First-Order Quasi-canonical Proof Systems

2019 ◽  
pp. 77-93
Author(s):  
Yotam Dvir ◽  
Arnon Avron
Keyword(s):  
First Order ◽  
Proof Systems

scholarly journals A duality between proof systems for cyclic term graphs

2007 ◽  
Vol 17 (3) ◽  
pp. 439-484 ◽  
Author(s):  
CLEMENS GRABMAYER
Keyword(s):  
Close Connection ◽  
Proof System ◽  
Consistency Checking ◽  
Duality Result ◽  
First Order ◽  
Proof Systems ◽  
Similar Proof ◽  
Equational Specifications ◽  
Cyclic Term

This paper presents a proof-theoretic observation about two kinds of proof systems for bisimilarity between cyclic term graphs.First we consider proof systems for demonstrating that μ term specifications of cyclic term graphs have the same tree unwinding. We establish a close connection between adaptations for μ terms over a general first-order signature of the coinductive axiomatisation of recursive type equivalence by Brandt and Henglein (Brandt and Henglein 1998) and of a proof system by Ariola and Klop (Ariola and Klop 1995) for consistency checking. We show that there exists a simple duality by mirroring between derivations in the former system and formalised consistency checks, which are called ‘consistency unfoldings', in the latter. This result sheds additional light on the axiomatisation of Brandt and Henglein: it provides an alternative soundness proof for the adaptation considered here.We then outline an analogous duality result that holds for a pair of similar proof systems for proving that equational specifications of cyclic term graphs are bisimilar.

scholarly journals Proof systems for Geometric theories (PROGEO)

2020 ◽  
Author(s):  
Elaine Pimentel
Keyword(s):  
Projective Geometry ◽  
Automated Deduction ◽  
First Order ◽  
Systematic Procedure ◽  
Proof Systems ◽  
Conservative Extensions

We plan to study the problem of finding conservative extensions of first order logics. In this project we intend to establish a systematic procedure for adding geometric theories in both intuitionistic and classical logics, as well as to extend this procedure to bipolar axioms, a generalization of the set of geometric axioms. This way, we obtain proof systems for several mathematical theories, such as lattices, algebra and projective geometry, being able to reason about such theories using automated deduction.

Introduction and Overview

2017 ◽  
Author(s):  
Neil Tennant
Keyword(s):  
Scientific Method ◽  
Proof Theory ◽  
Valid Argument ◽  
First Order ◽  
Proof Systems ◽  
Formal Systems ◽  
Deductive Logic ◽  
Order Language ◽  
Foundational Work ◽  
The Right

This is a foundational work, written not just for philosophers of logic, but for logicians and foundationalists generally. Like Frege we seek to deal with the formal first-order language of mathematics. We revisit Gentzen’s proof theory in order to build relevance into proofs, while leaving intact all the logical power one is entitled to expect of a deductive logic for mathematics and for scientific method generally. Proof systems are constituted by particular choices of rules of inference. We raise the issue of the reflexive stability of any argument for a particular choice of logic as the ‘right’ logic. We examine the question of pluralism v. absolutism in choice of logic, and suggest that the informal notion of valid argument is stable and robust enough for us to be able to ‘get it right’ with our formal systems of proof for both constructive and non-constructive reasoning.

Propositional proof systems, the consistency of first order theories and the complexity of computations

1989 ◽  
Vol 54 (3) ◽  
pp. 1063-1079 ◽  
Author(s):  
Jan Krajíček ◽  
Pavel Pudlák
Keyword(s):  
First Order ◽  
Proof Systems ◽  
Propositional Proof Systems ◽  
Propositional Proof

AbstractWe consider the problem about the length of proofs of the sentences saying that there is no proof of contradiction in S whose length is < n. We show the relation of this problem to some problems about propositional proof systems.

scholarly journals Reasoning Inside The Box: Deduction in Herbrand Logics

10.29007/kx2m ◽  
2018 ◽  
Author(s):  
Liron Cohen ◽  
Yoni Zohar
Keyword(s):  
Automated Reasoning ◽  
Proof System ◽  
First Order ◽  
Proof Systems ◽  
The Rich ◽  
Order Structures ◽  
Effective Proof

Herbrand structures are a subclass of standard first-order structures commonly used in logic and automated reasoning due to their strong definitional character. This paper is devoted to the logics induced by them: Herbrand and semi-Herbrand logics, with and without equality. The rich expressiveness of these logics entails that there is no adequate effective proof system for them. We therefore introduce infinitary proof systems for Herbrand logics, and prove their completeness. Natural and sound finitary approximations of the infinitary systems are also presented.

scholarly journals A note on first-order reasoning for minimum models

10.29007/5bvp ◽  
2018 ◽  
Author(s):  
David Rydeheard ◽  
Jesús Héctor Domínguez Sánchez
Keyword(s):  
Logical Framework ◽  
Computing Systems ◽  
First Order ◽  
Proof Systems

In this note, we consider a notion of minimum model suitable for formulating a semantics of evolvable computing systems using a revision-based logic. We explore a range of proof systems for reasoning in the logic of minimum models and consider their application to the simulation of evolvable systems. Finally, we outline how these proof systems may be implemented in a logical framework.

Second- and higher-order logics

2018 ◽  
Author(s):  
Shaughan Lavine
Keyword(s):  
Predicate Logic ◽  
Higher Order ◽  
Second Order ◽  
Order Logic ◽  
First Order Logic ◽  
Third Order ◽  
First Order ◽  
Proof Systems ◽  
The Universe ◽  
Second Order Logic

In first-order predicate logic there are symbols for fixed individuals, relations and functions on a given universe of individuals and there are variables ranging over the individuals, with associated quantifiers. Second-order logic adds variables ranging over relations and functions on the universe of individuals, and associated quantifiers, which are called second-order variables and quantifiers. Sometimes one also adds symbols for fixed higher-order relations and functions among and on the relations, functions and individuals of the original universe. One can add third-order variables ranging over relations and functions among and on the relations, functions and individuals on the universe, with associated quantifiers, and so on, to yield logics of even higher order. It is usual to use proof systems for higher-order logics (that is, logics beyond first-order) that include analogues of the first-order quantifier rules for all quantifiers. An extensional n-ary relation variable in effect ranges over arbitrary sets of n-tuples of members of the universe. (Functions are omitted here for simplicity: remarks about them parallel those for relations.) If the set of sets of n-tuples of members of a universe is fully determined once the universe itself is given, then the truth-values of sentences involving second-order quantifiers are determined in a structure like the ones used for first-order logic. However, if the notion of the set of all sets of n-tuples of members of a universe is specified in terms of some theory about sets or relations, then the universe of a structure must be supplemented by specifications of the domains of the various higher-order variables. No matter what theory one adopts, there are infinitely many choices for such domains compatible with the theory over any infinite universe. This casts doubt on the apparent clarity of the notion of ‘all n-ary relations on a domain’: since the notion cannot be defined categorically in terms of the domain using any theory whatsoever, how could it be well-determined?

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