By Krister Segerberg

This paintings kinds the author’s Ph.D. dissertation, submitted to Stanford collage in 1971. The author’s total function is to offer in an equipped type the idea of relational semantics (Kripke semantics) in modal propositional common sense, in addition to the extra basic neighbourhood semantics (Montague-Scott semantics), after which to use those systematically to the exam of a variety of person modal logics. He restricts himself to propositional modal logics; quantified modal logics will not be thought of. the writer brings jointly lower than one hide a good many effects that have been already identified in scattered shape in journals, in addition to others from oral communications; he systematizes those effects, relates them to one another, and refines them; he presents new proofs of many elderly theorems, developing, for instance, demonstrations through relational types for theorems formerly recognized simply via algebraic tools; and he additionally contributes a magnificent variety of new effects to the sphere. those works demonstrated a few notational and terminological conventions which were lasting. for example, the time period body used to be utilized in position of version structure.

In the 1st quantity the writer units out a few initial notions, introduces the assumption of neighbourhood semantics, establishes a number of uncomplicated consistency and completeness theorems when it comes to such semantics, introduces relational semantics and relates them to neighbourhood semantics, and starts off a learn of p-morphisms and filtrations of relational and neighbourhood types. within the moment quantity he applies those semantic strategies to an in depth learn of transitive relational versions and linked logics. within the 3rd quantity he adapts the notions and strategies built within the first so that it will conceal modal logics which are quasi-normal or quasi-regular, within the feel of together with the least common [regular] modal good judgment with no inevitably being themselves basic [regular]. [From the assessment by way of David Makinson.]

Filtration used to be used broadly via Segerberg to turn out completeness theorems. this system could be powerful in facing logics whose canonical version doesn't fulfill a few wanted estate, and springs into its personal whilst trying to axiomatise logics outlined by means of a few situation on finite frames. this technique used to be utilized in ``Essay'' to axiomatise a complete diversity of logics, together with these characterized via the periods of finite partial orderings, finite linear orderings (both irreflexive and reflexive), and the modal and demanding logics of the constructions of N, Z, Q, R, with the relation "more", "less", or their reflexive opposite numbers. [Taken from R.Goldblatt, Mathematical modal common sense: A view of its evolution, J. of utilized common sense, vol.1 (2003), 309-392.]

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It is an interesting question whether there are unnatural logics; none has yet been discovered. A somewhat related ques tion is whether there are logics that are not determined by any class of frames. The latter is probably the out standing question in this area of modal logic at the present time. A concept which will be important later is the fol lowing. A model 'll will be called distinguishable if and only if for all u, v in 'll , if u 4 v then there is some formula A such that j=A and Evidently, canonical models are always distinguishable.

1L «* __ 0 . Then K. ^ G P q > Hq O P ^ Moreover, □ C( PQ this violates R. suppose > and V Finally, (Pq A 2t P ^); a this violates P1), and >^O(P0 ; V is a nodel for EK'. For A and B are formulas such that for some if {0, I, 2,'3], hj[u3 (A >B) , There are two cases to consider. Then {2, 3) £ |J A . (1) ||A|| = [0, 1], — »,B|| . s absurd. (2) jjAj| = i0, 2]. Then {i, 3j Q ||A — > B |j . __

D4Alt^ : R is serial transitive, and U has 3 n+1 elements. v. K4EAlt^ : R is transitive euclidean. and U has s n+1 elements. vi. D4EAltn : R ------is serial-----------------transitive euclidean. — » and U has £ n+1 elements. vii. S4A1t : R is reflexive transitive, and U has g n elements. viii. S5Alt^ : R is universal. and U has gn elements. 4 some what: in the cases (ii) , (iv) , (vi)-(viii) the £ sign can be replaced by **. One way of establishing this is via filtrations (see Section 7). 5 THEOREM.