96 research outputs found
Program Semantics and Classical Logic
In the tradition of Denotational Semantics one usually lets program
constructs take their denotations in reflexive domains, i.e. in domains
where self-application is possible. For the bulk of programming
constructs, however, working with reflexive domains is an
unnecessary complication. In this paper we shall use the domains
of ordinary classical type logic to provide the semantics of a
simple programming language containing choice and recursion. We prove
that the rule of {\em Scott Induction\/} holds in this new setting, prove
soundness of a Hoare calculus relative to our semantics, give a
direct calculus on programs, and prove that the denotation of
any program in our semantics is equal to the union of the denotations
of all those programs such that follows from in our calculus
and does not contain recursion or choice
A Relational Formulation of the Theory of Types
This paper developes a relational---as opposed to a functional---theory of types. The theory is based on Hilbert and Bernays' eta operator plus the identity symbol, from which Church's lambda and the other usual operators are then defined. The logic is intended for use in the semantics of natural language
Tense and the Logic of Change
In this paper it is shown how the DRT (Discourse Representation Theory) treatment of temporal anaphora can be formalized within a version of Montague Semantics that is based on classical type logic
Intensional Models for the Theory of Types
In this paper we define intensional models for the classical theory of types,
thus arriving at an intensional type logic ITL. Intensional models generalize
Henkin's general models and have a natural definition. As a class they do not
validate the axiom of Extensionality. We give a cut-free sequent calculus for
type theory and show completeness of this calculus with respect to the class of
intensional models via a model existence theorem. After this we turn our
attention to applications. Firstly, it is argued that, since ITL is truly
intensional, it can be used to model ascriptions of propositional attitude
without predicting logical omniscience. In order to illustrate this a small
fragment of English is defined and provided with an ITL semantics. Secondly, it
is shown that ITL models contain certain objects that can be identified with
possible worlds. Essential elements of modal logic become available within
classical type theory once the axiom of Extensionality is given up.Comment: 25 page
A Theory of Names and True Intensionality
Standard approaches to proper names, based on Kripke's views, hold
that the semantic values of expressions are (set-theoretic)
functions from possible worlds to extensions and that names are
rigid designators, i.e.\ that their values are \emph{constant}
functions from worlds to entities. The difficulties with these
approaches are well-known and in this paper we develop an
alternative. Based on earlier work on a higher order logic that is
\emph{truly intensional} in the sense that it does not validate the
axiom scheme of Extensionality, we develop a simple theory of names
in which Kripke's intuitions concerning rigidity are accounted for,
but the more unpalatable consequences of standard implementations of
his theory are avoided. The logic uses Frege's distinction between
sense and reference and while it accepts the rigidity of names it
rejects the view that names have direct reference. Names have
constant denotations across possible worlds, but the semantic value
of a name is not determined by its denotation
Hyperfine-Grained Meanings in Classical Logic
This paper develops a semantics for a fragment of English that is based on the idea of `impossible possible worlds'. This idea has earlier been formulated by authors such as Montague, Cresswell, Hintikka, and Rantala, but the present set-up shows how it can be formalized in a completely unproblematic logic---the ordinary classical theory of types. The theory is put to use in an account of propositional attitudes that is `hyperfine-grained', i.e. that does not suffer from the well-known problems involved with replacing expressions by logical equivalents
Anaphora and the Logic of Change
This paper shows how the dynamic interpretation of natural language introduced in work by Hans Kamp and Irene Heim can be modeled in classical type logic. This provides a synthesis between Richard Montague's theory of natural language semantics and the work by Kamp and Heim
Context Update for Lambdas and Vectors
Vector models of language are based on the contextual aspects of words
and how they co-occur in text. Truth conditional models focus on the
logical aspects of language, the denotations of phrases, and their
compositional properties. In the latter approach the denotation of a
sentence determines its truth conditions and can be taken to be a
truth value, a set of possible worlds, a context change
potential, or similar. In this short paper, we develop a vector
semantics for language based on the simply typed lambda calculus. Our
semantics uses techniques familiar from the truth conditional tradition
and is based on a form of dynamic interpretation inspired by
Heim's context updates
Description Theory, LTAGs and Underspecified Semantics
An attractive way to model
the relation between an underspecified syntactic representation and
its completions is to let the underspecified representation correspond
to a logical description and the completions to the
models of that description. This approach, which underlies the
Description Theory of (Marcus et al. 1983) has been integrated
in (Vijay-Shanker 1992) with a pure unification approach to
Lexicalized Tree-Adjoining
Grammars (Joshi et al.\ 1975, Schabes 1990). We generalize
Description Theory by integrating semantic
information, that is, we propose to tackle both syntactic and
semantic underspecification using descriptions
New directions in type-theoretic grammars
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