Logic

An Essay in Classical Modal Logic by Krister Segerberg

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 model the speculation of relational semantics (Kripke semantics) in modal propositional good judgment, in addition to the extra basic 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 below one hide an excellent 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 recognized merely through 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 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 numerous simple consistency and completeness theorems when it comes to such semantics, introduces relational semantics and relates them to neighbourhood semantics, and starts a examine of p-morphisms and filtrations of relational and neighbourhood versions. within the moment quantity he applies those semantic concepts to a close examine of transitive relational types and linked logics. within the 3rd quantity he adapts the notions and methods constructed within the first with the intention to hide modal logics which are quasi-normal or quasi-regular, within the experience of together with the least common [regular] modal common sense with no inevitably being themselves general [regular]. [From the overview by means of David Makinson.]

Filtration used to be used generally via Segerberg to turn out completeness theorems. this method might 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 via a few on finite frames. this technique was once utilized in ``Essay'' to axiomatise a complete diversity 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 demanding 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 good judgment: A view of its evolution, J. of utilized good judgment, vol.1 (2003), 309-392.]

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An Essay in Classical Modal Logic

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Extra resources for An Essay in Classical Modal Logic

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

Then 9* V •*

The rule RR is_ derivable in ER. Proof. " Then our argument may be represented in the following manner: -46- X• A — hB 2. A ^ >B A A 3. □ A □ B LEMMA. 2 of R and 4 ; b y TF . i n EM. by h yp o th e sis d e r iv a b le i n EN ; f r c m 1 b y TF ; T <-~>A 3. □ T <-»DA fro m 4. □ T N ; 5. □ A fro m 2 b y RE 3 and 4 : by TF .

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