Finite nest structures and propositional logic

1966 ◽  
Vol 31 (3) ◽  
pp. 322-324 ◽  
Author(s):  
Raymond M. Smullyan
Keyword(s):  
Propositional Logic ◽  
Decision Procedure ◽  
Natural Deduction ◽  
Complete System ◽  
Deduction System ◽  
Quantification Theory ◽  
Natural Deduction System

Our terminology and notation is the same as that of [1], of which this note is a sequel.We wish to show that if we take the natural deduction system (N) described in [1], and delete the rules for the quantifiers, we obtain a complete system for propositional logic. [Of course we now construe “X”, “Y”, “Z” as syntactic variables ranging over formulas of propositional logic, rather than sentences of quantification theory.] Moreover the system serves as a neat decision procedure.

Completeness of an ancient logic

1972 ◽  
Vol 37 (4) ◽  
pp. 696-702 ◽  
Author(s):  
John Corcoran
Keyword(s):  
Present Article ◽  
Propositional Logic ◽  
Natural Deduction ◽  
Deductive System ◽  
Axiomatic System ◽  
Valid Argument ◽  
Deduction System ◽  
Strong Completeness ◽  
Natural Deduction System ◽  
System Review

In previous articles ([4], [5]) it has been shown that the deductive system developed by Aristotle in his “second logic” (cf. Bochenski [2, p. 43]) is a natural deduction system and not an axiomatic system as previously had been thought [6]. It was also pointed out that Aristotle's logic is self-sufficient in two senses: First, that it presupposed no other logical concepts, not even those of propositional logic; second, that it is (strongly) complete in the sense that every valid argument formable in the language of the system is demonstrable by means of a formal deduction in the system. Review of the system makes the first point obvious. The purpose of the present article is to prove the second. Strong completeness is demonstrated for the Aristotélian system.

scholarly journals An Alternative Natural Deduction for the Intuitionistic Propositional Logic

2016 ◽  
Vol 45 (1) ◽  
Author(s):  
Mirjana Ilić
Keyword(s):  
Intuitionistic Logic ◽  
Propositional Logic ◽  
Natural Deduction ◽  
Sequent Calculus ◽  
Intuitionistic Propositional Logic ◽  
Deduction System ◽  
Natural Deduction System

A natural deduction system NI, for the full propositional intuitionistic logic, is proposed. The operational rules of NI are obtained by the translation from Gentzen’s calculus LJ and the normalization is proved, via translations from sequent calculus derivations to natural deduction derivations and back.

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.

scholarly journals Sequent Systems without Improper Derivations

2021 ◽  
Author(s):  
Katsumi Sasaki
Keyword(s):  
Propositional Logic ◽  
Natural Deduction ◽  
Inference Rules ◽  
Classical Propositional Logic ◽  
Deduction System ◽  
Structural Rules ◽  
Sequent Systems ◽  
Natural Deduction System ◽  
Sequent System

In the natural deduction system for classical propositional logic given by G. Gentzen, there are some inference rules with assumptions discharged by the rule. D. Prawitz calls such inference rules improper, and others proper. Improper inference rules are more complicated and are often harder to understand than the proper ones. In the present paper, we distinguish between proper and improper derivations by using sequent systems. Specifically, we introduce a sequent system \(\vdash_{\bf Sc}\) for classical propositional logic with only structural rules, and prove that \(\vdash_{\bf Sc}\) does not allow improper derivations in general. For instance, the sequent \(\Rightarrow p \to q\) cannot be derived from the sequent \(p \Rightarrow q\) in \(\vdash_{\bf Sc}\). In order to prove the failure of improper derivations, we modify the usual notion of truth valuation, and using the modified valuation, we prove the completeness of \(\vdash_{\bf Sc}\). We also consider whether an improper derivation can be described generally by using \(\vdash_{\bf Sc}\).

scholarly journals COMPLETENESS VIA CORRESPONDENCE FOR EXTENSIONS OF THE LOGIC OF PARADOX

2012 ◽  
Vol 5 (4) ◽  
pp. 720-730 ◽  
Author(s):  
BARTELD KOOI ◽  
ALLARD TAMMINGA
Keyword(s):  
Correspondence Analysis ◽  
Natural Deduction ◽  
Correspondence Theory ◽  
Truth Table ◽  
Deduction System ◽  
Natural Deduction System ◽  
Logic Of Paradox

AbstractTaking our inspiration from modal correspondence theory, we present the idea of correspondence analysis for many-valued logics. As a benchmark case, we study truth-functional extensions of the Logic of Paradox (LP). First, we characterize each of the possible truth table entries for unary and binary operators that could be added to LP by an inference scheme. Second, we define a class of natural deduction systems on the basis of these characterizing inference schemes and a natural deduction system for LP. Third, we show that each of the resulting natural deduction systems is sound and complete with respect to its particular semantics.

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