Truth Constants Research Articles

Rational Pavelka logic: The best among three worlds?

This comparative survey explores three formal approaches to reasoning with partly true statements and degrees of truth, within the family of Łukasiewicz logic. These approaches are represented by infinite-valued Łukasiewicz logic (Ł), Rational Pavelka logic (RPL) and a logic with graded formulas that we refer to as Graded Rational Pavelka logic (GRPL). Truth constants for all rationals between 0 and 1 are used as a technical means to represent degrees of truth. Łukasiewicz logic ostensibly features no truth constants except 0 and 1; Rational Pavelka logic includes constants in the basic language, with suitable axioms; Graded Rational Pavelka logic works with graded formulas and proofs, following the original intent of Pavelka, inspired by Goguen's work. Historically, Pavelka's papers precede the definition of GRPL, which in turn precedes RPL; retrieving these steps, we discuss how these formal systems naturally evolve from each other, and we also recall how this process has been a somewhat contentious issue in the realm of Łukasiewicz logic. This work can also be read as a case study in logics, their fragments, and the relationship of the fragments to a logic.

Hyperresolution for Gödel logic with truth constants

In the paper, we generalise the well-known hyperresolution principle to the general first-order Gödel logic with explicit partial truth. We propose a hyperresolution calculus suitable for automated deduction in a useful expansion of Gödel logic by intermediate truth constants and the equality, ≖, strict order, ≺, projection, Δ, operators. We solve the deduction problem of a formula from a countable theory in this expansion. We expand Gödel logic by a countable set of intermediate truth constants of the form c¯, c∈(0,1). Our approach is based on translation of a formula to an equivalent satisfiable finite order clausal theory, consisting of order clauses. An order clause is a finite set of order literals of the form ε1⋄ε2 where εi is an atom or a truth constant or a quantified atom, and ⋄ is a connective ≖, equality, or ≺, strict order. ≖ and ≺ are interpreted by the equality and standard strict linear order on [0,1], respectively. We shall investigate the canonical standard completeness, where the semantics of Gödel logic is given by the standard G-algebra, and truth constants are interpreted by ‘themselves’. The hyperresolution calculus is refutation sound and complete for a countable order clausal theory under a certain condition for the set of truth constants occurring in the theory. As an interesting consequence, we get an affirmative solution to the open problem of recursive enumerability of unsatisfiable formulae in Gödel logic with truth constants and the equality, ≖, strict order, ≺, projection, Δ, operators.

Implicit definability of truth constants in Łukasiewicz logic

In the framework of propositional Łukasiewicz logic, a suitable notion of implicit definability, tailored to the intended real-valued semantics and referring to the elements of its domain, is introduced. Several variants of implicitly defining each of the rational elements in the standard semantics are explored, and based on that, a faithful interpretation of theories in Rational Pavelka logic in theories in Łukasiewicz logic is obtained. Some of these results were already presented in Hajek (Metamathematics of fuzzy logic, 1998) as technical statements. A connection to the lack of (deductive) Beth property in Łukasiewicz logic is drawn. Moreover, while irrational elements of the standard semantics are not implicitly definable by finitary means, a parallel development is possible for them in infinitary Łukasiewicz logic. As an application of definability of the rationals, it is shown how computational complexity results for Rational Pavelka logic can be obtained from analogous results for Łukasiewicz logic. The complexity of the definability notion itself is studied as well. Finally, we review the import of these results for the precision/vagueness discussion for fuzzy logic, and for the general standing of truth constants in Łukasiewicz logic.

Infinite-Valued First-Order Łukasiewicz Logic: Hypersequent Calculi Without Structural Rules and Proof Search for Sentences in the Prenex Form

The rational first-order Pavelka logic is an expansion of the infinite-valued first-order Łukasiewicz logic Ł∀ by truth constants. For this logic, we introduce a cumulative hypersequent calculus G1Ł∀ and a noncumulative hypersequent calculus G2Ł∀ without structural inference rules. We compare these calculi with the Baaz–Metcalfe hypersequent calculus GŁ∀ with structural rules. In particular, we show that every GŁ∀-provable sentence is G1Ł∀-provable and a Ł∀-sentence in the prenex form is GŁ∀-provable if and only if it is G2Ł∀-provable. For a tableau version of the calculus G2Ł∀, we describe a family of proof search algorithms that allow us to construct a proof of each G2Ł∀-provable sentence in the prenex form.

On strong standard completeness in some MTL $$_\Delta $$ Δ expansions

In this paper, inspired by the previous work of Franco Montagna on infinitary axiomatizations for standard $$\mathsf {BL}$$BL-algebras, we focus on a uniform approach to the following problem: given a left-continuous t-norm $$*$$ź, find an axiomatic system (possibly with infinitary rules) which is strongly complete with respect to the standard algebra [InlineEquation not available: see fulltext.] This system will be an expansion of Monoidal t-norm-based logic. First, we introduce an infinitary axiomatic system $$\mathsf {L}_*^\infty $$Lźź, expanding the language with $$\Delta $$Δ and countably many truth constants, and with only one infinitary inference rule, that is inspired in Takeuti---Titani density rule. Then we show that $$\mathsf {L}_*^\infty $$Lźź is indeed strongly complete with respect to the standard algebra [InlineEquation not available: see fulltext.]. Moreover, the approach is generalized to axiomatize expansions of these logics with additional operators whose intended semantics over [0, 1] satisfy some regularity conditions.

Displaying updates in logic

Abstract We generalize the language of substructural logics interpreted over the ternary relational semantics. This is motivated by our intention to capture within the logical framework of substructural logics various logic-based formalisms dealing with common sense reasoning and logical dynamics. This initiative is based on the key observation that an update can be represented abstractly by the ternary relation of the substructural framework: the first argument of the ternary relation represents an initial situation, the second an informative event and the third the resulting situation after the occurrence of the informative event. Refining the language of substructural logics may allow us to account for the fine-grained reasoning present in common sense reasoning. Thus, we introduce three triples of connectives that are interconnected by means of cyclic permutations. The usual fusion, implication and co-implication connectives form one of these triples. We define a cut-free display calculus for our language that generalizes the display calculus for modal logic. We do not resort to structural connectives for truth constants and we show how we can obtain our display rules using Gaggle Theory. We prove the soundness and strong completeness of our display calculus via a Henkin-style construction. Using correspondence results from substructural logics, we also obtain sound and complete display calculi for a wide variety of classical and substructural logics. Then, we define dual substructural connectives and we provide a display calculus for our language extended with these dual connectives. Finally, we focus on the specific case of bi-intuitionistic logic for which we provide a novel sound and strongly complete display calculus. In a companion article (Aucher, 2016, J. Logic Comput.), we provide a sequent calculus for update logic that generalizes the non-associative Lambek calculus and a sequent calculus for dynamic epistemic logic that extends this sequent calculus for update logic.

On the relationship between fuzzy autoepistemic logic and fuzzy modal logics of belief

Autoepistemic logic is an important formalism for nonmonotonic reasoning originally intended to model an ideal rational agent reflecting upon his own beliefs. Fuzzy autoepistemic logic is a generalization of autoepistemic logic that allows to represent an agent's rational beliefs on gradable propositions. It has recently been shown that, in the same way as autoepistemic logic generalizes answer set programming, fuzzy autoepistemic logic generalizes fuzzy answer set programming as well. Besides being related to answer set programming, autoepistemic logic is also closely related to several modal logics. To investigate whether a similar relationship holds in a fuzzy logical setting, we firstly generalize the main modal logics for belief to the setting of finitely-valued Łukasiewicz logic with truth constants Łkc, and secondly we relate them with fuzzy autoepistemic logics. Moreover, we show that the problem of satisfiability checking in these logics is NP-complete. Finally, we generalize Levesque's results on stable expansions, belief sets, and “only knowing” operators to our setting, and provide a complete axiomatization for a logic of “only knowing” in the Łkc framework.

A note on axiomatizations of Pavelka-style complete fuzzy logics

Pavelka-style completeness, a property relating degrees of provability and truth, was previously studied mainly in the context of logics with continuous connectives. It is known that in some other logics one can use infinitary deduction rule(s) to retain this form of completeness. The present paper offers a systematic study of this idea for fuzzy logics which expand MTL and are given by a fixed standard algebra. We explore the structure of the class of all ‘reasonable’ expansions of any such logic by rational truth constants and, for several prominent cases, provide axiomatizations of particular expansions enjoying the Pavelka-style completeness.

Expanding Basic Fuzzy Logic with truth constants for component delimiters

In this paper we investigate the propositional logic of standard algebras for Hájek's Basic Fuzzy Logic BL in a language expanded with propositional constants for the idempotent elements delimiting the Ł-, G-, and Π-components of the algebra. We start from a fixed standard BL-algebra; introduce new propositional constants; present a suitable set of axioms; investigate completeness results; and give some complexity results.

Dualities for Algebras of Fitting's Many-Valued Modal Logics

Stone-type duality connects logic, algebra, and topology in both conceptual and technical senses. This paper is intended to be a demonstration of this slogan. In this paper we focus on some versions of Fitting's L-valued logic and L-valued modal logic for a finite distributive lattice L. Building upon the theory of natural dualities, which is a universal algebraic theory of categorical dualities, we establish a Jonsson-Tarski-style duality for algebras of L-valued modal logic, which encompasses Jonsson-Tarski duality for modal algebras as the case L = 2. We also discuss how the dualities change when the algebras are enriched by truth constants. Topological perspectives following from the dualities provide compactness theorems for the logics and the effective classification of categories of algebras involved, which tells us that Stone-type duality makes it possible to use topology for logic and algebra in significant ways. The author is grateful to Professor Susumu Hayashi for his encouragement, to Shohei Izawa for his comments and discussions, and to Kentaro Sato for his suggesting a similar result to Theorem 2.5 for the category of algebras of Lukasiewicz n-valued logic.

Standard Gödel Modal Logics

We prove strong completeness of the □-version and the ◊-version of a Godel modal logic based on Kripke models where propositions at each world and the accessibility relation are both infinitely valued in the standard Godel algebra [0,1]. Some asymmetries are revealed: validity in the first logic is reducible to the class of frames having two-valued accessibility relation and this logic does not enjoy the finite model property, while validity in the second logic requires truly fuzzy accessibility relations and this logic has the finite model property. Analogues of the classical modal systems D, T, S4 and S5 are considered also, and the completeness results are extended to languages enriched with a discrete well ordered set of truth constants.

Fuzzy Topology and Łukasiewicz Logics from the Viewpoint of Duality Theory

This paper explores relationships between many-valued logic and fuzzy topology from the viewpoint of duality theory. We first show a fuzzy topological duality for the algebras of Łukasiewicz n-valued logic with truth constants, which generalizes Stone duality for Boolean algebras to the n-valued case via fuzzy topology. Then, based on this duality, we show a fuzzy topological duality for the algebras of modal Łukasiewicz n-valued logic with truth constants, which generalizes Jónsson-Tarski duality for modal algebras to the n-valued case via fuzzy topology. We emphasize that fuzzy topological spaces naturally arise as spectrums of algebras of many-valued logics.

Fuzzy Description Logics and t-norm based fuzzy logics

Description Logics (DLs) are knowledge representation languages built on the basis of classical logic. DLs allow the creation of knowledge bases and provide ways to reason on the contents of these bases. Fuzzy Description Logics (FDLs) are natural extensions of DLs for dealing with vague concepts, commonly present in real applications. Hájek proposed to deal with FDLs taking as basis t-norm based fuzzy logics with the aim of enriching the expressive possibilities in FDLs and to capitalize on recent developments in the field of Mathematical Fuzzy Logic. From this perspective we define a family of description languages, denoted by ALC ∗ ( S ) , which includes truth constants for representing truth degrees. Having truth constants in the language allows us to define the axioms of the knowledge bases as sentences of a predicate language in much the same way as in classical DLs. On the other hand, taking advantage of the expressive power provided by these truth constants, we define a graded notion of satisfiability, validity and subsumption of DL concepts as the satisfiability, validity and subsumption of evaluated formulas. In the last section we summarize some results concerning fuzzy logics associated with these new description languages, we analyze aspects relative to general and canonical semantics, and we prove some results relative to canonical standard completeness for some FDLs considered in the paper.

Reduced Implicate Tries with Updates

The reduced implicate trie, introduced in Murray and Rosenthal (2005, Vol. 3702 of Lecture Notes in Artifical Intelligence, pp. 231–244), is a data structure that may be used as a target language for knowledge compilation. It has the property that, even when large, it guarantees fast response to queries. Specifically, a query can be processed in time linear in the size of the query regardless of the size of the compiled knowledge base. The knowledge compilation paradigm typically assumes that the ‘intractable part’ of the processing be done once, during compilation. This assumption could render updating the knowledge base infeasible if recompilation is required. The ability to install updates without recompilation may therefore considerably widen applicability. In this article, several update operations not requiring recompilation are developed. These include disjunction, substitution of truth constants, conjunction with unit clauses, reordering of variables and conjunction with arbitrary clauses.

Computational complexity of t-norm based propositional fuzzy logics with rational truth constants

If a continuous t-norm on [ 0 , 1 ] maps pairs of rationals into rationals then the corresponding fuzzy propositional calculus can be extended by rational truth constants and “bookkeeping” axioms for them. (Łukasiewicz t-norm is the classical example.) Computational complexity of such logics is studied. Consequences for fuzzy description logic are formulated.

Graded tableaux for Rational Pavelka Logic: Research Articles

In this article, a sound and complete tableau system for Rational Pavelka Logic (RPL) is introduced. Extended formulas are used as the counterpart of the graded formulas. In this calculus, if we want to show that the graded formula (x, r) is tableau provable (in the finite fuzzy theory F, respectively), we develop a tableau for the extended formula [r, x] (for the set of extended formulas l[r, x], [x1, a1],…, [xn, an] r, respectively). If this tableau closes we claim that (x, r) is tableau provable (in the fuzzy theory F, respectively). We claim also that x is valid at the degree equal to the l.u.b. that allows the closure of the tableaux. Our tableaux are a first step toward efficient procedures of automated deduction in narrow fuzzy logic with truth constants. © 2005 Wiley Periodicals, Inc. Int J Int Syst 20: 1273–1285, 2005.

Graded tableaux for Rational Pavelka Logic

In this article, a sound and complete tableau system for Rational Pavelka Logic (RPL) is introduced. Extended formulas are used as the counterpart of the graded formulas. In this calculus, if we want to show that the graded formula (x, r) is tableau provable (in the finite fuzzy theory F, respectively), we develop a tableau for the extended formula [r, x] (for the set of extended formulas {[r, x], [x1, a1],…, [xn, an] }, respectively). If this tableau closes we claim that (x, r) is tableau provable (in the fuzzy theory F, respectively). We claim also that x is valid at the degree equal to the l.u.b. that allows the closure of the tableaux. Our tableaux are a first step toward efficient procedures of automated deduction in narrow fuzzy logic with truth constants. © 2005 Wiley Periodicals, Inc. Int J Int Syst 20: 1273–1285, 2005.

Rational Pavelka predicate logic is a conservative extension of Łukasiewicz predicate logic

AbstractRational Pavelka logic extends Łukasiewicz infinitely valued logic by adding truth constants r̄ for rationals in [0. 1]. We show that this is a conservative extension. We note that this shows that provability degree can be defined in Łukasiewicz logic. We also give a counterexample to a soundness theorem of Belluce and Chang published in 1963.