Combining Supervaluation and Degree Based Reasoning Under Vagueness

Abstract

Two popular approaches to formalize adequate reasoning with vague propositions are usually deemed incompatible: On the one hand, there is supervaluation with respect to precisification spaces, which consist in collections of classical interpretations that represent admissible ways of making vague atomic statements precise. On the other hand, t-norm based fuzzy logics model truth functional reasoning, where reals in the unit interval [0, 1] are interpreted as degrees of truth. We show that both types of reasoning can be combined within a single logic SŁ, that extends both: Łukasiewicz logic Ł and (classical) S5, where the modality corresponds to '. . . is true in all complete precisifications'. Our main result consists in a game theoretic interpretation of SŁ, building on ideas already introduced by Robin Giles in the 1970s to obtain a characterization of Ł in terms of a Lorenzen style dialogue game combined with bets on the results of binary experiments that may show dispersion. In our case the experiments are replaced by random evaluations with respect to a given probability distribution over permissible precisifications. upwards and downwards Löwenheim-Skolem, and recursive axiomatizability fail for 'natural' supervaluation based consequence relations.

Undead argument: the truth-functionality objection to fuzzy theories of vagueness

“…Cf. also Fermüller and Kosik (2006). connectives and vague predicates does not require responding in any particular way to compound sentences involving vague predicates in contexts involving borderline cases of those predicates.…”

Dialogue Games for Many-Valued Logics — an Overview

“…Most importantly for our current purpose, it is demonstrated in [25] that the evaluation of formulas of S L can be characterized by a dialogue game extending Giles's game for L, where 'dispersive elementary experiments' (see Section 2) are replaced by 'indeterministic evaluations' over precisification spaces. The dialogue rule for the supertruth modality involves a relativization to specific precisification points:…”

“…In [25], we have worked out a dialogue game based attempt to reconcile supervaluation and t-norm based ('fuzzy') evaluation within a common formal framework. To this aim we interpret 'supertruth' as a modal operator and define a logic S L that extends both, Lukasiewicz logic L, as well as the classical modal logic S5.…”

“…[25]). A formula F is valid in S L if and only if for every precisification space Π I have a winning strategy for the game starting with my assertion of F .…”

Giles’s Game and the Proof Theory of Łukasiewicz Logic

“…This variant of the game is discussed in more detail in [13] where also an interpretation of the evaluation of elementary states in terms of supervaluations with respect to precisification spaces is offered. The latter approach is discussed further in [15] where Lukasiewicz logic is enriched by a modal operator modelling 'supertruth' and a corresponding Giles-style characterization is presented.…”