Semantic Tableau Versions of Some Normal Modal Systems with Propositional Quantifiers.

Attention was directed to modal systems in which ‘necessarily α’ is interpreted as ‘α. is and always will be the case’ by Prior in his John Locke Lectures of 1956. The present paper shows that S4.3, the extension of S4 withALCLpLqLCLqLp,is complete with respect to this interpretation when time is taken to be continuous, and that D, the extension of S4.3 withALNLpLCLCLCpLpLpLp,is complete with respect to this interpretation when time is taken to be discrete. The method employed depends upon the application of an algebraic result of Garrett Birkhoff's to the models for these systems, in the sense of Tarski.A considerable amount of work on S4.3 and D precedes this paper. The original model with discrete time is given in Prior's [7] (p. 23, but note the correction in [8]); that taking time to be continuous yields a weaker system is pointed out by him in [9]. S4.3 and D are studied in [3] of Dummett and Lemmon, where it is shown that D includes S4.3 andCLCLCpLpLpCMLpLp.While in Oxford in 1963, Kripke proved that these were in fact sufficient for D, using semantic tableaux. A decision procedure for S4.3, using Birkhoff's result, is given in my [2]. Dummett conjectured, in a conversation, that taking time to be continuous yielded S4.3. Thus the originality of this paper lies in giving a suitable completeness proof for S4.3, and in the unified algebraic treatment of the systems. It should be emphasised that the credit for first axiomatising D belongs to Kripke.

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E. J. Lemmon. New foundations for Lewis modal systems. The journal of symbolic logic, vol. 22 no. 2 (1957), pp. 176–186.

Journal of Symbolic Logic ◽

10.2307/2964301 ◽

1958 ◽

Vol 23 (3) ◽

pp. 346-347

Author(s):

Naoto Yonemitsu

Keyword(s):

Symbolic Logic ◽

New Foundations ◽

Modal Systems

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Higher-Order Illative Combinatory Logic

Journal of Symbolic Logic ◽

10.2178/jsl.7803080 ◽

2013 ◽

Vol 78 (3) ◽

pp. 837-872 ◽

Cited By ~ 2

Author(s):

Łukasz Czajka

Keyword(s):

Strong Consistency ◽

Predicate Logic ◽

Higher Order ◽

Second Order ◽

Model Construction ◽

Partial Answer ◽

Combinatory Logic ◽

First Order ◽

Propositional Quantifiers

AbstractWe show a model construction for a system of higher-order illative combinatory logic thus establishing its strong consistency. We also use a variant of this construction to provide a complete embedding of first-order intuitionistic predicate logic with second-order propositional quantifiers into the system of Barendregt, Bunder and Dekkers, which gives a partial answer to a question posed by these authors.

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On finite and infinite modal systems

Journal of Symbolic Logic ◽

10.2307/2267611 ◽

1938 ◽

Vol 3 (2) ◽

pp. 77-82 ◽

Cited By ~ 4

Author(s):

C. West Churchman

Keyword(s):

Modal Logic ◽

Symbolic Logic ◽

Modal Operator ◽

Image Position ◽

Modal Systems ◽

Complex Modal

In Oskar Becker's Zur Logik der Modalitäten four systems of modal logic are considered. Two of these are mentioned in Appendix II of Lewis and Langford's Symbolic logic. The first system is based on A1–8 plus the postulate,From A7: ∼◊p⊰∼p we can prove the converse of C11 by writing ∼◊p for p, and hence deriveThe addition of this postulate to A1–8, as Becker points out, allows us to “reduce” all complex modal functions to six, and these six are precisely those which Lewis mentions in his postulates and theorems: p, ∼p, ◊p, ∼◊p, ∼◊∼p, and ◊∼p This reduction is accomplished by showingwhere ◊n means that the modal operator ◊ is repeated n times; e.g., ◊3p = ◊◊◊p. Then it is shown thatBy means of (1), (2), and (3) any complex modal function whatsoever may be reduced to one of the six “simple” modals mentioned above.It might be asked whether this reduction could be carried out still further, i.e., whether two of the six “irreducible” modals could not be equated. But such a reduction would have to be based on the fact that ◊p = p which is inconsistent with the set B1–9 of Lewis and Langford's Symbolic logic and independent of the set A1–8. Hence for neither set would such a reduction be possible.

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