A Constructive Approach to Intensional Contexts : Remarks on the Metaphysics of Model Theory

The basic distinction between extensional and intensional contexts is one of different denotation conditions (truth conditions). This difference in denotation conditions has been related to a fundamental question of semantic theory, namely: what do expressions of natural 'language denote? Most authors assume that in ex tensional contexts expressions denote 'real objects.' Since the rules of substitutivity of identicals and existential generalization do not hold in intensional contexts (cf. section 1), they are thus forced to postulate that in intensional contexts expressions denote something else. Frege, for example, assumes a de notational ambiguity between 'Bedeutung' and 'Sinn', Russell between 'primary occurrences' and 'secondary occurrences', Quine between 'proper occurrences' and 'accidental occurrences', and Montague between 'extensions' and 'intensions

Constraining Montague Grammar for computational applications

This work develops efficient methods for the implementation of Montague Grammar on a computer. It covers both the syntactic and the semantic aspects of that task. Using a simplified but adequate version of Montague Grammar it is shown how to translate from an English fragment to a purely extensional first-order language which can then be made amenable to standard automatic theorem-proving techniques. Translating a sentence of Montague English into the first-order predicate calculus usually proceeds via an intermediate translation in the typed lambda calculus which is then simplified by lambda-reduction to obtain a first-order equivalent. If sufficient sortal structure underlies the type theory for the reduced translation to always be a first-order one then perhaps it should be directly constructed during the syntactic analysis of the sentence so that the lambda-expressions never come into existence and no further processing is necessary. A method is proposed to achieve this involving the unification of meta-logical expressions which flesh out the type symbols of Montague's type theory with first-order schemas. It is then shown how to implement Montague Semantics without using a theorem prover for type theory. Nothing more than a theorem prover for the first-order predicate calculus is required. The first-order system can be used directly without encoding the whole of type theory. It is only necessary to encode a part of second-order logic and this can be done in an efficient, succinct, and readable manner. Furthermore the pseudo-second-order terms need never appear in any translations provided by the parser. They are vital just when higher-order reasoning must be simulated. The foundation of this approach is its five-sorted theory of Montague Semantics. The objects in this theory are entities, indices, propositions, properties, and quantities. It is a theory which can be expressed in the language of first-order logic by means of axiom schemas and there is a finite second-order axiomatisation which is the basis for the theorem-proving arrangement. It can be viewed as a very constrained set theory

Logical ambiguity

The thesis presents research in the field of model theoretic semantics on the problem of ambiguity, especially as it arises for sentences that contain junctions (and,or) and quantifiers (every man, a woman). A number of techniques that have been proposed are surveyed, and I conclude that these ought to be rejected because they do not make ambiguity 'emergent': they all have the feature that subtheories would be able to explain all syntactic facts yet would predict no ambiguity. In other words these accounts have a special purpose mechanism for generating ambiguities.It is argued that categorial grammars show promise for giving an 'emergent' account. This is because the only way to take a subtheory of a particular categorial grammar is by changing one of the small number of clauses by which the categorial grammar axiomatises an infinite set of syntactic rules, and such a change is likely to have a wider range of effects on the coverage of the grammar than simply the subtraction of ambiguity.Of categorial grammars proposed to date the most powerful is Lambek Categorial Grammar, which defines the set of syntactic rules by a notational variant of Gentzen's sequent calculus for implicational propositional logic, and which defines meaning assignment by using the Curry- Howard isomorphism between Natural Deduction proofs in implicational propositional logic and terms of typed lambda calculus. It is shown that no satisfactory account of the junctions and quantifiers is possible in Lambek categorial grammar.I introduce then a framework that I call Polymorphic Lambek Categorial Grammar, which adds variables and their universal quantification, to the language of categorisation. The set of syntac¬ tic rules is specified by a notational variant of Gentzen's sequent calculus for quantified proposi¬ tional logic, and which defines meaning assignment by using Girard's Extended Curry-Howard isomorphism between Natural Deduction proofs in quantified implicational propositional logic and terms of 2nd order polymorphic lambda calculus. It is shown that this allows an account of the junctions and quantifiers, and one which is 'emerg

New Equations for Neutral Terms: A Sound and Complete Decision Procedure, Formalized

The definitional equality of an intensional type theory is its test of type compatibility. Today's systems rely on ordinary evaluation semantics to compare expressions in types, frustrating users with type errors arising when evaluation fails to identify two `obviously' equal terms. If only the machine could decide a richer theory! We propose a way to decide theories which supplement evaluation with `$\nu$-rules', rearranging the neutral parts of normal forms, and report a successful initial experiment. We study a simple -calculus with primitive fold, map and append operations on lists and develop in Agda a sound and complete decision procedure for an equational theory enriched with monoid, functor and fusion laws

Transpension: The Right Adjoint to the Pi-type

Presheaf models of dependent type theory have been successfully applied to model HoTT, parametricity, and directed, guarded and nominal type theory. There has been considerable interest in internalizing aspects of these presheaf models, either to make the resulting language more expressive, or in order to carry out further reasoning internally, allowing greater abstraction and sometimes automated verification. While the constructions of presheaf models largely follow a common pattern, approaches towards internalization do not. Throughout the literature, various internal presheaf operators ($\surd$, $\Phi/\mathsf{extent}$, $\Psi/\mathsf{Gel}$, $\mathsf{Glue}$, $\mathsf{Weld}$, $\mathsf{mill}$, the strictness axiom and locally fresh names) can be found and little is known about their relative expressivenes. Moreover, some of these require that variables whose type is a shape (representable presheaf, e.g. an interval) be used affinely. We propose a novel type former, the transpension type, which is right adjoint to universal quantification over a shape. Its structure resembles a dependent version of the suspension type in HoTT. We give general typing rules and a presheaf semantics in terms of base category functors dubbed multipliers. Structural rules for shape variables and certain aspects of the transpension type depend on characteristics of the multiplier. We demonstrate how the transpension type and the strictness axiom can be combined to implement all and improve some of the aforementioned internalization operators (without formal claim in the case of locally fresh names)

Introducing Continuations

This working paper introduces CONTINUATIONS (a concept borrowed from computer science) as a new technique for characterizing certain aspects of the semantics of a natural language. I should emphasize at the outset that this is just an introduction, and that more a rigorous and thorough treatment is under development (see Barker (ms)). In the meantime, this paper mentions certain formal results without proving them, and describes certain new empirical generalizations without exploring them. What it will do is provide an explicit account of a range of familiar phenomena relat­ed to quantification, including quantifier scope ambiguity, NP as a scope island, and generalized coordination. What makes the account noteworthy is that it provides a fully and strictly compositional analysis of quantification and generalized coordina­tion that does not rely on syntactic movement operations such as Quantifier Move­ment, auxiliary storage mechanisms such as Cooper Storage, or type ambiguity as in Hendriks' Flexible Types system