A Preliminary Study of MV-Algebras with Two Quantifiers Which Commute

In this paper we investigate the class of MV-algebras equipped with two quantifiers which commute as a natural generalization of diagonal-free two-dimensional cylindric algebras (see Henkin et al., in Cylindric algebras, 1985). In the 40s, Tarski first introduced cylindric algebras in order to provide an algebraic apparatus for the study of classical predicate calculus. The diagonal–free two-dimensional cylindric algebras are special cylindric algebras. The treatment here of MV-algebras is done in terms of implication and negation. This allows us to simplify some results due to Di Nola and Grigolia (Ann Pure Appl Logic 128(1-3):125–139, 2004) related to the characterization of a quantifier in terms of some special sub-algebra associated to it. On the other hand, we present a topological duality for this class of algebras and we apply it to characterize the congruences of one algebra via certain closed sets. Finally, we study the subvariety of this class generated by a chain of length n + 1 (n <  ω). We prove that the subvariety is semisimple and we characterize their simple algebras. Using a special functional algebra, we determine all the simple finite algebras of this subvariety.     

Obligation as Optimal Goal Satisfaction

Formalising deontic concepts, such as obligation, prohibition and permission, is normally carried out in a modal logic with a possible world semantics, in which some worlds are better than others. The main focus in these logics is on inferring logical consequences, for example inferring that the obligation O q is a logical consequence of the obligations O p and O (p → q). In this paper we propose a non-modal approach in which obligations are preferred ways of satisfying goals expressed in first-order logic. To say that p is obligatory, but may be violated, resulting in a less than ideal situation s, means that the task is to satisfy the goal p ∨ s, and that models in which p is true are preferred to models in which s is true. Whereas, in modal logic, the preference relation between possible worlds is part of the semantics of the logic, in this non-modal approach, the preference relation between first-order models is external to the logic. Although our main focus is on satisfying goals, we also formulate a notion of logical consequence, which is comparable to the notion of logical consequence in modal deontic logic. In this formalisation, an obligation O p is a logical consequence of goals G, when p is true in all best models of G. We show how this non-modal approach to the treatment of deontic concepts deals with problems of contrary-to-duty obligations and normative conflicts, and argue that the approach is useful for many other applications, including abductive explanations, defeasible reasoning, combinatorial optimisation, and reactive systems of the production system variety.     

Grades of Probability Modality in the Law of Evidence

The paper presents an infinite hierarchy PR m [m = 1, 2, . . . ] of sound and complete axiomatic systems for modal logic with graded probabilistic modalities, which are to reflect what I have elsewhere called the Bolding-Ekelöf degrees of evidential strength as applied to the establishment of matters of fact in law-courts. Our present approach is seen to differ from earlier work by the author in that it treats the logic of these graded modalities not only from a semantical or model-theoretic viewpoint but from a prooftheoretical and axiomatic stance as well. A paramount feature of the approach is its use of so-called systematic frame constants as labels of diverse grades of probability. Apart from this novel feature our approach can be seen to go back to pioneering work by Lou Goble in 1970.     

The Structure Group of a Generalized Orthomodular Lattice

Orthomodular lattices with a two-valued Jauch–Piron state split into a generalized orthomodular lattice (GOML) and its dual. GOMLs are characterized as a class of L-algebras, a quantum structure which arises in the theory of Garside groups, algebraic logic, and in connections with solutions of the quantum Yang–Baxter equation. It is proved that every GOML X embeds into a group G(X) with a lattice structure such that the right multiplications in G(X) are lattice automorphisms. Up to isomorphism, X is uniquely determined by G(X), and the embedding \(X\hookrightarrow G(X)\) is a universal group-valued measure on X.     

The Mereological Foundation of Megethology

In Mathematics is megethology (Lewis (1993). Philosophia Mathematica, 1(1), 3–23) David K. Lewis proposes a structuralist reconstruction of classical set theory based on mereology. In order to formulate suitable hypotheses about the size of the universe of individuals without the help of set-theoretical notions, he uses the device of Boolos’ plural quantification for treating second order logic without commitment to set-theoretical entities. In this paper we show how, assuming the existence of a pairing function on atoms, as the unique assumption non expressed in a mereological language, a mereological foundation of set theory is achievable within first order logic. Furthermore, we show how a mereological codification of ordered pairs is achievable with a very restricted use of the notion of plurality without plural quantification.     

Congruence Lattices of Semilattices with Operators

The duality between congruence lattices of semilattices, and algebraic subsets of an algebraic lattice, is extended to include semilattices with operators. For a set G of operators on a semilattice S, we have \({{\rm Con}(S,+,0,G) \cong^{d} {{\rm S}_{p}}(L,H)}\), where L is the ideal lattice of S, and H is a corresponding set of adjoint maps on L. This duality is used to find some representations of lattices as congruence lattices of semilattices with operators. It is also shown that these congruence lattices satisfy the Jónsson–Kiefer property.     

The logic of how-questions

Philosophers and scientists are concerned with the why and the how of things. Questions like the following are so much grist for the philosopher’s and scientist’s mill: How can we be free and yet live in a deterministic universe?, How do neural processes give rise to conscious experience?, Why does conscious experience accompany certain physiological events at all?, How is a three-dimensional perception of depth generated by a pair of two-dimensional retinal images?. Since Belnap and Steel’s pioneering work on the logic of questions, Van Fraassen has managed to apply their approach in constructing an account of the logic of why-questions. Comparatively little, by contrast, has been written on the logic of how-questions despite the apparent centrality of questions such as How is it possible for us to be both free and determined? to philosophical enterprise.¹ In what follows I develop a logic for how-questions of various sorts including how-questions of cognitive resolution, how-questions of manner, how-questions of method, of means, and of mechanism.     

On fragments of Medvedev's logic

Medvedev's intermediate logic (MV) can be defined by means of Kripke semantics as the family of Kripke frames given by finite Boolean algebras without units as partially ordered sets. The aim of this paper is to present a proof of the theorem: For every set of connectivesΦ such that \(\{ \to , \vee , \urcorner \} \not \subseteq \Phi \subseteq \{ \to , \wedge , \urcorner \} \) theΦ-fragment ofMV equals theΦ fragment of intuitionistic logic. The final part of the paper brings the negative solution to the problem set forth by T. Hosoi and H. Ono, namely: is an intermediate logic based on the axiom (?a→b∨c) →(?a→b)∨(?a → c) separable?     

A Temporal Semantics for Basic Logic

In the context of truth-functional propositional many-valued logics, Hájek’s Basic Fuzzy Logic BL [14] plays a major rôle. The completeness theorem proved in [7] shows that BL is the logic of all continuous t-norms and their residua. This result, however, does not directly yield any meaningful interpretation of the truth values in BL per se. In an attempt to address this issue, in this paper we introduce a complete temporal semantics for BL. Specifically, we show that BL formulas can be interpreted as modal formulas over a flow of time, where the logic of each instant is ?ukasiewicz, with a finite or infinite number of truth values. As a main result, we obtain validity with respect to all flows of times that are non-branching to the future, and completeness with respect to all finite linear flows of time, or to an appropriate single infinite linear flow of time. It may be argued that this reduces the problem of establishing a meaningful interpretation of the truth values in BL logic to the analogous problem for ?ukasiewicz logic.     

ExpTime Tableau Decision Procedures for Regular Grammar Logics with Converse

Grammar logics were introduced by Fariñas del Cerro and Penttonen in 1988 and have been widely studied. In this paper we consider regular grammar logics with converse (REG ^c logics) and present sound and complete tableau calculi for the general satisfiability problem of REG ^c logics and the problem of checking consistency of an ABox w.r.t. a TBox in a REG ^c logic. Using our calculi we develop ExpTime (optimal) tableau decision procedures for the mentioned problems, to which various optimization techniques can be applied. We also prove a new result that the data complexity of the instance checking problem in REG ^c logics is coNP-complete.     

Second-order Logic and the Power Set

Ignacio Jane has argued that second-order logic presupposes some amount of set theory and hence cannot legitimately be used in axiomatizing set theory. I focus here on his claim that the second-order formulation of the Axiom of Separation presupposes the character of the power set operation, thereby preventing a thorough study of the power set of infinite sets, a central part of set theory. In reply I argue that substantive issues often cannot be separated from a logic, but rather must be presupposed. I call this the logic-metalogic link. There are two facets to the logic-metalogic link. First, when a logic is entangled with a substantive issue, the same position on that issue should be taken at the meta- level as at the object level; and second, if an expression has a clear meaning in natural language, then the corresponding concept can equally well be deployed in a formal language. The determinate nature of the power set operation is one such substantive issue in set theory. Whether there is a determinate power set of an infinite set can only be presupposed in set theory, not proved, so the use of second-order logic cannot be ruled out by virtue of presupposing one answer to this question. Moreover, the legitimacy of presupposing in the background logic that the power set of an infinite set is determinate is guaranteed by the clarity and definiteness of the notions of all and of subset. This is also exactly what is required for the same presupposition to be legitimately made in an axiomatic set theory, so the use of second-order logic in set theory rather than first-order logic does not require any new metatheoretic commitments.     

Reductio ad contradictionem: An Algebraic Perspective

We introduce a novel expansion of the four-valued Belnap–Dunn logic by a unary operator representing reductio ad contradictionem and study its algebraic semantics. This expansion thus contains both the direct, non-inferential negation of the Belnap–Dunn logic and an inferential negation akin to the negation of Johansson’s minimal logic. We formulate a sequent calculus for this logic and introduce the variety of reductio algebras as an algebraic semantics for this calculus. We then investigate some basic algebraic properties of this variety, in particular we show that it is locally finite and has EDPC. We identify the subdirectly irreducible algebras in this variety and describe the lattice of varieties of reductio algebras. In particular, we prove that this lattice contains an interval isomorphic to the lattice of classes of finite non-empty graphs with loops closed under surjective graph homomorphisms.     

The Modal Logic of Gödel Sentences

The modal logic of Gödel sentences, termed as GS, is introduced to analyze the logical properties of ‘true but unprovable’ sentences in formal arithmetic. The logic GS is, in a sense, dual to Grzegorczyk’s Logic, where modality can be interpreted as ‘true and provable’. As we show, GS and Grzegorczyk’s Logic are, in fact, mutually embeddable. We prove Kripke completeness and arithmetical completeness for GS. GS is also an extended system of the logic of ‘Essence and Accident’ proposed by Marcos (Bull Sect Log 34(1):43–56, 2005). We also clarify the relationships between GS and the provability logic GL and between GS and Intuitionistic Propositional Logic.     

Factors Influencing Adolescents’ Self-control According to Family Structure

We compared factors influencing adolescents’ self-control according to their family structure. Participants were 944 adolescents in five cities in South Korea (115 from single-parent families, 65 from grandparent-led families, and 764 from two-parent families). Data were collected using self-report questionnaires containing items on self-control, stress, parenting attitude, parent–adolescent communication, and family cohesion. Data were analyzed using stepwise multiple regressions with SPSS program. The factors influencing adolescents’ self-control differed across the three family structure groups. For single-parent families, stress and parental attitudes were significantly related to adolescents’ self-control (adjusted R²?=?0.37, p?<?0.001). In contrast, for grandparent-led families, family cohesion and parental attitude were significantly related to adolescents’ self-control (adjusted R²?=?0.31, p?<?0.01), while for two-parent families, stress, parental attitude, and parent–adolescent communication were related to the outcome (adjusted R²?=?0.24, p?<?0.001). Parental attitude was thus a common factor relating to self-control, regardless of family structure. On the other hand, the main factors influencing adolescents with low self-control were gender and stress. Our results confirm that adolescents’ self-control is not only affected by personal factors but also by parental and family factors. It is important to improve individual program to improve adolescents’ self-control according to family structure. The results of study may act as a base for improving individual intervention programs aimed at promoting adolescents’ self-control by factoring in family structure.     

Quasivarieties of Modules Over Path Algebras of Quivers

Let FΛ be a finite dimensional path algebra of a quiver Λ over a field F. Let L and R denote the varieties of all left and right FΛ-modules respectively. We prove the equivalence of the following statements.

• The subvariety lattice of L is a sublattice of the subquasivariety lattice of L.
• The subquasivariety lattice of R is distributive.
• Λ is an ordered forest.

     

Contextual-Hierarchical Reconstructions of the Strengthened Liar Problem

In this paper we shall introduce two types of contextual-hierarchical (from now on abbreviated by ‘ch’) approaches to the strengthened liar problem. These approaches, which we call the ‘standard’ and the ‘alternative’ ch-reconstructions of the strengthened liar problem, differ in their philosophical view regarding the nature of truth and the relation between the truth predicates T r ⁿ and T r ⁿ⁺¹ of different hierarchy-levels. The basic idea of the standard ch-reconstruction is that the T r ⁿ⁺¹-schema should hold for all sentences of \(\mathcal {L}^{n}\). In contrast, the alternative ch-reconstruction, for which we shall argue in section four, is motivated by the idea that T r ⁿ and T r ⁿ⁺¹ are coherent in the sense that the same sentences of \(\mathcal {L}^{n}\) should be true according to T r ⁿ and T r ⁿ⁺¹. We show that instances of the standard ch-reconstruction can be obtained by iterating Kripke’s strong Kleene jump operator. Furthermore, we will demonstrate how instances of the alternative ch-reconstruction can be obtained by a slight modification of the iterated axiom system KF and of the iterated strong Kleene jump operator.     

On Varieties of Biresiduation Algebras

A biresiduation algebra is a 〈/,\,1〉-subreduct of an integral residuated lattice. These algebras arise as algebraic models of the implicational fragment of the Full Lambek Calculus with weakening. We axiomatize the quasi-variety B of biresiduation algebras using a construction for integral residuated lattices. We define a filter of a biresiduation algebra and show that the lattice of filters is isomorphic to the lattice of B-congruences and that these lattices are distributive. We give a finite basis of terms for generating filters and use this to characterize the subvarieties of B with EDPC and also the discriminator varieties. A variety generated by a finite biresiduation algebra is shown to be a subvariety of B. The lattice of subvarieties of B is investigated; we show that there are precisely three finitely generated covers of the atom. Mathematics Subject Classification (2000): 03G25, 06F35, 06B10, 06B20 Dedicated to the memory of Willem Johannes Blok     

BK-lattices. Algebraic Semantics for Belnapian Modal Logics

Earlier algebraic semantics for Belnapian modal logics were defined in terms of twist-structures over modal algebras. In this paper we introduce the class of BK-lattices, show that this class coincides with the abstract closure of the class of twist-structures, and it forms a variety. We prove that the lattice of subvarieties of the variety of BK-lattices is dually isomorphic to the lattice of extensions of Belnapian modal logic BK. Finally, we describe invariants determining a twist-structure over a modal algebra.