Intuitionistic completeness for first order classical logic

In the past sixty years or so, a real forest of intuitionistic models for classical theories has grown. In this paper we will compare intuitionistic models of first order classical theories according to relevant issues, like completeness (w.r.t. first order classical provability), consistency, and relationship between a connective and its interpretation in a model. We briefly consider also intuitionistic models for classical ω-logic. All results included here, but a part of the proposition (a) below, are new. This work is, ideally, a continuation of a paper by McCarty, who considered intuitionistic completeness mostly for first order intuitionistic logi

Simple proof of the completeness theorem for second order classical and intuitionictic logic by reduction to first-order mono-sorted logic

International audienceWe present a simpler way than usual to deduce the completeness theorem for the second-oder classical logic from the first-order one. We also extend our method to the case of second-order intuitionistic logic

Kripke Semantics and Proof Systems for Combining Intuitionistic Logic and Classical Logic

International audienceWe combine intuitionistic logic and classical logic into a new, first-order logic called Polarized Intuitionistic Logic. This logic is based on a distinction between two dual polarities which we call red and green to distinguish them from other forms of polarization. The meaning of these polarities is defined model-theoretically by a Kripke-style semantics for the logic. Two proof systems are also formulated. The first system extends Gentzen's intuitionistic sequent calculus LJ. In addition, this system also bears essential similarities to Girard's LC proof system for classical logic. The second proof system is based on a semantic tableau and extends Dragalin's multiple-conclusion version of intuitionistic sequent calculus. We show that soundness and completeness hold for these notions of semantics and proofs, from which it follows that cut is admissible in the proof systems and that the propositional fragment of the logic is decidable

Term-forming Operators In First Order Logic

The two main accomplishments of this thesis are that it provides the first adequate semantics for Hilbert\u27s epsilon-operator and that it describes a general semantics for term forming operators (often called variable binding term operators of vbto\u27s ) more flexible than any in the literature.;The epsilon-operator was introduced by David Hilbert in the 1920s as a term forming operator in first order logic. The semantics so far available for epsilon has been designed for classical two-valued logic, and has required that additional extensionality assumptions be made. This thesis provides complete semantics for epsilon in classical extensional, classical non-extensional, Boolean valued, and intuitionistic first order systems. The natural step to generalizing the technique used in the epsilon case to get a general theory of term forming operators which handles the non-extensional and non-classical cases is then taken.;The thesis proceeds as follows. Chapter One gives a historical discussion of term forming operators. A brief, self-contained presentation of the untyped lambda-calculus, which illustrates the inevitable differences between lambda and any possible operator in first order logic, follows. A chapter is devoted to solving the syntactical difficulties involved in introducing a variable binding term forming operator to standard languages for first order logic. The semantics for epsilon, and in the intuitionistic case also for another of Hilbert\u27s creatures, tau, takes up the next several chapters. The discussion includes several new completeness and soundness results, and some new results about the extra strength these operators add to intuitionistic logic, including some new independence results. The final chapter includes an argument to the effect that the results earlier in the thesis show that we need a more general theory of term forming operators than any in the literature, and indicates the shape such a theory should take

Kripke Models for Classical Logic

We introduce a notion of Kripke model for classical logic for which we constructively prove soundness and cut-free completeness. We discuss the novelty of the notion and its potential applications

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

Focusing and Polarization in Intuitionistic Logic

A focused proof system provides a normal form to cut-free proofs that structures the application of invertible and non-invertible inference rules. The focused proof system of Andreoli for linear logic has been applied to both the proof search and the proof normalization approaches to computation. Various proof systems in literature exhibit characteristics of focusing to one degree or another. We present a new, focused proof system for intuitionistic logic, called LJF, and show how other proof systems can be mapped into the new system by inserting logical connectives that prematurely stop focusing. We also use LJF to design a focused proof system for classical logic. Our approach to the design and analysis of these systems is based on the completeness of focusing in linear logic and on the notion of polarity that appears in Girard's LC and LU proof systems