By Krister Segerberg
This paintings kinds the author’s Ph.D. dissertation, submitted to Stanford college in 1971. The author’s total objective is to give in an geared up type the idea of relational semantics (Kripke semantics) in modal propositional common sense, in addition to the extra common 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 usually are not thought of. the writer brings jointly below one hide 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 offers new proofs of many elderly theorems, developing, for instance, demonstrations through relational types for theorems formerly identified merely via algebraic equipment; and he additionally contributes a powerful variety of new effects to the sector. those works tested a few notational and terminological conventions which have been 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 belief of neighbourhood semantics, establishes numerous simple 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 concepts to an in depth learn of transitive relational versions and linked logics. within the 3rd quantity he adapts the notions and strategies constructed 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 basic [regular] modal good judgment with no inevitably being themselves common [regular]. [From the overview by means of David Makinson.]
Filtration used to be used greatly via Segerberg to end up completeness theorems. this system could be potent in facing logics whose canonical version doesn't fulfill a few wanted estate, and is derived 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 variety of logics, together with these characterized by way of the sessions of finite partial orderings, finite linear orderings (both irreflexive and reflexive), and the modal and stressful logics of the buildings 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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Satisfies the condition for Q. From these remarks we easily derive completeness results for ET, EQ, and ES. 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.
R is piecewise strongly connected if uRx & uRy implies that xRy or yRx. R is euclidean if uRx & uRy implies that xRy. 1 THEOREM. Suppose L is_ a. normal logic. Then the fol- 1swing is true of the canonical model for L. “ (U^, R^, V^> -49- tU 111~^ l■ ID KD c L, - j** - if KT 11. u ill* L> K4 c L, linen 1 ■ iy. if KG(D c L , Then RL u Y* KG £ L, then verreen c. If KLern^ g. L, then R^ is piecewise connected, vi v n , If KLem c L, then RT is piecewise strongly La connected. viii. If. KE c L, then is euclidean.
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. Every normal sublogic of S5 is_ infinite. Proof. Let L be any normal logic. then there is some finite frame Assume that L is finite 3? determining be an integer such that every element of J L. Let k has at -54- most k alternatives. Then Alt^ is valid in Alt^ is a derivable schema in L.
An Essay in Classical Modal Logic by Krister Segerberg