Logic and Philosophy of Logic, Miscellaneous

This work examines the structure of paradoxes, phenomena characterized by the apparent derivation of contradictions or counter-intuitive conclusions from acceptable premises using acceptable reasoning. Motivated by the recurrence of certain features across diverse paradoxes, a specific structural hypothesis regarding their generation is proposed and analyzed. The hypothesis posits that paradoxes frequently arise from the interaction of three components: (1) concepts that establish or interact with boundaries (such as those pertaining to scope, classification, truth, state, or definition); (2) the reflexive application (...) of these concepts or related rules (involving self-reference, self-inclusion, recursion, or causal/definitional circularity); and (3) the involvement of specific operational triggers – negation, limit (including infinity, extremes, or boundary values), or transformation (change of state, identity, position, or logical status) – within the reflexive structure. The study first provides a detailed conceptual analysis of these components – boundaries, reflexivity, and the triggers – outlining their definitions and potential roles based on discussions in logic, philosophy, and related fields. Subsequently, the hypothesis is examined against a wide range of paradoxes drawn from logic, set theory, semantics, physics, metaphysics, epistemology, and decision theory. This examination involves analyzing the structure of each paradox to identify the presence and interaction of the hypothesized components. The analysis indicated a recurrent correspondence between the structures of the examined paradoxes and the proposed pattern involving boundaries, reflexivity, and triggers. Further analysis explored the representation of this structural pattern using elements of logical and mathematical formalism, sketching how the interaction of boundary predicates/sets, reflexive constructions, and operators corresponding to negation, limits, or transformations can lead to derivations of inconsistency (e.g., P ↔ ¬P or P ∧ ¬P) within systems adhering to standard logical principles. The discussion also addressed the context and limitations surrounding the hypothesis, including the meaning of "often arise," the relationship between the identified structure and strategies for paradox resolution (which often involve modifying the system to restrict or manage components of the pattern), the distinction between the structure and its outcome in different systems, and the acknowledgment that the pattern may not be the sole source of all paradoxes and that its identification can involve interpretation. Finally, potential implications arising from the identification of this recurring structure were considered. These include the possibility of providing a unifying framework for paradox analysis across disciplines, offering insights relevant to the design of consistent conceptual and formal systems by highlighting potential points of instability, suggesting connections between the structure and characteristics of natural language or thought involving self-reference and boundary definition, and noting structural resonances with limitative results in mathematical logic, such as Gödel's incompleteness theorems. Potential directions for future work, including further categorization, analysis of non-paradoxical structures, cross-disciplinary applications, and exploration of cognitive aspects, were also outlined. (shrink)

The propositional connectives – and, or, not, if–then, etc. – are fundamental building blocks in formal as well as natural languages. In the Western tradition, they were first studied as such by the Stoics, and Propositional Logic is the fundament of practically all current systems of logic; every beginning logic course starts with it. Still, the proof theory and semantics of systems of propositional logic are far from trivial, and have been studied intensely by logicians in the last one and (...) a half centuries, not least in recent decades. It is actually a vast area of research, as witnessed by Lloyd Humberstone’s 1500 page tome The Connectives (2011), which overviews much of that research. Perhaps the most familiar recent work in this area concerns conditionals in formal and natural languages. The 4th Tsinghua Interdisciplinary Workshop on Logic, Language, and Meaning (TLLM 2024) also focused on the apparently simpler connectives expressing (various versions of) conjunction, disjunction, and negation. -/- Researchers working from a cross-linguistic perspective also study how the connectives are encoded in different languages, and ask whether classical logic is capable of capturing the variations and universals exhibited. Even in well-studied languages such as English, there are intricate phenomena that remain challenging for classical logic, including free choice disjunction, non-Boolean conjunction, and metalinguistic negation, to name just a few. There is also growing interest in the acquisition and processing of natural language connectives. In the context of the hotly discussed Large Language Models (LLMs), understanding connectives presents novel challenges that deserve in-depth exploration. (shrink)

One question that arises in connection with Ockhamism, and that perhaps has not yet received the attention it deserves, is how a coherent formal account of truth ascriptions can be provided by using a suitable truth predicate in the object language. We address this question and show its implications for some semantic issues that have been discussed in the literature on future contingents. Arguably, understanding how truth ascriptions work at the formal level helps to gain a deeper insight into Ockhamism (...) itself. (shrink)

Abstract – Resolving Ramanujan: A Structured Resonance Completion of His Intuitive Mathematics -/- This paper reframes the legacy of Srinivasa Ramanujan through the paradigm of structured resonance rather than probabilistic mathematics. Using the CODES framework (Chirality of Dynamic Emergent Systems), we demonstrate that Ramanujan’s so-called “intuition” was not mystical anomaly but high-coherence pattern recognition—perceiving mathematical structure through phase-locked emergence rather than formal derivation. We construct a new class of resonance-based models for partition functions, mock theta functions, prime gaps, and continued (...) fractions, showing that these outperform classical probabilistic models in coherence, compression, and structural fidelity. -/- The core contribution is the Resonance Coherence Lemma, which proves that for large n, CODES-derived functions yield higher structural coherence than any entropy-based approximation. We introduce coherence scores, chirality-based symmetry groups, and symbolic resonance engines to reinterpret Ramanujan’s lost notebook not as incomplete, but as a partially decoded harmonic field. The final sections outline the Ramanujan Resonance Engine (RRE), a symbolic AI system capable of completing intuitive mathematics using phase-space logic rather than statistical inference. -/- This work resolves longstanding mysteries in number theory, modular forms, and mathematical epistemology by replacing probability with resonance as the foundation of mathematical emergence. Ramanujan wasn’t ahead of his time—he was in tune with a deeper frequency. We phase-lock with it now. (shrink)

This paper clarifies and defends the global approach to defining logical validity for meta- and higher-level inferences. This is contrary to an emerging consensus in favour of local validity. Prevalent recent arguments claim that global validity is either superfluous in virtue of collapsing into local, or else untenable because it overgenerates validities, compromises the formality of logic, or breaks symmetry with regular validity. Accordingly, the literature on higher inferential logic has come to focus almost exclusively on local validity. Many key (...) philosophical takeaways that have since been drawn from the field are effectively based on local monism: the view that local is the One True Validity Criterion. The present paper argues that this is a mistake, resulting largely from an imprecise understanding of global validity. I untangle some common confusions about how global generalizes beyond the metalevel, and submit that its proper formulation is schematic. These clarifications help dispel each of the aforementioned arguments against global, leaving local monism without any justification. Finally, I outline how much of the philosophical literature around higher-level inferences has been built on local monism, and sketch out the sweeping consequences of rejecting it. (shrink)

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)

This article provides a computational example of a mathematical explanation within science, concerning computational equivalence of programs. In addition, it outlines the logical structure of the reasoning involved in explanations in applied mathematics. It concludes with a challenge that the nominalist provide a nominalistic explanation for the computational equivalence of certain programs.

We present a study of unpublished fragments of Jan F. Drewnowski’s manuscript from the years 1922–1928, which contains his own axiomatics for mereology. The sources are transcribed and two versions of mereology are reconstructed from them. The first one is given by Drewnowski. The second comes from Leśniewski and was known to Drewnowski from Leśniewski’s lectures. Drewnowski’s version is expressed in the language of ontology enriched with the primitive concept of a (proper) part, and its key axiom expresses the so-called (...) weak super-supplementation principle, which was named by Drewnowski “the postulate of the existence of subtractions”. Leśniewski’s axiomatics with the primitive concept of an ingrediens contains the axiom expressing the strong super-supplementation principle. In both systems the collective class of objects from the range of a given non-empty concept is defined as the upper bound of that range. From a historical point of view it is interesting to notice that the presented version of Leśniewski’s axiomatics has not been published yet. The same applies to Drewnowski’s approach. We reconstruct the proof of the equivalence of these two systems. Finally, we discuss questions stemming from their equivalence in frame of elementary mereology formulated in a modern way. (shrink)

The goal of this paper is to explain the nature of philosophy as a distinct science with its own subject-matter. This is achieved through a comparative analysis of mathematical and philosophical knowledge that reveals a profound similarity between mathematics and philosophy as mutually complementary sciences exploring the field of abstract entities that can be comprehended only by purely a priori theoretical inquiry. By considering this complementarity, a general definition of philosophy can be obtained by dualizing the traditional Aristotelian definition of (...) mathematics as the “science of quantity”. Philosophy should thus be interpreted as an a priori science of the pure qualitative attributes of being. (shrink)

In “A neglected reply to Prior’s dilemma” Beall [2012] presents a Weak Kleene framework where Prior’s dilemma for Hume’s no-ought-fromis thesis fails. It fails in the framework because addition, the inference rule that one of its horns relies on, is invalid. In this paper, we show that a more general result is necessary for the viability of Beall’s proposal – a result, which implies that Hume’s thesis holds in the proposed framework. We prove this result and thus show that Beall’s (...) proposal is indeed viable. (shrink)

Some singular terms are referentially empty by necessity. Oliver and Smiley have recently introduced the term ‘zilch’ for illustrating this kind of emptiness. The emptiness of ‘zilch’ is supposed to arise from the fact that its extension has been defined by a logically unsatisfiable condition. Casati and Fujikawa disagree with this explanation and claim that ‘zilch’ refers to some null thing. In this paper, I argue that neither of these positions is correct, since, for different reasons, they both misinterpret the (...) phenomenon of referential emptiness. As an alternative, I propose a representationalist account of emptiness, which can explain the properties of ‘zilch’ and similar terms more effectively. (shrink)

Accounts of analogical inference are usually categorized into four broad groups: abductive, deductive, inductive and sui generis. The purpose of this paper is to defend a sui generis model of analogical inference. It focuses on the sui generis account, as developed by Juthe [2005, 2009, 2015, 2016] and Botting’s [2017] criticism of it. This paper uses the pragmadialectical theory of argumentation as the methodological framework for analyzing and reconstructing argumentation. The paper has two main points. First, that Juthe’s arguments against (...) a deductive interpretation of prima facie analogy argumentation remain unaffected by Botting’s criticism, which means that many of the reasons against deductive reformulation of analogy argumentation still stand. The additional argument, which Botting himself brings up, that a deductive interpretation cannot account for the cumulative effect of analogies, just provides further reason to reject deductivism. The second main point of this paper is that an inductive interpretation of analogical inference also fails. There are constitutional differences between inductive and analogical inference that cannot be bridged. The result is a firm defense of the sui generis view of analogical inference. (shrink)

The paper consists of two parts. The first critically analyses Meyer’s [2005] version of the triviality objection to presentism (according to which, presentism is either trivial or untenable), and tries to show that his argument is untenable because – contrary to what he claimed – he did not take into account the entire possible spectrum of interpretations of the presentist’s thesis. In the second, positive part of the paper, it is shown that a leading form of tensed theory of time (...) postulates the same ontology as presentism and that it avoids the triviality problem which means that it can be used to generate an alternative formulation of presentism which is no longer vulnerable to the triviality objection. (shrink)

Béziau has recently argued that the logic LP commits dialetheists to trivialism and Martin has pointed out very clearly the main problems with that alleged result. My sole purpose here is to make the spirit of Martin’s reply more concise, exhibiting as clearly as possible the logical defects in Béziau’s reasoning. Additionally, I want to make some remarks on LP qua logic and not only as an interpreted language.

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)

This paper argues for the significance of Kaplan's logic LD in two ways: first, by looking at how logic got along before we had LD, and second, by using it to bring out the similarity between David Hume's thesis that one cannot deduce claims about the future on the basis of premises only about the past, and the so‐called "essentiality" of the indexical.

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)

No artigo A Negação Fregeana do Número 2 (2008), Greimann argumenta que a negação fregeana do número 2, presente no final do sexto parágrafo de Grundgesetze (1893, 1903), não envolve nenhum absurdo ainda que, segundo ele, seja uma negação do número 2 como um termo singular. Meu ponto aqui é mostrar que, na passagem em questão, Frege não necessariamente nega o número 2 como um termo singular, mas apenas exemplifica a possibilidade de se negar o emprego do número 2 como (...) uma classificação de outro objeto numa sentença falsa. (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)