Logic and Philosophy of Logic, Miscellaneous

The present paper investigates whether strictly classical inferences contribute to the formalization of (genuine) paradoxes within natural deduction. Tennant's criterion for paradoxicality relies on the generation of an infinite reduction sequence, which distinguishes genuine paradoxes from mere inconsistencies. His methodological conjecture posits that genuine paradoxes are never strictly classical and can be derived without classical inferences such as the Law of Excluded Middle, Dilemma, Classical Reductio, and Double Negation Elimination. -/- It appears that there were two reasons for Tennant's proposal (...) of the methodological conjecture. The one is that strictly classical inferences hinder the generation of an infinite reduction sequence and the other is that strictly classical inferences have no role in the formalization of genuine paradoxes. -/- This paper raises questions about these two reasons. Focusing on the liar paradox, it will be argued that strictly classical inferences do not interfere with the generation of an infinite reduction sequence and that the liar sentence may implicitly entail strictly classical inferences. Should this analysis hold, it would call into question not only Tennant’s motivation for advancing the methodological conjecture, but also challenge his contention that genuine paradoxes—exemplified by the liar paradox—are never constructed with reliance on strictly classical inferences. (shrink)

I argue for a transformative revival or actualization of the very core of an integrative, methodologically secured form of intellect‑mystical asceticism. This approach draws on traditional sources that are re‑examined from a systematic—synthetic and transcultural—philosophical perspective and in light of the multi‑civilizational global environment of the 21st century. The main traditional points of reference in this paper are provided by Nicolaus de Cusa and Ibn Sīnā, and I refer toa few others, such as Attar of Nishapur, in passing. I begin (...) by developing a basic concept of intellect‑mystical asceticism. It is distinguished from mystification, science, scientism, and modes of everyday communication and cognition. Then, I make the case for an updated, transcultural approach to intellect‑mysticism that can foster the internal (social) and external (environmental) reintegration of the human noosphere and technosphere in future planetary development. In this context, a modern intellect‑mystical philosophical notion of “knowing non‑knowing” (wissendes Nichtwissen, docta ignorantia) is developed. It is inspired by Nicolaus de Cusa and contextualized from a systematic transcultural angle at the same time. Finally, I discuss the problem of the practical, or rather ascetic, realization of the related possibilities of intellect‑mystical self‑enfolding. Here, the preceding steps of the reflection are mapped onto an outline regarding distinct developmental stages of such a transformative intellect‑mystical practice in Ibn Sīnā’s Remarks and Admonitions (al‑Ishārāt wat‑Tanbīhāt). (shrink)

Knowledge representation (KR) is actually more than representation: It involves also inference, namely inference of “new” knowledge, i.e. new facts. Logic programming is a suitable KR medium, but more often than not discussions on this programming paradigm focus on aspects other than KR. In this paper, I elaborate on the general theory of logic programming and give the essentials of two of its main implementations, to wit, Prolog and Datalog, from the viewpoint of deductive computing over knowledge bases, which includes (...) deductive programming. (shrink)

Traditional debates between semantic temporalists and eternalists appeal to the efficacy of temporal operators and the intuitive (in)validity of instances of temporal reasoning. In this paper, I argue that such debates are inconclusive at best and that under-explored arguments concerning perceptual experience, memory, and knowledge offer more productive means of advancing debates between temporalists and eternalists and rendering salient several significant potential costs and benefits of these views.

Ālāpa Paddhati, composed by Ācārya Devasena (c. tenth century, Vikrama Samvat) is a Jaina text primarily on the topics of the standpoints (naya) and the secondary-standpoints (upanaya). It also delves into the substances (dravya), their qualities or attributes (guṇa), modes (paryāya), and nature (svabhāva). It is true that without appreciating the import and applicability of the individual standpoints (naya), one may get lost in the complex maze of the standpoints and cause great harm to one’s understanding, and even to one’s (...) mental balance. Ālāpa Paddhati is essential reading for every serious reader of the Dravyānuyoga that spreads the light of right (scriptural) knowledge by illuminating the Reality of substances – soul (jīva) and non-soul (ajīva), merit (puṇya) and demerit (pāpa), bondage (bandha) and liberation (mokṣa), including influx (āsrava), stoppage (saṃvara), and gradual dissociation (nirjarā). The book has a useful Appendix that explains the doctrines of non-absolutism (anekāntavāda) and conditional predication (syādvāda). This slim volume needs to be read over and over again to internalize the underlying concepts fully. (shrink)

Complex logic is a novel logical framework, which formalizes the semantics of the categories of matter, space, and time in a system of logic that operates with complex logical objects. A complex logical object represents a superposition of a logical statement and its logical negation positioning any statement co-relatively to its logical negation. In the system of logical notations, where S is a logical statement and Not S is its logical negation, complex logic includes co-relative logical positions of S and (...) Not S with the probabilities of their truth within the scale 0... 1, excluding the boundary values of 0 and 1, into a single logical superposition. Thus, the logical superposition, summing up the probabilistic positions of S and Not S, has an invariant truth value equal to 1. In this context, complex logic proves to be invariant to reality, offering a unified logical interpretation of reality, spanning from quantum to cosmological scales. By defining the nature of matter and reality as a complex two-format processual essence with corresponding logical consequences, complex logic enhances our concept and understanding of reality. (shrink)

‘Feminist logic’ may sound like an impossible, incoherent, or irrelevant project, but it is none of these. We begin by delineating three categories into which projects in feminist logic might fall: philosophical logic, philosophy of logic, and pedagogy. We then defuse two distinct objections to the very idea of feminist logic: the irrelevance argument and the independence argument. Having done so, we turn to a particular kind of project in feminist philosophy of logic: Valerie Plumwood's feminist argument for a relevance (...) logic. Plumwood's work serves as our primary case study as we turn to the project of considering three different ways we might understand her argument and revisionist arguments like it: as a priori theorizing, as ameliorative conceptual engineering, or as instances of anti-exceptionalist approaches to logic. After arguing that the anti-exceptionalist approach seems to provide the most promising means of understanding the kind of project undertaken in a feminist challenge to classical logic, we briefly address the consequences that this might have for logic instruction. Here, we argue for the perhaps unexpected conclusion that feminist programs ought to offer more, not less, instruction in logic for those who take interest. (shrink)

The goal of this short note is to expand the concepts of ‘pluralism’, ‘neutrosophy’, ‘refined neutrosophy’, ‘refined neutrosophic set’, ‘multineutrosophic set’, and ‘plithogeny’ (Smarandache 2002, 2013, 2017, 2019, 2021, 2023a, 2023b, 2023c), into a larger category that I will refer to as MultiAlism (or MultiPolar). As a straightforward generalization, I propose the conceptualization of a MultiPolar System (different from a PluriPolar System), which is formed not only by multiple elements that might be random, or contradictory, or adjuvant, but also by (...) accepting features from more than one basic system (UniPolar, BiPolar, TriPolar, or PluriPolar systems). PluriAlism is a closed dynamic system without neutralities nor indeterminacies, while MultiAlism is an open dynamic system with neutralities and indeterminacies. PluriAlism is a uni-system (formed by elements from a single system), while MultiAlism is a MultiSystem (formed by elements from many systems). (shrink)

Historical Essays. HISTORY OF A FALLACY The logical fallacy that I am going to discuss here is one that it is quite easy to see by common sense in simple ...

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)

It is universally acknowledged that, for St. Thomas, there is a distinction between human persons or supposits and their natures or essences. But it is usually thought that there is no parallel distinction between the angelic person or supposit and its nature. Yet, as this paper argues, Aquinas consistently puts forward just such a distinction. This paper surveys Aquinas’s arguments for the unique identity of God with his essence and the corresponding distinctions between created persons and their essences, showing in (...) the process how the distinction found in angels differs from that found in material substances. It is important to recognize the distinction between supposit and nature in angels not only for its own sake as it touches on his understanding of created persons—human or angelic—but also insofar as it sheds light on Aquinas’s understanding of divine simplicity and of other act-potency compositions in creatures. (shrink)

Special issue in honour of Landon Rabern. This special issue of Discrete Mathematics is dedicated to his memory, as a tribute to his many research achievements. It contains 10 new articles written by his collaborators, friends, and colleagues that showcase his interests.

Several authors maintain that we are capable of having inconsistent intentional states, either in cases of illusion, in certain cases of imagination, or because the observable world is (partly) inconsistent and we perceive it as such. These views are all premised on the assumption that inconsistent intentional states—even if acknowledged—are peculiar and have nothing essential to do with our perceptual capacities. In the present article, I would like to present, and argue for, a much stronger thesis: that inconsistent intentional states (...) underlie the possibility of having intentional content in mind. I argue for this thesis based on a Husserlian phenomenological analysis of our grasp of objects, which I formulate in terms of incompatibility semantics. (shrink)

It is widely assumed that the identity relation is, among other things, transitive. Some have proposed that the identity relation might hold between objects contingently or occasionally. If, on those proposals, identity is shown to not be transitive, then there is reason to reject such proposals. One such argument attempts to show that the identity relation on such proposals violates transitivity in cases of ‘simultaneous’ fissions and fusion. I argue that, even in those cases, contingent identity and occasional identity are (...) transitive. (shrink)

The monograph contains three works on research on the concept of a rejected sentence. This research, conducted under the supervision of Prof. Jerzy Słupecki by U. Wybraniec-Skardowska (1) "Theory of rejected sentences" and G. Bryll (2) "Some supplements of theory of rejected sentences" and (3) "Logical relations between sentences of empirical sciences" led to the construction of a theory rejected sentences and made it possible to formalize certain issues in the methodology of empirical sciences. The concept of a rejected sentence (...) was introduced by J. Łukasiewicz in the course of his inquiry into Aristotelian Syllogistic. It was essentialy generelized by J. Słupecki who, with reference to Tarski's axiomatic theory of deductive systems (1930), gave the formal definition of the rejection function Cn' according to which a sentence y is rejected on the base of a set of sentences X iff at least one sentence which belongs to X can be deduced from the sentence y. Słupecki proved (1959) that the function Cn', which assignes to the set X the set of sentences rejected on the basis of X, is additive and satisfies the axioms of Tarski's general theory of deductive systems. Work (1) uses not only Tarski's general theory of systems but also his the so-called enlarged theory of systems (1930). It constructs a theory T of rejected sentences (propositions), which is created from Tarski's richer theory of systems by adding one axiom and a number of definitions; among them the definition of the rejection function Cn' is fundamental one. Cn' is a function which assignes to every set X of false sentences the set Cn'X whose members are exclusively false senteces. A number of theorems which describes properties of defined concepts have been proved in T. Some of them are necessary for conducting considertions of the concluding section of (1) devoted to the empirical sentences. Section 2 of (1) deals with the theory of the unit consequenses. It is characterized by an axiom from which follow all axioms of Tarski's general theory of deductive systems, additivity and the property that the unit consequence of the set X of sentences is a unit consequence only one sentence of X. An example of unit consequence is the rejection consequence Cn'. In section 3 of (1) the dual theory T' equivalent to T is presented. In T' the primitive concept is the rejection function Cn'. The fundamental axiom of T' states that Cn' is a unit consequence. The concluding section brings a few examples of applications of the definitions and theorems of T in the domain of the scientific methodology. Work (2) corresonds to J. Słupecki paper (1959) and the theory T of rejected propositions given in work (1). In this paper some suplements of the theory T are given. Work (3) is an extension of the theory of rejected propositios given in the paper (1). It is an attempt of formalization some problems of methodology of the empirical sciences which concern to such concepts as: empirical sentence, similar sentence, generalization, inductive conclusion etc. (shrink)

This chapter introduces inferential role semantics (IRS) and some of the challenges it faces. It also introduces inferentialism and places it into the wider context of contemporary philosophy of language. The chapter focuses on what is standardly considered both the most important test case for and the most natural application of IRS: logical inferentialism, the view that the meanings of the logical expressions are fully determined by the basic rules for their correct use, and that to understand a logical expression (...) is to use it in accordance with the appropriate rules. It discusses some of the benefits of logical inferentialism, chiefly with regard to the epistemology of logic, and considers a number of objections. The chapter critically examines Robert Brandom's inferentialism about linguistic and conceptual content in general. Finally, it considers a number of general objections to IRS and possible responses on the inferentialist's behalf. (shrink)

In Baghdad in the mid twelfth century Abū al-Barakāt proposes a radical new procedure for finding the conclusions of premise-pairs in syllogistic logic, and for identifying those premise-pairs that have no conclusions. The procedure makes no use of features of the standard Aristotelian apparatus, such as conversions or syllogistic figures. In place of these al-Barakāt writes out pages of diagrams consisting of labelled horizontal lines. He gives no instructions and no proof that the procedure will yield correct results. So the (...) reader has to work out what his procedure is and whether it is correct. The procedure turns out to be insightful and entirely correct, but this paper may be the first study to give a full description of the procedure and a rigorous proof of its correctness. (shrink)

I discuss a neglected solution to the skeptical problem introduced by Lewis Carroll’s “What the Tortoise Said to Achilles” (1895) in terms of a self-citational inferential license. I then consider some responses to this solution. The most significant response on behalf of the skeptic utilizes the familiar distinction between two ways of accepting a rule: as action-guiding and as a mere truth. I argue that this is ultimately unsatisfactory and conclude by opting for an alternative conception of rules as representations (...) of behavior deployed for various purposes, some theoretical and others practical. This alternative conception does not allow the skeptical problem to get off the ground. (shrink)

Livro em homenagem ao filósofo brasileiro Luiz Henrique Lopes, um dos maiores expoentes da filosofia analítica. Neste livro grandes nomes da filosofia brasileira discorrem sobre a filosofia analítica e vários assuntos da filosofia contemporânea.

In the first of the Insolubles in Chapter 8 of his Sophismata, Buridan contends that the inference Omnis propositio est affirmativa; ergo, nulla propositio est negativa (PS) is valid, even though it appeals to the self-reference in the conclusion to show that what we (following Read 2001) call the classical conception of validity (CCV) fails. This requires that we accept that there are good inferences in which a false conclusion follows from true premises. Partially following Hughes’ proposal (1982), we argue (...) that the First Sophism (PS) involves three different notions of validity. Two of them correspond to the ones described by Hughes (1982, 80–86), who calls them Theory A and Theory B. The third one—that will we call Theory C—is not mentioned by Hughes; instead, it is suggested by Buridan himself in the first three arguments in favor of the validity of PS. We show that: a) from what Buridan says in his Theory C it follows that PS is a formal and material consequence, and hence, a valid one. Then we show that: b) the rejection of CCV and the acceptance of Nulla propositio est negativa (NPN) as a (formal) consequence of Omnis propositio est affirmativa (OPA) leads to a paradox that bears similarities with the one put forward by Pseudo Scotus—which has been studied by Read (2001) and is related to Curry’s paradox. However, there are enough differences to merit considering this paradox separately, especially in relation to the so-called validity paradoxes. Interestingly, our work suggests that Buridan was aware of these problems, which explains why he introduced a new criterion for validity, one that is not based on truth-preservation but on what Spade (1988) calls firmness, and Klima (2016) correspondence. (shrink)

The Logic of Analogy is a study of the valid logical forms of qualitative and quantitative analogical argument, and the rules pertaining to them. It investigates equally valid conflicting arguments, statistics-based arguments and their utility in science, arguments from precedent used in law-making or law-application, and examines subsumption in analogical terms. Included for purposes of illustration is a large section on Talmudic use of analogical reasoning.

After some generalities about connections between functions and relations in Sections 1 and 2 recalls the possibility of taking the semantic values of ‐ary Boolean connectives as ‐ary relations among truth‐values rather than as ‐ary truth functions. Section 3, the bulk of the paper, looks at correlates of these truth‐value relations as applied to formulas, and explores in a preliminary way how their properties are related to the properties of “logical relations” among formulas such as equivalence, implication (entailment) and contrariety (...) (logical incompatibility), concentrating for illustrative purposes on binary logical relations such as those just listed. To avoid an excess of footnotes, some points have been deferred to an Appendix as “Longer Notes”. (shrink)

Western contemporary logic has been used to advance the field of Islamic philosophical theology, which historically utilised Aristotelian-Avicennian logic, on grounds of there being an inherent normativity in logic. This is in spite of the surrounding controversy on the status of logic in the Islamic theological tradition. The normative authority of logic means that it influences the content of what we ought to believe and how we ought to revise those beliefs. This paper seeks to demonstrate that, notwithstanding the incompatible (...) differences between the two systems, the underlying feature of both Western contemporary logic and Aristotelian- Avicennian logic is logical normativity. It then argues that an inherent normativity of logic in the Islamic theological/philosophical tradition is unmotivated. Instead, it proposes to reinstate logic as anti-exceptional within the Islamic theological/philosophical tradition as a viable alternative. (shrink)

We solve two problems from a work of Haskel and Pillay concerning maximal stable quotients of groups ∧-definable in NIP theories. The first result says that if G is a ∧-definable group in a distal theory, then Gst=G00 (where Gst is the smallest ∧-definable subgroup with G∕Gst stable, and G00 is the smallest ∧-definable subgroup of bounded index). In order to get it, we prove that distality is preserved under passing from T to the hyperimaginary expansion Theq. The second result (...) is an example of a group G definable in a nondistal NIP theory for which G=G00, but Gst is not an intersection of definable groups. Our example is a saturated extension of (R,+,[0,1]). Moreover, we make some observations on the question whether there is such an example which is a group of finite exponent. We also take the opportunity and give several characterizations of stability of hyperdefinable sets involving continuous logic. (shrink)

This paper summarises the contributions to our Topical Collection on indeterminacy and underdetermination. The collection includes papers in ethics, metaethics, logic, metaphysics, epistemology, philosophy of science, philosophy of language and philosophy of computation.

After introducing the debate between substance philosophy and process philosophy, and clarifying the relevance of the category of ‘substance’ in Peirce’s thought, the present paper reconstructs the role of ‘substance’ and ‘being’ from Peirce’s early works to his theory of the proposition, provided after his studies on the logic of relatives. If those two categories apparently disappear in Peirce’s writings from the mid-1890s onwards, the account of ‘subject’ and ‘copula’ in Peirce’s analysis of the proposition allows one to grasp the (...) reasons why Peirce omits ‘substance’ and ‘being’ in favor of his three categories (Firstness, Secondness, Thirdness), and to understand why his philosophy cannot be considered as a substance philosophy. (shrink)

El Anti-Excepcionalismo Lógico afirma que la Lógica es como cualquier otra ciencia. Si esta afirmación es cierta, entonces ella no sólo es revisable, sino que además todo lo que se puede decir sobre las ciencias aplica, mutatis mutandis, para la misma. El propósito de este artículo es explorar esta consecuencia del Anti-Excepcionalismo Lógico, acercando a la Filosofía de la Lógica el marco teórico de las Máquinas Nomológicas de Nancy Cartwright. De acuerdo con esta visión, lo que hay de verdadero en (...) las teorías científicas no está en el mundo sino en sus modelos: sistemas altamente controlados, estables, artificiales y específicos, dentro de los cuales se manifiestan regularidades sistematizables y enseñables. Mi afirmación es que lo mismo puede decirse de la Lógica: ella no captura las Leyes del Pensamiento o de la Razón, porque el pensamiento y la razón no están gobernados necesariamente por leyes. Sobre lo que sí puede decirse que versan las Leyes de la Lógica son metafísicas posibles: historias acerca de qué son las proposiciones, el pensamiento, la verdad y la validez. A fin de ilustrar y defender esta tesis, presento un ejemplo de Máquina Nomológica de la Lógica Clásica, basada en una lectura original del Tractatus de Wittgenstein, e investigo algunas de sus consecuencias. Mi conclusión es que esta imagen de la Lógica impacta en el problema de la revisión de la Lógica Clásica, en la medida en que muestra que dicha revisión no necesariamente responde a un proceso racional de adecuación de la teoría a los hechos porque los “hechos” relevantes no son anteriores a la teoría misma, sino que están moldeados y conducidos por ella. Palabras clave: Anti-Excepcionalismo Lógico, Abductivismo, Máquinas Nomológicas, Revisión Lógica, Filosofía de la Lógica. (shrink)

A teaching document I've used in my courses on truth and on incompleteness. Aimed at students who have a good grasp of basic logic, and decent math skills, it attempts to give them the background they need to understand a proper statement of the classic results due to Gödel and Tarski, and sketches their proofs. Topics covered include the notions of language and theory, the basics of formal syntax and arithmetization, formal arithmetic (Q and PA), representability, diagonalization, and the incompleteness (...) and undefinability theorems. (shrink)

In this paper, we present the history of and the research directions in relating logic. For this purpose we will describe Epstein's Programme, which postulates accounting for the content of sentences in logical research. We will focus on analysing the content relationship and Epstein's logics that are based on it, which are special cases of relating logic. Moreover, the set-assignment semantics will be discussed. Next, the Torunian Programme of Relating Semantics will be presented; this programme explores the various non-logical relationships (...) in logical research, including those which are content-related. We will present a general description of relating logic and semantics as well as the most prominent issues regarding the Torunian Programme, including some of its special cases and the results achieved to date. -/- . (shrink)

Answering a question of Cifú Lopes, we give a syntactic characterization of those continuous sentences that are preserved under reduced products of metric structures. In fact, we settle this question in the wider context of general structures as introduced by the second author.

Richard Heck has recently drawn attention on a new version of the Liar Paradox, one which relies on logical resources that are so weak as to suggest that it may not admit of any ‘‘truly satisfying, consistent solution’’. I argue that this conclusion is too strong. Heck’s Liar reduces to absurdity principles that are already rejected by consistent paracomplete theories of truth, such as Kripke’s and Field’s. Moreover, the new Liar gives us no reasons to think that these principles cannot (...) be consistently retained once the structural rule of contraction is restricted. I suggest that revisionary logicians have independent reasons for restricting such a rule. (shrink)

Originally published in 1934. This fourth edition originally published 1954., revised by C. W. K. Mundle. "It must be the desire of every reasonable person to know how to justify a contention which is of sufficient importance to be seriously questioned. The explicit formulation of the principles of sound reasoning is the concern of Logic". This book discusses the habit of sound reasoning which is acquired by consciously attending to the logical principles of sound reasoning, in order to apply them (...) to test the soundness of arguments. It isn't an introduction to logic but it encourages the practice of logic, of deciding whether reasons in argument are sound or unsound. Stress is laid upon the importance of considering language, which is a key instrument of our thinking and is imperfect. (shrink)

Originally published in 1962. A clear and simple account of the growth and structure of Mathematical Logic, no earlier knowledge of logic being required. After outlining the four lines of thought that have been its roots - the logic of Aristotle, the idea of all the parts of mathematics as systems to be designed on the same sort of plan as that used by Euclid and his Elements, and the discoveries in algebra and geometry in 1800-1860 - the book goes (...) on to give some of the main ideas and theories of the chief writers on Mathematical Logic: De Morgan, Boole, Jevons, Pierce, Frege, Peano, Whitehead, Russell, Post, Hilbert and Goebel. Written to assist readers who require a general picture of current logic, it will also be a guide for those who will later be going more deeply into the expert details of this field. (shrink)

Published in 1946, this volume does not purpose to be a treatise on logic. The author's contributions to the substance of logical doctrine have been made in his other works. What he has attempted in the studies that form this volume is an exploration of the periphery of logic, the relation of logic to the rest of the universe, the philosophical presuppositions which give logic its meaning and the applications which give it importance. It is his belief that formal logic (...) is the heart of philosophy, since the subject matter of logic is the formal aspect of all being. From this standpoint he explores the relation of logic to such fields of study as natural science, ethics, history and general philosophy. (shrink)

Skolem studied the germs at infinity of the smallest class of real valued functions on the positive real line containing the constant $1$, the identity function ${\mathbf {x}}$, and such that whenever f and g are in the set, $f+g,fg$ and $f^g$ are in the set. This set of germs is well ordered and Skolem conjectured that its order type is epsilon-zero. Van den Dries and Levitz computed the order type of the fragment below $2^{2^{\mathbf {x}}}$. Here we prove that (...) the set of asymptotic classes within any Archimedean class of Skolem functions has order type $\omega $. As a consequence we obtain, for each positive integer n, an upper bound for the fragment below $2^{n^{\mathbf {x}}}$. We deduce an epsilon-zero upper bound for the fragment below $2^{{\mathbf {x}}^{\mathbf {x}}}$, improving the previous epsilon-omega bound by Levitz. A novel feature of our approach is the use of Conway’s surreal number for asymptotic calculations. (shrink)

Our main result is that there exist structures which cannot be computably recovered from their tree of tuples. This implies that there are structures with no computable copies which nevertheless cannot code any information in a natural/functorial way.

A mass problem is a set of functions$\omega \to \omega $. For mass problems${\mathcal {C}}, {\mathcal {D}}$, one says that${\mathcal {C}}$is Muchnik reducible to${\mathcal {D}}$if each function in${\mathcal {C}}$is computed by a function in${\mathcal {D}}$. In this paper we study some highness properties of Turing oracles, which we view as mass problems. We compare them with respect to Muchnik reducibility and its uniform strengthening, Medvedev reducibility.For$p \in [0,1]$let${\mathcal {D}}(p)$be the mass problem of infinite bit sequencesy(i.e.,$\{0,1\}$-valued functions) such that for each (...) computable bit sequencex, the bit sequence$ x {\,\leftrightarrow\,} y$has asymptotic lower density at mostp(where$x {\,\leftrightarrow\,} y$has a$1$in positionniff$x(n) = y(n)$). We show that all members of this family of mass problems parameterized by a realpwith$0 p$for each computable setx. We prove that the Medvedev (and hence Muchnik) complexity of the mass problems${\mathcal {B}}(p)$is the same for all$p \in (0, 1/2)$, by showing that they are Medvedev equivalent to the mass problem of functions bounded by${2^{2}}^{n}$that are almost everywhere different from each computable function.Next, together with Joseph Miller, we obtain a proper hierarchy of the mass problems of type$\text {IOE}$: we show that for any order functiongthere exists a faster growing order function$h $such that$\text {IOE}(h)$is strictly above$\text {IOE}(g)$in the sense of Muchnik reducibility.We study cardinal characteristics in the sense of set theory that are analogous to the highness properties above. For instance,${\mathfrak {d}} (p)$is the least size of a setGof bit sequences such that for each bit sequencexthere is a bit sequenceyinGso that$\underline \rho (x {\,\leftrightarrow\,} y)>p$. We prove within ZFC all the coincidences of cardinal characteristics that are the analogs of the results above. (shrink)

We analyze the precise modal commitments of several natural varieties of set-theoretic potentialism, using tools we develop for a general model-theoretic account of potentialism, building on those of Hamkins, Leibman and Löwe [14], including the use of buttons, switches, dials and ratchets. Among the potentialist conceptions we consider are: rank potentialism, Grothendieck–Zermelo potentialism, transitive-set potentialism, forcing potentialism, countable-transitive-model potentialism, countable-model potentialism, and others. In each case, we identify lower bounds for the modal validities, which are generally either S4.2 or S4.3, (...) and an upper bound of S5, proving in each case that these bounds are optimal. The validity of S5 in a world is a potentialist maximality principle, an interesting set-theoretic principle of its own. The results can be viewed as providing an analysis of the modal commitments of the various set-theoretic multiverse conceptions corresponding to each potentialist account. (shrink)

A theory of magnitudes involves criteria for their equivalence, comparison and addition. In this article we examine these aspects from an abstract viewpoint, by focusing on the so-called De Zolt’s postulate in the theory of equivalence of plane polygons (“If a polygon is divided into polygonal parts in any given way, then the union of all but one of these parts is not equivalent to the given polygon”). We formulate an abstract version of this postulate and derive it from some (...) selected principles for magnitudes. We also formulate and derive an abstract version of Euclid’s Common Notion 5 (“The whole is greater than the part”), and analyze its logical relation to the former proposition. These results prove to be relevant for the clarification of some key conceptual aspects of Hilbert’s proof of De Zolt’s postulate, in his classicalFoundations of Geometry(1899). Furthermore, our abstract treatment of this central proposition provides interesting insights for the development of a well-behaved theory ofcompatiblemagnitudes. (shrink)

The Unified Foundational Ontology (UFO) was developed over the last two decades by consistently putting together theories from areas such as formal ontology in philosophy, cognitive science, linguistics, and philosophical logics. It comprises a number of micro-theories addressing fundamental conceptual modeling notions, including entity types and relationship types. The aim of this paper is to summarize the current state of UFO, presenting a formalization of the ontology, along with the analysis of a number of cases to illustrate the application of (...) UFO and facilitate its comparison with other foundational ontologies in this special issue. (The cases originate from the First FOUST Workshop – the Foundational Stance, an international forum dedicated to Foundational Ontology research.). (shrink)