Richard Zach

University of Calgary, Philosophy, Faculty Member

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Supervisors: Paolo Mancosu
Phone: +1 (403) 220-3170
Address: Department of Philosophy
University of Calgary
2500 University Drive NW
Calgary, AB T2N 1N4, Canada

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Grant R Y Schoenebeck

University of Michigan

Johannes Zachhuber

University of Oxford

Daniel D. Hutto

University of Wollongong

Ole T Hjortland

University of Bergen

Galen Strawson

The University of Texas at Austin

Eric S Nelson

Hong Kong University of Science and Technology

David Pierre Leibovitz

Carleton University

Giuseppe Primiero

Università degli Studi di Milano - State University of Milan (Italy)

Richard Bellamy

University College London

David Seamon

Kansas State University

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Papers by Richard Zach

Incompleteness of an infinite-valued first-order Gödel logic and of some temporal logics of programs

Computer Science Logic, 1996

MULTLOG: A system for axiomatizing many-valued logics

Lecture Notes in Computer Science, 1993

A system is presented which, given a first-order finitely-many valued logic by truth tables, prod... more A system is presented which, given a first-order finitely-many valued logic by truth tables, produces a sequent calculus, a natural deduction system, and a calculus for transformation to clausal form for many-valued resolution. The output can be in the form of a scientific paper��a LATEX document��which contains a presentation of the calculi and proves soundness and completeness for them.

MUltlog 1.0: Towards an expert system for many-valued logics

Lecture Notes in Computer Science, 1996

MUltlog is a system which takes as input the specification of a finitely-valued first-order logic... more MUltlog is a system which takes as input the specification of a finitely-valued first-order logic and produces a sequent calculus, a natural deduction system, and a calculus for transforming a many-valued formula to clauses suitable for many-valued resolution. All generated rules are optimized regarding their branching degree. The output is in the form of a scientific paper, written in LAT E X.

Heinrich Behmann’s 1921 Lecture on the Decision Problem and the Algebra of Logic

The Bulletin of Symbolic Logic, 2015

ABSTRACT Heinrich Behmann (1891-1970) obtained his Habilitation under David Hilbert in G\&quo... more ABSTRACT Heinrich Behmann (1891-1970) obtained his Habilitation under David Hilbert in G\&quot;ottingen in 1921 with a thesis on the decision problem. In his thesis, he solved-independently of L\&quot;owenheim and Skolem&#39;s earlier work-the decision problem for monadic second-order logic in a framework that combined elements of the algebra of logic and the newer axiomatic approach to logic then being developed in G\&quot;ottingen. In a talk given in 1921, he outlined this solution, but also presented important programmatic remarks on the significance of the decision problem and of decision procedures more generally. The text of this talk as well as a partial English translation are included.

MULTILOG: A System for Axiomatizing Many-valued Logics

Book Review: Michael Potter

Incompleteness of a first-order gödel logic and some temporal logics of programs

Lecture Notes in Computer Science, 1996

It is shown that the infinite-valued first-order Gödel logic G 0 based on the set of truth values... more It is shown that the infinite-valued first-order Gödel logic G 0 based on the set of truth values {1/k : k ∈ ω \ {0}} ∪ {0} is not r.e. The logic G 0 is the same as that obtained from the Kripke semantics for first-order intuitionistic logic with constant domains and where the order structure of the model is linear. From this, the unaxiomatizability of Kröger's temporal logic of programs (even of the fragment without the nexttime operator ) and of the authors' temporal logic of linear discrete time with gaps follows.

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Effective Finite-Valued Approximations of General Propositional Logics

Lecture Notes in Computer Science, 2008

Propositional logics in general, considered as a set of sentences, can be undecidable even if the... more Propositional logics in general, considered as a set of sentences, can be undecidable even if they have ��nice�� representations, eg, are given by a calculus. Even decidable propositional logics can be computationally complex (eg, already intuitionistic logic is PSPACE-complete). On the other hand, finite-valued logics are computationally relatively simple��at worst NP. Moreover, finite-valued semantics are simple, and general methods for theorem proving exist. This raises the question to what extent and under what circumstances propositional ...

Review of Leonard Bolc and Piotr Borowik: Many-valued Logics: 1. Theoretical Foundations

Journal of Applied Non-classical Logics, 1994

Godel’s First Incompleteness Theorem and Mathematical Instrumentalism

LATEX for Philosophers

The Development of Mathematical Logic from Russell to Tarski, 1900–1935 (with Richard Zach and Calixto Badesa)

The Adventure of Reason, 2010

Abstract: this paper that from the logicalpoint of view set theory is the proper foundation of th... more Abstract: this paper that from the logicalpoint of view set theory is the proper foundation of the mathematicalsciences. Thus, he adds, if one wants to give general definitional principlesthat hold for all of mathematics it is necessary to account for the definitionalprinciples of set theory. First, he begins his definitional analysiswith geometry. Relying on Pieri&#x27;s work on the foundations of geometry hestarts with two relations, xy and E(x, y , z). E(x, y , z) means that yand z are...

A Many-Valued Logic for Default Reasoning

RL EPSTEIN and WA CARNIELLI Computability. Computable functions, logic, and the foundations of mathematics

HISTORY AND PHILOSOPHY OF LOGIC, 2002

REVIEWS-Computability and Logic

Bulletin of Symbolic Logic, 2003

T. Franzen Godel's Theorem. An Incomplete Guide to its Use and Abuse

HISTORY AND PHILOSOPHY OF LOGIC, 2005

Early Writings

Note on calculi for a three-valued logic for logic programming

BULLETIN-EUROPEAN ASSOCIATION FOR THEORETICAL COMPUTER SCIENCE, 1992

Dual systems of sequents and tableaux for many-valued logics

Bulletin of the EATCS, 1993

The aim of this paper is to emphasize the fact that for all finitely-many-valued logics there is ... more The aim of this paper is to emphasize the fact that for all finitely-many-valued logics there is a completely systematic relation between sequent calculi and tableau systems. More importantly, we show that for both of these systems there are always two dual proof sytems (not just only two ways to interpret the calculi). This phenomenon may easily escape one's attention since in the classical (two-valued) case the two systems coincide.(In two-valued logic the assignment of a truth value and the exclusion of the opposite truth value describe ...

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Incompleteness of a first-order G��del logic and some temporal logics of programs

Computer Science Logic, 1996

It is shown that the infinite-valued first-order G��del logic G�� based on the set of truth value... more It is shown that the infinite-valued first-order G��del logic G�� based on the set of truth values {1/k: kw {0}} U {0} is not re The logic G�� is the same as that obtained from the Kripke semantics for first-order intuitionistic logic with constant domains and where the order structure of the model is linear. From this, the unaxiomatizability of Kr��ger's temporal logic of programs (even of the fragment without the nexttime operator O) and of the authors' temporal logic of linear discrete time with gaps follows.

Incompleteness of an infinite-valued first-order Gödel logic and of some temporal logics of programs

Computer Science Logic, 1996

MULTLOG: A system for axiomatizing many-valued logics

Lecture Notes in Computer Science, 1993

MUltlog 1.0: Towards an expert system for many-valued logics

Lecture Notes in Computer Science, 1996

Heinrich Behmann’s 1921 Lecture on the Decision Problem and the Algebra of Logic

The Bulletin of Symbolic Logic, 2015

MULTILOG: A System for Axiomatizing Many-valued Logics

Book Review: Michael Potter

Incompleteness of a first-order gödel logic and some temporal logics of programs

Lecture Notes in Computer Science, 1996

Download

Effective Finite-Valued Approximations of General Propositional Logics

Lecture Notes in Computer Science, 2008

Review of Leonard Bolc and Piotr Borowik: Many-valued Logics: 1. Theoretical Foundations

Journal of Applied Non-classical Logics, 1994

Godel’s First Incompleteness Theorem and Mathematical Instrumentalism

LATEX for Philosophers

The Development of Mathematical Logic from Russell to Tarski, 1900–1935 (with Richard Zach and Calixto Badesa)

The Adventure of Reason, 2010

A Many-Valued Logic for Default Reasoning

RL EPSTEIN and WA CARNIELLI Computability. Computable functions, logic, and the foundations of mathematics

HISTORY AND PHILOSOPHY OF LOGIC, 2002

REVIEWS-Computability and Logic

Bulletin of Symbolic Logic, 2003

T. Franzen Godel's Theorem. An Incomplete Guide to its Use and Abuse

HISTORY AND PHILOSOPHY OF LOGIC, 2005

Early Writings

Note on calculi for a three-valued logic for logic programming

BULLETIN-EUROPEAN ASSOCIATION FOR THEORETICAL COMPUTER SCIENCE, 1992

Dual systems of sequents and tableaux for many-valued logics

Bulletin of the EATCS, 1993

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Incompleteness of a first-order G��del logic and some temporal logics of programs

Computer Science Logic, 1996

Kurt Gödel and computability theory

Abstract. Although Kurt Gödel does not figure prominently in the history of computabilty theory, ... more Abstract. Although Kurt Gödel does not figure prominently in the history of computabilty theory, he exerted a significant influence on some of the founders of the field, both through his published work and through personal interaction. In particular, Gödel's 1931 paper on incompleteness and the methods developed therein were important for the early development of recursive function theory and the lambda calculus at the hands of Church, Kleene, and Rosser.

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Carnap's Linguistic Pluralism and Scientific Methodology

Abstract Carnap's doctrine of linguistic pluralism, enshrined in the "Principle of Tolerance" of ... more Abstract Carnap's doctrine of linguistic pluralism, enshrined in the "Principle of Tolerance" of his Logical Syntax of Language, became a cornerstone of Carnap's mature philosophy in general and conception of scientific method in particular. Not least because of widespread misunderstandings of Carnap's project on the basis of insufficient attention to this doctrine, it has become the subject of significant attention in recent years.

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Alan Turing and the Decision Problem

Many scientific questions are considered solved to the best possible degree when we have a method... more Many scientific questions are considered solved to the best possible degree when we have a method for computing a solution. This is especially true in mathematics and those areas of science in which phenomena can be described mathematically: one only has to think of the methods of symbolic algebra in order to solve equations, or laws of physics which allow one to calculate unknown quantities from known measurements. The crowning achievement of mathematics would thus be a systematic way to compute the solution to any mathematical problem. The hope that this was possible was perhaps first articulated by the 18th century mathematician-philosopher G. W. Leibniz. Advances in the foundations of mathematics in the early 20th century made it possible in the 1920s to first formulate the question of whether there is such a systematic way to find a solution to every mathematical problem. This became known as the decision problem, and it was considered a major open problem in the 1920s and 1930s. Alan Turing solved it in his first, groundbreaking paper "On computable numbers" (1936). In order to show that there cannot be a systematic computational procedure that solves every mathematical question, Turing had to provide a convincing analysis of what a computational procedure is. His abstract, mathematical model of computability is that of a Turing Machine. He showed that no Turing machine, and hence no computational procedure at all, could solve the Entscheidungsproblem.

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The epsilon calculus: An undervalued logical formalism

The epsilon operator is a term-forming operator which replaces quantifiers in ordinary predicate ... more The epsilon operator is a term-forming operator which replaces quantifiers in ordinary predicate logic. Specifically, in the calculus, a term εx A denotes some x satisfying A(x), if there is one. In Hilbert's Program, the epsilon terms play the role of ideal elements; the aim of Hilbert's finitistic consistency proofs is to give a procedure which removes such terms from a formal proof. The procedures by which this is to be carried out are based on Hilbert's epsilon substitution method. The epsilon calculus, however, has applications in other contexts as well. The first general application of the epsilon calculus was in Hilbert's epsilon theorems, which in turn provide the basis for the first correct proof of Herbrand's theorem. More recently, variants of the epsilon operator have been applied in linguistics and linguistic philosophy to deal with anaphoric pronouns and to give an account of “arbitrary objects.”

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An Introduction to Proof Theory: Normalization, Cut-Elimination, and Consistency Proofs

by Richard Zach, Paolo MANCOSU, and Sergio Galvan

Oxford University Press, 2021

An Introduction to Proof Theory provides an accessible introduction to the theory of proofs, with... more An Introduction to Proof Theory provides an accessible introduction to the theory of proofs, with details worked out and many examples and exercises. It also serves as a companion to reading the original pathbreaking articles by Gerhard Gentzen. The first half covers topics in structural proof theory, including the Gödel-Gentzen translation of classical into intuitionistic logic (and arithmetic), natural deduction and the normalization theorems (for both NJ and NK), the sequent calculus, including cut-elimination and mid-sequent theorems, and various applications of these results. The second half examines Gentzen’s consistency proof for first-order Peano Arithmetic. The theory of ordinal notations and other elements of ordinal proof theory are developed from scratch. The proof methods needed, especially proof by induction, are introduced in stages throughout the text.

Richard Zach

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