Nov#k, V.: On the Syntactico-Semantical Completeness of First--Order Fuzzy Logic. Part I --- Syntactical Aspects; Part II --- Main Results. Kybernetika 26(1990), 47--66; 134-- 154.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....fuzzy logic. The completeness theorem for propositional fuzzy logic was proved by J. Pavelka in [15] Later, some other variations appeared [18] proving, however, completeness only for formulas in the degree 1. Extension of Pavelka s proof to first order fuzzy logic was done by the author in [5, 6]. The proof, however, is quite complicated based on ultrafilter trick and on several tricky axioms. An open question remained, whether this proof can be done in a different way following classical Henkin construction of a model from the constats. This has been done by P. H#jek for fuzzy ....

....A)B, p A and (8x)A are formulas. A set of all the terms of a language J is denoted by M J and a set of all the well formed formulas by F J . The set of all terms without variables is denoted by MV . The common abbreviations of formulas :A, AB, AB, A B, A,B, 9x)A, A k are introduced (see [5, 6, 12, 13]) As explained in these works, syntax of fuzzy logic is evaluated by syntactic truth values. An evaluated formula is a couple a ffi A where A 2 F J and a 2 [0; 1] The (syntactic) truth value a is an evaluation of the formula A in the syntax of fuzzy logic. We introduce the following rules of ....

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Nov#k, V.: On the Syntactico-Semantical Completeness of First--Order Fuzzy Logic. Part I --- Syntactical Aspects; Part II --- Main Results. Kybernetika 26(1990), 47--66; 134-- 154.

....is denoted by M J and a set of all the well formed formulas by F J . The set of all terms without variables is denoted by MV . y) This is only auxiliary. As presented in [13] we can get rid of them. 2 The common abbreviations of formulas :A, AB, AB, A B, A,B, 9x)A, A k are introduced (see [6, 7, 12, 15]) Moreover, we will use also the abbreviation ArB defined by ArB : A :B) and call it Lukasiewicz disjunction. As explained in these works, syntax of fuzzy logic is evaluated by syntactic truth values. An evaluated formula is a couple a ffi A where A 2 F J and a 2 L. The (syntactic) ....

.... theory T in the language J of first order fuzzy logic (a fuzzy theory) is a triple T = hAL ; AS ; Ri where AL ae F J , and A S ae F J are fuzzy sets of logical and special axioms, respectively and R is a set of inference rules containing, at least, the rules RMP , r G and r Rb , b 2 L (cf. [6, 7, 14]) An evaluated proof (or shortly, a proof) of a formula A from a fuzzy set AS of special axioms is a sequence of evaluated formulas which are logical or special axioms or they are derived using some many valued inference rule. The provability degree in the fuzzy theory T = hAL ; AS ; Ri is given ....

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Nov#k, V.: On the Syntactico-Semantical Completeness of First--Order Fuzzy Logic. Part I --- Syntactical Aspects; Part II --- Main Results. Kybernetika 26(1990), 47--66; 134--154.

....formal theory in both fuzzy logics, demonstrate some of their basic properties and mutual connection of FLb and FLn. We will focus especially to syntactic aspects and specific questions of provability. However, we assume that the reader is, at least partly, acquainted with some of the cited works [7, 11, 16, 19, 23] where precise definitions of some concepts and proofs of some theorems, which are only recalled in this paper, can be found. 2 Formal theories in fuzzy logic in narrow sense 2.1 Truth values and consequence operation Recall that the set of truth values is considered to be the residuated ....

....important in FLb where we need them to accomplish interpretation of various natural language connectives, modifiers and, possibly, other linguistic phenomena. We will consider formal language J consisting of variables, constants, predicates, connectives and quantifiers, as defined, for example in [11, 19]. A specific feature is introducing symbols a for all truth values a 2 L. However, as demonstrated in [6, 8, 15] this is only a useful technical means. By F J we denote the set of all the well formed formulas (defined in a common way) and by M J sets of all terms in the language J . The basic ....

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Nov#k, V.: On the Syntactico-Semantical Completeness of First--Order Fuzzy Logic. Part I --- Syntactical Aspects; Part II --- Main Results. Kybernetika 26(1990), 47--66; 134--154.

....hand, by pointing out the advantages and disadvantages of the two concepts, on the other hand, by comparing their semantical strength. It should be noted that there are many other approaches to [0; 1] valued logics starting from different points of view, some of which are described in detail in [16, 26, 28]. For a rather extensive overview, see [11] Acknowledgement: The authors wish to thank Marie Demlov a, Petr H ajek and Dan Butnariu for numerous fruitful discussions on the topic of this paper. ....

V. Nov'ak. On the syntactico-semantical completeness of first-order fuzzy logic . Part I --- Syntactical aspects, Part II --- Main results. Kybernetika, 26:47--66, 134--154, 1990. 21

....R fuzzy logics seems to be an open field of research since we did not find any study of this subject in the literature. Finally, it should be noted that there are many other approaches to [0; 1] valued logics starting from different points of view, some of which are described in detail in [16, 27, 29]. For a rather extensive overview, see [11] Acknowledgement: The authors wish to thank Marie Demlov a, Petr H ajek and Dan Butnariu for numerous fruitful discussions on the topic of this paper. ....

V. Nov'ak. On the syntactico-semantical completeness of first-order fuzzy logic. Part I --- Syntactical aspects, Part II --- Main results. Kybernetika, 26:47--66, 134--154, 1990.

....bold multiplication) and residuation ( Lukasiewicz implication) respectively, given by a Omega b = 0 (a b Gamma 1) 2) a b = 1 (1 Gamma a b) a; b 2 [0; 1] 3) We also use the symbol u for an arbitrary t norm. The reasons for choosing (1 3) are manyfold and they were explained in [23, 13, 14, 21]) We will also deal with a formal language J which is the classical first order language extended by symbols for truth values and some additional connectives (for details see the previously cited works) By F J we denote a set of all well formed formulas in the language J . We will use the ....

....is a scheme [A 1 ; a 1 ] A n ; a n ] B; b] 28) where B = r syn (A 1 ; A n ) is a formula syntactically derived from A 1 ; A n and b = r sem (a 1 ; a n ) is its resulting evaluation. The functions r syn , r sem must fulfil reasonable conditions (cf. [13, 14, 23]) For example, a many valued rule of modus ponens has the form [A; a] A)B; b] B; a Omega b] 29) The properties of inference rules have been extensively discussed in [14, 23] Since in approximate resoning, we work with vague facts, an inference rule has the following form R : A 1 ; ....

[Article contains additional citation context not shown here]

Nov'ak, V., On the Syntactico-Semantical Completeness of First--Order Fuzzy Logic. Part I --- Syntactical Aspects; Part II --- Main Results. Kybernetika 26(1990), 47--66; 134--154.

....Although the definition of the notion of a model or an interpretation (i.e. the semantical part) of such a language is straight forward, the rules for logical deduction have to be modified and new rules have to be added for the sake of completeness. Examples for such extensions can be found in [21, 16, 17, 18, 19]. The completeness results of these papers are obtained for the price of a complex deduction mechanism, that guarantees completeness, but does not provide efficient methods for finding proofs. Therefore, these approaches are very valuable from a theoretical point of view, but are subject to ....

V. Nov'ak, On the Syntactico--Semantical Completeness of First--Order Fuzzy Logic, Part II: Main Results. Kybernetika 26 (1990), 134--154.

....Although the definition of the notion of a model or an interpretation (i.e. the semantical part) of such a language is straight forward, the rules for logical deduction have to be modified and new rules have to be added for the sake of completeness. Examples for such extensions can be found in [21, 16, 17, 18, 19]. The completeness results of these papers are obtained for the price of a complex deduction mechanism, that guarantees completeness, but does not provide efficient methods for finding proofs. Therefore, these approaches are very valuable from a theoretical point of view, but are subject to ....

....steps according to the supremum in Definition 2.3. Therefore, it is possible that the value Th (a) can only be approximated (with arbitrary exactness) when we only allow a finite number of deduction steps. The possibility of an infinite number of deduction steps is also considered in [16, 21]. But a number of additional axiom schemata and inference rules is needed for the completeness results in these papers. 3 A Probabilistic Interpretation for Prolog Extensions The previous section was devoted to a purely formal approach to [0; 1] valued Prolog without giving an interpretation of ....

V. Nov'ak, On the Syntactico--Semantical Completeness of First--Order Fuzzy Logic, Part I: Syntax and Semantics. Kybernetika 26 (1990), 47-- 66.

....corresponding predicate calculus is not recursively axiomatizable, i.e. the set of its 1 tautologies is not recursively enumerable. On the other hand, both Lukasiewicz s propositional and predicate logic has a graded version, originally formulated and investigated by Pavelka [14] and Nov ak [12]; for very simplified versions see [7] 8] Godel s logic has, besides the connectives above, also the disjunction with maximumas truth function as a primitive connective; Godel s propositional logic is completely axiomatized by the axioms of intuitionistic propositional logic plus the linearity ....

Nov' ak, V. On the syntactico-semantical completeness of first-order fuzzy logic I, II. Kybernetika 26 (1990), 47-26, 134-152.

....T r is Pi 2 ; there is a recursive theory T such that P r(T; 1) is Pi 2 complete. See again [16] Thus RPL8 is an elegant fuzzy predicate calculus with truth degree equal to provability degree; on the other hand, it badly undecidable. For details see [16] and its predecessors, in particular, [30]. 3.5 Godel predicate logic This logic is, in contradistinction to Lukasiewicz logic, recursively axiomatizable. Logical axioms are those of Godel propositional logic (see 2.5) plus the axioms (81) 82) 83) 91) 92) of BL8 (see above) Deduction rules are modus ponens and ....

Nov' ak, V. On the syntactico-semantical completeness of first-order fuzzy logic I, II. Kybernetika 26 (1990), 47-26, 134-152.

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