220 research outputs found
Topos Semantics for Higher-Order Modal Logic
We define the notion of a model of higher-order modal logic in an arbitrary
elementary topos . In contrast to the well-known interpretation of
(non-modal) higher-order logic, the type of propositions is not interpreted by
the subobject classifier , but rather by a suitable
complete Heyting algebra . The canonical map relating and
both serves to interpret equality and provides a modal
operator on in the form of a comonad. Examples of such structures arise
from surjective geometric morphisms , where . The logic differs from non-modal higher-order
logic in that the principles of functional and propositional extensionality are
no longer valid but may be replaced by modalized versions. The usual Kripke,
neighborhood, and sheaf semantics for propositional and first-order modal logic
are subsumed by this notion
Lewis meets Brouwer: constructive strict implication
C. I. Lewis invented modern modal logic as a theory of "strict implication".
Over the classical propositional calculus one can as well work with the unary
box connective. Intuitionistically, however, the strict implication has greater
expressive power than the box and allows to make distinctions invisible in the
ordinary syntax. In particular, the logic determined by the most popular
semantics of intuitionistic K becomes a proper extension of the minimal normal
logic of the binary connective. Even an extension of this minimal logic with
the "strength" axiom, classically near-trivial, preserves the distinction
between the binary and the unary setting. In fact, this distinction and the
strong constructive strict implication itself has been also discovered by the
functional programming community in their study of "arrows" as contrasted with
"idioms". Our particular focus is on arithmetical interpretations of the
intuitionistic strict implication in terms of preservativity in extensions of
Heyting's Arithmetic.Comment: Our invited contribution to the collection "L.E.J. Brouwer, 50 years
later
Kripke Semantics for a Logical Framework
We present a semantics (using Kripke lambda models) for a logical framework (minimal implicational predicate logic with quantification over all higher types). We apply the semantics to obtain straightforward adequacy proofs for encodings of logics in the framework. 1 Introduction There has been much recent interest in the development and use of logical frameworks. A logical framework is a formal system within which many different logics can be easily represented. It is hoped that such frameworks will facilitate the rapid development of proof assistants for the wide variety of different logics used in computer science and other fields. In this paper we give a semantic analysis (using Kripke lambda models) of the use of minimal implicational predicate logic (with quantification over all higher types) as a logical framework. We choose this framework because it is relatively straightforward to give it a useful semantics. The use of such a logic as a framework is not new. Similar logics ha..
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