Gentzenian Logical Calculi

2019 ◽  
pp. 46-58
Author(s):  
Jaroslav Peregrin
Keyword(s):  
Logical Calculi

ON THE AXIOMATIZATION OF FINITE-VALUED LOGICAL CALCULI

1985 ◽  
Vol 51 (2) ◽  
pp. 473-491 ◽  
Author(s):  
O M Anshakov ◽  
S V Rychkov
Keyword(s):  
Logical Calculi

Algebraic Structures for Logical Calculi

1999 ◽  
pp. 15-60
Author(s):  
Vilém Novák ◽  
Irina Perfilieva ◽  
Jiří Močkoř
Keyword(s):  
Algebraic Structures ◽  
Logical Calculi

On a class of functionally complete multi-valued logical calculi

Studia Logica ◽  
1978 ◽  
Vol 37 (2) ◽  
pp. 205-212 ◽  
Author(s):  
Václav Pinkava
Keyword(s):  
Logical Calculi

A minimum calculus for logic

1944 ◽  
Vol 9 (4) ◽  
pp. 89-94 ◽  
Author(s):  
Frederic B. Fitch
Keyword(s):  
Truth Table ◽  
Additional Property ◽  
Basic Logic ◽  
Double Negation ◽  
Logical Calculus ◽  
Logical Calculi

A logical calculus will be presented which not only is a formulation of a “basic logic” in the sense of the writer's previous papers, but which has the additional property that no weaker calculus can be a formulation of a basic logic. A sort of minimum logical calculus is thus attained, which has nothing superfluous about it for achieving the purpose for which it is designed.In the case of some logical calculi the question can arise as to whether certain of the postulates are really logically valid and necessary. Sometimes a test is available, such as the truth-table test, enabling us to distinguish between logically valid sentences and others, but often no such test is available, especially where quantifiers are involved. Is or is not the axiom of infinity, for example, to be regarded as logically valid? Or is the principle of double negation really acceptable, even though it satisfies the truth-table test?

scholarly journals On Finite-Valued Propositional Logical Calculi

1995 ◽  
Vol 36 (4) ◽  
pp. 606-629 ◽  
Author(s):  
O. Anshakov ◽  
S. Rychkov
Keyword(s):  
Logical Calculi

BUNDER’S PARADOX

2019 ◽  
Vol 13 (4) ◽  
pp. 829-844
Author(s):  
MICHAEL CAIE
Keyword(s):  
Prima Facie ◽  
Short Paper ◽  
Logical Constants ◽  
Standard Construction ◽  
Logical Calculi

AbstractSystems of illative logic are logical calculi formulated in the untyped λ-calculus supplemented with certain logical constants.1 In this short paper, I consider a paradox that arises in illative logic. I note two prima facie attractive ways of resolving the paradox. The first is well known to be consistent, and I briefly outline a now standard construction used by Scott and Aczel that establishes this. The second, however, has been thought to be inconsistent. I show that this isn’t so, by providing a nonempty class of models that establishes its consistency. I then provide an illative logic which is sound and complete for this class of models. I close by briefly noting some attractive features of the second resolution of this paradox.

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.

Complexity of interpolation and related problems in positive calculi

2002 ◽  
Vol 67 (1) ◽  
pp. 397-408 ◽  
Author(s):  
Larisa Maksimova
Keyword(s):  
Modal Logic ◽  
Heyting Algebras ◽  
Logical System ◽  
Complexity Bounds ◽  
Np Completeness ◽  
Interpolation Problems ◽  
Finite Set ◽  
Modal Logic S4 ◽  
Closure Algebras ◽  
Logical Calculi

AbstractWe consider the problem of recognizing important properties of logical calculi and find complexity bounds for some decidable properties. For a given logical system L, a property P of logical calculi is called decidable over L if there is an algorithm which for any finite set Ax of new axiom schemes decides whether the calculus L + Ax has the property P or not. In [11] the complexity of tabularity, pre-tabularity. and interpolation problems over the intuitionistic logic Int and over modal logic S4 was studied, also we found the complexity of amalgamation problems in varieties of Heyting algebras and closure algebras.In the present paper we deal with positive calculi. We prove NP-completeness of tabularity, DP-hardness of pretabularity and PSPACE-completeness of interpolation problem over Int+. In addition to above-mentioned properties, we consider Beth's definability properties. Also we improve some complexity bounds for properties of superintuitionistic calculi.

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