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Many-valued modal propositional calculi. (English) Zbl 0661.03011
This paper discusses three modal many-valued propositional calculi, viz. $${}_ nT$$, $${}_ nS4$$ and $${}_ nS5$$ (n$$\in {\mathbb{N}})$$. Semantical and syntactical characterizations of these logics are given.
An n-valued Kripke model is a triple $$<W,R,val>$$, where W is a nonempty set, the elements of which are called possible worlds, R is a binary relation on W, called the accessibility relation, and val assigns to any world w and to any propositional variable p an element of $$\{0,1/(n- 1),...,k/(n-1),...,1\}.$$ The value val(w,f) of a formula f in a world w of a given n-valued Kripke model is defined recursively as follows: $$val(w,\sim f)=1-val(w,f),$$ $$val(w,f\to g)=\min \{1,1- val(w,f)+val(w,g)\},$$ $$val(w,\square f)=\min \{val(w',f)| \quad wRw'\}.$$
A formula is valid in $${}_ nT$$ if its value is 1 in any world of any n- valued Kripke model with R reflexive. $${}_ nT$$ is syntactically characterized by
AO: the axioms and rules of classical propositional calculus
A1: $$\square (p\to q)\to (\square p\to \square q)$$
A2: $$(\square p+\square q)\to \square (p+q)$$, where $$p+q:=\sim p\to q$$
A3: $$\square (p+p)\to (\square p+\square p)$$
A4: $$(\diamond p+\diamond p)\to \diamond (p+p)$$
A5: $$\square p\to p$$
and the inference rule of necessitation: if f is formally provable (in $${}_ nT)$$, then so is $$\square f.$$
The completeness of $${}_ nT$$ with respect to n-valued Kripke semantics is proved in a constructive way. First a decision procedure for $${}_ nT$$ is described. Then the following two theorems are shown: 1. If f is a valid formula, then the procedure applied to f succeeds (the proof is by contraposition). 2. If the procedure applied to a formula f succeeds, then f is a theorem.
The definitions and results obtained for $${}_ nT$$ can easily be extended to systems like $${}_ nS4$$, $${}_ nS5$$, $${}_ nQ$$, $${}_ nB$$, and so on.
Reviewer: H.C.M.de Swart

##### MSC:
 03B45 Modal logic (including the logic of norms) 03B50 Many-valued logic
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