An Essay in Classical Modal Logic by Krister Segerberg

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 offer in an equipped 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 are usually not thought of. the writer brings jointly below one disguise a superb 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 types for theorems formerly identified in simple terms by way of algebraic equipment; and he additionally contributes a magnificent variety of new effects to the sector. those works proven a few notational and terminological conventions which have been 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 belief of neighbourhood semantics, establishes a number of simple consistency and completeness theorems when it comes to such semantics, introduces relational semantics and relates them to neighbourhood semantics, and starts off a examine of p-morphisms and filtrations of relational and neighbourhood versions. within the moment quantity he applies those semantic recommendations to a close learn of transitive relational versions and linked logics. within the 3rd quantity he adapts the notions and methods constructed within the first on the way to disguise modal logics which are quasi-normal or quasi-regular, within the feel of together with the least general [regular] modal common sense with out inevitably being themselves basic [regular]. [From the assessment by means of David Makinson.]

Filtration used to be used largely by way of Segerberg to end up completeness theorems. this method could be potent in facing logics whose canonical version doesn't fulfill a few wanted estate, and springs into its personal whilst trying to axiomatise logics outlined through a few 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 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 common sense, vol.1 (2003), 309-392.]

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Thrs ends the proof. A propositional function n of one variable is called a modality if n has a base {A} x-jhich satisfies the fol­ lowing conditions: i. A » ii. iii. ,*... o, P , for soma k, n $ Nat ; k n ’ ’ , •••> o,t S,€ {-i, □, o) ; if i ¥ 1, then o^ . Notice that although n is not uniquely defined, the are. 's (In a way, this definition would have been more pleasant had -n and 0 also been primitive. ) 4 If the modality is affirmative, otherwise negative. l/c agree to write D m and <>m for strings of m necessity operators and m possibility operators respectively (m = 0).

A) . (A word on the names of these schemata. ) We introduce some definitions. Let

Ii. iii, iv. By the pre­ We recursively for each formula C: & For each n e Nat, P ■ B v . ’ n n U1 ± * - _L. V Bn u. n * (C — * D) D (□ C ) ' « □ c . ) n n ' * We see that C is simply the formula resulting from simultaneous substitution in C of P for P , for all n n n * such that P occurs in C. n We now claim that for all u «■ U and all C, f=C if apd only if . The proof is by induction on the length of C. , u" ) n iff u - u1^ or ... or u ■ uj n iff j*’’B •u n U1 or ... or B n ui n V 3 n iff iff ru lL ★ p Ml ‘u n The inductive step is trivial, Thus, since A is rejected by y Since A is derivable in L, A , A* is rejected by Li.

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