Premise Semantics

Origin. David Lewis, "Adverbs of quantification" (in Edward Keenan ed., Formal Semantics of Natural Language, Cambridge University Press 1975, 3–15), argued that no plausible semantics for a two-place conditional connective interacts compositionally with always, usually, sometimes to give the right truth conditions, and that if instead marks the restriction of the adverb. Angelika Kratzer generalized the restriction idea to every conditional and built the modal theory it operates in: "What 'must' and 'can' must and can mean" (Linguistics and Philosophy 1, 1977, 337–355), "The notional category of modality" (in Hans-Jürgen Eikmeyer and Hannes Rieser eds., Words, Worlds, and Contexts: New Approaches in Word Semantics, de Gruyter 1981, 38–74), "Partition and revision: the semantics of counterfactuals" (Journal of Philosophical Logic 10, 1981, 201–216), and "Conditionals" (CLS 22, 1986, 1–15). After Irene Heim (1982) applied the analysis to donkey anaphora, Barbara Partee (1991) fixed the name "Lewis/Kratzer/Heim analysis", and it became the dominant treatment of conditionals in linguistics.

Models. Modals are quantifiers over possible worlds, and context supplies their domain twice over: a modal base f assigns each world the premises settled there — what is known, what the circumstances are — and an ordering source g assigns the premises that rank the worlds the modal base admits — what the law demands, what is normal, what is wanted. One lexical entry for must then yields epistemic, deontic, bouletic, and circumstantial readings as f and g vary, so the apparent polysemy of the modals is context-dependence (Kratzer 1977, 1981). The ordering source is forced by deontic cases: if must quantified over all law-conforming worlds, "Robin must pay a fine" would be false after Robin double-parks, since no law-conforming world contains the offence; quantifying over the law-best worlds among the circumstance-admitted ones gets it right. Conditionals enter because if is not a connective: an if-clause restricts the modal base of whatever operator is present — "the history of the conditional is the story of a syntactic mistake" (Kratzer 1986) — so indicative, counterfactual, modalized, and adverbially quantified conditionals are one construction, differing in the operator restricted. Where Conditional Logic takes A > C as a two-place operator over a similarity parameter, premise semantics derives the counterfactual from premise sets — consistent combinations of the antecedent with the ranked background premises — and Lewis proved the two pictures equivalent. Update Semantics moves the modal machinery into context change; premise semantics stays truth-conditional, with context entering only through the parameters f and g.

Formalism.

Conversational backgrounds: f, g: functions from worlds to sets of propositions. "In view of what is known": f(w) = the propositions known in w. Modal base f: admits the worlds in ⋂f(w). Ordering source g: ranks them.

Induced ordering: u ≤_{g(w)} v iff {p ∈ g(w) : v ∈ p} ⊆ {p ∈ g(w) : u ∈ p}. u is at least as good as v: u satisfies every g(w)-premise v satisfies. A preorder, and not total in general.

Modals: must φ, with the limit assumption: every ≤{g(w)}-best world in ⋂f(w) is a φ-world. Without it: for every u ∈ ⋂f(w) there is v ≤{g(w)} u such that every z ≤_{g(w)} v is a φ-world. can φ: the dual. Epistemic: epistemic f, stereotypical g. Deontic: circumstantial f, g from the law. Bouletic: circumstantial f, g from desires. Graded notions — good possibility, slight possibility, comparative possibility — from the same ordering, no new primitives.

Restriction: If-clause: adds the antecedent to the modal base. [if A][must φ]: must quantifies over the best worlds in ⋂(f(w) ∪ {A}). [if A][usually φ]: the adverb quantifies over A-cases (Lewis 1975). Bare conditionals: a covert operator — epistemic necessity or generic frequency — restricted by the if-clause (Kratzer 1986).

Counterfactuals by premise sets (Kratzer 1981): Premise set for A at w: a consistent set containing A, the modal-base facts, and as many g(w)-premises as consistency allows. A □→ C true at w iff every such premise set entails C. Quantification over premise sets, not over closest worlds: no limit assumption required.

Symbols.

Metatheory. Lewis, "Ordering semantics and premise semantics for counterfactuals" (Journal of Philosophical Logic 10, 1981, 217–234), proved the equivalence theorem: premise semantics and ordering semantics are intertranslatable, with premise sets corresponding exactly to ordering semantics without the totality requirement — a set of premises induces the preorder ≤_{g(w)} defined above, which need not be total — so the counterfactual logic is that of ordering semantics on orderings that need not be total (see Conditional Logic). The restrictor analysis's distinctive prediction is Lewis's 1975 result: the restrictor reading hands the if-clause to the adverb as its domain and gets "if it is sunny, we usually play soccer" right, while a connective reading of if can be forced only with a third truth value, a far-fetched interpretation of the connective, and a permanent restriction on the admissible cases (Lewis 1975, fn. 4). The limit assumption is dispensable on both sides: Lewis's completeness proof for ordering semantics avoids it, and Kratzer's truth conditions — the double quantification for must, the quantification over premise sets for the counterfactual — are stated so that antecedent worlds without best elements cause no difficulty. Heim (1982) showed the same restriction mechanism handles donkey anaphora, which is what carried the analysis through linguistics.

Applies to. The semantics of modal auxiliaries — epistemic, deontic, bouletic, circumstantial readings from one entry. Conditionals of every mood. Adverbs of quantification and donkey sentences. Graded and comparative modality. Deontic paradoxes turning on settled facts.

Limitations. The conversational backgrounds are parameters the theory does not fix: as with the similarity ordering of Conditional Logic, the semantics states how f and g determine truth conditions and says nothing about which f and g a context supplies, so particular predictions require an account of context the framework does not contain. The covert modal in bare conditionals is the contested point: on the restrictor analysis a bare conditional's only propositional content is the covertly modalized one, yet "Probably so" in reply to "If he didn't tell Harry, he told Tom" embeds an unmodalized conditional under the probability operator (von Fintel 2003, 2007), speakers with different evidence assess what seems to be one conditional rather than each their own epistemic claim, and nothing settles that the covert operator is epistemic necessity rather than another modal. Separating conditionals by the operators they contain loses the predictions the shared if…then form once made: how much indicative and counterfactual conditionals have in common becomes a question about their covert and overt modals. And rival analyses that give the if-clause content of its own remain live — plural definite descriptions of worlds (Schein 2003; Schlenker 2004; Bhatt and Pancheva 2006), and the three-valued connective going back to Belnap (1970, 1973), on which operators quantify over the cases where the conditional is defined.

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