The completeness of the modal logic S4 for all topological spaces as well as for the real line R, the n-dimensional Euclidean space R and the segment (0, 1) etc. (with 2 interpreted as interior) was proved by McKinsey and Tarski in 1944. Several simplified proofs contain gaps. A new proof presented here combines the ideas published later by G. Mints and M. Aiello, J. van Benthem, G. Bezhanishvi...

1. S.L. Bloom, The hyperprojective hierarchy, Notices of the AMS, Vol. 15(1968), 343. 2. S.L. Bloom, Theories of kind W, Notices of the AMS, 16(1969), 507. 3. S.L. Bloom, D. Brown, Classical abstract logics I, Notices of the AMS, 18(1971), 827. 4. S.L. Bloom, D. Brown, Classical abstract logics II, Notices of the AMS, 18(1971), 968. 5. S.L. Bloom, R. Suszko, D. Brown, Structurality, substitutio...

This work gives a natural deduction presentations to some of the sublogic of predicate logic vB94] and also to some of Alechina and J. van Benthem's sublogics. These presentations are based on BM92, Ben95, Gab94] and Gabbay's Labelled Deductive System LDS Gab94]. It also proves some correspondence results for the sublogics presented in vB94] and some completeness theorem to the classes of model...

Although there are machine-certified proofs of correctness of Alg. W and Alg. J, the correctness proof of Wand’s type reconstruction has never been machine checked. We give here a brief description of our attempt at machine-certified proof of correctness of Wand’s algorithm. The correctness is essentially given in terms of completeness and soundness with respect to the Hindley-Milner type syste...

We discuss some aspects of recent research as well as more general issues about motivation, useful methods and open problems in the field of cosmological singularities. In particular, we review some of the approaches to the general area and include discussions of the method of asymptotic splittings, singularity and completeness theorems and the use of the Bel-Robinson energy to prove completene...

We present a simplified proof of Japaridze’s arithmetical completeness theorem for the well-known polymodal provability logic GLP. The simplification is achieved by employing a fragment J of GLP that enjoys a more convenient Kripke-style semantics than the logic considered in the papers by Ignatiev and Boolos. In particular, this allows us to simplify the arithmetical fixed point construction a...

In Conradie, Goranko, and Vakarelov (2006, Logical Methods in Computer Science, 2) we introduced a new algorithm, SQEMA, for computing first-order equivalents and proving the canonicity of modal formulae of the basic modal language. Here we extend SQEMA, first to arbitrary and reversive polyadic modal languages, and then to hybrid polyadic languages too. We present the algorithm, illustrate it ...