A non-classical logic for physics

Bridges Between Contextual Linguistic Models of Vagueness and T-Norm Based Fuzzy Logic

“…On the linguistic side context and precisification based approaches suggested, e.g., by Kennedy [16], Kyburg and Moreau [18], and already earlier by Pinkal [25] and Bosch [3] are certainly worth investigating from this perspective. On the fuzzy logic side we just mention similarity semantics [27,17,30], Robin Giles's dialogue and betting game based characterisation of Lukasiewicz logic [9,8] (extended to other logics in [4,6]), acceptability semantics [22], rerandomising semantics [13,11], and approximation semantics [2,23] as alternative candidates for corresponding bridge heads. We plan to explore at least some of these options in future work.…”

Randomized game semantics for semi-fuzzy quantifiers

“…The restrictions of L and L w to the propositional part will be denoted by L p and L w p , respectively. In order to transfer H-games into a many-valued setting we borrow an idea of Giles [10,9] and reformulate the winning condition in a way that will lead to an interesting interpretation of intermediate truth values in terms of expected risks of payments. We conceive of the evaluation of the atomic formula A at the final state of an H-game as a (binary) experiment E A that either fails, meaning v M (A) = 0, or succeeds, meaning v M (A) = 1.…”

“…3 From H-games to G-games Already in the 1970s Robin Giles [10,9] introduced an evaluation game that was intended to provide 'tangible meaning' to reasoning about statements with dispersive semantic tests as they appear in physics. For the logical rules of his game Giles referred not to Henkin or Hintikka, but to Lorenzen's dialogue game semantics for intuitionistic logic [15].…”

“…Relative to the proof of Theorem 4 (see [10,9,7] and it remains to show that my optimal way to reduce the exhibited quantified formula to instances as required by rule (R Π k m ) results in a risk that corresponds to the specified truth function. In order to do so remember that the principle of limited liability is in place.…”

“…quantifiers with classical (bi-valued) scope, and proposes an axiomatically defined scheme for lifting these to fully fuzzy quantification in a later step. Here we want to embed certain semi-fuzzy quantifiers into Lukasiewizc logic, which is not only one of most important formalisms of deductive (mathematical) fuzzy logics in the sense of Hájek [12,13,5], but also allows for a particularly interesting semantic characterization in terms of dialogue games combined with bets on dispersive experiments, introduced by Giles [10,9]. In a sense, this reacts to the mentioned embarrassment of riches by suggesting a bottom-up approach: we start with a convincing and well explored semantic framework for standard first-order Lukasiewicz logic and extend it in a rather lean manner that remains close to its spirit.…”