Abstract
We introduce a game for (extended) Gödel logic where the players’ interaction stepwise reduces claims about the relative order of truth degrees of complex formulas to atomic truth comparison claims. Using the concept of disjunctive game states this semantic game is lifted to a provability game, where winning strategies correspond to proofs in a sequents-of-relations calculus.
Highlights
Fuzzy logics, by which we mean logics where the connectives are interpreted as functions of the unit interval [0, 1], come in many variants
If we take the minimum, min(x, y), as t-norm modeling conjunction ∧, the corresponding residuum as truth function for implication →, and define the negation by ¬A = A → ⊥,1 we arrive at Godel logic, where every formula
We suggest that the sketched transformation of a semantic game into a provability game via moving from single states into disjunctive states can be seen as a general principle, rather than a trick that works only for propositional CL
Summary
By which we mean logics where the connectives are interpreted as functions of the unit interval [0, 1], come in many variants. Even if we restrict attention to t-norm based logics, where a left continuous t-norm ◦ serves as truth function for conjunction and the (unique) residuum of ◦ models implication, there are still infinitely many different fuzzy logics to choose from. Almost all of these logics feature truth functions that yield values that are in general different from 0 and 1, and different from each argument value. Christian Fermuller: Research supported by FWF Project P 32684.
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