J. Michael Dunn and Gary M. Hardegree. Algebraic methods in philosophical logic. Oxford logic guides, no. 41. Clarendon Press, Oxford University Press, Oxford, New York, etc., 2001, xv + 470 pp.

The Infinite-Valued Łukasiewicz Logic and Probability

The paper concerns the algebraic structure of the set of cumulative distribution functions as well as the relationship between the resulting algebra and the infinite-valued Łukasiewicz algebra. The paper also discusses interrelations holding between the logical systems determined by the above algebras. Zadanie „ Wdrożenie platformy Open Journal System dla czasopisma „ Bulletin of the Section of Logic” finansowane w ramach umowy 948/P-DUN/2016 ze środków Ministra Nauki i Szkolnictwa Wyższego przeznaczonych na działalność upowszechniającą naukę.

Weakly algebraizable logics

AbstractIn the paper we study the class of weakly algebraizable logics, characterized by the monotonicity and injectivity of the Leibniz operator on the theories of the logic. This class forms a new level in the non-linear hierarchy of protoalgebraic logics.

Protoalgebraicity and the Deduction Theorem

This chapter is intended as an introduction to the Deduction Theorem and to applications of this theorem in metalogic.

Freedom and Enforcement in Action : A Study in Formal Action Theory

Stit Frames as Action Systems

Stit semantics gives an account of action from a certain perspective: actions are seen not as operations performed in action systems and yielding new states of affairs, but rather as selections of preexistent trajectories of the system in time. Main problems of stit semantics are recapitulated. The interrelations between stit semantics and the approach based on ordered action systems are discussed more fully.

Ordered Action Systems

Ordered action systems are systems whose sets of states are partially ordered. A typology of ordered systems depending on the properties of the component order relation is presented. A significant role of fixed points of the transition relation in ordered systems is emphasized. Numerous examples illustrate the theory.

Additivity of the Equationally-Defined Commutator and Relatively Congruence-Distributive Subquasivarieties

Modularity and Related Topics

Epistemic Aspects of Action Systems

The theory of action conventionally distinguishes real actions and doxastic (or epistemic) actions. Real actions (or as we put it—praxeological actions) bring about changes in material objects of the environment external to the agent. Epistemic actions concern mental states of agents—they bring about changes of agents’ knowledge or beliefs about the environment as well as about other beliefs. Some logical issues concerning knowledge, action, truth, and the epistemic status of agents are discussed. In this context the frame and ramification problems are also analyzed. The key issue raised in this chapter is that of non-monotonicity of reasoning. A reasoning is non-monotonic if some conclusio…

Quasivarieties of Algebras

This chapter plays a twofold role in the book. Firstly, the chapter surveys basic facts about quasivarieties of algebras. These facts are widely utilised in the subsequent chapters devoted to algebraizable logics. Secondly, the chapter shows how the methods initially elaborated for protoalgebraic sentential logics in the first part can be also applied in the area of equational logic. Most of the results presented in this chapter are proved by way of adapting the purely consequential methods of sentential logic to the needs of the (quasi) equational systems associated with quasivarieties of algebras.

Triangular irreducibility of congruences in quasivarieties

Certain forms of irreducibility as well as of equational definability of relative congruences in quasivarieties are investigated. For any integer \({m \geqslant 3}\) and a quasivariety Q, the notion of an m-triangularily meet-irreducible Q-congruence in the algebras of Q is defined. In Section 2, some characterizations of finitely generated quasivarieties involving this notion are provided. Section 3 deals with quasivarieties with equationally definable m-triangular meets of relatively principal congruences. References to finitely based quasivarieties and varieties are discussed.

Commutator Laws in Finitely Generated Quasivarieties

Additivity of the Equationally-Defined Commutator

In this chapter we are concerned with the problem of additivity of the equationally defined commutator.

Elementary Action Systems

This chapter expounds basic notions. An elementary action system is a triple consisting of the set of states, the transition relation between states, and a family of binary relations defined on the set of states. The elements of this family are called atomic actions. Each pair of states belonging to an atomic action is a possible performance of this action. This purely extensional understanding of atomic actions is close to dynamic logic. Compound actions are defined as sets of finite sequences of atomic actions. Thus compound actions are regarded as languages over the alphabet whose elements are atomic actions. This chapter is concerned with the problem of performability of actions and the…

Rasiowa–Sikorski Sets and Forcing

The paper is concerned with the problem of building models for first-order languages from the perspective of the classic paper of Rasiowa and Sikorski (1950). The central idea, due to Rasiowa and Sikorski and developed in this paper, is constructing first-order models from individual variables. The notion of a Rasiowa–Sikorski set of formulas of an arbitrary language L is introduced. Investigations are confined to countable languages. Each Rasiowa–Sikorski set defines a countable model for L. Conversely, each countable model for L is determined, up to isomorphism, by some Rasiowa–Sikorski set. Consequences of these facts are investigated.

AC is Equivalent to the Coherence Principle. Corrigendum to my Paper "Induction Principles for Sets"

Theorem 3.7 of [1] is corrected. Two coherence principles and the ultrafilter property for partial functions contained in a relation are formulated. The equivalence of the coherent principles with AC and the equivalence of the ultrafilter property with BPI is shown.

Probabilistic Interpretations of Predicates

In classical logic, any m-ary predicate is interpreted as an m-argument two-valued relation defined on a non-empty universe. In probability theory, m-ary predicates are interpreted as probability measures on the mth power of a probability space. m-ary probabilistic predicates are equivalently semantically characterized as m-dimensional cumulative distribution functions defined on \(\mathbb {R}^m\). The paper is mainly concerned with probabilistic interpretations of unary predicates in the algebra of cumulative distribution functions defined on \(\mathbb {R}\). This algebra, enriched with two constants, forms a bounded De Morgan algebra. Two logical systems based on the algebra of cumulative…

Regularly Algebraizable Logics

A sentential logic (S, C) is regularly algebraizable (alias 1-algebraizable) if it possesses a non-empty system E(p, q) of equivalence sentences such that E(p, q) ⊆ C(p, q).

Don Pigozzi on Abstract Algebraic Logic, Universal Algebra, and Computer Science

This book celebrates the work of Don Pigozzi on the occasion of his 80th birthday. In addition to articles written by leading specialists and his disciples, it presents Pigozzi’s scientific output and discusses his impact on the development of science. The book both catalogues his works and offers an extensive profile of Pigozzi as a person, sketching the most important events, not only related to his scientific activity, but also from his personal life. It reflects Pigozzi's contribution to the rise and development of areas such as abstract algebraic logic (AAL), universal algebra and computer science, and introduces new scientific results. Some of the papers also present chronologically o…

Deduction theorems within RM and its extensions

AbstractIn [13], M. Tokarz specified some infinite family of consequence operations among all ones associated with the relevant logic RM or with the extensions of RM and proved that each of them admits a deduction theorem scheme. In this paper, we show that the family is complete in a sense that if C is a consequence operation with CRM ≤ C and C admits a deduction theorem scheme, then C is equal to a consequence operation specified in [13]. In algebraic terms, this means that the only quasivarieties of Sugihara algebras with the relative congruence extension property are the quasivarieties corresponding, via the algebraization process, to the consequence operations specified in [13].

The Equationally-Defined Commutator in Quasivarieties Generated by Two-Element Algebras

The notion of the equationally-defined commutator was introduced and thoroughly investigated in (Czelakowski, 2015). In this work the properties of the equationally-defined commutator in quasivarieties generated by two-element algebras are examined. It is proved: If a quasivariety Q is generated by a finite set of two-element algebras, then the equationally-defined commutator of Q is additive (Theorem 3.1) Moreover it satisfies the associativity law (Theorem 3.6). The second result is strengthened if the quasivariety is generated by a single two-element algebra 2: If Q = SP(2), then the equationally-defined commutator of Q universally validates one of the following laws: [x,y] = x^y or [x,y…

Deontology of Compound Actions

This paper, being a companion to the book [2] elaborates the deontology of sequential and compound actions based on relational models and formal constructs borrowed from formal linguistics. The semantic constructions presented in this paper emulate to some extent the content of  [3] but are more involved. Although the present work should be regarded as a sequel of [3] it is self-contained and may be read independently. The issue of permission and obligation of actions is presented in the form of a logical system . This system is semantically defined by providing its intended models in which the role of actions of various types (atomic, sequential and compound ones) is accentuated. Since the…

Commutator Equations and the Equationally-Defined Commutator

If \(\underline{s} = s_{1},\ldots,s_{m}\), and \(\underline{t} = t_{1},\ldots,t_{m}\) are sequences of terms (both sequences of the same length m) and X is a set of equations then $$\displaystyle{\underline{s} \approx \underline{ t} \in X}$$ abbreviates the fact that s i ≈ t i ∈ X for i = 1, …, m.)

Relative principal congruences in congruence-modular quasivarieties

The problem of definability of relative principal congruences in relatively congruence modular (RCM) quasivarieties is investigated. The RCM quasivarieties are characterized in terms of parameterized families of finite sets of pairs of terms which define relative principal congruences.

Basic Properties of Quasivarieties

This chapter supplies basic facts concerning quasivarieties and the equational systems associated with quasivarieties. Many of these facts are of syntactical character. An equational logic is an extension of the familiar Birkhoff’s logic. The narrative structure of the book is strictly linked with the properties of lattices of theories of equational logics. Examining these lattice requires formal tools. They are introduced in this part; some of them are new.

Situational Action Systems

Situational aspects of action are discussed. The presented approach emphasizes the role of situational contexts in which actions are performed. These contexts influence the course of an action; they are determined not only by the current state of the system but also shaped by other factors as time, the previously undertaken actions and their succession, the agents of actions and so on. The distinction between states and situations is explored from the perspective of action systems. The notion of a situational action system is introduced and its theory is expounded. Numerous examples illustrate the reach of the theory.

Action and Deontology

This chapter is concerned with the deontology of actions. According to the presented approach, actions and not propositions are deontologically loaded. Norms direct actions and define the circumstances in which actions are permitted, prohibited, or mandated. Norms are therefore viewed as deontological rules of conduct. The definitions of permission, prohibition, and obligatoriness of an action are formulated in terms of the relation of transition of an action system. A typology of atomic norms is presented. To each atomic norm a proposition is associated and called the normative proposition corresponding to this norm. A logical system, the basic deontic logic, is defined and an adequate sem…

The Equationally-Defined Commutator

More on Finitely Generated Quasivarieties

We begin with the following observation concerning arbitrary finitely generated quasivarieties

Performability of Actions

AbstractAction theory may be regarded as a theoretical foundation of AI, because it provides in a logically coherent way the principles of performing actions by agents. But, more importantly, action theory offers a formal ontology mainly based on set-theoretic constructs. This ontology isolates various types of actions as structured entities: atomic, sequential, compound, ordered, situational actions etc., and it is a solid and non-removable foundation of any rational activity. The paper is mainly concerned with a bunch of issues centered around the notion of performability of actions. It seems that the problem of performability of actions, though of basic importance for purely practical ap…

Basic Definitions and Facts

Symbol is treated here as a primitive entity as point or line in geometry. Let Con = {f α : α < β} be a well-ordered set of symbols called a language type. β is an ordinal number. The elements of the above set are called connectives. To each connective f α a natural number α(α) ∈ w called the rank of f α or the arity of f α is assigned. The arity α(α) defines the number of arguments of f α . Thus we speak of nullary, unary, or binary connectives, etc. In the sequel Con is assumed to be fixed but arbitrary.

Monotone Relations, Fixed Points and Recursive Definitions

The paper is concerned with reflexive points of relations. The significance of reflexive points in the context of indeterminate recursion principles is shown.

Logics and operators

Two connectives are of special interest in metalogical investigations — the connective of implication which is important due to its connections to the notion of inference, and the connective of equivalence. The latter connective expresses, in the material sense, the fact that two sentences have the same logical value while in the strict sense it expresses the fact that two sentences are interderivable on the basis of a given logic. The process of identification of equivalent sentences relative to theories of a logic C defines a class of abstract algebras. The members of the class are called Lindenbaum-Tarski algebras of the logic C. One may abstract from the origin of these algebras and exa…

Forcing for First-Order Languages from the Perspective of Rasiowa–Sikorski Lemma

The paper is concerned with the problem of building models for first-order languages from the perspective of the classic paper of Rasiowa and Sikorski [9]. The central idea, developed in this paper, consists in constructing first-order models from individual variables. The key notion of a Rasiowa–Sikorski set of formulas for an arbitrary countable language L is examined. Each Rasiowa–Sikorski set defines a countable model for L . Conversely, every countable model for L is determined by a Rasiowa–Sikorski set. The focus is on constructing Rasiowa–Sikorski sets by applying forcing techniques restricted to Boolean algebras arising from the subsets of the set of atomic formulas of L .