scholarly journals The Method of Socratic Proofs Meets Correspondence Analysis

2019 ◽  
Vol 48 (2) ◽  
pp. 99-116
Author(s):  
Dorota Leszczyńska-Jasion ◽  
Yaroslav Petrukhin ◽  
Vasilyi Shangin
Keyword(s):  
Correspondence Analysis ◽  
Boolean Functions ◽  
Propositional Logic ◽  
Sequent Calculus ◽  
Sequent Calculi ◽  
Classical Propositional Logic ◽  
Logic Of Questions ◽  
Proof Systems

The goal of this paper is to propose correspondence analysis as a technique for generating the so-called erotetic (i.e. pertaining to the logic of questions) calculi which constitute the method of Socratic proofs by Andrzej Wiśniewski. As we explain in the paper, in order to successfully design an erotetic calculus one needs invertible sequent-calculus-style rules. For this reason, the proposed correspondence analysis resulting in invertible rules can constitute a new foundation for the method of Socratic proofs. Correspondence analysis is Kooi and Tamminga's technique for designing proof systems. In this paper it is used to consider sequent calculi with non-branching (the only exception being the rule of cut), invertible rules for the negation fragment of classical propositional logic and its extensions by binary Boolean functions.

scholarly journals Functional Completeness in CPL via Correspondence Analysis

2019 ◽  
Vol 48 (1) ◽  
Author(s):  
Dorota Leszczyńska-Jasion ◽  
Yaroslav Petrukhin ◽  
Vasilyi Shangin ◽  
Marcin Jukiewicz
Keyword(s):  
Correspondence Analysis ◽  
Boolean Functions ◽  
Propositional Logic ◽  
Sequent Calculus ◽  
Functional Completeness ◽  
Sequent Calculi ◽  
Classical Propositional Logic ◽  
Proof Systems ◽  
Sequent Systems ◽  
Branching Rule

Kooi and Tamminga's correspondence analysis is a technique for designing proof systems, mostly, natural deduction and sequent systems. In this paper it is used to generate sequent calculi with invertible rules, whose only branching rule is the rule of cut. The calculi pertain to classical propositional logic and any of its fragments that may be obtained from adding a set (sets) of rules characterizing a two-argument Boolean function(s) to the negation fragment of classical propositional logic. The properties of soundness and completeness of the calculi are demonstrated. The proof of completeness is conducted by Kalmár's method. Most of the presented sequent-calculus rules have been obtained automatically, by a rule-generating algorithm implemented in Python. Correctness of the algorithm is demonstrated. This automated approach allowed us to analyse thousands of possible rules' schemes, hundreds of rules corresponding to Boolean functions, and to nd dozens of those invertible. Interestingly, the analysis revealed that the presented proof-theoretic framework provides a syntactic characteristics of such an important semantic property as functional completeness.

scholarly journals Deduction in Non-Fregean Propositional Logic SCI

Axioms ◽  
2019 ◽  
Vol 8 (4) ◽  
pp. 115 ◽  
Author(s):  
Joanna Golińska-Pilarek ◽  
Magdalena Welle
Keyword(s):  
Propositional Logic ◽  
Sequent Calculus ◽  
Classical Propositional Logic ◽  
Tableau System ◽  
Soundness And Completeness

We study deduction systems for the weakest, extensional and two-valued non-Fregean propositional logic SCI . The language of SCI is obtained by expanding the language of classical propositional logic with a new binary connective ≡ that expresses the identity of two statements; that is, it connects two statements and forms a new one, which is true whenever the semantic correlates of the arguments are the same. On the formal side, SCI is an extension of classical propositional logic with axioms characterizing the identity connective, postulating that identity must be an equivalence and obey an extensionality principle. First, we present and discuss two types of systems for SCI known from the literature, namely sequent calculus and a dual tableau-like system. Then, we present a new dual tableau system for SCI and prove its soundness and completeness. Finally, we discuss and compare the systems presented in the paper.

Gentzen sequent calculi for some intuitionistic modal logics

2019 ◽  
Vol 27 (4) ◽  
pp. 596-623
Author(s):  
Zhe Lin ◽  
Minghui Ma
Keyword(s):  
Propositional Logic ◽  
Sequent Calculus ◽  
Modal Logics ◽  
Cut Elimination ◽  
Sequent Calculi ◽  
Intuitionistic Propositional Logic ◽  
Propositional Language ◽  
Intuitionistic Modal Logics ◽  
Subformula Property ◽  
Gentzen Sequent Calculus

Abstract Intuitionistic modal logics are extensions of intuitionistic propositional logic with modal axioms. We treat with two modal languages ${\mathscr{L}}_\Diamond $ and $\mathscr{L}_{\Diamond ,\Box }$ which extend the intuitionistic propositional language with $\Diamond $ and $\Diamond ,\Box $, respectively. Gentzen sequent calculi are established for several intuitionistic modal logics. In particular, we introduce a Gentzen sequent calculus for the well-known intuitionistic modal logic $\textsf{MIPC}$. These sequent calculi admit cut elimination and subformula property. They are decidable.

The deduction rule and linear and near-linear proof simulations

1993 ◽  
Vol 58 (2) ◽  
pp. 688-709 ◽  
Author(s):  
Maria Luisa Bonet ◽  
Samuel R. Buss
Keyword(s):  
Propositional Logic ◽  
Natural Deduction ◽  
Sequent Calculus ◽  
Upper Bounds ◽  
Proof System ◽  
Proof Systems ◽  
Linear Speedup ◽  
Frege Systems ◽  
Deduction Rule ◽  
Gentzen Sequent Calculus

AbstractWe introduce new proof systems for propositional logic, simple deduction Frege systems, general deduction Frege systems, and nested deduction Frege systems, which augment Frege systems with variants of the deduction rule. We give upper bounds on the lengths of proofs in Frege proof systems compared to lengths in these new systems. As applications we give near-linear simulations of the propositional Gentzen sequent calculus and the natural deduction calculus by Frege proofs. The length of a proof is the number of lines (or formulas) in the proof.A general deduction Frege proof system provides at most quadratic speedup over Frege proof systems. A nested deduction Frege proof system provides at most a nearly linear speedup over Frege system where by “nearly linear” is meant the ratio of proof lengths is O(α(n)) where α is the inverse Ackermann function. A nested deduction Frege system can linearly simulate the propositional sequent calculus, the tree-like general deduction Frege calculus, and the natural deduction calculus. Hence a Frege proof system can simulate all those proof systems with proof lengths bounded by O(n . α(n)). Also we show that a Frege proof of n lines can be transformed into a tree-like Frege proof of O(n log n) lines and of height O(log n). As a corollary of this fact we can prove that natural deduction and sequent calculus tree-like systems simulate Frege systems with proof lengths bounded by O(n log n).

PEIRCE’S CALCULI FOR CLASSICAL PROPOSITIONAL LOGIC

2018 ◽  
Vol 13 (3) ◽  
pp. 509-540 ◽  
Author(s):  
MINGHUI MA ◽  
AHTI-VEIKKO PIETARINEN
Keyword(s):  
Propositional Logic ◽  
Natural Deduction ◽  
Sequent Calculus ◽  
Theoretic Analysis ◽  
Classical Propositional Logic ◽  
Deduction System ◽  
Natural Deduction System ◽  
Logical Algebra ◽  
Graphical System ◽  
Logical Calculi

AbstractThis article investigates Charles Peirce’s development of logical calculi for classical propositional logic in 1880–1896. Peirce’s 1880 work on the algebra of logic resulted in a successful calculus for Boolean algebra. This calculus, denoted byPC, is here presented as a sequent calculus and not as a natural deduction system. It is shown that Peirce’s aim was to presentPCas a sequent calculus. The law of distributivity, which Peirce states in 1880, is proved using Peirce’s Rule, which is a residuation, inPC. The transitional systems of the algebra of the copula that Peirce develops since 1880 paved the way to the 1896 graphical system of the alpha graphs. It is shown how the rules of the alpha system reinterpret Boolean algebras, answering Peirce’s statement that logical graphs supply a new system of fundamental assumptions to logical algebra. A proof-theoretic analysis is given for the connection betweenPCand the alpha system.

RELEVANCE LOGIC AND THE CALCULUS OF RELATIONS

2010 ◽  
Vol 3 (1) ◽  
pp. 41-70 ◽  
Author(s):  
ROGER D. MADDUX
Keyword(s):  
Propositional Logic ◽  
Relevance Logic ◽  
Binary Relations ◽  
Classical Propositional Logic

Sound and complete semantics for classical propositional logic can be obtained by interpreting sentences as sets. Replacing sets with commuting dense binary relations produces an interpretation that turns out to be sound but not complete for R. Adding transitivity yields sound and complete semantics for RM, because all normal Sugihara matrices are representable as algebras of binary relations.

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