Lewis’s symbolisation is unappealing because it doesn’t explain “if … might” in terms of “if” and “might” separately, and instead treats it as an unanalysed idiom – specifically, the dual of the “would” conditional.
  • Stalnaker thinks we can analyse “might” as describing epistemic possibility, and then “if” is the standard Stalnaker counterfactual.
  • One objection might be that the surface grammar suggests “might” should have narrow scope relative to “if”, so Stalnaker’s symbolisation is ad-hoc.
    • But actually, we accept wide scope readings in analogous English sentences – e.g. “if he is a bachelor, he must be unmarried” should be formalised as the de dicto claim “necessarily, if you’re a bachelor you’re unmarried” rather than de re “if you’re a bachelor, you’re necessarily unmarried”.
• Overall, seems better to favour Stalnaker. Our natural use of “might” and “would” suggests that they aren’t duals.
  • It would be strange to deny “would” while affirm “might”: e.g., “He wouldn’t have come if invited, although he might have come if invited” seems malformed.
    • Stalnaker can capture this: it’s Moore-paradoxical to claim something (he wouldn’t have come) but say it’s compatible with your knowledge that the opposite holds (he might have).
    • Lewis can’t: denying necessity (he needn’t come) while affirming possibility (but he might) is totally normal.

Limit

• Lewis’s one-inch line.
  • Suppose a line is just under an inch. The sentence “If this line were more than one inch long, it would be 100 miles long” is false, intuitively and under both theories. But they disagree about why.
  • Lewis says limit assumption fails: there simply is no closest more-than-one-inch world.
    • His counterfactual still can correctly evaluate the statement, though: there’s always a closer more-than-inch world where the line isn’t 100 miles long, so the Lewis truth-condition makes the statement false.
  • If limit assumption did hold, then there would need to be a particular closest more-than-inch world – say, 1.347in. But then the sentence “if the line were more than one inch, it would be 1.347in” is true, which seems arbitrary and wrong.
• Stalnaker’s response
  • The closest-phi-world selection function picks out a world that makes a minimal change in relevant respects. In most contexts, a millimetre extra length isn’t relevant.
    • So, it’s indeterminate which such world is selected, but that’s fine: supervaluations handle the indeterminacy (see below).
  • Moreover, there’s a bizarre consequence of Lewis’s view: for every real number x, “if the line had been more than one inch long, it would not have been x inches long” is true.
    • So there is no length the line might have had – even though the line might’ve been more than an inch long and would’ve had some length.
  • Stalnaker thinks that the selection function should be taken as a primitive. Rather than starting with a notion of similarity then deriving the selection function, it’s the other way round.
    • If the similarity relation were primitive, then limit is a substantive claim about it, and Lewis can reject the plausibility of that.
    • But taking the selection function to be primitive, limit holds automatically, by construction of the function.
    • This isn’t very satisfying; Stalnaker seems to admit this. Similarity seems to be much more a feature of the structure of the world than selection function is.

Uniqueness / antisymmetry

• Stalnaker concedes that antisymmetry is “a grossly implausible assumption” about how we actually use the English counterfactual, because of ties in similarity.
  • But, it’s just meant to be an “idealising assumption” (like how first-order logic idealises vague predicates), and can be made realistic by adding in supervaluations.
• Supervaluations are brought in by Stalnaker, because he thinks we can treat the problem of ties and vagueness about possible worlds as an instance of the general problem of semantic indeterminacy.
  • This preserves CEM, and addresses the uniqueness problem.
  • But Lewis objects:
    • Stalnaker conditionals still rely on the false limit assumption. In reality, there are cases with no closest worlds (like the one-inch line) – so no selection function should be admissible. SV can handle many closest worlds, but if there’s no closest world then every sentence would be vacuously supertrue.
    • Even using SV, Stalnaker conditionals don’t deal with the difference between “might” and “would”, because (under Lewis’s symbolisation) they behave identically. Although, Stalnaker can just use his alternative epistemic reading of “might”.
• Somewhat separately, note that in an SC- (or LC-) model, there’s no constraint at all on the relationship between ⪯ᵢ and ⪯ⱼ when i ≠ j.

Conclusions

• Overall: CEM is a genuine virtue, and the costs of validating it are manageable.
  • Stalnaker has a better analysis of epistemics & can handle might vs would.
  • Supervaluations deal with vagueness perfectly well.
  • Limit assumption defensible if you take selection function as primitive, but this is weak.
• Might counterfactuals aren’t a decisive reason to prefer Lewis over Stalnaker. In fact, they favour Stalnaker.
• Antisymmetry is one vice of Stalnaker’s semantics, but actually limit seems more insidious.
• On balance, Stalnaker’s theory with supervaluations seems most preferable.