Jan Łukasiewicz. W sprawie aksjomatyki implikacyjnego rachunku zdań (Concerning an axiom system of the imphcational propositional calculus). VI Zjazd Matematyków Polskich, Warszawa 20-23 IX 1948, supplement to Annates de la Société Polonaise de Mathématique, vol. 22, Kraków1950, pp. 87–92.

1955 ◽  
Vol 20 (2) ◽  
pp. 173-174
Author(s):  
Bolesław Sobociński
Keyword(s):  
Propositional Calculus ◽  
Axiom System

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.

A system of implicit quantification

1968 ◽  
Vol 32 (4) ◽  
pp. 480-504 ◽  
Author(s):  
J. Jay Zeman
Keyword(s):  
Subject Matter ◽  
Traditional Method ◽  
Natural Deduction ◽  
Propositional Calculus ◽  
Axiom System ◽  
Symbolic Logic ◽  
Quantification Theory ◽  
The Subject

The “traditional” method of presenting the subject-matter of symbolic logic involves setting down, first of all, a basis for a propositional calculus—which basis might be a system of natural deduction, an axiom system, or a rule concerning tautologous formulas. The next step, ordinarily, consists of the introduction of quantifiers into the symbol-set of the system, and the stating of axioms or rules for quantification. In this paper I shall propose a system somewhat different from the ordinary; this system has rules for quantification and is, indeed, equivalent to classical quantification theory. It departs from the usual, however, in that it has no primitive quantifiers.

Logical works, by Mordchaj Wajsberg. Edited and with an introduction by Stanisław J. Surma. ZakВad Narodowy imienia Ossolińskich, Wydawnictwo Polskiej Akademii Nauk, Wrocław etc. 1977, 216 pp. - Stanisław J. Surma. Mordchaj Wajsberg. Life and work. Pp. 7–11. - Mordchaj Wajsberg. Axiomatization of the three-valued propositional calculus. Pp. 12–29. A reprint of XXXV 442(15) (English translation by B. Gruchman and S. McCall of 4371). - Mordchaj Wajsberg. On the axiom system of propositional calculus. Pp. 30–36. English translation of 4372. - Mordchaj Wajsberg. A new axiom of propositional calculus in Sheffer's sbmbols. Pp. 37–39. English translation of 4373. - Mordchaj Wajsberg. Investigations of functional calculus for finite domain of individuals. Pp. 40–49. English translation of 4374. - Mordchaj Wajsberg. An extended class calculus. Pp. 50–61. English translation of 4375. - Mordchaj Wajsberg. A contribution to metamathematics. Pp. 62–88. English translation of 4376. - Mordchaj Wajsberg. Contributions to meta-calculus of propositions I. Pp. 89–106. English translation of 4377. - Mordchaj Wajsberg. On the matrix method of independence proofs. Pp. 107–131. English translation of I 75. - Mordchaj Wajsberg. On A. Heyting's propositional calculus. Pp. 132–171. English translation of III 169. - Mordchaj Wajsberg. Contributions to metalogic. Pp. 172–200. Areprint of XXXV 442(16) (English translation by S. McCall and P. Woodruff of II 93). - Mordchaj Wajsberg. Contributions to metalogic II. Pp. 201–214. A reprint of XXXV 442(17) (English translation by S. McCall of V 31). - Mordchaj Wajsberg. Review of Mihailescu's Recherches sur les formes normalespar rapport à l'equivalence et la disjonction, dans le calcul des propositions (IV 91). Pp. 215–216. English translation of pp. 91–92 of The journal of symbolic logic, vol. 4 (1939).

1983 ◽  
Vol 48 (3) ◽  
pp. 873-874
Author(s):  
Storrs McCall
Keyword(s):  
English Translation ◽  
Functional Calculus ◽  
Matrix Method ◽  
Propositional Calculus ◽  
Axiom System ◽  
Finite Domain ◽  
Symbolic Logic ◽  
The Matrix ◽  
Extended Class

Hermann Dishkant. The first order predicate calculus based on the logic of quantum mechanics. Reports on mathematical logic, no. 3 (1974), pp. 9–17. - G. N. Georgacarakos. Orthomodularity and relevance. Journal of philosophical logic, vol. 8 (1979), pp. 415–432. - G. N. Georgacarakos. Equationally definable implication algebras for orthomodular lattices. Studia logica, vol. 39 (1980), pp. 5–18. - R. J. Greechie and S. P. Gudder. Is a quantum logic a logic?Helvetica physica acta, vol. 44 (1971), pp. 238–240. - Gary M. Hardegree. The conditional in abstract and concrete quantum logic. The logico-algehraic approach to quantum mechanics, volume II, Contemporary consolidation, edited by C. A. Hooker, The University of Western Ontario series in philosophy of science, vol. 5, D. Reidel Publishing Company, Dordrecht, Boston, and London, 1979, pp. 49–108. - Gary M. Hardegree. Material implication in orthomodular (and Boolean) lattices. Notre Dame journal of formal logic, vol. 22 (1981), pp. 163–182. - J. M. Jauch and C. Piron. What is “quantum-logic”?Quanta, Essays in theoretical physics dedicated to Gregor Wentzel, edited by P. G. O. Freund, C. J. Goebel, and Y. Nambu, The University of Chicago Press, Chicago and London1970, pp. 166–181. - Jerzy Kotas. An axiom system for the modular logic. English with Polish and Russian summaries. Studia logica, vol. 21 (1967), pp. 17–38. - P. Mittelstaedt. On the interpretation of the lattice of subspaces of the Hilbert space as a propositional calculus. Zeitschrift für Naturforschung, vol. 27a no. 8–9 (1972), pp. 1358–1362. - J. Jay Zeman. Generalized normal logic. Journal of philosophical logic, vol. 7(1978), pp. 225–243.

1983 ◽  
Vol 48 (1) ◽  
pp. 206-208 ◽  
Author(s):  
Alasdair Urquhart
Keyword(s):  
Quantum Mechanics ◽  
Quantum Logic ◽  
Propositional Calculus ◽  
Axiom System ◽  
Theoretical Physics ◽  
Reidel Publishing Company ◽  
Philosophical Logic ◽  
Orthomodular Lattices ◽  
Implication Algebras ◽  
The University

Corrections for "On the lengths of proofs in the propositional calculus preliminary version"

1974 ◽  
Vol 6 (3) ◽  
pp. 15-22 ◽  
Author(s):  
Stephen Cook ◽  
Robert Reckhow
Keyword(s):  
Propositional Calculus ◽  
Preliminary Version

CAPITAL ALLOCATION WITH MULTIVARIATE RISK MEASURES: AN AXIOMATIC APPROACH

2019 ◽  
Vol 34 (2) ◽  
pp. 297-315
Author(s):  
Linxiao Wei ◽  
Yijun Hu
Keyword(s):  
Standard Deviation ◽  
Portfolio Management ◽  
Risk Measures ◽  
Risk Measure ◽  
Axiomatic Approach ◽  
Capital Allocation ◽  
Axiom System ◽  
Multivariate Risk ◽  
Multivariate Risk Measures ◽  
Multivariate Risk Measure

AbstractCapital allocation is of central importance in portfolio management and risk-based performance measurement. Capital allocations for univariate risk measures have been extensively studied in the finance literature. In contrast to this situation, few papers dealt with capital allocations for multivariate risk measures. In this paper, we propose an axiom system for capital allocation with multivariate risk measures. We first recall the class of the positively homogeneous and subadditive multivariate risk measures, and provide the corresponding representation results. Then it is shown that for a given positively homogeneous and subadditive multivariate risk measure, there exists a capital allocation principle. Furthermore, the uniqueness of the capital allocation principe is characterized. Finally, examples are also given to derive the explicit capital allocation principles for the multivariate risk measures based on mean and standard deviation, including the multivariate mean-standard-deviation risk measures.

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