25 research outputs found
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
Mechanizing Principia Logico-Metaphysica in Functional Type Theory
Principia Logico-Metaphysica contains a foundational logical theory for
metaphysics, mathematics, and the sciences. It includes a canonical development
of Abstract Object Theory [AOT], a metaphysical theory (inspired by ideas of
Ernst Mally, formalized by Zalta) that distinguishes between ordinary and
abstract objects.
This article reports on recent work in which AOT has been successfully
represented and partly automated in the proof assistant system Isabelle/HOL.
Initial experiments within this framework reveal a crucial but overlooked fact:
a deeply-rooted and known paradox is reintroduced in AOT when the logic of
complex terms is simply adjoined to AOT's specially-formulated comprehension
principle for relations. This result constitutes a new and important paradox,
given how much expressive and analytic power is contributed by having the two
kinds of complex terms in the system. Its discovery is the highlight of our
joint project and provides strong evidence for a new kind of scientific
practice in philosophy, namely, computational metaphysics.
Our results were made technically possible by a suitable adaptation of
Benzm\"uller's metalogical approach to universal reasoning by semantically
embedding theories in classical higher-order logic. This approach enables one
to reuse state-of-the-art higher-order proof assistants, such as Isabelle/HOL,
for mechanizing and experimentally exploring challenging logics and theories
such as AOT. Our results also provide a fresh perspective on the question of
whether relational type theory or functional type theory better serves as a
foundation for logic and metaphysics.Comment: 14 pages, 6 figures; preprint of article with same title to appear in
The Review of Symbolic Logi
Theory of Concepts
UID/FIL/00183/2013authorsversionpublishe
Analytic Tableaux for Simple Type Theory and its First-Order Fragment
We study simple type theory with primitive equality (STT) and its first-order
fragment EFO, which restricts equality and quantification to base types but
retains lambda abstraction and higher-order variables. As deductive system we
employ a cut-free tableau calculus. We consider completeness, compactness, and
existence of countable models. We prove these properties for STT with respect
to Henkin models and for EFO with respect to standard models. We also show that
the tableau system yields a decision procedure for three EFO fragments
Predication and cognitive context: Between minimalism and contextualism
In this paper, we suggest a strategy for modelling cognitive context within a truth\u2010conditional semantics, using Asher's model of predication. This allows us to introduce the notion of type presupposition intended as a lexical constraint to the composition of the truth\u2010conditional content. More specifi\u2010cally, we suggest that this model of predication produces a notion of truth\u2010conditional meaning where the cognitive context fixes a set of lexical restrictions and forced modifi\u2010cations. We conclude that this model might offer an inter\u2010mediate position between Minimalism and Contextualism: an account that provides intuitive truth conditions within a formal semantic theory
Faithful Semantical Embedding of a Dyadic Deontic Logic in HOL
A shallow semantical embedding of a dyadic deontic logic by Carmo and Jones
in classical higher-order logic is presented. This embedding is proven sound
and complete, that is, faithful.
The work presented here provides the theoretical foundation for the
implementation and automation of dyadic deontic logic within off-the-shelf
higher-order theorem provers and proof assistants.Comment: 23 pages, 3 figure