Proof Theory

The philosophy of alternative logics

Full operational set theory with unbounded existential quantification and power set

This paper deals with some structural properties of the sequent calculus and describes strong symmetries between cut-free derivations and derivations, which do not make use of identity axioms. Both of them are discussed from a semantic and syntactic point of view. Identity axioms and cuts are closely related to the treatment of negation in the sequent calculus, so the results of this article explain some nice symmetries of negation.

/pdf/about-some-symmetries-of-negation-2774jo5uu7.pdf

About some symmetries of negation

Godel’s modal logic approach to analyzing provability attracted a great deal of attention and eventually led to two distinct mathematical models. The first is the modal logic GL, also known as the Provability Logic, which was shown in 1979 by Solovay to be the logic of the formal provability predicate. The second is Godel’s original modal logic of provability S4, together with its explicit counterpart, the Logic of Proofs LP, which was shown in 1995 by Artemov to provide an exact provability semantics for S4. These two models complement each other and cover a wide range of applications, from traditional proof theory to λ-calculi and formal epistemology.

On Two Models of Provability

computable functions are defined by abstract finite deterministic algorithms on many-sorted algebras. We show that there exist finite universal algebraic specifications that specify uniquely (up to isomorphism) (i) all abstract computable functions on any many-sorted algebra; (ii) all functions effectively approximable by abstract computable functions on any metric algebra. We show that there exist universal algebraic specifications for all the classically computable functions on the set ℝ of real numbers. The algebraic specifications used are mainly bounded universal equations and conditional equations. We investigate the initial algebra semantics of these specifications, and derive situations where algebraic specifications precisely define the computable functions.

/pdf/abstract-computability-and-algebraic-specification-7l6n7nw6lg.pdf

Proof Theory

Citations

The philosophy of alternative logics

Full operational set theory with unbounded existential quantification and power set

About some symmetries of negation

On Two Models of Provability

Abstract computability and algebraic specification

Related Papers (5)

Untersuchungen über das logische Schließen. II

Introduction to Metamathematics

The collected papers of Gerhard Gentzen

Basic proof theory

Linear logic