The paper begins with a more carefully stated version of ontologically neutral (ON) logic, originally introduced in (Hailperin, 1997). A non-infinitistic semantics which includes a definition of potential infinite validity follows. It is shown, without appeal to the actual infinite, that this notion provides a necessary and sufficient condition for provability in ON logic.

In this manuscript, published here for the first time, Tarski explores the concept of logical notion. He draws on Klein's Erlanger Programm to locate the logical notions of ordinary geometry as those invariant under all transformations of space. Generalizing, he explicates the concept of logical notion of an arbitrary discipline.

We try to explain Tarski's conception of logical notions, as it emerges from alecture of his, delivered in 1966 and published posthumously in 1986 (Historyand Philosophy of Logic 7, 143–154), a conception based on the idea ofinvariance. The evaluation of Tarski's proposal leads us to consider an interesting(and neglected) reply to Skolem in which Tarski hints at his own point of view onthe foundations of set theory. Then, comparing the lecture of 1966 with Tarski'slast work and with an (...) earlier paper written with Lindenbaum, it is shown thatTarski's conception of logical notions, with its essentially type-theoretic character,did not undergo any significant modifications throughout his life. A remark onTarski's prudential attitude on the topic in the famous paper on the concept oflogical consequence (and elsewhere) concludes our paper. (shrink)

We try to explain Tarski's conception of logical notions, as it emerges from alecture of his, delivered in 1966 and published posthumously in 1986 (Historyand Philosophy of Logic7, 143–154), a conception based on the idea ofinvariance. The evaluation of Tarski's proposal leads us to consider an interesting(and neglected) reply to Skolem in which Tarski hints at his own point of view onthe foundations of set theory. Then, comparing the lecture of 1966 with Tarski'slast work and with an earlier (...) paper written with Lindenbaum, it is shown thatTarski's conception of logical notions, with its essentially type-theoretic character,did not undergo any significant modifications throughout his life. A remark onTarski's prudential attitude on the topic in the famous paper on the concept oflogical consequence (and elsewhere) concludes our paper. (shrink)

During the past decade there has been a very effective campaign against any explanation of remembering whose basic concept is that of a causally mediating trace. This paper attempts to provide such an explanation by presenting an explicit deductive argument for the existence of the memory trace. The conclusion is shown to follow from reasonable, empirical assumptions of which the most interesting is a spatiotemporal contiguity thesis. Set-theoretic techniques are used to provide a framework of analysis and probabilistic definitions of (...) some causal notions, as that of a causal chain, are presented. (shrink)

An intrinsic property is roughly a property things have in virtue of how they are, as opposed to how they are related to things outside of them. This paper argues that it is not possible to give a definition of 'intrinsic' that involves only logical, modal and mereological notions, and does not depend on any special assumptions about either properties or possible worlds.

We discuss a theory presented in a posthumous paper by Alfred Tarski entitled “What are logical notions?”. Although the theory of these logical notions is something outside of the main stream of logic, not presented in logic textbooks, it is a very interesting theory and can easily be understood by anybody, especially studying the simplest case of the four basic logical notions. This is what we are doing here, as well as introducing a challenging (...) fifth logical notion. We first recall the context and origin of what are here called Tarski-Lindenbaum logical notions. In the second part, we present these notions in the simple case of a binary relation. In the third part, we examine in which sense these are considered as logical notions contrasting them with an example of a nonlogical relation. In the fourth part, we discuss the formulations of the four logical notions in natural language and in first-order logic without equality, emphasizing the fact that two of the four logical notions cannot be expressed in this formal language. In the fifth part, we discuss the relations between these notions using the theory of the square of opposition. In the sixth part, we introduce the notion of variety corresponding to all non-logical notions and we argue that it can be considered as a logical notion because it is invariant, always referring to the same class of structures. In the seventh part, we present an enigma: is variety formalizable in first-order logic without equality? There follow recollections concerning Jan Woleński. This paper is dedicated to his 80th birthday. We end with the bibliography, giving some precise references for those wanting to know more about the topic. (shrink)

In contemporary philosophy, it is tempting to apply the metaphysics of properties to the specific case of truth, in the hope of making progress on the investigation of the latter. In this paper, I argue that a different approach, mostly independent from the metaphysics of properties and based on the naturalness, in Lewis’ sense, of semantic nations, is often a better alternative, both in general and in some specific cases. In particular, adopting the new perspective, I present a new problem (...) of combining logical validity and strong truth pluralism, and offer a way to sharply distinguish deflationism and primitivism about truth. The main original upshot of the paper is offering a perspective on philosophy of truth that sheds new light on the general problem of truth and on some particular issues. (shrink)

This paper claims that there is no such thing as the correct answer to the question of what is logical form: two significantly different notions of logical form are needed to fulfil two major theoretical roles that pertain respectively to logic and semantics. The first part of the paper outlines the thesis that a unique notion of logical form fulfils both roles, and argues that the alleged best candidate for making it true is unsuited for one (...) of the two roles. The second part spells out a considerably different notion which is free from that problem, although it does not fit the other role. As it will be suggested, each of the two notions suits at most one role, so the uniqueness thesis is ungrounded. (shrink)

The word ‘logic’ has many senses. Here we will understand it as meaning an account of what follows from what and why. With contemporary methodology, logic in this sense – though it may not always have been thought of in this way – is a branch of applied mathematics. This has various implications for how one understands a number of issues concerning validity. In this paper I will explain this perspective of logic, and explore some of its consequences with respect (...) to the notion of logical form. (shrink)

The main objective of the paper is to provide a conceptual apparatus of a general logical theory of language communication. The aim of the paper is to outline a formal-logical theory of language in which the concepts of the phenomenon of language communication and language communication in general are defined and some conditions for their adequacy are formulated. The theory explicates the key notions of contemporary syntax, semantics, and pragmatics. The theory is formalized on two levels: token-level (...) and type-level. As such, it takes into account the dual – token and type – ontological character of linguistic entities. The basic notions of the theory: language communication, meaning and interpretation are introduced on the second, type-level of formalization, and their required prior formalization of some of the notions introduced on the first, token-level; among others, the notion of an act of communication. Owing to the theory, it is possible to address the problems of adequacy of both empirical acts of communication and of language communication in general. All the conditions of adequacy of communication discussed in the presented paper, are valid for one-way communication (sender-recipient); nevertheless, they can also apply to the reverse direction of language communication (recipient-sender). Therefore, they concern the problem of two-way understanding in language communication. (shrink)

We consider countable so‐called rich subsemigroups of ; each such semigroup T gives a variety CPEAT that is axiomatizable by a finite schema of equations taken in a countable subsignature of that of ω‐dimensional cylindric‐polyadic algebras with equality where substitutions are restricted to maps in T. It is shown that for any such T, if and only if is representable as a concrete set algebra of ω‐ary relations. The operations in the signature are set‐theoretically interpreted like in polyadic equality set (...) algebras, but such operations are relativized to a union of cartesian spaces that are not necessarily disjoint. This is a form of guarding semantics. We show that CPEAT is canonical and atom‐canonical. Imposing an extra condition on T, we prove that atomic algebras in CPEAT are completely representable and that CPEAT has the super amalgamation property. If T is rich and finitely represented, it is shown that CPEAT is term definitionally equivalent to a finitely axiomatizable Sahlqvist variety. Such semigroups exist. This can be regarded as a solution to the central finitizability problem in algebraic logic for first order logic with equality if we do not insist on full fledged commutativity of quantifiers. The finite dimensional case is approached from the view point of guarded and clique guarded (relativized) semantics of fragments of first order logic using finitely many variables. Both positive and negative results are presented. (shrink)

Henkin and Tarski proved that an atomic cylindric algebra in which every atom is a rectangle must be representable . This theorem and its analogues for quasi-polyadic algebras with and without equality are formulated in Henkin, Monk and Tarski [13]. We introduce a natural and more general notion of rectangular density that can be applied to arbitrary cylindric and quasi-polyadic algebras, not just atomic ones. We then show that every rectangularly dense cylindric algebra is representable, and we extend this result (...) to other classes of algebras of logic, for example quasi-polyadic algebras and substitution-cylindrification algebras with and without equality, relation algebras, and special Boolean monoids. The results of op. cit. mentioned above are special cases of our general theorems. We point out an error in the proof of the Henkin-Monk-Tarski representation theorem for atomic, equality-free, quasi-polyadic algebras with rectangular atoms. The error consists in the implicit assumption of a property that does not, in general, hold. We then give a correct proof of their theorem. Henkin and Tarski also introduced the notion of a rich cylindric algebra and proved in op. cit. that every rich cylindric algebra of finite dimension satisfying certain special identities is representable. We introduce a modification of the notion of a rich algebra that, in our opinion, renders it more natural. In particular, under this modification richness becomes a density notion. Moreover, our notion of richness applies not only to algebras with equality, such as cylindric algebras, but also to algebras without equality. We show that a finite dimensional algebra is rich iff it is rectangularly dense and quasi-atomic; moreover, each of these conditions is also equivalent to a very natural condition of point density . As a consequence, every finite dimensional rich algebra of logic is representable. We do not have to assume the validity of any special identities to establish this representability. Not only does this give an improvement of the Henkin-Tarski representation theorem for rich cylindric algebras, it solves positively an open problem in op. cit. concerning the representability of finite dimensional rich quasi-polyadic algebras without equality. (shrink)

The most widespread models of rational reasoners (the model based on modal epistemic logic and the model based on probability theory) exhibit the problem of logical omniscience. The most common strategy for avoiding this problem is to interpret the models as describing the explicit beliefs of an ideal reasoner, but only the implicit beliefs of a real reasoner. I argue that this strategy faces serious normative issues. In this paper, I present the more fundamental problem of logical omnipotence, (...) which highlights the normative content of the problem of logical omniscience. I introduce two developments of the notion of implicit belief (accessible and stable belief ) and use them in two versions of the most common strategy applied to the problem of logical omnipotence. (shrink)

We present a logic of agency called LAE whose language includes propositional constants for actions and expectations. The logic is based on Von Wright’s theory of agency in general and his analysis of instrumentality in particular. An axiomatization of the logic, including an independence of agents axiom, is provided and soundness and completeness are shown with respect to its intended class of frames. The framework of LAE will allow us to formally define a manifold of concepts involved in agency theories, (...) including Von Wright’s four elementary forms of action, the notion of forbearance and notions of instrumentality that make reference to an agent’s expectations. (shrink)

The aim of the paper is twofold. First, we want to recapture the genesis of the logics of order. The origin of this notion is traced back to the work of Jerzy Kotas, Roman Suszko, Richard Routley and Robert K. Meyer. A further development of the theory of logics of order is presented in the papers of Jacek K. Kabziński. Quite contemporarily, this notion gained in significance in the papers of Carles Noguera and Petr Cintula. Logics of order are named (...) there _logics of weak implications_. They play a crucial role in their monograph (Noguera and Cintula _Logic and Implication. An Introduction to the General Algebraic Study of Non-Classical Logics_, Trends in Logic 57, Springer, Berlin, 2021). But, more importantly, the other goal is to define some subclasses of the logics of order in reference to later results of Jacek K. Kabziński and Michael Dunn. The original conception of implication is due to Kabziński. Implication is a stronger notion than the notion of the connective of order aka weak implication. As a result, the three subclasses of logics of order are isolated: logics of implication, logics of symmetry, and tonoidal logics. These notions are uniformly defined and investigated from various viewpoints in terms of consequence operations. The emphasis is put on their semantics. (shrink)

This paper distinguishes two versions of Ryle's notion of 'logical geography'. Logical geography: The network of relationships between current uses of a collection of concepts. (Probably what Ryle meant by the term.) Logical topography Features of the portion of reality, or types of portions of reality, related to a given set of concepts, where the reality may be capable of being divided up in different ways using different networks of relationships between concepts. -/- Studying/analysing logical topography (...) includes evaluating the alternative divisions in terms of the theories or applications that are supported by the different sets of concepts, or perhaps some other uses of the concepts. -/- On re-reading Appendices III and IV of my Oxford DPhil thesis "Knowing and Understanding Relations between meaning and truth, meaning and necessary truth, meaning and synthetic necessary truth",, online here, I found to my surprise that several of the topics discussed below had been discussed in those two Appendices, and that the distinction between logical topography and logical geography was made in Appendix IV, discussing conceptual analysis, even though I did not use these words. (shrink)

In Poggiolesi we have introduced a rigorous definition of the notion of complete and immediate formal grounding; in the present paper our aim is to construct a logic for the notion of complete and immediate formal grounding based on that definition. Our logic will have the form of a calculus of natural deduction, will be proved to be sound and complete and will allow us to have fine-grained grounding principles.

Many known tools for proving expressibility bounds for first-order logic are based on one of several locality properties. In this paper we characterize the relationship between those notions of locality. We note that Gaifman's locality theorem gives rise to two notions: one deals with sentences and one with open formulae. We prove that the former implies Hanf's notion of locality, which in turn implies Gaifman's locality for open formulae. Each of these implies the bounded degree property, which is (...) one of the easiest tools for proving expressibility bounds. These results apply beyond the first-order case. We use them to derive expressibility bounds for first-order logic with unary quantifiers and counting. We also characterize the notions of locality on structures of small degree. (shrink)

In To be is to be the object of a possible act of choice the authors defended Boolos’ thesis that plural quantification is part of logic. To this purpose, plural quantification was explained in terms of plural reference, and a semantics of plural acts of choice, performed by an ideal team of agents, was introduced. In this paper, following that approach, we develop a theory of concepts that—in a sense to be explained—can be labeled as a theory of logical (...) concepts. Within this theory, we propose a new logicist approach to natural numbers. Then, we compare our logicism with Frege’s traditional logicism. (shrink)

Intermediate prepositional logics we consider here describe the setI() of regular informational types introduced by Yu. T. Medvedev [7]. He showed thatI() is a Heyting algebra. This algebra gives rise to the logic of infinite problems from [13] denoted here asLM 1. Some other definitions of negation inI() lead to logicsLM n (n ). We study inclusions between these and other systems, proveLM n to be non-finitely axiomatizable (n ) and recursively axiomatizable (n ). We also show that formulas in (...) one variable do not separateLM from Heyting's logicH, andLM n (n ) from Scott's logic (H+S). (shrink)

In this thesis I investigate the notion of experience in German Idealist Philosophy. I focus on the exploration of an alternative to the transcendental model notion of experience through Schelling's insight into the notion of logogrif. The structural division of this project into two sections reflects the two theoretical standpoints of this project, namely the logic and the logogrif of experience. The first section - the logic of experience - explores the notion of experience provided in Kant's Critique of Pure (...) Reason, Critique of Judgement and Fichte's Science of Knowledge. I argue that Kant's fundamental question about the possibility of synthetic a priori judgements succeeds in thematising the aporia of cognitive experience but results in a subject-oriented, representational model which radically confines the notion of experience to the constitutive laws of the understanding or to the normative precepts of Reason. Experience is founded upon a sharp division between faith and knowledge, will and logic, desire and reflection, absolute and finitude. Fichte's endeavour to articulate a non-representational account of experience, does not succeed in extricating itself from the representational model, so long as experience is reduced to the ever-producing deeds of the self-positing ego. Despite the serious differences between Kant's and Fichte's notions of experience, both accounts, so long as they unfold from a transcendental standpoint, attempt to resolve experience into conceptual laws or determinations of the ego's absolute will. Experience is transformed into an object of the subject's cognitive or volitional faculties. The paradoxes of man's interaction with the world are intended to be accommodated either by the law-giving spontaneity of the understanding and the Architectonic of Pure Reason or by the overpowerful primordial act of the self-positing ego. This implies the conceptualisation of the self in terms of constant identity-through-time, or sheer self-determination. However, this conceptualisation remains at the normative or prescriptive level, which in turn is projected upon the world. The latter, though appears as the subject's property, essentially remains alien and opaque, confirming the radical limitations of the ego rather than its order-giving authority. Moreover, this notion of experience is ultimately founded upon a radical expulsion of the divine from the world, the de-spiritualisation of the sensual and the de-sensualisation of the spiritual, the sharp juxtaposition between absolute and finitude. This results in a self-defeating subjectivity, whose firm identity and rule-giving authority does not rescue it from its perennial unattainability to 'organise the conditioned' or 'conquer the unconditioned'. In Kant's and Fichte's thought, however, I detect elements that potentially transgress their transcendental account of experience. These are found in Kant's concept of spontaneity and free play between understanding and imagination, and Fichte's concept of productivity. I argue that these elements lose their potential dynamism, so long as they are absorbed by the transcendental demands for the solution of the aporias of logic. However, these elements point to the need for a radical re-conceptualisation of the notion of experience. This is provided by means of Schelling's logogriflic approach, which constitutes the theme of the second section. The second section - the logogrif of experience - attempts to articulate a different approach towards the notion of experience, through an exploration of Schelling's versatile and provocative thought. This section focuses on Schelling's original insight into the notion and act of logogrif, which opens the dialogue between logos and mythos, cosmic becoming and human soul, cosmic imagination and human reflection, faith and knowledge. This section attempts to illuminate Schelling's fascinating philosophical investigations and discoveries that have been rather overlooked, possibly, due to Hegel's overwhelming critique. This section, after a brief critical examination of the Identity Philosophy, attempts to elucidate Schelling's notion of experience through his middle works, Of Human Freedom, Ages of the World, The Deities of Samothrace, which are treated as a self-developing trilogy. Schelling re-addresses the aporias of logic not as part of Reason's self-interrogation but as part of the cosmic paradoxes and living experiences. In this way, Schelling resets the scene of the debate on the conditions of possibility for cognitive experience by putting on the stage the enigmas of the cosmos and life rather than the Tribunal of Reason. Logic itself is conceived as a potency in the cosmic becoming, and consequently can no longer attempt to establish the transcendental conditions for the possibility of cognitive experience. Cosmic becoming, in which man is an active part, is conceived as the process of the movement, the interaction, the transformations and transmutations of multiple potencies. These, far beyond any mechanical conceptualisation, appear as self-moving and yet interdependent, unknown yet familiar, inscrutable and yet manifest powers, describing the mystery of life itself. The latter is depicted as an ever-recurrent act of longing for self-expression as active unity. Experience is conceived as the lived process of a network of living potencies, which may not only resist rational powers but may also puzzle and seize them. In this context, reflection acquires a plastic dimension, as opposed to its rigidity in the representational model of experience. Reflection depicts cosmic longing's self-formation, whose man is part. This self-bending formation partially illuminates the nature of longing, and from this standpoint is the logic of the longing. However, this formation is movable, transmutable and mostly ineffable, and from this standpoint is the logic of a riddle: a logogrif. Logogrif is the transitive term that attempts to describe the transition of experience from its enacted phase to its allusive conceptual utterance, and in this sense the term itself participates in both phases, as both form of thought and form of life. The logogrific approach to experience in turn transposes us as from the realm of pure concepts to the realm of the mystery of life, from pure thought to acts of longing, from the Architectonic of Pure Reason to Cosmic Theurgy. The latter term attempts to grasp the paradox and dynamism of cosmic and non-cosmic becoming by means of multiple, vanishing and ever-recurring, transmutable potencies, or in Schelling's terms 'the magic of insoluble life'. Schelling's logogrific account consists in a powerful voice for the re-enchantment of the world, the introduction into the notion of experience of the imminence of the divine. This is not suggested in terms of the adoption of old religious doctrines but by means of the discovery and re-discovery of the theurgy of life, through the intensification of our artistic mood, the creative expansion of our deeds. This notion of experience allows for the reconsideration of the notion of the self, in terms of a dynamic, conflictual process between conscious and unconscious powers and the critical revaluation of the accounts of subjectivity which reduce it to the sphere of self-consciousness. The thesis concludes with the need for an investigation into the relation between logos and mythos, which only tangentially has been introduced by the present project. In this context it will be possible to re-appraise the potential that the logogrific approach opens for an alternative to both logical and traditional mythological patterns of thinking. (shrink)

In this paper the authors develop a logic of concepts within a mathematical linguistic theory. In the set of concepts defined in a belief system, the order relationship and Boolean algebra of the concepts are considered. This study is designed to obtain a tool, which is the metatheoretical base of this type of theory.

Tarski presented his definition of consequence operator to explain the most important notions which any logical consequence concept must contemplate. A Tarski space is a pair constituted by a nonempty set and a consequence operator. This structure characterizes an almost topological space. This paper presents an algebraic view of the Tarski spaces and introduces a modal propositional logic which has as a model exactly the closed sets of a Tarski space.

Tarski presented his definition of consequence operator to explain the most important notions which any logical consequence concept must contemplate. A Tarski space is a pair constituted by a nonempty set and a consequence operator. This structure characterizes an almost topological space. This paper presents an algebraic view of the Tarski spaces and introduces a modal propositional logic which has as a model exactly the closed sets of a Tarski space. • DOI:10.5007/1808-1711.2010v14n1p47.

Compactness is an important property of classical propositional logic. It can be defined in two equivalent ways. The first one states that simultaneous satisfiability of an infinite set of formulae is equivalent to the satisfiability of all its finite subsets. The second one states that if a set of formulae entails a formula, then there is a finite subset entailing this formula as well. In propositional many-valued logic, we have different degrees of satisfiability and different possible definitions of entailment, hence (...) the questions of compactness is more complex. In this paper we will deal with compactness of Gödel, GödelΔ, and Gödel∼ logics. (shrink)

The philosophy of logical empiricism has largely determined the direction and the range of problems of the philosophy, which later became known as analytic philosophy. Philosophers of the Vienna Circle and their followers had to dissociate their program from other philosophies predominant in the early twentieth century (particularly from neo-Kantianism and neo-Hegelianism). As a part of this task the revision of the concepts of classical epistemology, including the concept of apriori was carried out. The paper examines how, in the (...) framework of logical empiricism, apriority became identified with analyticity, and discusses alternative theories of the a priori, which resulted of this identification. (shrink)

According to a popular interpretation, Carnap’s interpretation of probability had evolved from a logical towards a subjective conception. However Carnap himself insisted that his basic philosophical view of probability was always the same. I address this apparent clash between Carnap's self-identification and the subsequent interpretations of his work. Following its original intentions, I reconstruct inductive logic as an explication. The emerging picture is of a versatile linguistic framework, whose main function is not the discovery of objective logical relations (...) in the object language, but the stipulation of conceptual possibilities. Within this representation, I map out the changes that the project went through. Seen from such an explication-based perspective, inductive logic becomes quite hard to categorize using the standard labels. (shrink)

Tarski apresentou sua definição de operador de consequência com a intenção de expor as concepções fundamentais da consequência lógica. Um espaço de Tarski é um par ordenado determinado por um conjunto não vazio e um operador de consequência sobre este conjunto. Esta estrutura matemática caracteriza um espaço quase topológico. Este artigo mostra uma visão algébrica dos espaços de Tarski e introduz uma lógica proposicional modal que interpreta o seu operador modal nos conjuntos fechados de algum espaço de Tarski. DOI:10.5007/1808-1711.2010v14n1p47.

Recent years have seen the appearance of many English-language hand books of logic and numerous monographs on topical discoveries in the foundations of mathematics. These publications on the foundations of mathematics as a whole are rather difficult for the beginners or refer the reader to other handbooks and various piecemeal contribu tions and also sometimes to largely conceived "mathematical fol klore" of unpublished results. As distinct from these, the present book is as easy as possible systematic exposition of the now (...) classical results in the foundations of mathematics. Hence the book may be useful especially for those readers who want to have all the proofs carried out in full and all the concepts explained in detail. In this sense the book is self-contained. The reader's ability to guess is not assumed, and the author's ambition was to reduce the use of such words as evident and obvious in proofs to a minimum. This is why the book, it is believed, may be helpful in teaching or learning the foundation of mathematics in those situations in which the student cannot refer to a parallel lecture on the subject. This is also the reason that I do not insert in the book the last results and the most modem and fashionable approaches to the subject, which does not enrich the essential knowledge in founda tions but can discourage the beginner by their abstract form. A. G. (shrink)

In Transparent Intensional Logic we can recognize two distinct notions of computation that loosely correspond to term rewriting and term interpretation as known from lambda calculus. Our goal will be to further explore these two notions and examine some of their properties.