By Krister Segerberg
This paintings types the author’s Ph.D. dissertation, submitted to Stanford collage in 1971. The author’s total function is to offer in an prepared style the speculation of relational semantics (Kripke semantics) in modal propositional good judgment, in addition to the extra common neighbourhood semantics (Montague-Scott semantics), after which to use those systematically to the exam of quite a lot of person modal logics. He restricts himself to propositional modal logics; quantified modal logics aren't thought of. the writer brings jointly lower than one disguise a 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 presents new proofs of many elderly theorems, developing, for instance, demonstrations through relational versions for theorems formerly identified merely via algebraic tools; and he additionally contributes a magnificent variety of new effects to the sector. those works proven 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 a research of p-morphisms and filtrations of relational and neighbourhood types. within the moment quantity he applies those semantic ideas to a close research of transitive relational types and linked logics. within the 3rd quantity he adapts the notions and methods constructed within the first so one can hide modal logics which are quasi-normal or quasi-regular, within the experience of together with the least basic [regular] modal common sense with no inevitably being themselves general [regular]. [From the overview via David Makinson.]
Filtration used to be used generally by means of Segerberg to turn out completeness theorems. this method could be potent in facing logics whose canonical version doesn't fulfill a few wanted estate, and is derived into its personal while trying to axiomatise logics outlined via a few situation on finite frames. this technique was once utilized in ``Essay'' to axiomatise a complete diversity of logics, together with these characterized by means of the periods 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 common sense: A view of its evolution, J. of utilized good judgment, vol.1 (2003), 309-392.]
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A. i r,l v ... vA. t 1,1-1 vA. i,m-l .. Then B. < v. if anc' only if iy i . J This contradicts the fact that, by the new schema Alt 2- J n □ B1 v □ (Bj^ —> B2 ) v ... V □ (B ^ a ... a —+ B^+1 ) e u. 4 THEOREM. The following logics are determined by the following conditions on frames (U, R) : i. KAlt n each point has g n alternatives. -53- ii. DAIt : R is serial, and each point has = n alternatives. iii. K4Alt^ : R is transitive. and U has g n+1 elements. iv. D4Alt^ : R is serial transitive, and U has 3 n+1 elements.
ETQ and EQS are of course the incon sistent logic, and EST =* ES. Notice that ES is the logic determined by the class of singular frames. Completeness results of the kind we have new exempli fied may not be very striking. However, via the filtra tion technique to be presented below we may use them to conclude that these logics are decidable and even have the finite model property. The completeness results also permit us to conclude that the logics studied are distinct. We shall exemplify the latter contention in connection with the next group of schemata to be studied.
7- the propositional letter? in A in order of appearance. Then since A. , n m (Pj1 » •'** Pj ^ A » J1 Jn , [P. , ... , P. ] “ A. Clearly, Jl>'• • »Jn n m Jn (P .... p. ) e A. Hence A is a finite base for 12. ji ’ jn 1 Conversely, suppose that 12 is logically finite. Then there is some finite set X2^ that is a base for XI . There is no loss of generality if we assume that XI Qq c (for if this is not already true, we can replace each formula A in X2 ^ - X2 ^ A — >A')- by a formula A' c XI such that Hence there, are at most finitely many bases for propositional functions in at most n variables.
An Essay in Classical Modal Logic by Krister Segerberg