conditional excluded middle and might conditionals

simon goldstein | rutgers university

The literature on conditionals has agreed that it is absurd to accept both of the following principles: conditional excluded middle duality φ → ♢ψ

(φ → ψ ) ∨ (φ → ¬ψ ).

¬(φ → ¬ψ ).

The first principle, cem, says that conditionals go in for excluded middle. This means that when we conditionally suppose some claim φ, we can always either infer ψ or infer ¬ψ. Plausible or not, there are many arguments for cem in the literature, often involving the behavior of conditionals under higher operators like only or under quantifiers.1 The second principle, Duality, relates ordinary conditionals to a special subvariety—might conditionals. Among its many merits, Duality makes sense of the `inescapable clash' involved in asserting q if p and might not q if p: (1)

??If I flip a fair coin, it will land heads; but it also might land tails if I flip it.2

Together, these two principles imply the absurd result that ordinary conditionals are logically equivalent to might conditionals: collapse φ → ♢ψ

φ → ψ.

Collapse looks absurd. After all, might conditionals are used very differently in thought and talk than their ordinary counterparts. To take just one example, ordinary conditionals go in for Modus Ponens, as in (2); but when we substitute the ordinary conditional for its might counterpart we get an invalid argument: (2)

If Mary is at the party, so is Sally. Mary is at the party. So: Sally is at the party.

(3)

If Mary is at the party, Sally might be. Mary is at the party. ??So: Sally is at the party.

Yet if the first premise of (2) is logically equivalent to the first premise of (3), it's hard to see how one argument could be good and the other bad. In this paper, I validate both cem and Duality. So Collapse turns out to be valid. To explain why Collapse isn't as bad as we thought, we will rely on a dynamic semantics where meaning is a lot more fine grained than logical equivalence. So even though ordinary conditionals co-entail their might counterparts, the two constructions have very different meanings. These differences in meanings have important effects. Among these effects is that Modus Ponens will be valid for ordinary conditionals, but invalid for might conditionals (as in (2) and (3) above). This leads to a very surprising result. Two logically equivalent sentences will fail to be preserve validity in arguments. That is, even though φ → ♢φ, and φ; φ → ψ ψ, it just ain't so that φ; φ → ♢ψ ψ. This means that φ→ψ our consequence relation will not obey the classic rule of Cut, which says that whenever φ entails ψ, we can replace ψ with φ as a premise in any argument. Much of the basic idea here is already present in von Fintel 1997, where conditionals carry homogeneity presuppositions. But I will implement these ideas in a dynamic semantics, building on the dynamic conditional in Gillies 2004. Going this way allows us to predict which kinds of conditionals presuppose homogeneity, which von Fintel 1997 must instead stipulate lexically. 1 For

some of these arguments, see von Fintel and Iatridou 2002 and von Fintel 1997. sorts of arguments date back to Lewis 1973. See DeRose 1999 for an attempted pragmatic explanation of (1) that opponents of Duality might appeal to. For another argument for Duality, see Gillies 2010, who suggests that φ → ψ is equivalent to φ → □φ. Duality follows quickly from this, given the duality of ♢ and □. 2 These

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simon goldstein | rutgers university

conditional excluded middle and might conditionals

The background framework is update semantics (Veltman 1996). In this framework, the meaning of a sentence isn't its truth conditions, but is instead its context change potential. Identifying contexts with sets of worlds, a sentence φ's meaning [φ] is a function from one set of worlds s to a potentially distinct set of worlds s′ . A context s supports a sentence φ iff s is a fixed point of [φ]. Not every CCP is defined in every context. Sometimes a sentence just can't be sensibly interpreted in a context, because it presupposes that the context already support some other claim. Entailment only cares about the contexts where the argument is defined. In that case, entailment is just a matter of preserving support: Definition 1 (Support, Entailment). s supports φ (s |= φ) iff s[φ] = s. Γ entails δ (Γ δ) iff for every context s where s[δ ] and s[γ] are defined for every γ ∈ Γ, if s |= γ for every γ ∈ Γ, then s |= δ. Our semantics will conservatively extend Veltman 1996 with a new conditional: Definition 2 (Update Semantics). 1. s[α] = {w ∈ s | w(α) = 1}

4. s[φ ∨ ψ ] = s[φ] ∪ s[ψ ]

2. s[¬φ] = s − s[φ]

5. s[♢φ] = {w ∈ s | s[φ] ̸= ∅}

3. s[φ ∧ ψ ] = s[φ][ψ ] Against this backdrop, I will add a new conditional. The conditional is built from two parts. The first part is a dynamic conditional from Gillies 2004, which says that φ → ψ tests a context to see whether updating with φ supports ψ. If so, it leaves the context the same; otherwise, it returns the empty state. The second part is a homogeneity presupposition from von Fintel 1997. In updatology: updating with φ → ψ is only defined in contexts s where s[φ] supports either ψ or ¬ψ: Definition 3. s[φ → ψ ] is defined only if s[φ] |= ψ or s[φ] |= ¬ψ. If defined, s[φ → ψ ] = {w ∈ s | s[φ] |= ψ } The testy, dynamic part of the conditional gets us Duality. For φ → ♢ψ is supported just in case updating with φ creates a context that is consistent with ψ. But this means that ¬ψ is not supported, which in turn means that ¬(φ → ¬ψ ) is. The homogeneity presupposition gets us cem. For when we zoom into the defined contexts, either φ → ψ or φ → ¬ψ will pass their test. All of this means we get Collapse. For when we zoom into the contexts where φ → ψ is defined, we are left with information that is homogenous with respect to ψ when it learns φ. In these homogenous spaces, ♢φ and φ have the same effect. But Collapse ain't as bad as its chalked up to be. For [φ → φ] ̸= [φ → ♢φ]. While φ; φ → ψ ψ, we don't have that φ; φ → ♢ψ ψ. This is because the second argument doesn't require that φ → ψ is defined. references Keith DeRose. Can it be that it would have been even though it might not have been? Philosophical Perspectives, 13:387--413, 1999. Anthony S. Gillies. Epistemic conditionals and conditional epistemics. Nous, 38(4):585--616, 2004. Anthony S. Gillies. Iffiness. Semantics and Pragmatics, 3(4):1--42, January 2010. doi: 10.3765/sp.3.4. David Lewis. Counterfactuals. Blackwell, 1973. Frank Veltman. Defaults in update semantics. Journal of Philosophical Logic, 25(3):221--261, 1996. Kai von Fintel. Bare plurals, bare conditionals, and Only. Journal of Semantics, 14:1--56, 1997. Kai von Fintel and Sabine Iatridou. If and when `if '-clauses can restrict quantifiers. 2002.

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