Abstract
A hyperintensional semantics for natural language is proposed which is agnostic about the question of whether propositions are sets of worlds or worlds are (maximal consistent) sets of propositions. Montague’s theory of intensional senses is replaced by a weaker theory, written in standard classical higher-order logic, of fine-grained senses which are in a many-to-one correspondence with intensions; Montague’s theory can then be recovered from the proposed theory by identifying the type of propositions with the type of sets of worlds and adding an axiom to the effect that each world is the set of propositions which are true there. Senses are compositionally assigned to linguistic expressions by a categorial grammar with only two rule schemas, based on the implicative fragment of intuitionistic linear propositional logic, and a fully explicit grammar fragment is provided that illustrates the compositional assignment of sense to a variety of constructions, including dummy-subject constructions, infinitive complements, predicative adjectives and nominals, raising to subject, ‘tough-movement’, and quantifier scope ambiguities. Notably, the grammar and the derivations that it licenses never make reference to either worlds or to the extensions of senses.
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Notes
More precisely, Montague’s intensions assigned extensions not to worlds but rather to world-time pairs; but for the sake of expository simplicity, we ignore temporality altogether.
Being a procedure need not preclude being a structured meaning; Tichý’s constructions have been characterized as ‘structured procedures’ by (Duži et al. (2010), p. 3).
For details of one such implementation, see Pollard (2008).
Note that we cannot simply identify worlds with ultrafilters, because ultrafilters have type \(\mathrm {p}\rightarrow \mathrm {t}\) and so, in any Henkin model of the theory, the interpretation of \(\mathrm {w}\) would be required to be a subset of the interpretation of \(\mathrm {p}\rightarrow \mathrm {t}\). Pollard (2008) got around this problem by working in a higher order logic—the boolean version of Lambek and Scott’s (1986) categorical logic—with a topos interpretation, so that subtypes corresponding to the kernel of a definable predicate (e.g. being an ultrafilter) are always available. In that version of the theory, the morphism that interprets \(@_{\mathrm {p}}\) is (the categorical generalization of) the Stone-dual mapping on a (pre-)boolean algebra (Pollard 2011).
They are mutually inconsistent because in MS not every ultrafilter corresponds to a world, only the principal ones do; see Sect. 4.7.2.
In fact, worlds are absent from Thomason’s system, though he tentatively introduces them toward the end of his exposition.
Predicate adjectives like easy also permit dummy-subject constructions (it is easy for Mary to please John), and ‘tough-movement’ constructions (John is easy for Mary to please), as discussed below.
Here I depart from both Montague and Frege in taking the determination of sense, but not of reference, to be compositional. This obviates the need to stipulate that, in certain contexts, the reference of an expression is its customary sense, or to apply a Montague-style \(\hat{}\)-operator to the arguments of a predicate.
There are also nonlogical axioms, corresponding to lexical entries.
The idea that phenos are string terms rather than strings is prefigured by Montague’s use of the ‘syntactic variables’ \(\mathbf {he}_n\), which are replaced by ‘real’ strings when the rules of quantification T14—T16 are applied. In Oehrle’s, and LCG’s, pheno-terms, the ‘syntactic variables’ are eliminated in favor of authentic variables of the higher-order pheno theory.
The double occurrences of \(A\) and \(B\) in the lexical entries \(\mathbf {is}\) and \(\mathbf {to}\) correspond to the fact that auxiliary verbs are ‘raising-to-subject’ predicates, i.e. the unrealized subject of the complement is identified with the subject of the auxiliary.
This technique of simulating transformational movement by applying a string function to the null string is due to Muskens (2007).
The movement operation in question was called ‘tough-movement’ because it was occasioned only in sentences containing certain predicates, such as tough, easy, a bitch, etc.
Here \(\mathsf {b}\) is the sense component in the lexical entry for the name Justin Bieber, and \(\mathsf {equals}: \mathrm {e}\rightarrow \mathrm {e}\rightarrow \mathrm {p}\) is a constant axiomatized as follows:
$$\begin{aligned} \vdash \forall _{wxy}. (x\ \mathsf {equals}\ y)@w \leftrightarrow (x = y) \end{aligned}$$This constant is introduced in the lexical entry for the predicativization sign that combines with an \(\mathrm {NP}\) to form a predicate:
$$\begin{aligned} \vdash \lambda _s.s; \mathrm {NP}\multimap \mathrm {Prd}; \lambda _{xy}.y\ \mathsf {equals}\ x. \end{aligned}$$Frege’s (1892) problem is the analogous puzzle for definite descriptions (the Evening Star and the Morning Star).
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Acknowledgments
AHS is the result of joint work with Andy Plummer. The dynamic extension of the theory arises from a collaboration with Scott Martin. For corrections and helpful comments on an earlier version, thanks are due to Manjuan Duan, Murat Yasavul, and an anonymous referee.
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Pollard, C. Agnostic hyperintensional semantics. Synthese 192, 535–562 (2015). https://doi.org/10.1007/s11229-013-0373-2
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DOI: https://doi.org/10.1007/s11229-013-0373-2