Notre Dame Journal of Formal Logic

A Simple Embedding of T into Double S5

Steven Kuhn


The system obtained by adding full propositional quantification to S5 is known to be decidable, while that obtained by doing so for T is known to be recursively intertranslatable with full second-order logic. Recently it was shown that the system with two S5 operators and full propositional quantification is also recursively intertranslatable with second-order logic. This note establishes that the map assigning [1][2]p to \squarep provides a validity and satisfaction preserving translation between the T system and the double S5 system, thus providing an easier proof of the recent result.

Article information

Notre Dame J. Formal Logic, Volume 45, Number 1 (2004), 13-18.

First available in Project Euclid: 2 September 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03B45: Modal logic (including the logic of norms) {For knowledge and belief, see 03B42; for temporal logic, see 03B44; for provability logic, see also 03F45}

propositional quantifier 2S5


Kuhn, Steven. A Simple Embedding of T into Double S5. Notre Dame J. Formal Logic 45 (2004), no. 1, 13--18. doi:10.1305/ndjfl/1094155276.

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  • [1] Antonelli, G. A., and R. H. Thomason, "Representability in second-order propositional poly-modal logic", The Journal of Symbolic Logic, vol. 67 (2002), pp. 1039--1054.
  • [2] Fine, K., "Propositional quantifiers in modal logic", Theoria, vol. 36 (1970), pp. 336--46.