A vector logic for intensional formal semantics

Abstract

Formal semantics and distributional semantics are distinct approaches to linguistic meaning: the former models meaning as reference via model-theoretic structures; the latter as vectors in high-dimensional spaces shaped by usage. This paper establishes which part of intensional formal semantics admits a linear vector-space encoding. Kripke-style intensional models, with any finite collection of index sorts collected in a compound index space, embed injectively into vector spaces: primitive domains go to free carriers; intensions and other functions go to linear operators. Semantic functions lift to unique multilinear maps on the free carriers, and composition is preserved. The operator encoding of a function domain compresses its free carrier, and we characterize the functionals: a Boolean-valued functional of a power set acts linearly on operator encodings exactly when it is constant, an ultrafilter indicator, or the complement of one, so that on finite domains the nonconstant ones are Montague's individuals and their negations. Determiners over a restrictor of two or more elements, modal operators over two or more accessible indices, and attitude operators over two or more alternatives are outside the linear regime, and take, instead, the form of a linear accumulation followed by a decision. Modality is defined uniformly over measure frames, in which counting measure recovers Kripke semantics at every cardinality, and continuous measures make necessity truth almost everywhere; we give the correspondence conditions for the axioms D, T, B, 4, and 5 under measures, the Kronecker factorization of accessibility over compound indices, and the reading of measure-based modality as graded modality.