Some philosophers argue that a theory of logical validity should not interpret its own language, because a Russellian argument shows that self-applicability is inconsistent with the ability to capture all the interpretations of its own language. First, I set up a formal system to examine the Russellian argument. I then defend the need for self-applicability. I argue that self-applicability seems to be implied by generality, and that the Russellian argument rests on a test for meaning that is biased against self-applicability. (...) I propose an alternative test which is unproblematic for self-applicability. I conclude that self-applicability can be vindicated and the Russellian argument can be resisted. (shrink)

In the same spirit with the theme of last issue of this SGJ journal (“Ongoing creation”), this paper shortly reviews a plausible mechanism from Aether to become ordinary matter from the perspective of Neutrosophic Logic. We also discuss two other possible applications of Neutrosophic Logic, including a resolution of conflicting paradigms in medicine. We hope that some ideas as outlined herein will be proved useful in the near future.

Justification of theorems plays a vital role in any rational human activity. It is indispensable in science. The deductive method of justifying theorems is used in all sciences and it is the only method of justifying theorems in deductive disciplines. It is based on the notion of proof, thus it is a method of proving theorems. In the Warsaw School of Logic (WSL) – the famous branch of the Lvov-Warsaw School (LWS) – two types of the method: axiomatic deduction (...) method and natural deduction method were developed and practiced. In this paper, both of these methods are briefly discussed with an emphasis on their historical, groundbreaking significance for logic. The axiomatic method by means of rejection (proposed by Jan Łukasiewicz – a co-creator of the WSL), which is the method of the so-called rejection proof (rejection/refutation method) in logical systems and the proving method of generalized natural deduction, which is a hybrid deduction–refutation method of proving theorems, are also outlined in the paper. The author discusses their historical significance. This paper also contains a brief mention of the most significant results which the application of the discussed methods introduced into contemporary scientific research, not only logical one. (shrink)

Information-theoretic approaches to formal logic analyse the "common intuitive" concept of propositional implication (or argumental validity) in terms of information content of propositions and sets of propositions: one given proposition implies a second if the former contains all of the information contained by the latter; an argument is valid if the conclusion contains no information beyond that of the premise-set. This paper locates information-theoretic approaches historically, philosophically and pragmatically. Advantages and disadvantages are identified by examining such approaches in themselves (...) and by contrasting them with standard transformation-theoretic approaches. Transformation-theoretic approaches analyse validity (and thus implication) in terms of transformations that map one argument onto another: a given argument is valid if no transformation carries it onto an argument with all true premises and false conclusion. Model-theoretic, set-theoretic, and substitution-theoretic approaches, which dominate current literature, can be construed as transformation-theoretic, as can the so-called possible-worlds approaches. Ontic and epistemic presuppositions of both types of approaches are considered. Attention is given to the question of whether our historically cumulative experience applying logic is better explained from a purely information-theoretic perspective or from a purely transformation-theoretic perspective or whether apparent conflicts between the two types of approaches need to be reconciled in order to forge a new type of approach that recognizes their basic complementarity. (shrink)

This paper introduces the special issue on the Concept of God of the Journal of Applied Logics (College Publications). The issue contains the following articles: Logic and the Concept of God, by Stanisław Krajewski and Ricardo Silvestre; Mathematical Models in Theology. A Buber-inspired Model of God and its Application to “Shema Israel”, by Stanisław Krajewski; Gödel’s God-like Essence, by Talia Leven; A Logical Solution to the Paradox of the Stone, by Héctor Hernández Ortiz and Victor Cantero; No New (...) Solutions to the Logical Problem of the Trinity, by Beau Branson; What Means ‘Tri-’ in ‘Trinity’ ? An Eastern Patristic Approach to the ‘Quasi-Ordinals’, by Basil Lourié; The Éminence Grise of Christology: Porphyry’s Logical Teaching as a Cornerstone of Argumentation in Christological Debates of the Fifth and Sixth Centruies, by Anna Zhyrkova; The Problem of Universals in Late Patristic Theology, by Dirk Krasmüller; Intuitionist Reasoning in the Tri-unitrian Theology of Nicolas of Cues, by Antonino Drago. (shrink)

The book is a collection of papers and aims to unify the questions of syntax and semantics of language, which are included in logic, philosophy and ontology of language. The leading motif of the presented selection of works is the differentiation between linguistic tokens (material, concrete objects) and linguistic types (ideal, abstract objects) following two philosophical trends: nominalism (concretism) and Platonizing version of realism. The opening article under the title “The Dual Ontological Nature of Language Signs and the Problem (...) of Their Mutual Relations” provides a broad introduction into the problem area connected with this differentiation, while the logic-formal characteristics of the distinction are framed in the work entitled “On the Type-Token Relationships” (Chapter 1). The basic part of the book deals with issues relating to syntax (Chapters 2-4) and semantics of language (Chapters 5-6), as well as pertaining to syntactic-semantic pragmatic questions (Chapters 7-13). Throughout the book, language, categorial language, is characterized syntactically as generated by classical categorial grammar (Chapter 2) and formalized on two opposing levels: as language of expression-tokens (level of tokens) and language of expression-types (level of types). The author’s considerations contained in Chapters 2 and 4 lead to the important philosophical conclusion that in formal-logical syntactic studies on language the assumption that expression-types constitute the primary language layer while expression-tokens make the secondary one, can be neglected; thus, this speaks in favour of the opposing standpoint—the concretistic one—in the ontology of language syntax. In the works “Meaning and Interpretations”, Parts I and II (Chapters 5 and 6), it is underlined, however, that such semantic concepts as: meaning, denotation and interpretation are defined on the types level, yet their formal definitions require making use of notions of the tokens level. The semantic notions introduced in the above-mentioned articles are also used in the following works of the present selection, under the titles: “Three Principles of Compositionality” and “On Metaknowledge and Truth” (Chapters 7 and 8). They formalize two principles of compositionality that are well known in the literature on the subject, deriving from Frege, i.e. those of meaning and of denotation; they are related to the syntactic principle of compositionality which was introduced by the author. All the three principles are, at the same time, three conditions of homomorphism of categorial language algebra into three kinds of non-standard models of language (one syntactic and two semantic ones: intensional and extensional), which allows introducing three definitions of truthfulness into these models. The next two works in the collection, entitled: “On Language Adequacy” and “What is the Sense in Logic and Philosophy of Language” (Chapters 9 and 10) concern adequacy of categorial language syntax along with its dual semantics: intensional and extensional, and categorial compatibility of any of its syntactic categories with two corresponding semantic categories: intensional and extensional, based on the compatibility the syntactic category of each language expression with the ontological category assigned to its denotatum. The well-known problem of categorial compatibility for first-order quantifiers finds its solution in the paper “Categories of First-Order Quantifiers” (Chapter 11). In the work “Logic and Ontology of Language” (Chapter 12), being in a sense a summary of the considerations presented in the preceding chapters of the book, language is treated as an ontological being, characterized in compliance with the logical conception of language proposed by Ajdukiewicz. Application—like throughout the book—of tools of classical logic and set theory has resulted in emergence of a general formal logical theory of syntax, semantics and pragmatics of language, which takes into account duality in the understanding of linguistic expressions as tokens (concretes) and types (abstract objects). In terms that take into account a functional approach to language itself, there comes out an ontological neutrality of logic with respect to existential assumptions relating to the ontological nature of linguistic expressions and their extra-linguistic ontological counterparts. The issues connected with applying logic while explaining the manner of using linguistic tokens and linguistic types to determine notions of language communication are raised and illustrated in the last chapter of the work, bearing the title “A Logical Conceptualization of Knowledge on the Notion of Language Communication”. (shrink)

Logical abductivism is the epistemic view about logic according to which logical theories are justified by abduction (or Inference to the Best Explanation), that is on how well they explain the relevant evidence, so that the correct logical theory turns out to be the one that explains it best. Arguably, this view should be equally applied to both deductive and non-deductive logics, abduction included. But while there seems to be nothing wrong in principle in using abduction to determine (...) the correct logical theories of deduction and induction, things might be more complicated regarding logical theories of abduction. We may wonder whether allowing for an abductive justification of a theory of abduction is an epistemically legitimate move, since here circularity casts its shadow and makes the situation darker. This is the issue to which this work is devoted. I will defend that, to be effective, an abductive justification for a theory of abduction calls for a justification of abduction in advance, which we do not yet have. (shrink)

In this paper I propose and motivate a logic of the interdefined concepts of making true and control, understood as intensional propositional operators to be indexed to an agent. While bearing a resemblance to earlier logics in the tradition, the motivations, semantics, and object language theory differ on crucial points. Applying this logic to widespread formal theories of agency, I use it as a framework to argue against the ubiquitous assumption that the strongest actions or options available to (...) a given agent must always be pairwise incompatible. The conclusion is that this assumption conflicts with failures of higher order control of agents over their degree or precision of control, failures exhibited by such imperfect agents as ourselves. I discuss models in this setting for understanding such imperfectly self-controlling agents. In an appendix, I prove several relevant results about the logic described, including soundness and completeness both for it and for certain natural extensions. (shrink)

A logic is called 'paraconsistent' if it rejects the rule called 'ex contradictione quodlibet', according to which any conclusion follows from inconsistent premises. While logicians have proposed many technically developed paraconsistent logical systems and contemporary philosophers like Graham Priest have advanced the view that some contradictions can be true, and advocated a paraconsistent logic to deal with them, until recent times these systems have been little understood by philosophers. This book presents a comprehensive overview on paraconsistent logical systems (...) to change this situation. The book includes almost every major author currently working in the field. The papers are on the cutting edge of the literature some of which discuss current debates and others present important new ideas. The editors have avoided papers about technical details of paraconsistent logic, but instead concentrated upon works that discuss more 'big picture' ideas. Different treatments of paradoxes takes centre stage in many of the papers, but also there are several papers on how to interpret paraconistent logic and some on how it can be applied to philosophy of mathematics, the philosophy of language, and metaphysics. (shrink)

In some recent works, Crupi and Iacona have outlined an analysis of ‘if’ based on Chrysippus’ idea that a conditional holds whenever the negation of its consequent is incompatible with its antecedent. This paper presents a sound and complete system of conditional logic that accommodates their analysis. The soundness and completeness proofs that will be provided rely on a general method elaborated by Raidl, which applies to a wide range of systems of conditional logic.

We present a framework for epistemic logic, modeling the logical aspects of System 1 and System 2 cognitive processes, as per dual process theories of reasoning. The framework combines non-normal worlds semantics with the techniques of Dynamic Epistemic Logic. It models non-logically-omniscient, but moderately rational agents: their System 1 makes fast sense of incoming information by integrating it on the basis of their background knowledge and beliefs. Their System 2 allows them to slowly, step-wise unpack some of the (...) logical consequences of such knowledge and beliefs, by paying a cognitive cost. The framework is applied to three instances of limited rationality, widely discussed in cognitive psychology: Stereotypical Thinking, the Framing Effect, and the Anchoring Effect. (shrink)

Deontic logic is devoted to the study of logical properties of normative predicates such as permission, obligation and prohibition. Since it is usual to apply these predicates to actions, many deontic logicians have proposed formalisms where actions and action combinators are present. Some standard action combinators are action conjunction, choice between actions and not doing a given action. These combinators resemble boolean operators, and therefore the theory of boolean algebra offers a well-known athematical framework to study the properties of (...) the classic deontic operators when applied to actions. In his seminal work, Segerberg uses constructions coming from boolean algebras to formalize the usual deontic notions. Segerberg’s work provided the initial step to understand logical properties of deontic operators when they are applied to actions. In the last years, other authors have proposed related logics. In this chapter we introduce Segerberg’s work, study related formalisms and investigate further challenges in this area. (shrink)

I distinguish two ways of developing anti-exceptionalist approaches to logical revision. The first emphasizes comparing the theoretical virtuousness of developed bodies of logical theories, such as classical and intuitionistic logic. I'll call this whole theory comparison. The second attempts local repairs to problematic bits of our logical theories, such as dropping excluded middle to deal with intuitions about vagueness. I'll call this the piecemeal approach. I then briefly discuss a problem I've developed elsewhere for comparisons of logical theories. Essentially, (...) the problem is that a pair of logics may each evaluate the alternative as superior to themselves, resulting in oscillation between logical options. The piecemeal approach offers a way out of this problem andthereby might seem a preferable to whole theory comparisons. I go on to show that reflective equilibrium, the best known piecemeal method, has deep problems of its own when applied to logic. (shrink)

Epistemic two-dimensional semantics is a theory in the philosophy of language that provides an account of meaning which is sensitive to the distinction between necessity and apriority. While this theory is usually presented in an informal manner, I take some steps in formalizing it in this paper. To do so, I define a semantics for a propositional modal logic with operators for the modalities of necessity, actuality, and apriority that captures the relevant ideas of epistemic two-dimensional semantics. I also (...) describe some properties of the logic that are interesting from a philosophical perspective, and apply it to the so-called nesting problem. (shrink)

Demonstrative logic, the study of demonstration as opposed to persuasion, is the subject of Aristotle's two-volume Analytics. Many examples are geometrical. Demonstration produces knowledge (of the truth of propositions). Persuasion merely produces opinion. Aristotle presented a general truth-and-consequence conception of demonstration meant to apply to all demonstrations. According to him, a demonstration, which normally proves a conclusion not previously known to be true, is an extended argumentation beginning with premises known to be truths and containing a chain of reasoning (...) showing by deductively evident steps that its conclusion is a consequence of its premises. In particular, a demonstration is a deduction whose premises are known to be true. Aristotle's general theory of demonstration required a prior general theory of deduction presented in the Prior Analytics. His general immediate-deduction-chaining conception of deduction was meant to apply to all deductions. According to him, any deduction that is not immediately evident is an extended argumentation that involves a chaining of intermediate immediately evident steps that shows its final conclusion to follow logically from its premises. To illustrate his general theory of deduction, he presented an ingeniously simple and mathematically precise special case traditionally known as the categorical syllogistic. (shrink)

W. V. Quine famously defended two theses that have fallen rather dramatically out of fashion. The first is that intensions are “creatures of darkness” that ultimately have no place in respectable philosophical circles, owing primarily to their lack of rigorous identity conditions. However, although he was thoroughly familiar with Carnap’s foundational studies in what would become known as possible world semantics, it likely wouldn’t yet have been apparent to Quine that he was fighting a losing battle against intensions, due in (...) large measure to developments stemming from Carnap’s studies and culminating in the work of Kripke, Hintikka, and Bayart. These developments undermined Quine’s crusade against intensions on two fronts. First, in the context of possible world semantics, intensions could after all be given rigorous identity conditions by defining them (in the simplest case) as functions from worlds to appropriate extensions, a fact exploited to powerful and influential effect in logic and linguistics by the likes of Kaplan, Montague, Lewis, and Cresswell. Second, the rise of possible world semantics fueled a strong resurgence of metaphysics in contemporary analytic philosophy that saw properties and propositions widely, fruitfully, and unabashedly adopted as ontological primitives in their own right. This resurgence — happily, in my view — continues into the present day. -/- For a time, at any rate, Quine experienced somewhat better success with his second thesis: that higher-order logic is, at worst, confused and, at best, a quirky notational alternative to standard first-order logic. However, Quine notwithstanding, a great deal of recent work in formal metaphysics transpires in a higher-order logical framework in which properties and propositions fall into an infinite hierarchy of types of (at least) every finite order. Initially, the most philosophically compelling reason for embracing such a framework since Russell first proposed his simple theory of types was simply that it provides a relatively natural explanation of the paradoxes. However, since the seminal work of Prior there has been a growing trend to consider higher-order logic to be the most philosophically natural framework for metaphysical inquiry, many of the contributors to this volume being among the most important and influential advocates of this view. Indeed, this is now quite arguably the dominant view among formal metaphysicians. -/- In this paper, and against the current tide, I will argue in §1 that there are still good reasons to think that Quine’s second battle is not yet lost and that the correct framework for logic is first-order and type-free — properties and propositions, logically speaking, are just individuals among others in a single domain of quantification — and that it arises naturally out of our most basic logical and semantical intuitions. The data I will draw upon are not new and are well-known to contemporary higher-order metaphysicians. However, I will try to defend my thesis in what I believe is a novel way by suggesting that these basic intuitions ground a reasonable distinction between “pure” logic and non-logical theory, and that Russell-style semantic paradoxes of truth and exemplification arise only when we move beyond the purely logical and, hence, do not of themselves provide any strong objection to a type-free conception of properties and propositions. -/- Most of my arguments in §1 are largely independent of any specific account of the nature of properties and propositions beyond their type-freedom. However, I will in addition argue that there are good reasons to take propositions, at least, to be very fine-grained. My arguments are thus bolstered significantly if it can be shown that there are in fact well-defined examples of logics that are not only type-free but which comport with such a conception of propositions. It is the purpose of §2 to lay out a logic of this sort in some detail, drawing especially upon work by George Bealer and related work of my own. With the logic in place, it will be possible to generalize the line of argument noted above regarding Russell-style paradoxes and, in §3, apply it to two propositional paradoxes — the Prior-Kaplan paradox and the Russell-Myhill paradox — that are often taken to threaten the sort of account developed here. (shrink)

The program put forward in von Wright's last works defines deontic logic as ``a study of conditions which must be satisfied in rational norm-giving activity'' and thus introduces the perspective of logical pragmatics. In this paper a formal explication for von Wright's program is proposed within the framework of set-theoretic approach and extended to a two-sets model which allows for the separate treatment of obligation-norms and permission norms. The three translation functions connecting the language of deontic logic with (...) the language of the extended set-theoretical approach are introduced, and used in proving the correspondence between the deontic theorems, on one side, and the perfection properties of the norm-set and the ``counter-set'', on the other side. In this way the possibility of reinterpretation of standard deontic logic as the theory of perfection properties that ought to be achieved in norm-giving activity has been formally proved. The extended set-theoretic approach is applied to the problem of rationality of principles of completion of normative systems. The paper concludes with a plaidoyer for logical pragmatics turn envisaged in the late phase of Von Wright's work in deontic logic. (shrink)

In the *Science of Logic*, Hegel states unequivocally that the category of “life” is a strictly logical, or pure, form of thinking. His treatment of actual life – i.e., that which empirically constitutes nature – arises first in his *Philosophy of Nature* when the logic is applied under the conditions of space and time. Nevertheless, many commentators find Hegel’s development of this category as a purely logical one especially difficult to accept. Indeed, they find this development only (...) comprehensible as long as one simultaneously assumes that Hegel breaks his promise to let the logic do the leading. However, if Hegel were to in fact allow the logical development to be led by biological analogies at this point, problems would ensue. Not only would it contradict his own speculative method, which should secure the necessity of the categories, but it would also endanger the ontological generality of the category of life itself. Beyond undermining his method and the logical integrity of the category, however, I will argue that such a reading makes the transition to the next category of “cognition” unintelligible and problematic. My aim in the first part of this paper is to argue how logical life can be read as a pure category. I then argue in the second part how my reconstruction makes the transition to cognition intelligible without resorting to profane or supernatural interpretations. (shrink)

Classical logic is usually interpreted as the logic of propositions. But from Boole's original development up to modern categorical logic, there has always been the alternative interpretation of classical logic as the logic of subsets of any given (nonempty) universe set. Partitions on a universe set are dual to subsets of a universe set in the sense of the reverse-the-arrows category-theoretic duality--which is reflected in the duality between quotient objects and subobjects throughout algebra. Hence the (...) idea arises of a dual logic of partitions. That dual logic is described here. Partition logic is at the same mathematical level as subset logic since models for both are constructed from (partitions on or subsets of) arbitrary unstructured sets with no ordering relations, compatibility or accessibility relations, or topologies on the sets. Just as Boole developed logical finite probability theory as a quantitative treatment of subset logic, applying the analogous mathematical steps to partition logic yields a logical notion of entropy so that information theory can be refounded on partition logic. But the biggest application is that when partition logic and the accompanying logical information theory are "lifted" to complex vector spaces, then the mathematical framework of quantum mechanics is obtained. Partition logic models indefiniteness (i.e., numerical attributes on a set become more definite as the inverse-image partition becomes more refined) while subset logic models the definiteness of classical physics (an entity either definitely has a property or definitely does not). Hence partition logic provides the backstory so the old idea of "objective indefiniteness" in QM can be fleshed out to a full interpretation of quantum mechanics. (shrink)

This study concerns logical systems considered as theories. By searching for the problems which the traditionally given systems may reasonably be intended to solve, we clarify the rationales for the adequacy criteria commonly applied to logical systems. From this point of view there appear to be three basic types of logical systems: those concerned with logical truth; those concerned with logical truth and with logical consequence; and those concerned with deduction per se as well as with logical truth and (...) logical consequence. Adequacy criteria for systems of the first two types include: effectiveness, soundness, completeness, Post completeness, "strong soundness" and strong completeness. Consideration of a logical system as a theory of deduction leads us to attempt to formulate two adequacy criteria for systems of proofs. The first deals with the concept of rigor or "gaplessness" in proofs. The second is a completeness condition for a system of proofs. An historical note at the end of the paper suggests a remarkable parallel between the above hierarchy of systems and the actual historical development of this area of logic. (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)

Suppose that a sign at the entrance of a hotel reads: “Don’t enter these premises unless you are accompanied by a registered guest”. You see someone who is about to enter, and you tell her: “Don’t enter these premises if you are an unaccompanied registered guest”. She asks why, and you reply: “It follows from what the sign says”. It seems that you made a valid inference from an imperative premise to an imperative conclusion. But it also seems that imperatives (...) cannot be true or false, so what does it mean to say that your inference is valid? It cannot mean that the truth of its premise guarantees the truth of its conclusion. One is thus faced with what is known as “Jørgensen’s dilemma” (Ross 1941: 55-6): it seems that imperative logic cannot exist because logic deals only with entities that, unlike imperatives, can be true or false, but it also seems that imperative logic must exist. It must exist not only because inferences with imperatives can be valid, but also because imperatives (like “Enter” and “Don’t enter”) can be inconsistent with each other, and also because one can apply logical operations to imperatives: “Don’t enter” is the negation of “Enter”, and “Sing or dance” is the disjunction of “Sing” and “Dance”. A standard reaction to this dilemma consists in basing imperative logic on analogues of truth and falsity. For example, the imperative “Don’t enter” is satisfied if you don’t enter and is violated if you enter, and one might say that an inference from an imperative premise to an imperative conclusion is valid exactly if the satisfaction (rather than the truth) of the premise guarantees the satisfaction of the conclusion. But before getting into the details, more needs to be said on what exactly imperatives are. (shrink)

We study logical reduction (factorization) of relations into relations of lower arity by Boolean or relative products that come from applying conjunctions and existential quantifiers to predicates, i.e. by primitive positive formulas of predicate calculus. Our algebraic framework unifies natural joins and data dependencies of database theory and relational algebra of clone theory with the bond algebra of C.S. Peirce. We also offer new constructions of reductions, systematically study irreducible relations and reductions to them and introduce a new characteristic of (...) relations, ternarity, that measures their ‘complexity of relating’ and allows to refine reduction results. In particular, we refine Peirce’s controversial reduction thesis, and show that reducibility behaviour is dramatically different on finite and infinite domains. (shrink)

When one thinks—knows, believes, imagines—that something is the case, one’s thought has a topic: it is about something, towards which one’s mind is directed. What is the logic of thought, so understood? This book begins to explore the idea that, to answer the question, we should take topics seriously. It proposes a hyperintensional account of the propositional contents of thought, arguing that these are individuated not only by the set of possible worlds at which they are true, but also (...) by their topic: what they are about. The book then builds epistemic, doxastic, probabilistic, and conditional logics based on this view. It applies them to issues ranging from dogmatism, scepticism, and epistemic fallibilism, to imagination and suppositional reasoning, belief revision, framing effects, and the acceptability of indicative conditionals. (shrink)

In the paper, original formal-logical conception of syntactic and semantic: intensional and extensional senses of expressions of any language L is outlined. Syntax and bi-level intensional and extensional semantics of language L are characterized categorically: in the spirit of some Husserl’s ideas of pure grammar, Leśniewski-Ajukiewicz’s theory syntactic/semantic categories and in accordance with Frege’s ontological canons, Bocheński’s famous motto—syntax mirrors ontology and some ideas of Suszko: language should be a linguistic scheme of ontological reality and simultaneously a tool of its (...) cognition. In the logical conception of language L, its expressions should satisfy some general conditions of language adequacy. The adequacy ensures their unambiguous syntactic and semantic senses and mutual, syntactic, and semantic compatibility, correspondence guaranteed by the acceptance of a postulate of categorial compatibility syntactic and semantic categories of expressions of L. From this postulate, three principles of compositionality follow: one syntactic and two semantic already known to Frege. They are treated as conditions of homomorphism partial algebra of L into algebraic models of L: syntactic, intensional, and extensional. In the paper, they are applied to some expressions with quantifiers. Language adequacy connected with the logical senses described in the logical conception of language L is, of course, an idealization, but only expressions with high degrees of precision of their senses, after due justification, may become theorems of science. (shrink)

Logic is about reasoning, or so the story goes. This thesis looks at the concept of logic, what it is, and what claims of correctness of logics amount to. The concept of logic is not a settled matter, and has not been throughout the history of it as a notion. Tools from conceptual analysis aid in this historical venture. Once the unsettledness of logic is established we see the repercussions in current debates in the philosophy of (...) logic. Much of the battle over the ‘one true logic’ is conceptually talking past each other. The theory of logics-as-formalizations is presented as a conceptually open theory of logic which is Carnapian in flavour and grounding. Rudolf Carnap’s notions surrounding ‘external’ and ‘pseudo-questions’ about linguistic frameworks apply to formalizations, thus logics, as well. An account of what formalizations are, a more structured sub-set of modelling, is given to ground the claim that logics are formalizations. Finally, a novel account of correctness, the COFE framework, is developed which allows the notions of logical monism, pluralism and nihilism to be more precisely formulated than they currently are in the discourse. (shrink)

Modal logic is one of philosophy’s many children. As a mature adult it has moved out of the parental home and is nowadays straying far from its parent. But the ties are still there: philosophy is important to modal logic, modal logic is important for philosophy. Or, at least, this is a thesis we try to defend in this chapter. Limitations of space have ruled out any attempt at writing a survey of all the work going on (...) in our field—a book would be needed for that. Instead, we have tried to select material that is of interest in its own right or exemplifies noteworthy features in interesting ways. Here are some themes that have guided us throughout the writing: • The back-and-forth between philosophy and modal logic. There has been a good deal of give-and-take in the past. Carnap tried to use his modal logic to throw light on old philosophical questions, thereby inspiring others to continue his work and still others to criticise it. He certainly provoked Quine, who in his turn provided—and continues to provide—a healthy challenge to modal logicians. And Kripke’s and David Lewis’s philosophies are connected, in interesting ways, with their modal logic. Analytic philosophy would have been a lot different without modal logic! • The interpretation problem. The problem of providing a certain modal logic with an intuitive interpretation should not be conflated with the problem of providing a formal system with a model-theoretic semantics. An intuitively appealing model-theoretic semantics may be an important step towards solving the interpretation problem, but only a step. One may compare this situation with that in probability theory, where definitions of concepts like ‘outcome space’ and ‘random variable’ are orthogonal to questions about “interpretations” of the concept of probability. • The value of formalisation. Modal logic sets standards of precision, which are a challenge to—and sometimes a model for—philosophy. Classical philosophical questions can be sharpened and seen from a new perspective when formulated in a framework of modal logic. On the other hand, representing old questions in a formal garb has its dangers, such as simplification and distortion. • Why modal logic rather than classical (first or higher order) logic? The idioms of modal logic—today there are many!—seem better to correspond to human ways of thinking than ordinary extensional logic. (Cf. Chomsky’s conjecture that the NP + VP pattern is wired into the human brain.) In his An Essay in Modal Logic (1951) von Wright distinguished between four kinds of modalities: alethic (modes of truth: necessity, possibility and impossibility), epistemic (modes of being known: known to be true, known to be false, undecided), deontic (modes of obligation: obligatory, permitted, forbidden) and existential (modes of existence: universality, existence, emptiness). The existential modalities are not usually counted as modalities, but the other three categories are exemplified in three sections into which this chapter is divided. Section 1 is devoted to alethic modal logic and reviews some main themes at the heart of philosophical modal logic. Sections 2 and 3 deal with topics in epistemic logic and deontic logic, respectively, and are meant to illustrate two different uses that modal logic or indeed any logic can have: it may be applied to already existing (non-logical) theory, or it can be used to develop new theory. (shrink)

Information-theoretic approaches to formal logic analyze the "common intuitive" concepts of implication, consequence, and validity in terms of information content of propositions and sets of propositions: one given proposition implies a second if the former contains all of the information contained by the latter; one given proposition is a consequence of a second if the latter contains all of the information contained by the former; an argument is valid if the conclusion contains no information beyond that of the premise-set. (...) This paper locates information-theoretic approaches historically, philosophically, and pragmatically. Advantages and disadvantages are identified by examining such approaches in themselves and by contrasting them with standard transformation-theoretic approaches. Transformation-theoretic approaches analyze validity (and thus implication) in terms of transformations that map one argument onto another: a given argument is valid if no transformation carries it onto an argument with all true premises and false conclusion. Model-theoretic, set-theoretic, and substitution-theoretic approaches, which dominate current literature, can be construed as transformation-theoretic, as can the so-called possible-worlds approaches. Ontic and epistemic presuppositions of both types of approaches are considered. Attention is given to the question of whether our historically cumulative experience applying logic is better explained from a purely information-theoretic perspective or from a purely transformation-theoretic perspective or whether apparent conflicts between the two types of approaches need to be reconciled in order to forge a new type of approach that recognizes their basic complementarity. (shrink)

The Art of Logic by Avi Sion is a collection of recent essays on various topics in logic theory and in applied logic. The same faculty and art of logic is called for in formulating theoretical logic and in applying its findings to diverse fields. The essays here collected deal with some very important issues in logic, philosophy, and spirituality, which he had not previously treated in as much detail if at all.

In the present paper we propose a system of propositional logic for reasoning about justification, truthmaking, and the connection between justifiers and truthmakers. The logic of justification and truthmaking is developed according to the fundamental ideas introduced by Artemov. Justifiers and truthmakers are treated in a similar way, exploiting the intuition that justifiers provide epistemic grounds for propositions to be considered true, while truthmakers provide ontological grounds for propositions to be true. This system of logic is then (...) applied both for interpreting the notorious definition of knowledge as justified true belief and for advancing a new solution to Gettier counterexamples to this standard definition. (shrink)

This brief note corrects an error in one of the reduction steps in my paper 'Normalisation for Bilateral Classical Logic with some Philosophical Remarks' published in the Journal of Applied Logics 8/2 (2021): 531-556.

It would be good to have a Bayesian decision theory that assesses our decisions and thinking according to everyday standards of rationality — standards that do not require logical omniscience (Garber 1983, Hacking 1967). To that end we develop a “fragmented” decision theory in which a single state of mind is represented by a family of credence functions, each associated with a distinct choice condition (Lewis 1982, Stalnaker 1984). The theory imposes a local coherence assumption guaranteeing that as an agent's (...) attention shifts, successive batches of "obvious" logical information become available to her. A rule of expected utility maximization can then be applied to the decision of what to attend to next during a train of thought. On the resulting theory, rationality requires ordinary agents to be logically competent and to often engage in trains of thought that increase the unification of their states of mind. But rationality does not require ordinary agents to be logically omniscient. (shrink)

We are much better equipped to let the facts reveal themselves to us instead of blinding ourselves to them or stubbornly trying to force them into preconceived molds. We no longer embarrass ourselves in front of our students, for example, by insisting that “Some Xs are Y” means the same as “Some X is Y”, and lamely adding “for purposes of logic” whenever there is pushback. Logic teaching in this century can exploit the new spirit of objectivity, humility, (...) clarity, observationalism, contextualism, and pluralism. Besides the new spirit there have been quiet developments in logic and its history and philosophy that could radically improve logic teaching. This lecture expands points which apply equally well in first, second, and third courses, i.e. in “critical thinking”, “deductive logic”, and “symbolic logic”. (shrink)

We present a number of modal logics to reason about group norms. As a preliminary step, we discuss the ontological status of the group to which the norms are applied, by adapting the classification made by Christian List of collective attitudes into aggregated, common, and corporate attitudes. Accordingly, we shall introduce modality to capture aggregated, common, and corporate group norms. We investigate then the principles for reasoning about those types of modalities. Finally, we discuss the relationship between group norms (...) and types of collective responsibility. (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)

To reconcile the standard possible worlds model of knowledge with the intuition that ordinary agents fall far short of logical omniscience, a Stalnakerian strategy appeals to two components. The first is the idea that mathematical and logical knowledge is at bottom metalinguistic knowledge. The second is the idea that non-ideal minds are often fragmented. In this paper, we investigate this Stalnakerian reconciliation strategy and argue, ultimately, that it fails. We are not the first to complain about the Stalnakerian strategy. But (...) in contrast to existing complaints, we want to cause trouble for the strategy directly on its home turf. That is, we will advance our objection while granting both the plausibility of the fragmentation component—save for an extreme version of it—and that of the metalinguistic component. Once our central objection to the Stalnakerian strategy is in place, we will show how it negatively affects Adam Elga and Augustín Rayo’s recent attempt to apply the Stalnakerian strategy in the context of Bayesian decision theory. (shrink)

This paper contributes to explaining the rise of logical empiricism in mid-twentieth century (North) America and to a better understanding of American philosophy of science before the dominance of logical empiricism. We show that, contrary to a number of existing histories, philosophy of science was already a distinct subfield of philosophy, one with its own approaches and issues, even before logical empiricists arrived in America. It was a form of speculative philosophy with a concern for speculative metaphysics, normative issues relating (...) to science and society and issues which later were associated with logical empiricist philosophy of science, issues such as confirmation, scientific explanation, reductionism and laws of nature. Further, philosophy of science was not primarily pragmatist in orientation. We also show, with the help of our historical characterization, that a recent account of the emergence of analytic philosophy applies to the rise of logical empiricism. It has been argued that the emergence of American analytic philosophy is partly explained by analytic philosophers’ use of key institutions, including of journals, to marginalize speculative philosophy and promote analytic philosophy. We argue that this use of institutions included the marginalization of speculative and value-laden philosophy of science and the promotion of logical empiricism. (shrink)

We have now mapped the set of analogies Ai,j to conceptual and mechanical meanings. This allows us to recognize how the Group Algebraic System G decomposes into five smaller subsystems, each of which relate to well-known symbolic systems. Furthermore, by recognizing the algorithmic transformations between these subsystems, we can apply each representing a single component of the Group Algebraic System G, or model how algorithms are used in mathematics, by mapping its meaning onto the corresponding transformation steps between the subsystems. (...) Thus, we have transformed a Group Algebraic System G into five simpler subsystems (namely, symbolic analogic, lateral algebraic, calculus of infinity tensors, perturbations in waves, and algorithmic formation of symbols), each of which may also be represented algorithmically. This mapping process serves as a the basis for studying the algebraic machinery used in mathematics and logic to manipulate symbols, such as the classical logical systems or algebraic systems, using algorithms. (shrink)

I show that intuitive and logical considerations do not justify introducing Leibniz’s Law of the Indiscernibility of Identicals in more than a limited form, as applying to atomic formulas. Once this is accepted, it follows that Leibniz’s Law generalises to all formulas of the first-order Predicate Calculus but not to modal formulas. Among other things, identity turns out to be logically contingent.

Many philosophers have been attracted to the idea of using the logical form of a true sentence as a guide to the metaphysical grounds of the fact stated by that sentence. This paper looks at a particular instance of that idea: the widely accepted principle that disjunctions are grounded in their true disjuncts. I will argue that an unrestricted version of this principle has several problematic consequences and that it’s not obvious how the principle might be restricted in order to (...) avoid them. My suggestion is that, instead of trying to restrict the principle, we should distinguish between metaphysical and conceptual grounds and take the principle to apply exclusively to the latter. This suggestion, if correct, carries over to other prominent attempts at using logical form as a guide to ground. (shrink)

In this paper, I attempt to clarify the heart of Dewey’s philosophy: his method (denotative method (DM) / pattern of inquiry (PI)). Despite the traditional understanding of Dewey as anti-foundationalist, I want to show that Dewey did have metaphysical foundations for his method: the principle of continuity or theory of emergentism. I also argue that Dewey’s metaphysical position is better named as ‘cultural emergentism’, rather than his own term ‘cultural naturalism’. What Dewey called ‘common sense’ in his Logic, Husserl (...) termed as the ‘life-world’ in his Crisis. I compare two perspectives of dealing with the phenomenon and conclude that for Dewey, the difference between natural sciences and the common sense inquiry is that of subject-matter but not of method. Thus, the goal is to find the unified method to be applied in both domains. Whereas Husserl was more pessimistic: for him, the difference was not only in subject-matter, but in the very methods. Following that discussion, I also attempt to reformulate the hard problem of consciousness in Deweyan terms. In the end, I compare Dewey’s DM / PI with Popper’s understandings of scientific method and conclude that there is no significant difference between the two and that Dewey’s method could also be looked at as hypothetic-deductive method, with the only difference in emphases. (shrink)

Logical reasoning is not only a component of religious faith (cf., for instance, the "Golden rule"), but, in addition, the religious faith itself can be conceived as a logical pragmatic function applied to sentences and their meanings. Pragmatic role of religious faith is shown on the examples of the analogy of seed and spoken word (e.g., Mt 13:3-23) and on the degrees of faith described in the episode about Nicodemus (John 3). Pragmatics adds (different grades of) perseverance to the (...) correctness of reasoning and to the truth-preservation of the consequence relation. Correspondingly, religious faith is described by formally defined model and satisfaction relation, which are applied to and examined on the text of John 3. (shrink)

I address a type of circularity threat that arises for the view that we employ general basic logical principles in deductive reasoning. This type of threat has been used to argue that whatever knowing such principles is, it cannot be a fully cognitive or propositional state, otherwise deductive reasoning would not be possible. I look at two versions of the circularity threat and answer them in a way that both challenges the view that we need to apply general logical principles (...) in deductive reasoning and defuses the threat to a cognitivist account of knowing basic logical principles. (shrink)

Multialgebras have been much studied in mathematics and in computer science. In 2016 Carnielli and Coniglio introduced a class of multialgebras called swap structures, as a semantic framework for dealing with several Logics of Formal Inconsistency that cannot be semantically characterized by a single finite matrix. In particular, these LFIs are not algebraizable by the standard tools of abstract algebraic logic. In this paper, the first steps towards a theory of non-deterministic algebraization of logics by swap structures are given. (...) Specifically, a formal study of swap structures for LFIs is developed, by adapting concepts of universal algebra to multialgebras in a suitable way. A decomposition theorem similar to Birkhoff’s representation theorem is obtained for each class of swap structures. Moreover, when applied to the 3-valued algebraizable logics J3 and Ciore, their classes of algebraic models are retrieved, and the swap structures semantics become twist structures semantics. This fact, together with the existence of a functor from the category of Boolean algebras to the category of swap structures for each LFI, suggests that swap structures can be seen as non-deterministic twist structures. This opens new avenues for dealing with non-algebraizable logics by the more general methodology of multialgebraic semantics. (shrink)

Although the cognition is significant in strategic reasoning, its role has been weakly analyzed, because only the average intelligence is usually considered. For example, prisoner's dilemma in game theory, would have different outcomes for persons with different intelligence. I show how various levels of intelligence influence the quality of reasoning, decision, or the probability of psychosis. I explain my original methodology developed for my MA thesis in clinical psychology in 1998, and grant research in 1999, demonstrating the bias of the (...) classic IQ method, and how the intelligence limits thinking. Based on that I defined Personality Model, providing insight into understanding of psychosis (schizophrenia, bi-polar), which has not been explained yet by psychology or psychiatry. In addition, it enables to analyze and assess non-linear problems, utilizable in computer programming, visualization (animation) or other fields including Baduk game. I've already applied some principles in complex information system www.each.co.uk, and video-animations exhibited in London, Germany, Tokyo. I need to mention my experience in chess composition between 1994 and 2000, winning a few international prizes and inventing a special class of fairy rules redefining the mate. The chess composition principles or patterns show the way to organize logical series to higher advanced mechanisms (like calculus), applicable to other fields. One of such principles is a logical aesthetic innovation: new strategy, defined by Italian composers. Finally I show how the simple redefinition of the classic utility concept links economics and psychology to explain irrational / destructive behavior. All presented results (from the research) can be repeated. (shrink)

The logic of qualia is different than classical logic. We take the first steps in defining it and applying it to the Hard Problem.

In his essay ‘“Wang’s Paradox”’, Crispin Wright proposes a solution to the Sorites Paradox (in particular, the form of it he calls the ‘Paradox of Sharp Boundaries’) that involves adopting intuitionistic logic when reasoning with vague predicates. He does not give a semantic theory which accounts for the validity of intuitionistic logic (and the invalidity of stronger logics) in that area. The present essay tentatively makes good the deficiency. By applying a theorem of Tarski, it shows that intuitionistic (...) logic is the strongest logic that may be applied, given certain semantic assumptions about vague predicates. The essay ends with an inconclusive discussion of whether those semantic assumptions should be accepted. (shrink)

Causal models show promise as a foundation for the semantics of counterfactual sentences. However, current approaches face limitations compared to the alternative similarity theory: they only apply to a limited subset of counterfactuals and the connection to counterfactual logic is not straightforward. This paper addresses these difficulties using exogenous interventions, where causal interventions change the values of exogenous variables rather than structural equations. This model accommodates judgments about backtracking counterfactuals, extends to logically complex counterfactuals, and validates familiar principles of (...) counterfactual logic. This combines the interventionist intuitions of the causal approach with the logical advantages of the similarity approach. (shrink)

We present two defeasible logics of norm-propositions (statements about norms) that (i) consistently allow for the possibility of normative gaps and normative conflicts, and (ii) map each premise set to a sufficiently rich consequence set. In order to meet (i), we define the logic LNP, a conflict- and gap-tolerant logic of norm-propositions capable of formalizing both normative conflicts and normative gaps within the object language. Next, we strengthen LNP within the adaptive logic framework for non-monotonic reasoning in (...) order to meet (ii). This results in the adaptive logics LNPr and LNPm, which interpret a given set of premises in such a way that normative conflicts and normative gaps are avoided ‘whenever possible’. LNPr and LNPm are equipped with a preferential semantics and a dynamic proof theory. (shrink)