K. Fine. Normal forms in modal logic. Notre Dame Journ. of Formal Logic, 16:(2), pp. 229--237, 1975.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....) be the submodel of (F n k , # n k ) generated by y k = x n k . For a non principal ultrafilter U in #, consider the ultrabouqet X = # U X n and the valuation # = # U # n . Then X = L, by Lemma 3.7, and X,#,u = S by Lemma 3.6. # Corollary 3. 9 Every uniform logic (in the sense of [5]) containing D is S Ncomplete. PROOF. The normal form construction from [5] cf. also [3] shows that every uniform logic above D is determined by a class of finite acyclic frames. # In particular, the famous logic M = K ##p # ##p which is very secondorder in Kripke semantics [2] becomes ....

....ultrafilter U in #, consider the ultrabouqet X = # U X n and the valuation # = # U # n . Then X = L, by Lemma 3.7, and X,#,u = S by Lemma 3.6. # Corollary 3.9 Every uniform logic (in the sense of [5] containing D is S Ncomplete. PROOF. The normal form construction from [5] (cf. also [3] shows that every uniform logic above D is determined by a class of finite acyclic frames. # In particular, the famous logic M = K ##p # ##p which is very secondorder in Kripke semantics [2] becomes better in neighbourhood semantics. 8 4 S N incompleteness Now let us ....

K. FINE. Normal forms in modal logic. Notre Dame Journ. of Formal Logic, 16:(2), pp. 229--237, 1975.

....completeness applied here can be modi ed to an alternative completeness proof for that logic, by extending the notion of canonical forms to all formulas. We note that this method can also be adapted to prove completeness of modal logic itself. In fact, that was essentially done quite awhile ago in [Fine, 1975], where the use of normal forms in modal logic were promoted. Finally, the completeness of the full Game Logic introduced in [Parikh, 1985] is still open. We hope that the method applied here can be extended to the game language with iteration and give a handle to solving that problem, too. 9 ....

K. Fine, Normal Forms in Modal Logic, Notre Dame Journal of Formal Logic, vol XVI (2), 1975, 229-237.

....as a theorem p :p, or Sigmap: p. The latter is often used as a charactistic axiom of K:Alt 1 . Since p Sigma : Sigmap is a theorem of K, we have shown the lower right equivalence. The lower left equivalence is similar. These equivalences allow to create very simple normal forms (see [ Fine, 1975a ] for the general case) Recall from standard propositional logic that we can transform any formula into disjunctive normal form. For each formula OE there exists an equivalent formula which is a disjunction of formulae OE i such that each OE i is a conjunction of either a variable or its ....

Kit Fine. Normal forms in modal logic. Notre Dame Journal of Formal Logic, 16:31 -- 42, 1975.

....due to the implicit or explicit presence of accessibility relations and in many cases deduction can become quite unmanageable. Different from tableaux systems for first order logic, modal tableaux systems are special purpose systems with the characteristic properties of accessibility built in. Fine (1975) uses a reduction to normal form from which finite models can be constructed. Other practical methods use resolution calculi. For example, Mints (1986, 1989, 1990) proposes a resolution calculus adapted and extended for modal propositional formulae. We use a resolution calculus too. But in ....

Fine, K. (1975), Normal forms in modal logic, Notre Dame J. Formal Logic 16, 229--237.

....canonical. In this paper we advance our understanding of this concept of neighborhood canonicity by identifying a large class of easily recognised logics for which the finite model property will imply neighborhood canonicity. In fact, this class includes the so called uniform logics of KIT FINE [4] and so we will quickly get the canonicity of FINE s uniform class since he showed that these logics have the finite model property. As FINE points out, the McKinsey logic is in this class and so we will have demonstrated that neighborhood canonicity does not necessarily imply relational ....

....notation (A; I; v) ffl Sigma for Sigma a subset of S (P ) will mean that (A; I; v) ffl for each 2 Sigma. To complete this section we shall define a logic, state what it means for that logic to be even, provide some examples, and compare this notion to that of KIT FINE s uniform logics of [4]. Definition 2.5. A logic is a set L ae S (P ) with the property that it is closed under substitution (replacement of propositional letters by arbitrary formulae) modus ponens, and replacement by provable equivalents (replacing subformula of a thesis of L with another formula L equivalent 1 to ....

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FINE, KIT. "Normal Forms in Modal Logic." Notre Dame Journal of Formal Logic, vol. 16 (1975), no. 2, pp. 229--237.

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K. Fine. Normal forms in modal logic. Notre Dame Journ. of Formal Logic, 16:(2), pp. 229--237, 1975.

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K. Fine. Normal forms in modal logic. Notre Dame Journ. of Formal Logic, 16:(2), pp. 229--237, 1975.

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