S. K. Thomason, Reduction of second-order logic to modal logic, Zeitschrift f ur Mathematische Logik und Grundlagen der Mathematik, vol. 21 (1975), pp. 107--114.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the more properties a simulation preserves or re ects, the more useful it is. G odel s translation of intuitionistic logic in Grzegorczyk s logic, cf. 4] provides a wellknown early example of a simulation. Important results in modal logic were obtained by Thomason in the early seventies, cf. [16, 17]. Thomason showed how polymodal logics (that is, normal modal logics in a language with a number of diamonds or unary modalities) can be simulated by monomodal ones, and applies this result to prove certain (negative) results concerning monomodal logics. Thomason s approach was taken up and ....

S.K. Thomason. Reduction of second-order logic to modal logic. Zeitschrift fur mathemathische Logik und Grundlagen der Mathem atik, 21:107-114, 1975. 47

....the more properties a simulation preserves or re ects, the more useful it is. G odel s translation of intuitionistic logic in Grzegorczyk s logic, cf. 4] provides a wellknown early example of a simulation. Important results in modal logic were obtained by Thomason in the early seventies, cf. [16, 17]. Thomason showed how polymodal logics (that is, normal modal logics in a language with a number of diamonds or unary modalities) can be simulated by monomodal ones, and applies this result to prove certain (negative) results concerning monomodal logics. Thomason s approach was taken up and ....

S.K. Thomason. Reduction of second-order logic to modal logic. Zeitschrift fur mathemathische Logik und Grundlagen der Mathem atik, 21:107-114, 1975.

....by using Thomason s translations [9] 10] 6] But it in the transitive case such counterexamples are impossible, due to Theorem 3.1. This leads us again to an old question: Does there exist a Thomason style translation from polymodal logics (and consequently, from classical second order logic[11]) to monomodal K4 logics A common opinion among modal logicians has been that this question has an answer yes . However the previous observations seem to point at the answer no . ....

S.K.THOMASON. Reduction of second-order logic to modal logic. Zeitschrift f. math. Logik und Grundlagen der Mathematik, v. 21 (1975), 107--114. 11

....inclusion is proper: for example, in propositional logic it is easy to see that if p is a primitive proposition, then p ax v false is sound, whereas p v false is not sound. The second notion is that of nonschematic inference rules, i.e. rules where we do not consider substitution instances [Tho75a, Tho75b, Kap87]. 6 Thus, taking ns v to denote nonschematic validity inference, we have oe ns v if j= oe implies j= As we mentioned above, nonschematic validity inference is quite different from validity inference. For example, in propositional logic, p ns v false is sound, although p v ....

....the nonschematic case, as does the EXPTIME completeness result for structure inference, since we can reduce structure inference to validity. In the case of validity inference, it is not hard to show using Theorem 5.1 that the problem is PSPACE complete. However, it follows from results of Thomason [Tho75a, Tho75b] that the second order theory of a binary relation can be reduced to nonschematic frame inference. This tells us that the complexity of nonschematic frame inference is at least as high as that of full type theory. This means that the complexity is higher than any level of the arithmetic or ....

S. Thomason. Reduction of second-order logic to modal logic. Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik, 21:107--114, 1975.

....the Priorean language has some striking weaknesses. Now for the surprise. Although Priorean languages have these rst order limitations, they nonetheless succeed in stepping across the boundary into second 5 The detailed development of this answer is due to several logicians, including Thomason [48, 49, 50], Fine [23] Goldblatt [30] and van Benthem [3] this work is still some of the most interesting in modal logic. The rst frame incompleteness result (which signaled the presence of second order phenomena) was proved (by Thomason [48] in the setting of Priorean tense logic, but a second order ....

S. Thomason. Reduction of second-order logic to modal logic. Zeitschrift fur mathematische Logik und Grundlagen ser Mathematik, 21:107-114, 1975.

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S. K. Thomason, Reduction of second-order logic to modal logic, Zeitschrift f ur Mathematische Logik und Grundlagen der Mathematik, vol. 21 (1975), pp. 107--114.

No context found.

S. K. Thomason, Reduction of second-order logic to modal logic, Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik, vol. 21 (1975), pp. 107--114.

No context found.

S. K. Thomason. Reduction of second-order logic to modal logic. Zeitschrift fur Mathematische Logik und Grundlagen der Matematik, 21:107-114, 1975.

No context found.

S. Thomason. Reduction of second-order logic to modal logic. Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik, 21:107-114, 1975.

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