Gentzen, Gerhard Karl Erich (1909–45)

2018 ◽  
Author(s):  
Volker Peckhaus
Keyword(s):  
Intuitionistic Logic ◽  
Cut Elimination ◽  
German Mathematician ◽  
Consistency Proofs ◽  
Logical Calculi

The German mathematician and logician Gerhard Gentzen devoted his life to proving the consistency of arithmetic and analysis. His work should be seen as contributing to the post-Gödelian development of Hilbert’s programme. In this connection he developed several logical calculi. The main device used in his proofs was a theorem in which he proved the eliminability of the inference known as ‘cut’ from a variety of different kinds of proofs. This ‘cut-elimination theorem’ yields the consistency of both classical and intuitionistic logic, and the consistency of arithmetic without complete induction. His later work was aimed at providing consistency proofs for less restricted systems of arithmetic and analysis.

scholarly journals Normalization as a consequence of cut elimination

2009 ◽  
Vol 86 (100) ◽  
pp. 27-34
Author(s):  
Mirjana Borisavljevic
Keyword(s):  
Intuitionistic Logic ◽  
Natural Deduction ◽  
Cut Elimination ◽  
Deduction System ◽  
Normalization Theorem ◽  
Natural Deduction System

Pairs of systems, which consist of a system of sequents and a natural deduction system for some part of intuitionistic logic, are considered. For each of these pairs of systems the property that the normalization theorem is a consequence of the cut-elimination theorem is presented.

Gentzen's Proof Systems: Byproducts in a Work of Genius

2012 ◽  
Vol 18 (3) ◽  
pp. 313-367 ◽  
Author(s):  
Jan von Plato
Keyword(s):  
Natural Deduction ◽  
Sequent Calculus ◽  
Central Component ◽  
Cut Elimination ◽  
Complete Class ◽  
Tree Form ◽  
Proof Systems ◽  
Set Up ◽  
Logical Calculi ◽  
Subformula Property

AbstractGentzen's systems of natural deduction and sequent calculus were byproducts in his program of proving the consistency of arithmetic and analysis. It is suggested that the central component in his results on logical calculi was the use of a tree form for derivations. It allows the composition of derivations and the permutation of the order of application of rules, with a full control over the structure of derivations as a result. Recently found documents shed new light on the discovery of these calculi. In particular, Gentzen set up five different forms of natural calculi and gave a detailed proof of normalization for intuitionistic natural deduction. An early handwritten manuscript of his thesis shows that a direct translation from natural deduction to the axiomatic logic of Hilbert and Ackermann was, in addition to the influence of Paul Hertz, the second component in the discovery of sequent calculus. A system intermediate between the sequent calculus LI and axiomatic logic, denoted LIG in unpublished sources, is implicit in Gentzen's published thesis of 1934–35. The calculus has half rules, half “groundsequents,” and does not allow full cut elimination. Nevertheless, a translation from LI to LIG in the published thesis gives a subformula property for a complete class of derivations in LIG. After the thesis, Gentzen continued to work on variants of sequent calculi for ten more years, in the hope to find a consistency proof for arithmetic within an intuitionistic calculus.

Embedding theorems for LTL and its variants

2014 ◽  
Vol 25 (1) ◽  
pp. 83-134 ◽  
Author(s):  
NORIHIRO KAMIDE
Keyword(s):  
Intuitionistic Logic ◽  
Classical Logic ◽  
Linear Time ◽  
Cut Elimination ◽  
Embedding Theorems ◽  
Infinitary Logic ◽  
Spatial Logic ◽  
Linear Time Temporal Logic ◽  
Np Complete ◽  
Dynamic Topological Logic

In this paper, we prove some embedding theorems for LTL (linear-time temporal logic) and its variants:viz. some generalisations, extensions and fragments of LTL. Using these embedding theorems, we give uniform proofs of the completeness, cut-elimination and/or decidability theorems for LTL and its variants. The proposed embedding theorems clarify the relationships between some LTL-variations (for example, LTL, a dynamic topological logic, a fixpoint logic, a spatial logic, Prior's logic, Davies' logic and an NP-complete LTL) and some traditional logics (for example, classical logic, intuitionistic logic and infinitary logic).

On the unification of classical, intuitionistic and affine logics

2018 ◽  
Vol 29 (8) ◽  
pp. 1177-1216
Author(s):  
CHUCK LIANG
Keyword(s):  
Intuitionistic Logic ◽  
Classical Logic ◽  
Sequent Calculus ◽  
Linear Logic ◽  
Kripke Semantics ◽  
Cut Elimination ◽  
Structural Rules ◽  
Phase Semantics ◽  
Intuitionistic Logics

This article presents a unified logic that combines classical logic, intuitionistic logic and affine linear logic (restricting contraction but not weakening). We show that this unification can be achieved semantically, syntactically and in the computational interpretation of proofs. It extends our previous work in combining classical and intuitionistic logics. Compared to linear logic, classical fragments of proofs are better isolated from non-classical fragments. We define a phase semantics for this logic that naturally extends the Kripke semantics of intuitionistic logic. We present a sequent calculus with novel structural rules, which entail a more elaborate procedure for cut elimination. Computationally, this system allows affine-linear interpretations of proofs to be combined with classical interpretations, such as the λμ calculus. We show how cut elimination must respect the boundaries between classical and non-classical modes of proof that correspond to delimited control effects.

scholarly journals Full Cut Elimination and Interpolation for Intuitionistic Logic with Existence Predicate

2019 ◽  
Vol 48 (2) ◽  
pp. 137-158 ◽  
Author(s):  
Paolo Maffezioli ◽  
Eugenio Orlandelli
Keyword(s):  
Intuitionistic Logic ◽  
Direct Proof ◽  
Interpolation Property ◽  
Cut Elimination

In previous work by Baaz and Iemhoff, a Gentzen calculus for intuitionistic logic with existence predicate is presented that satisfies partial cut elimination and Craig's interpolation property; it is also conjectured that interpolation fails for the implication-free fragment. In this paper an equivalent calculus is introduced that satisfies full cut elimination and allows a direct proof of interpolation via Maehara's lemma. In this way, it is possible to obtain much simpler interpolants and to better understand and (partly) overcome the failure of interpolation for the implication-free fragment.

Which structural rules admit cut elimination? An algebraic criterion

2007 ◽  
Vol 72 (3) ◽  
pp. 738-754 ◽  
Author(s):  
Kazushige Terui
Keyword(s):  
Intuitionistic Logic ◽  
General Class ◽  
Residuated Lattices ◽  
Cut Elimination ◽  
Inference Rules ◽  
Lambek Calculus ◽  
Free Logic ◽  
Algebraic Semantics ◽  
Propagation Property ◽  
Structural Rules

AbstractConsider a general class of structural inference rules such as exchange, weakening, contraction and their generalizations. Among them, some are harmless but others do harm to cut elimination. Hence it is natural to ask under which condition cut elimination is preserved when a set of structural rules is added to a structure-free logic. The aim of this work is to give such a condition by using algebraic semantics.We consider full Lambek calculus (FL), i.e., intuitionistic logic without any structural rules, as our basic framework. Residuated lattices are the algebraic structures corresponding to FL. In this setting, we introduce a criterion, called the propagation property, that can be stated both in syntactic and algebraic terminologies. We then show that, for any set ℛ of structural rules, the cut elimination theorem holds for FL enriched with ℛ if and only if ℛ satisfies the propagation property.As an application, we show that any set ℛ of structural rules can be “completed” into another set ℛ*, so that the cut elimination theorem holds for FL enriched with ℛ*. while the provability remains the same.

scholarly journals Relations and Non-commutative Linear Logic

1991 ◽  
Vol 20 (372) ◽  
Author(s):  
Carolyn Brown ◽  
Doug Gurr
Keyword(s):  
Intuitionistic Logic ◽  
Arbitrary Number ◽  
Linear Logic ◽  
Expressive Power ◽  
Cut Elimination ◽  
Natural Class ◽  
Structural Rules ◽  
Restricted Sense ◽  
Intuitionistic Linear Logic ◽  
Natural Way

<p>Linear logic differs from intuitionistic logic primarily in the absence of the structural rules of weakening and contraction. Weakening allows us to prove a proposition in the context of irrelevant or unused premises, while contraction allows us to use a premise an arbitrary number of times. Linear logic has been called a ''resource-conscious'' logic, since the premises of a sequent must appear exactly as many times as they are used.</p><p>In this paper, we address this ''experimental nature'' by presenting a non-commutative intuitionistic linear logic with several attractive properties. Our logic discards even the limited commutativityof Yetter's logic in which cyclic permutations of formulae are permitted. It arises in a natural way from the system of intuitionistic linear logic presented by Girard and Lafont, and we prove a cut elimination theorem. In linear logic, the rules of weakening and contraction are recovered in a restricted sense by the introduction of the exponential modality(!). This recaptures the expressive power of intuitionistic logic. In our logic the modality ! recovers the non-commutative analogues of these structural rules. However, the most appealing property of our logic is that it is both sound and complete with respect to interpretation in a natural class of models which we call relational quantales.</p>

scholarly journals An Investigation into Intuitionistic Logic with Identity

2019 ◽  
Vol 48 (4) ◽  
Author(s):  
Szymon Chlebowski ◽  
Dorota Leszczyńska-Jasion
Keyword(s):  
Intuitionistic Logic ◽  
Sequent Calculus ◽  
Kripke Semantics ◽  
Cut Elimination ◽  
Propositional Identity

We define Kripke semantics for propositional intuitionistic logic with Suszko’s identity (ISCI). We propose sequent calculus for ISCI along with cut-elimination theorem. We sketch a constructive interpretation of Suszko’s propositional identity connective.

scholarly journals Schema Complexity in Propositional-Based Logics

Mathematics ◽  
2021 ◽  
Vol 9 (21) ◽  
pp. 2671
Author(s):  
Jaime Ramos ◽  
João Rasga ◽  
Cristina Sernadas
Keyword(s):  
Modal Logic ◽  
Intuitionistic Logic ◽  
Propositional Logic ◽  
Language Translation ◽  
Cut Elimination ◽  
Propositional Modal Logic ◽  
Schema Complexity

The essential structure of derivations is used as a tool for measuring the complexity of schema consequences in propositional-based logics. Our schema derivations allow the use of schema lemmas and this is reflected on the schema complexity. In particular, the number of times a schema lemma is used in a derivation is not relevant. We also address the application of metatheorems and compare the complexity of a schema derivation after eliminating the metatheorem and before doing so. As illustrations, we consider a propositional modal logic presented by a Hilbert calculus and an intuitionist propositional logic presented by a Gentzen calculus. For the former, we discuss the use of the metatheorem of deduction and its elimination, and for the latter, we analyze the cut and its elimination. Furthermore, we capitalize on the result for the cut elimination for intuitionistic logic, to obtain a similar result for Nelson’s logic via a language translation.

scholarly journals Frege and Wittgenstein: The Limits of the Analytic Style

Elements ◽  
2010 ◽  
Vol 6 (1) ◽  
Author(s):  
Christopher Sheridan
Keyword(s):  
Twentieth Century ◽  
Logical Analysis ◽  
Linguistic Meaning ◽  
Bertrand Russell ◽  
Gottlob Frege ◽  
Symbolic Logic ◽  
Scientific Language ◽  
German Mathematician ◽  
Broad Application ◽  
Analytic Tradition

The analytic tradition in philosophy stems from the work of German mathematician and logician Gottlob Frege. Bertrand Russell brough Frege's program to render language-particularly scientific language-in formal logical terms to the forefront of philosophy in the early twentieth century. The quest to clarify language and parse out genuine philosophical problems remains a cornerstone of analytic philosophy, but investigative programs involving the broad application of formal symbolic logic to language have largely been abandoned due to the influence of Ludwig Wittgenstein's later work. This article identifies the key philosophical moves that must be performed successfully in order for Frege's "conceptual notation" and other similar systems to adequately capture syntax and semantics. These moves ultimately fail as a result of the nature of linguistic meaning. The shift away from formal logical analysis of language and the emergence of the current analytic style becomes clearer when this failure is examined critically.

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