By Krister Segerberg

This paintings types the author’s Ph.D. dissertation, submitted to Stanford collage in 1971. The author’s total goal is to give in an geared up type the speculation 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 conceal a very good many effects that have been already recognized 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 offers new proofs of many aged theorems, developing, for instance, demonstrations through relational versions for theorems formerly identified in basic terms by means of algebraic equipment; and he additionally contributes a powerful variety of new effects to the sphere. those works verified a few notational and terminological conventions which were lasting. for example, the time period body was once 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 easy consistency and completeness theorems by way of such semantics, introduces relational semantics and relates them to neighbourhood semantics, and starts a research of p-morphisms and filtrations of relational and neighbourhood types. within the moment quantity he applies those semantic thoughts to an in depth research of transitive relational types and linked logics. within the 3rd quantity he adapts the notions and methods constructed within the first as a way to hide modal logics which are quasi-normal or quasi-regular, within the feel of together with the least basic [regular] modal good judgment with out unavoidably being themselves common [regular]. [From the overview through David Makinson.]

Filtration was once used broadly by way of Segerberg to turn out completeness theorems. this system might be potent in facing logics whose canonical version doesn't fulfill a few wanted estate, and springs into its personal while trying to axiomatise logics outlined by way of a few situation on finite frames. this system used to be utilized in ``Essay'' to axiomatise an entire diversity of logics, together with these characterized via the sessions of finite partial orderings, finite linear orderings (both irreflexive and reflexive), and the modal and annoying 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 good judgment: A view of its evolution, J. of utilized good judgment, vol.1 (2003), 309-392.]

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U , N, V> be this model: - {0, 1, 2, 3) ; N. = {{0}, (0, 1), {0, 2, 3)}, for i - 0, 1, 2, 3 f(0 , 1}, if n = 0 , V (n) - \ {0, 2), if n > 0 . — Then ^ p P 0 , tx>D(p0 PL ) > and >oD P l ’ which violates K'. Yet 2d is a model for EK since each der intersection and thus satisfies (k). 2 THEOREM. K 1 is_ not derivable in ER. Proof. Let 'll~ (U, N, V) be this model: is closed un- -44- 0 = {Q, 1,2,3 j , Ni - {{0, i], (0, 1, 2], {0, 1, 3}, i0, 2, 3 j , (0, 1, 2, 3}], for - 0, 1, 2, 3 ; i f { 0 , 1] , i f n - 0 , V (n) = P [{0, 2}, if n > 0 .

T , Rt Suppose L , Q l , ”L > be the Then for all formulas A and all points u e UT , ^ A if. and only if A c u . When we deal with normal logics later, all frames and models will be relational and normal. a burden always to have to put $ It would then be for Q in the structures, -26- and we will simply drop Q in those contexts. identify __. Thus we will 4>) with , and , V> with In particular, if L is a normal logic we will usually write for the relational canonical model for L. __

2 c L, then R is transitive and con vergent. v. L, then R is serial. transitive. and convergent. - vi. 2 c. L > £h£n R is reflexive. transi tive. and ssnxsimit • Yii. and connected. viii- If. 3 5L l , then R is §££1^1, m n s i l i v s , and connected. 3 q L, then R is reflexive. iransi- tive. and connected. If K4E c L, then R 2ii* If sii* If <,s transitive and eutlidesn* & L> £h§n R is, c£snsl£ixe Md §ymMUa>Js« d* e & i, then Ris snrisl, tran&iiiYA, and fiKtlidaan. xiU. If S3 £L L, then R is universal.