The Experts below are selected from a list of 108 Experts worldwide ranked by ideXlab platform
Petr Hájek - One of the best experts on this subject based on the ideXlab platform.
-
The sorites paradox and fuzzy logic
International Journal of General Systems, 2003Co-Authors: Petr Hájek, Vilém NovákAbstract:The sorites paradox (interpreted as the paradox of small natural numbers) is analyzed using mathematical fuzzy logic. In the first part, we present an extension of BL-fuzzy logic by a new Unary Connective At of almost true and the crisp Peano arithmetic extended by a fuzzy predicate of feasibility. Then we give examples of possible semantics of At and examples of semantics of feasible numbers. In the second part, we present an analysis of the sorites paradox within fuzzy logic with evaluated syntax and show that under a very natural assumption we obtain a consistent fuzzy theory. Thus, sorites is not paradoxical at all.
-
On very true
Fuzzy Sets and Systems, 2001Co-Authors: Petr HájekAbstract:Abstract The fuzzy truth value “very true” is formalized as a Unary Connective (hedge). A complete axiomatization is presented.
-
A HEDGE FOR GÖDEL FUZZY LOGIC
International Journal of Uncertainty Fuzziness and Knowledge-Based Systems, 2000Co-Authors: Petr Hájek, Dagmar HarmancovÁAbstract:We add to Gödel propositional fuzzy logic an Unary Connective originating in intuitionistic logic and having a natural meaning in fuzzy logic. We prove completeness with respect to a natural many-valued semantics.
-
Embedding Logics into Product Logic
Studia Logica, 1998Co-Authors: Matthias Baaz, Petr Hájek, David Švejda, Jan KrajíčekAbstract:We construct a faithful interpretation of Łukasiewicz's logic in product logic (both propositional and predicate). Using known facts it follows that the product predicate logic is not recursively axiomatizable. We prove a completeness theorem for product logic extended by a Unary Connective δ of Baaz [1]. We show that Gödel's logic is a sublogic of this extended product logic. We also prove NP-completeness of the set of propositional formulas satisfiable in product logic (resp. in Gödel's logic).
-
Embedding Logics Into Product Logic
1Co-Authors: Matthias Baaz, Petr Hájek, Jan Krajíček, David ŠvejdaAbstract:We construct a faithful interpretation of / Lukasiewicz's logic in the product logic (both propositional and predicate). Using known facts it follows that the product predicate logic is not recursively axiomatizable. We prove a completeness theorem for the product logic extended by a Unary Connective 4 of Baaz [1]. We show that Godel's logic is a sublogic of this extended product logic. We also prove NP-completeness of the set of propositional formulas satisfiable in product logic (resp. in Godel's logic). 1 Introduction We shall be concerned with many-valued logics in this paper; in particular, in / Lukasiewicz's logic / L, Godel's logic G and product logic P. Our aim is to obtain information about complexity of these logics in terms of recursive theory (in the case of predicate logic) or in terms of computational complexity theory (in the case of propositional logic). Scarpellini [13] and Mundici [9] provide such information for / Lukasiewicz's logic. Hence we shall concentrate on ..
Stepan Kuznetsov - One of the best experts on this subject based on the ideXlab platform.
-
WoLLIC - The Lambek Calculus with Iteration: Two Variants
Logic Language Information and Computation, 2017Co-Authors: Stepan KuznetsovAbstract:Formulae of the Lambek calculus are constructed using three binary Connectives, multiplication and two divisions. We extend it using a Unary Connective, positive Kleene iteration. For this new operation, following its natural interpretation, we present two lines of calculi. The first one is a fragment of infinitary action logic and includes an omega-rule for introducing iteration to the antecedent. We also consider a version with infinite (but finitely branching) derivations and prove equivalence of these two versions. In Kleene algebras, this line of calculi corresponds to the *-continuous case. For the second line, we restrict our infinite derivations to cyclic (regular) ones. We show that this system is equivalent to a variant of action logic that corresponds to general residuated Kleene algebras, not necessarily *-continuous. Finally, we show that, in contrast with the case without division operations (considered by Kozen), the first system is strictly stronger than the second one. To prove this, we use a complexity argument. Namely, we show, using methods of Buszkowski and Palka, that the first system is \(\varPi _1^0\)-hard, and therefore is not recursively enumerable and cannot be described by a calculus with finite derivations.
-
The Lambek calculus with iteration: two variants
arXiv: Logic, 2017Co-Authors: Stepan KuznetsovAbstract:Formulae of the Lambek calculus are constructed using three binary Connectives, multiplication and two divisions. We extend it using a Unary Connective, positive Kleene iteration. For this new operation, following its natural interpretation, we present two lines of calculi. The first one is a fragment of infinitary action logic and includes an omega-rule for introducing iteration to the antecedent. We also consider a version with infinite (but finitely branching) derivations and prove equivalence of these two versions. In Kleene algebras, this line of calculi corresponds to the *-continuous case. For the second line, we restrict our infinite derivations to cyclic (regular) ones. We show that this system is equivalent to a variant of action logic that corresponds to general residuated Kleene algebras, not necessarily *-continuous. Finally, we show that, in contrast with the case without division operations (considered by Kozen), the first system is strictly stronger than the second one. To prove this, we use a complexity argument. Namely, we show, using methods of Buszkowski and Palka, that the first system is $\Pi_1^0$-hard, and therefore is not recursively enumerable and cannot be described by a calculus with finite derivations.
-
Categories and Types in Logic, Language, and Physics - L-Completeness of the Lambek Calculus with the Reversal Operation Allowing Empty Antecedents
Lecture Notes in Computer Science, 2014Co-Authors: Stepan KuznetsovAbstract:In this paper we prove that the Lambek calculus allowing empty antecedents and enriched with a Unary Connective corresponding to language reversal is complete with respect to the class of models on subsets of free monoids (L-models).
Gemma Robles - One of the best experts on this subject based on the ideXlab platform.
-
Minimal non-relevant logics without the K axiom II. Negation introduced via the Unary Connective.
Reports on Mathematical Logic, 2010Co-Authors: Gemma RoblesAbstract:In the first part of this paper (RML No. 42) a spectrum of constructive logics without the K axiom is defined. Negation is introduced with a propositional falsity constant. The aim of this second part is to build up logics definitionally equivalent to those displayed in the first part, negatio
-
Negation introduced with the Unary Connective
Journal of Applied Non-Classical Logics, 2009Co-Authors: Gemma RoblesAbstract:In the first part of this paper (Mendez and Robles 2008) a minimal and an intuitionistic negation is introduced in a wide spectrum of relevance logics extending Routley and Meyer's basic positive logic B+. It is proved that although all these logics have the characteristic paradoxes of consistency, they lack the K rule (and so, the K axiom). Negation is introduced with a propositional falsity constant. The aim of this paper is to build up logics definitionally equivalent to those in the aforementioned paper, negation being now introduced with the Unary Connective. Relational ternary semantics are provided for the new logics and soundness and completeness results are proved.
Matthias Baaz - One of the best experts on this subject based on the ideXlab platform.
-
Gödel logics with an operator shifting truth values
2014Co-Authors: Matthias Baaz, Oliver FaschingAbstract:We consider GÃűdel logics extended by an operator whose semantics is given by I(o(A)) = min{1, r + I(A)}. The language of propositional GÃűdel logics L p consists of a countably infinite set Var of propositional variables and the Connectives ⊥, ⊃, ∧, ∨ with their usual arities. We will consider extensions by a Unary Connective o, by a Unary Connective △ or by both. For any r ∈ [0, 1], a GÃűdel r-interpretation I maps formulas to V such that I(⊥) = 0, I(A ∧ B) = min{I(A), I(B)}, I(A ∨ B) = max{I(A), I(B)}
-
LPAR short papers(Yogyakarta) - Gödel logics with an operator shifting truth values.
2010Co-Authors: Matthias Baaz, Oliver FaschingAbstract:We consider GAűdel logics extended by an operator whose semantics is given by I(o(A)) = min{1, r + I(A)}. The language of propositional GAűdel logics L consists of a countably infinite set Var of propositional variables and the Connectives ⊥, ⊃, ∧, ∨ with their usual arities. We will consider extensions by a Unary Connective o, by a Unary Connective 4 or by both. For any r ∈ [0, 1], a GAűdel r-interpretation I maps formulas to V such that I(⊥) = 0, I(A ∧B) = min{I(A), I(B)}, I(A ∨B) = max{I(A), I(B)}, I(A ⊃ B) = { 1 I(A) ≤ I(B), I(B) I(A) > I(B). If the language contains o resp. 4, we additionally require I(o(A)) = min{1, r + I(A)}, I(4(A)) = { 1 I(A) = 1 0 I(A) < 1. Let G be some Hilbert-Frege style proof calculus that is sound and complete for propositional GAűdel logics (without o and 4), e. g. take a proof system for intuitionistic logic, plus the schema of linearity (A ⊃ B) ∨ (B ⊃ A), see [3] or, alternatively, use one of the systems described in [4]. We prove that G enhanced by the axiom schemata (⊥ ≺ o⊥) ⊃ (A ≺ oA), (⊥ ↔ o⊥) ⊃ (A ↔ oA), and o(A ⊃ B) ↔ (oA ⊃ oB) is sound and complete w. r. t. the above semantics. Generalizing ideas from [2], we also give an algorithm that constructs a proof for any valid formula. However, this semantics fails to have a compact entailment. The above proof system can also be further combined with a proof system for 4, see [1], to yield a sound and complete calculus for the valid formulas in that language. While the propositional fragment has quite a simple structure, we will show that first order GAűdel logic enhanced by this ring operator is not recursively enumerable, using a technique by Scarpellini [5] employed for Łukasiewicz logic. This ring operator makes the borderline of similarities and contrasts between Łukasiewicz logic visible. The situation changes if one interprets o, more generally, as a function with certain monotonicity properties. ∗partially supported by Austrian Science Fund (FWF-P22416) A. Voronkov, G. Sutcliffe, M. Baaz, C. Fermuller (eds.), LPAR-17-short (EPiC Series, vol. 13), pp. 13–14 13 Godel logics with an operator shifting truth values M. Baaz, O. Fasching
-
Embedding Logics into Product Logic
Studia Logica, 1998Co-Authors: Matthias Baaz, Petr Hájek, David Švejda, Jan KrajíčekAbstract:We construct a faithful interpretation of Łukasiewicz's logic in product logic (both propositional and predicate). Using known facts it follows that the product predicate logic is not recursively axiomatizable. We prove a completeness theorem for product logic extended by a Unary Connective δ of Baaz [1]. We show that Gödel's logic is a sublogic of this extended product logic. We also prove NP-completeness of the set of propositional formulas satisfiable in product logic (resp. in Gödel's logic).
-
Embedding Logics Into Product Logic
1Co-Authors: Matthias Baaz, Petr Hájek, Jan Krajíček, David ŠvejdaAbstract:We construct a faithful interpretation of / Lukasiewicz's logic in the product logic (both propositional and predicate). Using known facts it follows that the product predicate logic is not recursively axiomatizable. We prove a completeness theorem for the product logic extended by a Unary Connective 4 of Baaz [1]. We show that Godel's logic is a sublogic of this extended product logic. We also prove NP-completeness of the set of propositional formulas satisfiable in product logic (resp. in Godel's logic). 1 Introduction We shall be concerned with many-valued logics in this paper; in particular, in / Lukasiewicz's logic / L, Godel's logic G and product logic P. Our aim is to obtain information about complexity of these logics in terms of recursive theory (in the case of predicate logic) or in terms of computational complexity theory (in the case of propositional logic). Scarpellini [13] and Mundici [9] provide such information for / Lukasiewicz's logic. Hence we shall concentrate on ..
Dana Šalounová - One of the best experts on this subject based on the ideXlab platform.
-
Truth values on generalizations of some commutative fuzzy structures
Fuzzy Sets and Systems, 2006Co-Authors: Jiří Rachůnek, Dana ŠalounováAbstract:Hajek introduced the logic BL"v"t enriching the logic BL by a Unary Connective vt which is a formalization of Zadeh's fuzzy truth value ''very true''. BL"v"t-algebras, i.e. BL-algebras with Unary operations, called vt-operators, which are among others subdiagonal, are an algebraic counterpart of BL"v"t. Residuated lattice ordered monoids (R@?-monoids) are common generalizations of BL-algebras and Heyting algebras. In the paper, we study algebraic properties of R@?"v"t-algebras (and consequently of BL"v"t-algebras) and of those that are enriched by derived superdiagonal operators which in the case of MV-algebras are the duals to vt-operators.
