Ambiguity-marking Kleene logic: adapting K3 for or and some ambiguity in natural language

Abstract

We start from the fact that or and some do not receive the same reading in every position in a sentence. Sometimes or allows both disjuncts to hold and sometimes it excludes them; sometimes some is compatible with all and sometimes it is not. What decides between one reading and the other is not so much the conversational situation as the monotonicity of the context in which the lexical item appears. Grice explains the phenomenon as an inference the hearer draws about what the speaker has chosen not to say. His explanation is cancellable, it rests on reasonable maxims, and it forces no change to the semantics. Nevertheless, we present a range of phenomena that escape it. These phenomena that escape the pragmatic explanation lead Gennaro Chierchia to argue that strengthening is not a conversational residue but a grammatical operation. We reconstruct his mechanism, which consists in a set of alternatives associated with each lexical item, and an exhaustification operator that can be anchored at any node of the syntactic structure, and illustrate it for or and some. Out of that discussion come four requirements any theory of the phenomenon would have to meet (lexical trigger, locality, sensitivity to monotonicity, and reversibility). We also show the mathematical requirements that his mechanism imposes: an intensional higher-order metalanguage, two-dimensional meanings with a composition rule of their own, contextual pruning, and repair strategies for the cases in which exhaustification turns out contradictory. With those requirements in mind, we propose an expansion of Strong Kleene, which we have called Ambiguity-marking Kleene logic (K3), intended to make the formal language itself record the ambiguity rather than resolve it. This calls for a connective that yields indeterminacy from determinate arguments, something no Strong Kleene connective does, and from it come the ambiguous disjunction ˜∨ and the ambiguous quantifier ˜∃. The algebraic study of the resulting logic shows that ˜∨ is the greatest lower bound, in the information order, of its two readings (the most informative value compatible with both), and it also shows how restrictive this design is: ˜∨ is not a lattice operation, it is not monotone in the truth order, it is not definable in Strong Kleene, and K3 has no theorems. We describe the formalisation of natural-language sentences as a function Φ that assigns to each sentence not a formula but a formula together with a set of admissible resolutions, over which a supervaluation is defined; polarity-sensitive pruning stages return the classical tautologies, idempotence and De Morgan, whereas absorption never returns. We show that the final stage, the hearer’s choice, cannot be a semantic operation, and that the point at which compositionality fails is the point at which semantics ends and pragmatics begins.

We conclude by comparing the two frameworks over the fragment of language for which they are designed. Chierchia’s operator strengthens the sentence and returns a determinate proposition, whereas ˜∨ commits itself only where the two readings agree. Our framework meets the four requirements, though locality comes out weaker and the polarity that drives sensitivity to monotonicity is enumerated rather than characterised, and it covers considerably less: with no modals there is no free choice, with no epistemic content there are no ignorance inferences, and the discourse asymmetry of or is invisible to a commutative connective.