The present paper is thought as a formal study of distributive closure systems which arise in the domain of sentential logics. Special stress is laid on the notion of a C-filter, playing the role analogous to that of a congruence in universal algebra. A sentential logic C is called filter distributive if the lattice of C-filters in every algebra similar to the language of C is distributive. Theorem IV.2 in Section IV gives a method (...) of axiomatization of those filter distributive logics for which the class Matr (C) prime of C-prime matrices (models) is axiomatizable. In Section V, the attention is focused on axiomatic strengthenings of filter distributive logics. The theorems placed there may be regarded, to some extent, as the matrix counterparts of Baker's well-known theorem from universal algebra [9, § 62, Theorem 2]. (shrink)

We show that a finitely generated protoalgebraic strict universal Horn class that is filter-distributive is finitely based. Equivalently, every protoalgebraic and filter-distributive multidimensional deductive system determined by a finite set of finite matrices can be presented by finitely many axioms and rules.

We show that a finitely generated protoalgebraic strict universal Horn class that is filter-distributive is finitely based. Equivalently, every protoalgebraic and filter-distributive multidimensional deductive system determined by a finite set of finite matrices can be presented by finitely many axioms and rules.

There exist important deductive systems, such as the non-normal modal logics, that are not proper subjects of classical algebraic logic in the sense that their metatheory cannot be reduced to the equational metatheory of any particular class of algebras. Nevertheless, most of these systems are amenable to the methods of universal algebra when applied to the matrix models of the system. In the present paper we consider a wide class of deductive systems of this kind called protoalgebraic logics. (...) These include almost all (non-pathological) systems of prepositional logic that have occurred in the literature. The relationship between the metatheory of a protoalgebraic logic and its matrix models is studied. The following results are obtained for any finite matrix model U of a filter-distributive protoalgebraic logic : (I) The extension U of is finitely axiomatized (provided has only finitely many inference rules); (II) U has only finitely many extensions. (shrink)

Assuming the consistency of the theory “ZFC + there exists a measurable cardinal”, we construct 1. a model in which the first cardinal κ, such that 2κ > κ+, bears a normal filter F whose associated boolean algebra is κ+-distributive ,2. a model where there is a measurable cardinal κ such that, for every regular cardinal ρ < κ, 2ρ = ρ++ holds,3. a model of “ZFC + GCH” where there exists a non-measurable cardinal κ bearing a normal (...) filter F whose associated boolean algebra is κ+-distributive. (shrink)

The logicality of language is the hypothesis that the language system has access to a ‘natural’ logic that can identify and filter out as unacceptable expressions that have trivial meanings—that is, that are true/false in all possible worlds or situations in which they are defined. This hypothesis helps explain otherwise puzzling patterns concerning the distribution of various functional terms and phrases. Despite its promise, logicality vastly over-generates unacceptability assignments. Most solutions to this problem rest on specific stipulations about the (...) properties of logical form—roughly, the level of linguistic representation which feeds into the interpretation procedures—and have substantial implications for traditional philosophical disputes about the nature of language. Specifically, contextualism and semantic minimalism, construed as competing hypotheses about the nature and degree of context-sensitivity at the level of logical form, suggest different approaches to the over-generation problem. In this paper, I explore the implications of pairing logicality with various forms of contextualism and semantic minimalism. I argue that to adequately solve the over-generation problem, logicality should be implemented in a constrained contextualist framework. (shrink)

Some aspects of the theory of Boolean algebras and distributive lattices–in particular, the Stone Representation Theorems and the properties of filters and ideals–are analyzed in a constructive setting.

This paper has two parts. The first is concerned with a variant of a family of games introduced by Holy and Schlicht, that we call Welch games. Player II having a winning strategy in the Welch game of length [Formula: see text] on [Formula: see text] is equivalent to weak compactness. Winning the game of length [Formula: see text] is equivalent to [Formula: see text] being measurable. We show that for games of intermediate length [Formula: see text], II winning implies (...) the existence of precipitous ideals with [Formula: see text]-closed, [Formula: see text]-dense trees. The second part shows the first is not vacuous. For each [Formula: see text] between [Formula: see text] and [Formula: see text], it gives a model where II wins the games of length [Formula: see text], but not [Formula: see text]. The technique also gives models where for all [Formula: see text] there are [Formula: see text]-complete, normal, [Formula: see text]-distributive ideals having dense sets that are [Formula: see text]-closed, but not [Formula: see text]-closed. (shrink)

Paraconsistent Weak Kleene logic is the 3-valued logic with two designated values defined through the weak Kleene tables. This paper is a first attempt to investigate PWK within the perspective and methods of abstract algebraic logic. We give a Hilbert-style system for PWK and prove a normal form theorem. We examine some algebraic structures for PWK, called involutive bisemilattices, showing that they are distributive as bisemilattices and that they form a variety, \, generated by the 3-element algebra WK; we (...) also prove that every involutive bisemilattice is representable as the Płonka sum over a direct system of Boolean algebras. We then study PWK from the viewpoint of AAL. We show that \ is not the equivalent algebraic semantics of any algebraisable logic and that PWK is neither protoalgebraic nor selfextensional, not assertional, but it is truth-equational. We fully characterise the deductive filters of PWK on members of \ and the reduced matrix models of PWK. Finally, we investigate PWK with the methods of second-order AAL—we describe the class \ of PWK-algebras, algebra reducts of basic full generalised matrix models of PWK, showing that they coincide with the quasivariety generated by WK—which differs from \—and explicitly providing a quasiequational basis for it. (shrink)

Structurally free logic LC was introduced in [4]. A natural extension of LC, in particular, in a sequent formulation, is by conjunction and disjunction that do not distribute over each other. We define a set theoretical semantics for these logics via constructing a representation of a lattice that we extend by intensional operations. Canonically, minimally overlapping filter-ideal pairs are used; this construction avoids the use of an equivalent of the axiom of choice and lends transparency to the structure. (...) We also discuss adapting the present semantics for substructural logics, as well as, the question of cannnonicity of LC+. (shrink)

Inspired by the definition of tense operators on distributive lattices presented by Chajda and Paseka in 2015, in this paper, we introduce and study the variety of tense distributive lattices with implication and we prove that these are categorically equivalent to a full subcategory of the category of tense centered Kleene algebras with implication. Moreover, we apply such an equivalence to describe the congruences of the algebras of each variety by means of tense 1-filters and tense centered deductive (...) systems, respectively. (shrink)

We provide tools for a concise axiomatization of a broad class of quantifiers in many-valued logic, so-called distribution quantifiers. Although sound and complete axiomatizations for such quantifiers exist, their size renders them virtually useless for practical purposes. We show that for quantifiers based on finite distributive lattices compact axiomatizations can be obtained schematically. This is achieved by providing a link between skolemized signed formulas and filters/ideals in Boolean set lattices. Then lattice theoretic tools such as Birkhoff's representation theorem for (...) finite distributive lattices are used to derive tableau-style axiomatizations of distribution quantifiers. (shrink)

This paper is a contribution to the study of the rôle of disjunction inAlgebraic Logic. Several kinds of (generalized) disjunctions, usually defined using a suitable variant of the proof by cases property, were introduced and extensively studied in the literature mainly in the context of finitary logics. The goals of this paper are to extend these results to all logics, to systematize the multitude of notions of disjunction (both those already considered in the literature and those introduced in (...) this paper), and to show several interesting applications allowed by the presence of a suitable disjunction in a given logic. (shrink)

The dissertation deals with problems in "logic", more precisely, it deals with particular formal systems aiming at capturing patterns of valid reasoning. Sequent calculi were proposed to characterize logical connectives via introduction rules. These systems customarily also have structural rules which allow one to rearrange the set of premises and conclusions. In the "structurally free logic" of Dunn and Meyer the structural rules are replaced by combinatory rules which allow the same reshuffling of formulae, and additionally introduce an explicit marker (...) for the application of the rule---the combinator. The dissertation gives syntactic extensions of such systems introducing conjunction and disjunction . Cut elimination is proved for these systems. ;Syntactic systems as a rule are accompanied by a semantics which interprets them. The extended systems are supplied with semantics using algebraic methods. First, the Lindenbaum algebras are formed, and then these algebras are represented in an algebra of sets. This gives a so called Kripke-style semantics for the systems. The representation for the distributive extension utilizes well-known techniques; it uses sets of prime filters and set theoretical operations on them to deal with conjunction and disjunction, plus a ternary accessibility relation to accommodate the intensional operators as fusion and implication. The representation theorem is proven generally, i.e., for arbitrary sets of combinators using the "squeeze lemma" of Routley and Meyer. ;The non-distributive calculi require a different representation since union and intersection are distributive. Two previous representations for lattices are overviewed, and a modification of the Hartonas-Dunn representation is successfully augmented with the intensional operations. Instead of sets of filters, sets of pairs of minimally inconsistent filters and ideals are used and the representation theorem is proven. The representation results provide soundness and completeness for the systems. The dissertation also investigates properties of dual combinators , proving a series of lemmas about dual combinatory systems not being Church-Rosser and being inconsistent. Combinatorially definable classes of lambda-terms are considered as well. (shrink)

Logics that do not have a deduction-detachment theorem (briefly, a DDT) may still possess a contextual DDT —a syntactic notion introduced here for arbitrary deductive systems, along with a local variant. Substructural logics without sentential constants are natural witnesses to these phenomena. In the presence of a contextual DDT, we can still upgrade many weak completeness results to strong ones, e.g., the finite model property implies the strong finite model property. It turns out that a finitary system has (...) a contextual DDT iff it is protoalgebraic and gives rise to a dually Brouwerian semilattice of compact deductive filters in every finitely generated algebra of the corresponding type. Any such system is filter distributive, although it may lack the filter extension property. More generally, filter distributivity and modularity are characterized for all finitary systems with a local contextual DDT, and several examples are discussed. For algebraizable logics, the well-known correspondence between the DDT and the equational definability of principal congruences is adapted to the contextual case. (shrink)

Gödel algebras form the locally finite variety of Heyting algebras satisfying the prelinearity axiom =. In 1969, Horn proved that a Heyting algebra is a Gödel algebra if and only if its set of prime filters partially ordered by reverse inclusion–i.e. its prime spectrum–is a forest. Our main result characterizes Gödel algebras that are free over some finite distributive lattice by an intrisic property of their spectral forest.

The paper consists of two parts. The first part begins with the problem of whether the original three-valued calculus, invented by J. Łukasiewicz, really conforms to his philosophical and semantic intuitions. I claim that one of the basic semantic assumptions underlying Łukasiewicz's three-valued logic should be that if under any possible circumstances a sentence of the form "X will be the case at time t" is true (resp. false) at time t, then this sentence must be already true (resp. false) (...) at present. However, it is easy to see that this principle is violated in Lukasiewicz's original calculus (as the cases of the law of excluded middle and the law of contradiction show). Nevertheless it is possible to construct (either with the help of the notion of "supervaluation", or purely algebraically) a different three-valued, semi-classical sentential calculus, which would properly incorporate Łukasiewicz's initial intuitions. Algebraically, this calculus has the ordinary Boolean structure, and therefore it retains all classically valid formulas. Yet because possible valuations are no longer represented by ultrafilters, but by filters (not necessarily maximal), the new calculus displays certain non-classical metalogical features (like, for example, nonextensionality and the lack of the metalogical rule enabling one to derive "p is true or q is true" from" 'p ∨ q' is true"). The second part analyses whether the proposed calculus could be useful in formalizing inferences in situations, when for some reason (epistemological or ontological) our knowledge of certain facts is subject to limitation. Special attention should be paid to the possibility of employing this calculus to the case of quantum mechanics. I am going to compare it with standard non-Boolean quantum logic (in the Jauch-Piron approach), and to show that certain shortcomings of the latter can be avoided in the former. For example, I will argue that in order to properly account for quantum features of microphysics, we do not need to drop the law of distributivity. Also the idea of "reading off" the logical structure of propositions from the structure of Hilbert space leads to some conceptual troubles, which I am going to point out. The thesis of the paper is that all we need to speak about quantum reality can be acquired by dropping the principle of bivalence and extensionality, while accepting all classically valid formulas. (shrink)

A systematic approach to the algebraic and Kripke semantics for logics with restricted structural rules, notably for logics on an underlying non-distributive lattice, is developed. We provide a new topological representation theorem for general lattices, using the filter space X. Our representation involves a galois connection on subsets of X, hence a closure operator $\Gamma$, and the image of the representation map is characterized as the collection of $\Gamma$-stable, compact-open subsets of the filter space . (...) The original lattice ${\cal L}$ is thus imbedded in the complete lattice of stable sets. The representation is shown to be functorial and is extended to a Stone-type duality. ;In the presence of additional operators, the extension problem is addressed in terms of what we call adjoint completions of normal operators. An operator f on a lattice L is normal if f: $L\sp{} \times \cdots \times L\sp{} \to L\sp{}$ is a monotone map, where each $L\sp{}$, $i \in n$ + 1, is either L or $L\sp{op}$ and f sends finite joins of $L\sp{}$ to joins in $L\sp{}$. Every familiar operator, such as $\circ$ , + , $\gets, \to, \neg, \square$ O is normal. By a uniform argument we show that normal operators $f\sb{i}$ on L extend to operators $F\sb{i}$ on $\Gamma X$ that distribute over arbitrary joins or meets and that \sb{i\in I})$ is a universal such extension. ;We also raise and resolve the question of representability of operators on stable sets by relations. We give an extensive number of examples of representation of specific operators and provide explicit constructions of canonical Kripke frames, based on our extension arguments, for a number of logical systems, notably for Intuitionistic and Classical Linear Logics, or Non-commutative Linear Logic and of fragments thereof. (shrink)

We give a decision procedure for the ∀∃-theory of the weak truth-table (wtt) degrees of the recursively enumerable sets. The key to this decision procedure is a characterization of the finite lattices which can be embedded into the r.e. wtt-degrees by a map which preserves the least and greatest elements: a finite lattice has such an embedding if and only if it is distributive and the ideal generated by its cappable elements and the filter generated by its cuppable (...) elements are disjoint. We formulate general criteria that allow one to conclude that a distributive upper semi-lattice has a decidable two-quantifier theory. These criteria are applied not only to the weak truth-table degrees of the recursively enumerable sets but also to various substructures of the polynomial many-one (pm) degrees of the recursive sets. These applications to the pm degrees require no new complexity-theoretic results. The fact that the pm-degrees of the recursive sets have a decidable two-quantifier theory answers a question raised by Shore and Slaman in [21]. (shrink)

In this paper we introduce a new natural deduction system for the logic of lattices, and a number of extensions of lattice logic with different negation connectives. We provide the class of natural deduction proofs with both a standard inductive definition and a global graph-theoretical criterion for correctness, and we show how normalisation in this system corresponds to cut elimination in the sequent calculus for lattice logic. This natural deduction system is inspired both by Shoesmith and Smiley's multiple conclusion systems (...) for classical logic and Girard's proofnets for linear logic. (shrink)

We construct an algebra of generalized functions endowed with a canonical embedding of the space of Schwartz distributions.We offer a solution to the problem of multiplication of Schwartz distributions similar to but different from Colombeau’s solution.We show that the set of scalars of our algebra is an algebraically closed field unlike its counterpart in Colombeau theory, which is a ring with zero divisors. We prove a Hahn–Banach extension principle which does not hold in Colombeau theory. We establish a connection between (...) our theory with non-standard analysis and thus answer, although indirectly, a question raised by Colombeau. This article provides a bridge between Colombeau theory of generalized functions and non-standard analysis. (shrink)

Urquhart works in several areas of logic where he has proved important results. Our paper outlines his topological lattice representation and attempts to relate it to other lattice representations. We show that there are different ways to generalize Priestley’s representation of distributive lattices—Urquhart’s being one of them, which tries to keep prime filters in the representation. Along the way, we also mention how semi-lattices and lattices figured into Urquhart’s work.

A filter of a sentential logic ? is Leibniz when it is the smallest one among all the ?-filters on the same algebra having the same Leibniz congruence. This paper studies these filters and the sentential logic ?+ defined by the class of all ?-matrices whose filter is Leibniz, which is called the strong version of ?, in the context of protoalgebraic logics with theorems. Topics studied include an enhanced Correspondence Theorem, characterizations of the weak algebraizability of (...) ?+ and of the explicit definability of Leibniz filters, and several theorems of transfer of metalogical properties from ? to ?+. For finitely equivalential logics stronger results are obtained. Besides the general theory, the paper examines the examples of modal logics, quantum logics and Łukasiewicz's finitely-valued logics. One finds that in some cases the existence of a weak and a strong version of a logic corresponds to well-known situations in the literature, such as the local and the global consequences for normal modal logics; while in others these constructions give an independent interest to the study of other lesser-known logics, such as the lattice-based many-valued logics. (shrink)

We introduce game-theoretic semantics for systems without the conveniences of either a De Morgan negation, or of distribution of conjunction over disjunction and conversely. Much of game playing rests on challenges issued by one player to the other to satisfy, or refute, a sentence, while forcing him/her to move to some other place in the game’s chessboard-like configuration. Correctness of the game-theoretic semantics is proven for both a training game, corresponding to Positive Lattice Logic and for more advanced games for (...) the logics of lattices with weak negation and modal operators. (shrink)

We use the language of categorical logic to construct generic saturated models of intuitionistic theories. Our main technique is the thorough study of the filter construction on categories with finite limits, which is the completion of subobject lattices under filtered meets. When restricted to coherent or Heyting categories, classifying categories of intuitionistic first-order theories, the resulting categories are filtered meet coherent categories, coherent categories with complete subobject lattices such that both finite disjunctions and existential quantification distribute over filtered meets. (...) Such categories naturally embed into Grothendieck toposes which then contain saturated models of the theory we started with. (shrink)

This article presents an approach to the semantics of non-distributive propositional logics that is based on a lattice representation theorem that delivers a canonical extension of the lattice. Our approach supports both a plain Kripke-style semantics and, by restriction, a general frame semantics. Unlike the framework of generalized Kripke frames, the semantic approach presented in this article is suitable for modeling applied logics, as it respects the intended interpretation of the logical operators. This is made possible by (...) restricting admissible interpretations. (shrink)

Let $$\textbf{K}$$ and $$\textbf{M}$$ be locally finite quasivarieties of finite type such that $$\textbf{K}\subset \textbf{M}$$. If $$\textbf{K}$$ is profinite then the filter $$[\textbf{K},\textbf{M}]$$ in the quasivariety lattice $$\textrm{Lq}(\textbf{M})$$ is an atomic lattice and $$\textbf{K}$$ has an independent quasi-equational basis relative to $$\textbf{M}$$. Applications of these results for lattices, unary algebras, groups, unary algebras, and distributive algebras are presented which concern some well-known problems on standard topological quasivarieties and other problems.

We study which [Formula: see text]-distributive forcing notions of size [Formula: see text] can be embedded into tree Prikry forcing notions with [Formula: see text]-complete ultrafilters under various large cardinal assumptions. An alternative formulation — can the filter of dense open subsets of a [Formula: see text]-distributive forcing notion of size [Formula: see text] be extended to a [Formula: see text]-complete ultrafilter.

We adjust the notion of finitary filter pair, which was coined for creating and analyzing finitary logics, in such a way that we can treat logics of cardinality $$\kappa $$, where $$\kappa $$ is a regular cardinal. The corresponding new notion is called $$\kappa $$ -filter pair. A filter pair can be seen as a presentation of a logic, and we ask what different $$\kappa $$ -filter pairs give rise to a fixed logic of (...) cardinality $$\kappa $$. To make the question well-defined we restrict to a subcollection of filter pairs and establish a bijection from that collection to the set of natural extensions of that logic by a set of variables of cardinality $$\kappa $$. Along the way we use $$\kappa $$ -filter pairs to construct natural extensions for a given logic, work out the relationships between this construction and several others proposed in the literature, and show that the collection of natural extensions forms a complete lattice. In an optional section we introduce and motivate the concept of a general filter pair. (shrink)

Hilbert algebras provide the equivalent algebraic semantics in the sense of Blok and Pigozzi to the implication fragment of intuitionistic logic. They are closely related to implicative semilattices. Porta proved that every Hilbert algebra has a free implicative semilattice extension. In this paper we introduce the notion of an optimal deductive filter of a Hilbert algebra and use it to provide a different proof of the existence of the free implicative semilattice extension of a Hilbert algebra as well as (...) a simplified characterization of it. The optimal deductive filters turn out to be the traces in the Hilbert algebra of the prime filters of the distributive lattice free extension of the free implicative semilattice extension of the Hilbert algebra. To define the concept of optimal deductive filter we need to introduce the concept of a strong Frink ideal for Hilbert algebras which generalizes the concept of a Frink ideal for posets. (shrink)

We characterize (both from a syntactic and an algebraic point of view) the normal K4-logics for which unification is filtering. We also give a sufficient semantic criterion for existence of most general unifiers, covering natural extensions of K4.2⁺ (i.e., of the modal system obtained from K4 by adding to it, as a further axiom schemata, the modal translation of the weak excluded middle principle).

Logics $L^F(M)$ are considered, in which $M$ ("most") is a new first-order quantifier whose interpretation depends on a given filter $F$ of subsets of $\omega$. It is proved that countable compactness and axiomatizability are each equivalent to the assertion that $F$ is not of the form $\{(\bigcap F) \cup X: |\omega - X| < \omega\}$ with $|\omega - \bigcap F| = \omega$. Moreover the set of validities of $L^F(M)$ and even of $L^F_{\omega_1\omega}(M)$ depends only on a few basic (...) properties of F. Similar characterizations are given of the class of filters $F$ for which $L^F(M)$ has the interpolation or Robinson properties. An omitting types theorem is also proved. These results sharpen the corresponding known theorems on weak models $(\mathfrak{U}, q)$, where the collection $q$ is allowed to vary. In addition they provide extensions of first-order logic which possess some nice properties, thus escaping from contradicting Lindstrom's Theorem [1969] only because satisfaction is not isomorphism-invariant (as it is tied to the filter $F$). However, Lindstrom's argument is applied to characterize the invariant sentences as just those of first-order logic. (shrink)

As COVID-19 and its variants spread across Australia at differing paces and intensity, the country’s response to the risk of infection and contagion revealed an intensification of bordering practices as a form of risk mitigation with disparate impacts on different segments of the Australian community. Australia’s international border was closed for both inbound and outbound travel, with few exceptions, while states and territories, Indigenous communities, and local government areas were subject to a patchwork of varying restrictions. By focusing on borders (...) at various levels, our research traces how the logics of medico-legal bordering have filtered down from the international to the intra-national, and indeed, into hyper-local spaces. This is not just apparent in the COVID-19 moment but in previous pandemics of 1918 to 1919 influenza and smallpox, in which practices of quarantine and lockdowns were both unevenly distributed and implemented on multiple scales of social organization. An interdisciplinary approach between history and law reveals that human movement during pandemic times in Australia has been regulated in a manner that sees mobility as a risk to public health capable of mitigation through the strict enforcement of borders as a technology of both confinement and exclusion. (shrink)

A formal language is positional if it involves a positional connecitve, i.e. a connective of realization to relate formulas to points of a kind, like points of realization or points of relativization. The connective in focus in this paper is the connective “R”, first introduced by Jerzy Łoś. Formulas [Rαφ] involve a singular name α and a formula φ to the effect that φ is satisfied relative to the position designated by α. In weak positional calculi no nested occurences of (...) the connective “R” are allowed. The distribution problem in weak positional logics is actually the problem of distributivity of the connective “R” over classical connectives, viz. the problem of relation between the occurences of classical connectives inside and outside the scope of the positional connective “R”. (shrink)

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. (shrink)

We extend the relational algebra of Chin and Tarski so that it is multisorted or, as we prefer, typed. Each type supports a local Boolean algebra outfitted with a converse operator. From Lyndon, we know that relation algebras cannot be represented as proper relation algebras where a proper relation algebra has binary relations as elements and the algebra is singly-typed. Here, the intensional conjunction, which was to represent relational composition in Chin and Tarski, spans three different local algebras, thus the (...) term distributed in the title. Since we do not rely on proper relation algebras, we are free to re-express the algebras as typed. In doing so, we allow many different intensional conjunction operators. We construct a typed logic over these algebras, also known as heterogeneous algebras of Birkhoff and Lipson. The logic can be seen as a form of relevance logic with a classical negation connective where the Routley-Meyer star operator is reified as a converse connective in the logic. Relevance logic itself is not typed but our work shows how it can be made so. Some of the properties of classical relevance logic are weakened from Routley-Meyer’s version which is too strong for a logic over relation algebras. (shrink)

Logics L F (M) are considered, in which M ("most") is a new first-order quantifier whose interpretation depends on a given filter F of subsets of ω. It is proved that countable compactness and axiomatizability are each equivalent to the assertion that F is not of the form $\{(\bigcap F) \cup X:|\omega - X| with $|\omega - \bigcap F| = \omega$ . Moreover the set of validities of L F (M) and even of L F ω 1 ω (...) (M) depends only on a few basic properties of F. Similar characterizations are given of the class of filters F for which L F (M) has the interpolation or Robinson properties. An omitting types theorem is also proved. These results sharpen the corresponding known theorems on weak models (U, q), where the collection q is allowed to vary. In addition they provide extensions of first-order logic which possess some nice properties, thus escaping from contradicting Lindstrom's Theorem [1969] only because satisfaction is not isomorphism-invariant (as it is tied to the filter F). However, Lindstrom's argument is applied to characterize the invariant sentences as just those of first-order logic. (shrink)

Public announcement logic is an extension of epistemic logic with dynamic operators that model the effects of all agents simultaneously and publicly acquiring the same piece of information. One of the extensions of PAL, group announcement logic, allows quantification over announcements made by agents. In GAL, it is possible to reason about what groups can achieve by making such announcements. It seems intuitive that this notion of coalitional ability should be closely related to the notion of distributed knowledge, the implicit (...) knowledge of a group. Thus, we study the extension of GAL with distributed knowledge, and in particular possible interaction properties between GAL operators and distributed knowledge. The perhaps surprising result is that, in fact, there are no interaction properties, contrary to intuition. We make this claim precise by providing a sound and complete axiomatisation of GAL with distributed knowledge. We also consider several natural variants of GAL with distributed knowledge, as well as some other related logic, and compare their expressive power. (shrink)

We present a new frame semantics for positive relevant and substructural propositional logics. This frame semantics is both a generalisation of Routley–Meyer ternary frames and a simplification of them. The key innovation of this semantics is the use of a single accessibility relation to relate collections of points to points. Different logics are modeled by varying the kinds of collections used: they can be sets, multisets, lists or trees. We show that collection frames on trees are sound and (...) complete for the basic positive distributive substructural logic $\mathsf {B}^+$, that collection frames on multisets are sound and complete for $\mathsf {RW}^+$ (the relevant logic $\mathsf {R}^+$, without contraction, or equivalently, positive multiplicative and additive linear logic with distribution for the additive connectives), and that collection frames on sets are sound for the positive relevant logic $\mathsf {R}^+$. The completeness of set frames for $\mathsf {R}^+$ is, currently, an open question. (shrink)

The paper shows that in the Art of Thinking (The Port Royal Logic) Arnauld and Nicole introduce a new way to state the truth-conditions for categorical propositions. The definition uses two new ideas: the notion of distributive or, as they call it, universal term, which they abstract from distributive supposition in medieval logic, and their own version of what is now called a conservative quantifier in general quantification theory. Contrary to the interpretation of Jean-Claude Parienté and others, the (...) truth-conditions do not require the introduction of a new concept of ‘indefinite’ term restriction because the notion of conservative quantifier is formulated in terms of the standard notion of term intersection. The discussion shows the following. Distributive supposition could not be used in an analysis of truth because it is explained in terms of entailment, and entailment in terms of truth. By abstracting from semantic identities that underlie distribution, the new concept of distributive term is definitionally prior to truth and can, therefore, be used in a non-circular way to state truth-conditions. Using only standard restriction, the Logic’s truth-conditions for the categorical propositions are stated solely in terms of (1) universal (distributive) term, (2) conservative quantifier, and (3) affirmative and negative proposition. It is explained why the Cartesian notion of extension as a set of ideas is in this context equivalent to medieval and modern notions of extension. (shrink)

The notion of contact algebra is one of the main tools in the region-based theory of space. It is an extension of Boolean algebra with an additional relation C called contact. There are some problems related to the motivation of the operation of Boolean complementation. Because of this operation is dropped and the language of distributive lattices is extended by considering as non-definable primitives the relations of contact, nontangential inclusion and dual contact. It is obtained an axiomatization of the (...) theory consisting of the universal formulas in the language true in all contact algebras. The structures in, satisfying the axioms in question, are called extended distributive contact lattices. In this paper we consider several logics, corresponding to EDCL. We give completeness theorems with respect to both algebraic and topological semantics for these logics. It turns out that they are decidable. (shrink)

Logics for ‘generally’ were introduced for handling assertions with vague notions, by non-standard generalized quantifiers, and to reason qualitatively about them . Filter logic is intended to address ‘most’. Here, we show that filter logic can be faithfully embedded into a classical first-order theory of certain predicates, called compatible. We also use representative predicates to enable elimination of the generalized quantifier. These devices permit using classical first-order methods to reason about consequence in filter logic and help (...) clarifying the role of such extended logics for ‘generally’. (shrink)

Bilattices, introduced by Ginsberg as a uniform framework for inference in artificial intelligence, are algebraic structures that proved useful in many fields. In recent years, Arieli and Avron developed a logical system based on a class of bilattice-based matrices, called logical bilattices, and provided a Gentzen-style calculus for it. This logic is essentially an expansion of the well-known Belnap–Dunn four-valued logic to the standard language of bilattices. Our aim is to study Arieli and Avron’s logic from the perspective of abstract (...) algebraic logic . We introduce a Hilbert-style axiomatization in order to investigate the properties of the algebraic models of this logic, proving that every formula can be reduced to an equivalent normal form and that our axiomatization is complete w.r.t. Arieli and Avron’s semantics. In this way, we are able to classify this logic according to the criteria of AAL. We show, for instance, that it is non-protoalgebraic and non-self-extensional. We also characterize its Tarski congruence and the class of algebraic reducts of its reduced generalized models, which in the general theory of AAL is usually taken to be the algebraic counterpart of a sentential logic. This class turns out to be the variety generated by the smallest non-trivial bilattice, which is strictly contained in the class of algebraic reducts of logical bilattices. On the other hand, we prove that the class of algebraic reducts of reduced models of our logic is strictly included in the class of algebraic reducts of its reduced generalized models. Another interesting result obtained is that, as happens with some implicationless fragments of well-known logics, we can associate with our logic a Gentzen calculus which is algebraizable in the sense of Rebagliato and Verdú . We also prove some purely algebraic results concerning bilattices, for instance that the variety of distributive bilattices is generated by the smallest non-trivial bilattice. This result is based on an improvement of a theorem by Avron stating that every bounded interlaced bilattice is isomorphic to a certain product of two bounded lattices. We generalize it to the case of unbounded interlaced bilattices. (shrink)

Certain distributivity results for Lukasiewicz’s infinite-valued logic Lℵ0 are proved axiomatically (for the first time) with the help of the automated reasoning program Otter [16]. In addition, non -distributivity results are established for a wide variety of positive substructural logics by the use of logical matrices discovered with the automated model findingprograms Mace [15] and MaGIC [25].

Certain distributivity results for Lukasiewicz’s infinite-valued logic Lℵ0..