Gödel-Dummett Predicate Logics and Axioms of Prenexability
نویسنده
چکیده
Gödel-Dummett logic in general is a multi-valued logic where a truth value of a formula can be any number from the real interval [0, 1] and where implication → is evaluated via the Gödel implication function. As to truth values, 0 (falsity) and 1 (truth) are the extremal truth values whereas the remaining truth values are called intermediate. Gödel implication function ⇒ is defined as follows: a ⇒ b = 1 if a ≤ b, and a ⇒ b = b otherwise. The truth functions of the remaining propositional symbols conjunction & and disjunction ∨ are the functions min and max respectively. Negation ¬A of a formula A is in Gödel-Dummett logic understood as A→⊥ where ⊥ is a constant for falsity with a truth value equal 0. Thus truth function of negation is the function a 7→ (a⇒ 0); speaking exactly, a⇒ 0 = 1 if a = 0 and a⇒ 0 = 0 for all a > 0. A particular Gödel-Dummett logic is obtained by restricting the range of possible truth values, i.e. by specifying a truth value set. More exactly, a logic T is based on a truth value set V where {0, 1} ⊆ V ⊆ [0, 1] if only the elements of V can be chosen as truth values of propositional atoms. Then a propositional formula A is a tautology of that logic T or a tautology of the set V if v(A) = 1 for each truth evaluation v based on V , i.e. for each truth evaluation v (a function defined on all propositional atoms and extendible uniquely to all propositional formulas) whose range is a subset of V . One can easily verify that (i) each truth value set V such that {0, 1} ⊆ V ⊆ [0, 1] is closed under all truth functions ⇒, min, and max, (ii) if V1 ⊆ V2 then all tautologies of the Gödel-Dummett logic based on V2 are simultaneously tautologies of the logic based on V1, and (iii) if two truth value sets are order isomorphic then the logics based on them are the same (equivalent). Also, to
منابع مشابه
On interplay of quantifiers in Gödel-Dummett fuzzy logics
Axiomatization of Gödel-Dummett predicate logics S2G, S3G, and PG, where PG is the weakest logic in which all prenex operations are sound, and the relationships of these logics to logics known from the literature are discussed. Examples of non-prenexable formulas are given for those logics where some prenex operation is not available. Inter-expressibility of quantifiers is explored for each of ...
متن کاملTranslating Labels to Hypersequents for Intermediate Logics with Geometric Kripke Semantics
We give a procedure for translating geometric Kripke frame axioms into structural hypersequent rules for the corresponding intermediate logics in Int/Geo that admit weakening, contraction and in some cases, cut. We give a procedure for translating labelled sequents in the corresponding logic to hypersequents that share the same linear models (which correspond to Gödel-Dummett logic). We prove t...
متن کاملFormal systems of fuzzy logic and their fragments
Formal systems of fuzzy logic (including the well-known à Lukasiewicz and Gödel-Dummett infinite-valued logics) are well-established logical systems and respected members of the broad family of the so-called substructural logics closely related to the famous logic BCK. The study of fragments of logical systems is an important issue of research in any class of nonclassical logics. Here we study ...
متن کاملQuantifier Elimination in Second-Order Predicate Logic
An algorithm is presented which eliminates second–order quantifiers over predicate variables in formulae of type ∃P1, . . . , Pnψ where ψ is an arbitrary formula of first– order predicate logic. The resulting formula is equivalent to the original formula – if the algorithm terminates. The algorithm can for example be applied to do interpolation, to eliminate the second–order quantifiers in circ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2007