sam carter & simon goldstein | rutgers university
disjunction and distributivity
1 introduction Suppose that a card is drawn from a standard deck. Under these conditions (1) appears true, and appears equivalent to (2): (1)
If the card is red, then either it must be a heart or it must be a diamond.
(2)
If the card is red, then either it is a heart or it is a diamond.
We argue that consideration of (1)-(2), along with analogous examples, motivates the acceptance of the following principle: conditional modal disjunction χ → (φ ∨ ψ )
χ → (□φ ∨ □ψ )
cmd is invalid on all standard accounts of ∨, →, and □. We thus investigate several new dynamic meanings for ∨ and → that validate cmd, and discuss their various merits and costs.
2 update semantics Our investigation will take place within update semantics (Veltman 1996), a type of dynamic semantics.1 In update semantics, the denotion [φ] of a sentence φ is its context change potential. Identifying contexts with sets of worlds, [φ] is a function from one set of worlds s to a potentially distinct set of worlds s′ . Adopting a standard account of support and entailment, let a context s support a sentence φ iff s is a fixed point of [φ]. Let a set of sentences Γ entail δ iff every fixed point of all of Γ is a fixed point of δ: Definition 1 (Support, Entailment). s supports φ (s |= φ) iff s[φ] = s. Γ entails δ (Γ context s, if s |= γ for every γ ∈ Γ, then s |= δ.
δ) iff for every
In update semantics, standard clauses for α, ∨, □, and → are as follows: Definition 2 (Update Semantics). 1. s[α] = {w ∈ s | w(α) = 1}
3. s[□φ] = {w ∈ s | s |= φ}
2. s[φ ∨ ψ ] = s[φ] ∪ s[ψ ]
4. s[φ → ψ ] = {w ∈ s | s[φ] |= ψ }2
But these standard dynamic meanings invalidate cmd.3
3 distributivity To validate cmd, we propose enriching update semantics with a special distributivity operator. Here is the big picture idea. cmd suggests that in a special environment (disjunctions in the consequent of conditionals), the modal claim □φ acts just like the non-modal claim φ. For φ ∨ ψ has the same effect there as □φ ∨ □ψ, and our intuitions about the truth of (1) suggest that both claims act like φ ∨ ψ. It turns out that there is an operator one can add to update semantics that converts the update of □φ into the update of φ. So we want this operator to activate exactly when □φ occurs in environments like that in cmd. 1 Stalnaker 1973; Karttunen 1974; Heim 1982; Heim 1983; Veltman 1985; Groenendijk and Stokhof 1990; Groenendijk and Stokhof 1991; and many others. 2 Gillies 2004 3 For let s = {w, v}, where w(α) = v(α) = 1, v( β) = 0 = 1 − w( β), and w(γ ) = 0 = 1 − v(γ ). Here s[α] = s, s[α][ β ∨ γ ] = s, and s[α][□ β ∨ □γ] = ∅. So s |= α → ( β ∨ γ) and s ̸|= α → (□ β ∨ □γ).
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sam carter & simon goldstein | rutgers university
disjunction and distributivity
Our operator will take any ordinary CCP and transform it into a distributive CCP by (i) zooming in to each world in the context, (ii) updating only this world with the CCP, and (iii) unioning the results: ∪ Definition 3 (Distributive Update Operation). ↓ (φ) = λs. w∈s {w}[φ]. We can now inject this operator into the meaning of various lexical items, to validate cmd. A first attempt could be to make disjunction an essentially distributive operation, so that it unions the result of updating distributively with each disjunct: Definition 4 (Distributive Disjunction). s[φ ∨ ψ ] = s ↓ (φ) ∪ s ↓ (ψ ) Combined with our earlier clauses, this validates cmd.4 Unfortunately, however, this solution is too powerful. It overgenerates to validate the stronger: collapse φ ∨ ψ
□φ ∨ □ψ .5
So we need a different application of distributivity. We propose to locate distributivity not in disjunction, but rather in the conditional. The conditional φ → ψ tests that updating with φ creates a context that distributively supports the conclusion: Definition 5 (Distributive Conditional). s[φ → ψ ] = {w ∈ s | s[φ] ↓ (ψ ) = s[φ]} This validates cmd, since ↓ (φ ∨ ψ ) =↓ (□φ ∨ □ψ ). It does not validate Collapse. It instead validates the following principle, which is plausible and has been defended in Gillies 2010: conditional modal substitution φ → ψ
φ → □ψ
In addition to the empirical interest of cmd, this paper offers a case study in modularity. We have found that a given operator (↓) can be injected into update semantics in different places, with predictably different effects. In principle, one could reach similar effects with slightly different operators. For example, rather than our distributive operator, one could instead rely on a maximality operator: ∪ Definition 6 (Maximizing Update Operation). ↑ (φ) = λs. {s′ ⊆ s | s′ |= φ}. Replacing ↓ with ↑ in all the above would generate quite similar predictions for our original data, but would make different predictions when it came to might, the dual of must. For example, ↓ (♢φ) =↓ (□φ) =↓ (φ), but [♢φ] =↑ (♢φ) ̸=↑ (□φ) =↑ (φ). references 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. Jeroen Groenendijk and Martin Stokhof. Dynamic montague grammar. In Laszlo Kalman and Laszlo Polos, editors, Papers from the Second Symposium on Logic and Language, pages 3--48, Budapest, 1990. Akademiai Kiado. Jeroen Groenendijk and Martin Stokhof. Dynamic predicate logic. Linguistics and Philosophy, 14(1):39--100, February 1991. Irene Heim. The Semantics of Definite and Indefinite Noun Phrases. PhD thesis, University of Massachusetts, Amherst, MA, 1982. Irene Heim. On the projection problem for presuppositions. WCCFL, 2:114--125, 1983. Lauri Karttunen. Presuppositions and linguistic context. Theoretical Linguistics, 1:181--194, 1974. Robert Stalnaker. Presuppositions. Journal of Philosophical Logic, 2:447--457, 1973. Frank Veltman. Logics for Conditionals. PhD thesis, University of Amsterdam, 1985. Frank Veltman. Defaults in update semantics. Journal of Philosophical Logic, 25(3):221--261, 1996. 4↓
(□φ) =↓ (φ). So [ χ → (φ ∨ ψ )] = [ χ → (□φ ∨ □ψ )]. ↓ (φ) =↓ (□φ), we have that [φ ∨ ψ ] = [□φ ∨ □ψ ].
5 Since
2
