Leo Simons. A reduction in the number of independent axiom schemata for S4. Notre Dame journal of formal logic, vol. 3 (1962), pp. 256–258. - Bolesław Sobociński. An axiom-system for {K; N}-propositional calculus related to Simons' axiomatization of S3.Notre Dame journal of formal logic, vol. 3 (1962), pp. 206–208.

1967 ◽  
Vol 32 (2) ◽  
pp. 245-245
Author(s):  
Naoto Yonemitsu
Keyword(s):  
Propositional Calculus ◽  
Axiom System ◽  
Formal Logic ◽  
Independent Axiom

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.

1970 ◽  
Vol 35 (3) ◽  
pp. 442-446
Author(s):  
Witold A. Pogorzelski
Keyword(s):  
English Translation ◽  
Propositional Logic ◽  
Axiom System ◽  
Formal Logic ◽  
History Of ◽  
Equivalential Calculus

A Very Independent Axiom System

1961 ◽  
Vol 68 (2) ◽  
pp. 159 ◽  
Author(s):  
Frank Harary
Keyword(s):  
Axiom System ◽  
Independent Axiom

Alternative postulate sets for Lewis's S5

1956 ◽  
Vol 21 (4) ◽  
pp. 347-349
Author(s):  
E. J. Lemmon
Keyword(s):  
Propositional Calculus ◽  
Formal Logic ◽  
Propositional Formula ◽  
Modal Operator ◽  
Special Rules ◽  
Definition Of ◽  
Derived Rules

Professor Prior (Formal logic (1955), pp. 305–307) gives alternative postulate sets for Lewis's S5, one substantially Lewis's own set (p. 305, 6.1) and another, considerably simpler, due to Prior himself (pp. 306–307, 6.5). This note aims at showing that the systems based on these two postulate sets are in fact equivalent. We label the system based on Lewis's postulates ‘S5’, as usual, and that on Prior's postulates ‘P’.The form of the postulate set for P considered here is as follows. We add to the full propositional calculus one definition:Df. M: M = NLN;and two special rules:L1: ⊦Cαβ → ⊦CLαβ;L2: ⊦Cαβ → ⊦CαLβ, if α is fully modalized (a propositional formula is fully modalized, if and only if all occurrences of propositional variables in it lie within the scope of a modal operator).Prior's text (pp. 201–205) indicates sufficiently how it could be shown that any thesis of S5 is a thesis of P and that the rules and definitions of S5 are obtainable as derived rules of P. We turn, therefore, to the problem of showing that any thesis of P is a thesis of S5 and that the rules and definition of P are obtainable as derived rules of S5.

A Gentzen- or Beth-type system, a practical decision procedure and a constructive completeness proof for the counterfactual logics VC and VCS

1983 ◽  
Vol 48 (1) ◽  
pp. 1-20 ◽  
Author(s):  
H. C. M. de Swart
Keyword(s):  
Decision Procedure ◽  
Type System ◽  
Propositional Calculus ◽  
Axiom System ◽  
Completeness Proof ◽  
Counterfactual Conditional ◽  
System A ◽  
Counterfactual Logic ◽  
Classical Propositional Calculus

In [1] and [2] D. Lewis formulates his counterfactual logic VC as follows. The language contains the connectives ∧, ∨, ⊃, ¬ and the binary connective ≤. A ≤ B is read as “A is at least as possible as B”. The following connectives are defined in terms of ≤.A < B: = ¬(B ≤ A) (it is more possible that A than that B).◊ A ≔ ¬(⊥ ≤ A) (⊥ is the false formula; A is possible).□ A ≔ ⊥ ≤ ¬A (A is necessary). (if A were the case, then B would be the case). (if A were the case, then B might be the case). and are two counterfactual conditional operators. (AB) iff ¬(A ¬B).The following axiom system VC is presented by D. Lewis in [1] and [2]: V: (1) Truthfunctional classical propositional calculus.

Another Very Independent Axiom System

1961 ◽  
Vol 68 (10) ◽  
pp. 992
Author(s):  
J. W. Ellis
Keyword(s):  
Axiom System ◽  
Independent Axiom
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