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14 results on '"Witold A. Pogorzelski"'

1. A Proof System for Classical Logic

Author

Witold A. Pogorzelski and Piotr Wojtylak

Subjects
Discrete mathematics, Logic, Computational logic, Classical logic, Statistics::Other Statistics, Intermediate logic, Higher-order logic, TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES, History and Philosophy of Science, Philosophy of logic, Many-valued logic, Calculus, Dynamic logic (modal logic), Rule of inference, Mathematics
Abstract

An operation on inferential rules, called H-operation, is used to minimize the axiom basis for classical logic.

Published
2005

2. [Untitled]

Author

Witold A. Pogorzelski and Piotr Wojtylak

Subjects
Propositional variable, Discrete mathematics, Logic, Well-formed formula, Zeroth-order logic, Atomic formula, Propositional calculus, Propositional formula, Algebra, TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES, History and Philosophy of Science, Computer Science::Logic in Computer Science, T-norm fuzzy logics, Autoepistemic logic, Mathematics
Abstract

We attempt to define the classical propositional logic by use of appropriate derivability conditions called Cn-definitions. The conditions characterize basic properties of propositional connectives.

Published
2001

3. Completeness Theory for Propositional Logics

Author

Witold A. Pogorzelski, Piotr Wojtylak, Witold A. Pogorzelski, and Piotr Wojtylak

Subjects
Completeness theorem
Abstract

Completeness is one of the most important notions in logic and the foundations of mathematics. Many variants of the notion have been de?ned in literature. We shallconcentrateonthesevariants,andaspects,of completenesswhicharede?ned in propositional logic. Completeness means the possibility of getting all correct and reliable sc- mata of inference by use of logical methods. The word ‘all', seemingly neutral, is here a crucial point of distinction. Assuming the de?nition as given by E. Post we get, say, a global notion of completeness in which the reliability refers only to syntactic means of logic and outside the correct schemata of inference there are only inconsistent ones. It is impossible, however, to leave aside local aspects of the notion when we want to make it relative to some given or invented notion of truth. Completeness understood in this sense is the adequacy of logic in relation to some semantics, and the change of the logic is accompanied by the change of its semantics. Such completeness was e?ectively used by J.?ukasiewicz and investigated in general terms by A. Tarski and A. Lindenbaum, which gave strong foundations for research in logic and, in particular, for the notion of consequence operation determined by a logical system. The choice of logical means, by use of which we intend to represent logical inferences, is also important. Most of the de?nitions and results in completeness theory were originally developed in terms of propositional logic. Propositional formal systems?nd many applications in logic and theoretical computer science.

Published
2008

4. A Minimal Implicational Logic

Author

Witold A. Pogorzelski

Subjects
Discrete mathematics, Deduction theorem, Computer Science::Logic in Computer Science, Minimal logic, Computer Science::Artificial Intelligence, Modus ponens, Axiom schema, Mathematics
Abstract

The system L c of purely implicational logic is constructed. It is weaker than the Hilbert’s implicational logic L H , but the deduction theorem and some kind of modus ponens rule are valid for L c . Hence L c can be considered as a system which characterizes the standard implication connective.

Published
1994

7. On the scope of the classical deduction theorem

Author

Witold A. Pogorzelski

Subjects
Philosophy, Deduction theorem, Fundamental theorem, Scope (project management), Logic, No-go theorem, Calculus, Sequent calculus, Mathematics
Abstract

§1. The set Cn(R, A + X) is the set of all expressions deducible, on the ground of formulas A, from the class X of formulas of the implicational propositional calculus, by means of rules belonging to the set R. In other words, Cn(R, A + X) is the set of expressions obtained from the set X by the help of a two-parameter function Cn[R, A]. The formula is called the classical deduction theorem.The classical deduction theorem is true for the system 〈R, A〉 (where R is the set of primitive rules, A is the set of axioms of the propositional calculus) if it holds for the function Cn[R, A].

Published
1968

14. Tadeusz Kotarbiński. Introduction. Notes on the development of formal logic in Poland in the years 1900–39. Polish logic 1920–1939, edited by Storrs McCall. The Clarendon Press, Oxford1967, pp. 1–14. - Jan Łukasiewicz. On the notion of possibility. English translation of the first half of 1864 by H. Hiż. Polish logic 1920–1939, edited by Storrs McCall. The Clarendon Press, Oxford1967, pp. 15–16. - Jan Łukasiewicz. On three-valued logic. English translation of the second half of 1864 by H. Hiż. Polish logic 1920–1939, edited by Storrs McCall. The Clarendon Press, Oxford1967, pp. 16–18. - Jan Łukasiewicz. On determinism. English translation of XXXIII 130 by Z. Jordan. Polish logic 1920–1939, edited by Storrs McCall. The Clarendon Press, Oxford1967, pp. 19–39. - Jan Łukasiewicz. Philosophical remarks on many-valued systems of propositional logic. English translation of 1868 by H. Weber. Polish logic 1920–1939, edited by Storrs McCall. The Clarendon Press, Oxford1967, pp. 40–65. - Jan Łukasiewicz. On the history of the logic of propositions. English translation of 18613 by S. McCall. Polish logic 1920–1939, edited by Storrs McCall. The Clarendon Press, Oxford1967, pp. 66–87. - Jan Łukasiewicz. The equivalential calculus. English translation of VI115 by P. Woodruff. Polish logic 1920–1939, edited by Storrs McCall. The Clarendon Press, Oxford1967, pp. 88–115. - Stanisław Leśniewski. Introductory remarks to the continuation of my article: ‘Grundzüge eines neuen Systems der Grundlagen der Mathematik.’ English translation of V 83 by W. Teichmann and S. McCall. Polish logic 1920–1939, edited by Storrs McCall. The Clarendon Press, Oxford1967, pp. 116–169. - Stanisław Leśniewski. On definitions in the so-called theory of deduction. English translation of 20213 by E. C. Luschei. Polish logic 1920–1939, edited by Storrs McCall. The Clarendon Press, Oxford1967, pp. 170–187. - Bolesław Sobociński. Successive simplifications of the axiom-system of Leiniewski's ontology. English translation of 4432 by Z. Jordan. Polish logic 1920–1939, edited by Storrs McCall. The Clarendon Press, Oxford1967, pp. 188–200

Author

Witold A. Pogorzelski

Subjects
Philosophy, Logic
Published
1970

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