The modal logic S4.2 is obtained by adding to some base for S4 containing a primitive rule of necessitation the axiom scheme CMLpLMp. Equivalently one ...
527 Notre Dame Journal of Formal Logic Volume XIII, Number 4, October 1972 NDJFAM
SEMANTICS FOR S4.2 ALLEN HAZEN
The modal logic S4.2 is obtained by adding to some base for S4 containing a primitive rule of necessitation the axiom scheme CMLpLMp. Equivalently one could add the transform, under G'όdel's interpretation of the intuitionistic propositional calculus in S4, of either ANpNNp or CNKpqANpNq. Since CMLpLMLp is valid in S4.2 and CLMLpLMp in S4, a natural deduction system for S4.2 may be obtained by adding to Fitch's system for S4 (cf. [2]) a rule to the effect that any formula of the form MLp may be reiterated as is into strict subproofs. The first cited formula guarantees the derivability of the rule and the second the derivation, in the natural deduction system, of the characteristic axiom. This logic has been studied algebraically (in [1]), but I have not found, anywhere in the literature cited in [3], a semantics for it in the style of Kripke (i.e. a semantic theory specifying that a model structure be a set of possible worlds over which a reflexive relation of accessibility is defined, that atomic formulas may take values at a given possible world without regard to their values at others, that the truth value of a truth-functional compound at a world be determined by the truth values at that world of its components in the usual way, that an L-(M-) statement be true at a world if and only if the formula which is the scope of its initial modal operator is true at all (some) of the worlds accessible to it, and that a formula is valid if and only if it is true at every world of every model structure). Such a semantics may be provided by adding to the requirements for an S4 model structure (that the accessibility relation be transitive) the stipulation that for any two worlds in a model structure there is a third world in the structure accessible from both of them. Our proofs of soundness and completeness will be parasitic on similar proofs for S4. Since S4.2 may be axiomatized with no primitive rules of inference other than those in some chosen axiomatization of S4, it suffices for soundness to prove the validity of the characteristic axiom. For its antecedent to be true at a world there must be some world at which the formula represented in the axiom scheme by Lp is true, and for the consequent to be false there must be one at which LNp is true. But there must be a world accessible to both of these.
Received July 21, 1971
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ALLEN HAZEN
We may adapt the Henkin-style completeness proof for S4 given in [3], as follows: call the system of maximal consistent sets "constructed" in accordance with the directions given for the S4 proof (but with the individual sets being maximal consistent with respect to S4.2) Class 1. Class 2 is formed by adding to Class 1 for each pair of maximal consistent sets in Class 1 a maximal consistent set containing all L-statements in either of the two sets. Class 3 is built up from Class 2 in accordance with the directions for building Class 1 from one of its subsets (the only difference being that sets may have to be indexed with transfinite ordinals), and so on. The model structure is given by Class N, which is the union of the Classes n for all natural numbers n. The only difficutly lies in showing that it is possible to have the sets needed to fill the even-numbered classes—that a set containing all the L-statements of two different sets will always be consistent. But suppose it were not. This could only arise if one set contained Lp and the other LNp for some formula p. But in this case the first set of Class 1, from which all the others are accessible, would contain both MLp and MLNp, and so, MLNp being equivalent to NLMp and MLp implying LMp, would not be consistent, contrary to the assumptions of the proof. I leave to more imaginative members of the philosophical community the task of finding an intuitive interpretation for this system.
REFERENCES [1] Dummett, M. A. E., and E. J. Lemmon, "Modal logics between S4 and S5," Zeitschriβ fur Mathematische Logik und Grundlagen der Mathematik, vol. 3 (1959). [2] Fitch, F. B., Symbolic Logic; an introduction, Ronald P r e s s , New York (1952). [3] Hughes, G. E., and M. J. Cresswell, An Introduction to Modal Logic, Methuen, London (1968). University of Pittsburgh Pittsburgh, Pennsylvania
