1,721,035 research outputs found
On the scope of some formulas defining additive connectives in fuzzy logics
Full text linkIn (Fuzzy Sets and Systems 149 (2005) 297) Wang et al. defined a new fuzzy logic called NMG. They also introduced new formulas to define the additive connectives from multiplicative conjunction, residuated implication and bottom in NMG. However, they did not study the scope of these formulas in the general framework of fuzzy logics. This is the aim of this paper. Therefore, we add the definability formulas to known fuzzy logics as new axioms, following the method used in (Beyond Two: Theory and Applications of Multiple-Valued Logic, 2003, 251.), and we obtain some families of logics presented in a simpler language. Finally, we discuss the standard completeness of these new logics. © 2005 Elsevier B.V. All rights reserved
LP and LP1\2: two fuzzy logics joining Lukasiewicz and Product logics
No full textIn this paper we provide a finite axiomatization (using two finitary rules only) for the propositional logic (called LΠ) resulting from the combination of Lukasiewicz and Product Logics, together with the logic obtained by from LΠ by the adding of a constant symbol and of a defining axiom for 1/2, called LΠ1/2. We show that LΠ1/2 contains all the most important propositional fuzzy logics: Lukasiewicz Logic, Product Logic, Gödel's Fuzzy Logic, Takeuti and Titani's Propositional Logic, Pavelka's Rational Logic, Pavelka's Rational Product Logic, the Lukasiewicz Logic with Δ, and the Product and Gödel's Logics with Δ and involution. Standard completeness results are proved by means of investigating the algebras corresponding to LΠ and LΠ1/2. For these algebras, we prove a theorem of subdirect representation and we show that linearly ordered algebras can be represented as algebras on the unit interval of either a linearly ordered field, or of the ordered ring of integers, Z
On rational weak Nilpotent Minimum logics
No full textIn this paper we investigate extensions of Gödel and Nilpotent Minimum logics by adding rational truth-values as truth constants in the language and by adding corresponding book-keeping axioms for the truth-constants. We also investigate the rational extensions of some parametric families of Weak Nilpotent Minimum logics, weaker than both Gödel and Nilpotent Minimum logics. Weak and strong standard completeness of these logics are studied in general and in particular when we restrict ourselves to formulas of the kind r̄ → φ, where r is a rational in [0, 1] and φ is a formula without rational truth-constants. © 2006 Old City Publishing, Inc
A logical approach to fuzzy truth hedges
Full text linkThe starting point of this paper are the works of Hájek and Vychodil on the axiomatization of truth-stressing and-depressing hedges as expansions of Hájek's BL logic by new unary connectives. They showed that their logics are chain-complete, but standard completeness was only proved for the expansions over Gödel logic. We propose weaker axiomatizations over an arbitrary core fuzzy logic which have two main advantages: (i) they preserve the standard completeness properties of the original logic and (ii) any subdiagonal (resp. superdiagonal) non-decreasing function on [0, 1] preserving 0 and 1 is a sound interpretation of the truth-stresser (resp. depresser) connectives. Hence, these logics accommodate most of the truth hedge functions used in the literature about of fuzzy logic in a broader sense. © 2013 Elsevier Inc. All rights reserved
On triangular norm based axiomatic extensions of the weak nilpotent minimum logic
Full text linkIn this paper we carry out an algebraic investigation of the weak nilpotent minimum logic (WNM) and its t-norm based axiomatic extensions. We consider the algebraic counterpart of WNM, the variety of WNM-algebras (double-struck Wdouble-struck Ndouble-struck M) and prove that it is locally finite, so all its subvarieties are generated by finite chains. We give criteria to compare varieties generated by finite families of WNM-chains, in particular varieties generated by standard WNM-chains, or equivalently t-norm based axiomatic extensions of WNM, and we study their standard completeness properties. We also characterize the generic WNM-chains, i. e. those that generate the variety double-struck Wdouble-struck Ndouble-struck M, and we give finite axiomatizations for some t-norm based extensions of WNM
On expansions of WNM t-norm based logics with truth-constants
Full text linkThis paper focuses on completeness results about generic expansions of propositional weak nilpotent minimum (WNM) logics with truth-constants. Indeed, we consider algebraic semantics for expansions of these logics with a set of truth-constants { over(r, -) | r ∈ C }, for a suitable countable C ⊆ [0, 1], and provide a full description of completeness results when: (i) the t-norm is a weak nilpotent minimum satisfying the finite partition property and (ii) the set of truth-constants covers all the unit interval in the sense that each interval of the partition contains values of C in its interior. © 2009 Elsevier B.V. All rights reserved
First-order t-norm based fuzzy logics with truth-constants: Distinguished semantics and completeness properties
Full text linkThis paper aims at being a systematic investigation of different completeness properties of first-order predicate logics with truth-constants based on a large class of left-continuous t-norms (mainly continuous and weak nilpotent minimum t-norms). We consider standard semantics over the real unit interval but also we explore alternative semantics based on the rational unit interval and on finite chains. We prove that expansions with truth-constants are conservative and we study their real, rational and finite chain completeness properties. Particularly interesting is the case of considering canonical real and rational semantics provided by the algebras where the truth-constants are interpreted as the numbers they actually name. Finally, we study completeness properties restricted to evaluated formulae of the kind over(r, -) → φ, where φ has no additional truth-constants. © 2009 Elsevier B.V. All rights reserved
Expanding the propositional logic of a t-norm with truth-constants: Completeness results for rational semantics
No full textIn this paper we consider the expansions of logics of a left-continuous t-norm with truth-constants from a subalgebra of the rational unit interval. From known results on standard semantics, we study completeness for these propositional logics with respect to chains defined over the rational unit interval with a special attention to the completeness with respect to the canonical chain, i.e. the algebra over [0,1] ∩ Q where each truth-constant is interpreted in its corresponding rational truth-value. Finally, we study rational completeness results when we restrict ourselves to deductions between the so-called evaluated formulae. © Springer-Verlag 2009
On some varieties of MTL-algebras
Full text linkThe study of perfect, local and bipartite IMTL-algebras presented in [29] is generalized in this paper to the general non-involutive case, i.e. to MTL-algebras. To this end we describe the radical of MTL-algebras and characterize perfect MTL-algebras as those for which the quotient by the radical is isomorphic to the two-element Boolean algebra, and a special class of bipartite MTL-algebras, BP0, as those for which the quotient by the radical is a Boolean algebra. We prove that BP0 is the variety generated by all perfect MTL-algebras and give some equational bases for it. We also introduce a new way to build MTL-algebras by adding a negation fixpoint to a perfect algebra and also by adding some set of points whose negation is the fixpoint. Finally, we consider the varieties generated by those algebras, giving equational bases for them, and we study which of them define a fuzzy logic with standard completeness theorem. © 2005 Oxford University Press
Generalized continuous and left-continuous t-norms arising from algebraic semantics for fuzzy logics
Full text linkThis paper focuses on the issue of how generalizations of continuous and left-continuous t-norms over linearly ordered sets should be from a logical point of view. Taking into account recent results in the scope of algebraic semantics for fuzzy logics over chains with a monoidal residuated operation, we advocate linearly ordered BL-algebras and MTL-algebras as adequate generalizations of continuous and left-continuous t-norms respectively. In both cases, the underlying basic structure is that of linearly ordered residuated lattices. Although the residuation property is equivalent to left-continuity in t-norms, continuous t-norms have received much more attention due to their simpler structure. We review their complete description in terms of ordinal sums and discuss the problem of describing the structure of their generalization to BL-chains. In particular we show the good behavior of BL-algebras over a finite or complete chain, and discuss the partial knowledge of rational BL-chains. Then we move to the general non-continuous case corresponding to left-continuous t-norms and MTL-chains. The unsolved problem of describing the structure of left-continuous t-norms is presented together with a fistful of construction-decomposition techniques that apply to some distinguished families of t-norms and, finally, we discuss the situation in the general study of MTL-chains as a natural generalization of left-continuous t-norms. © 2009 Elsevier Inc. All rights reserved
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