Logic in Philosophy

Logic has been a—disputed—ingredient in the emergence and development of the now very large field known as knowledge representation and reasoning. In this book (in progress), I select some central topics in this highly fruitful, albeit controversial, association (e.g., non-monotonic reasoning, implicit belief, logical omniscience, closed world assumption), identifying their sources and analyzing/explaining their elaboration in highly influential published work.

We can introduce singular terms for ordered pairs by means of an abstraction principle. Doing so proves useful for a number of projects in the philosophy of mathematics. However there is a question whether we can appeal to the abstraction principle in good faith, since a version of the Caesar Problem can be generated, posing the worry that abstraction fails to introduce expressions which refer determinately to the requisite sort of object. In this short paper I will pose the difficulty, (...) and then propose a solution. (shrink)

A brief but rigorous description of the logical structure of mathematical truth.

This paper introduces the Calculus of Qualia (CQ), a novel formal system that transcends the referential paradigm of traditional language by incorporating non-referential terms. These terms directly present phenomenal experiences, such as blackness, rather than referring to them, addressing longstanding limitations in expressing subjective consciousness. Building on insights from phenomenology and philosophy of mind, CQ defines qualations—expressions involving non-referential terms--qualia--with distinct logical behaviors like instance preservation and resistance to collapse. The framework offers new tools to formalize the hard problem of (...) consciousness and challenges classical thought experiments like the zombie argument, suggesting that qualia’s irreducibility demands a presentational rather than referential approach. This calculus bridges rigorous analysis and experiential meaning, opening pathways for rethinking language, consciousness, and their formal representation. (shrink)

When we count, we often count fractions, too. We contend that fractional counting involves partial entities, which are merely possible parts of entities of the counted kind. The size of these possible parts is measured with respect to the size of a possible member of that kind. Therefore, partialhood is mereomodal, and the logical form of fractional counting claims includes mereological predicates, modal operators, and a measurement functor. Different varieties of modality and forms of measurement are involved, depending on the (...) kinds of entities to be counted and the context. The mereomodal account validates the idea that fractional counting is a way of counting by identity, in continuity with logic-based accounts of non-fractional counting, albeit more complex than them. Such an account also explains why some kinds of entities are not involved in partialhood and cannot be fractionally counted, while others only have marginal involvement in these phenomena. In the last part, we discuss some difficult cases and show that an integrity condition for partial entities is required in the logical form of some fractional counting claims. (shrink)

This paper develops a formal theory of non-monotonic consequence which differs from most extant theories in that it assumes Contraposition as a basic principle of defeasible reasoning. We define a minimal logic that combines Contraposition with three uncontroversial inference rules, and we prove some key results that characterize this logic and its possible extensions.

Negation often reinforces problematic habits of othering, but rethinking negation can make good on feminist hopes for logic as a transformative space for inclusion. As Plumwood argues in her 1993 paper, not all uses of negation in the context of social identity are inherently problematic, but the widespread implicit use of classical negation has limited our options with respect to representing difference, ultimately reinforcing dualisms that essentialize social differences in problematic ways. In response to these limitations, I take inspiration from (...) Dembroff’s recent work on the metaphysics of genderqueer identity to build models of social identity using the Heyting-Brouwer logic developed by Rauszer in her 1974 paper. Ultimately, I argue that these models demonstrate both how classical negation reinforces problematic habits of othering and how alternative forms of negation can transform our treatment of social identity altogether. (shrink)

Definite descriptions are expressions of the form `the F'. The present entry begins by explaining Russell's theory of definite descriptions, according to which `The F is G' means `There is exactly one F and it is G'. It then discusses the alternative theories of Frege and Carnap, two influential criticisms by Strawson and Donnellan, and some formalisations of theories of definite descriptions building on Hintikka's and Lambert's work in free logic.

This paper critically examines two arguments against the generic multiverse, both of which are due to W. Hugh Woodin. Versions of the first argument have appeared a number of times in print, while the second argument is relatively novel. We shall investigate these arguments through the lens of two different attitudes one may take toward the methodology and metaphysics of set theory; and we shall observe that the impact of these arguments depends significantly on which of these attitudes is upheld. (...) Our examination of the second argument involves the development of a new model for Steel’s multiverse theory, which is delivered in the Appendix. (shrink)

A hundred years ago, Ramsey (1927) devised a new system of notation to more perspicuously represent the idea that ‘p’ and ‘¬¬p’ express the same fact. Recently, Monroy Pérez (2023) has argued that we can apply Ramsey’s insight to solve a recalcitrant problem for logical elitism — the thesis, most famously defended by Sider (2011), that the world’s fundamental metaphysical structure includes logical structure. Here, I discuss the general role of language in the framing of logical elitism and present four (...) arguments that show that the Ramseyan strategy for defending logical elitism fails. First, it does not eliminate all arbitrariness even for zeroth-order classical logic (§3). Second, it is a case of a more general strategy that exploits the fact that classical negation is an involution; then, choosing the Ramseyan proposal among the other cases is arbitrary (§4). Third, there are proposals to solve the riddle that are simpler than every case of the involutory strategy (§5). The discussion has consequences for logical elitism at large: in my fourth argument I contend, contrary to a fairly favoured opinion, that logical elitism should not be thought of as a thesis about languages, but as a thesis about joints in metaphysical structure (§6). (shrink)

Informal logic consists of logic and arguments for everyday life. Informal logic connects us to formal classical logic by extending, on some level, the rules of the formalized languages in classical logic to the languages we humans speak everyday across the world. Informal logic, perhaps more cogently than classical logic, can be useful, meaningful, and worthwhile in the “real” world for helping you distinguish good arguments from bad ones. In chapter 1, you’ll read about and discover these “real” world, everyday (...) arguments and their components, and contrast these with non-arguments. In chapter 2, you’ll endeavor to understand one type of everyday argument, deductive arguments, along with its concepts of validity and soundness. In chapter 3, you’ll come across other types of non-deductive everyday arguments, such as inductive arguments and inferences to the best explanation, and discover how they work in accordance with the concepts of strength, cogency, and the IBE clause, to hint at a few. In chapter 4, you’ll become more conscious and aware of how “real” life, everyday arguments can go wrong, by learning about the different types of informal fallacies and how to correct them according to certain rules or heuristics. Finally, in chapter 5, you’ll philosophize about how the different theories of meaning, theories of truth, and characteristics of facts can help you understand arguments in everyday language and provide a new perspective on the differences between good and bad arguments, thinking, and thought processes....Term logic represents ultimate clarity and precision when it comes to classical logic and thinking. In [chapter 6] on term logic, you’ll get an overview of this clarity and precision in section 6.1. In section 6.2, you’ll get into and explore the bare units of ultimate clear and precise thinking with categorical propositions. In sections 6.3, 6.4, and 6.5 (in the first edition of this textbook), you’ll multiply these categorical propositions into two category inferences, three category inferences, and even more numerous polysyllogisms and enthymemes, respectively. Finally, (in the first edition of this textbook), you’ll understand the characteristics of valid inferences in term logic, in section 6.6, and the potential pitfalls of fallacies in term logic in section 6.7. (shrink)

Buddhist philosophers have investigated the techniques and methodologies of debate and argumentation which are important aspects of Buddhist intellectual life. This was particularly the case in India, where Buddhism and Buddhist philosophy originated. But these investigations have also engaged philosophers in China, Japan, Korea and Tibet, and many other parts of the world that have been influenced by Buddhism and Buddhist philosophy. Several elements of the Buddhist tradition of philosophy are thought to be part of this investigation. -/- There are (...) interesting reasoning patterns discernible in the writings of Buddhist philosophers. For instance, the Mādhyamika philosopher Nāgārjuna presents arguments for the emptiness of all things in terms of catuṣkoṭi (four ‘corners’: roughly speaking, truth, false, both, neither). There are also the Indian vāda (debate) literature and Tibetan bsdus grwa (collected topics) that list techniques of debates. -/- The main interest here is the tradition of Buddhist philosophy, sometimes referred to as Pramāṇavāda, whose central figures are Dignāga (approx. 480–540 CE) and Dharmakīrti (6th– 7th CE). This philosophy is understood to be the Buddhist school of logic-epistemology. ‘Pramāṇavāda’ is not a doxographical term traditionally used to refer to a recognised school of Buddhist philosophy. It is a conventional term that is sometimes used in modern literature. Nevertheless, in what follows, we think of it as a tradition within Buddhist philosophy and ‘Buddhist logic’ refers to what is developed in this tradition. -/- Buddhist logicians have systematically analysed the kind of reasoning involved in acquiring knowledge. They hold that there are valid ways to reason that are productive of knowledge. In what follows, some of the main elements of their analyses will be described. Note, however, that, while exegetical studies of Buddhist texts are important, we must step back from them and consider what is involved in taking a Buddhist approach to logic in light of modern formal logic. This analysis will be done against the backdrop of the contemporary literature on logic and related subjects in contemporary philosophy to make sense of the logical studies by Buddhist logicians. (shrink)

One of the most fundamental tasks for philosophers is to comprehend the nature of human activities. Philosophers of language investigate the structure and functioning of linguistic life, while philosophers of science and epistemologists examine the principles that underpin human scientific and epistemic life. Ethicists, philosophers of art, and philosophers of religion explore the foundations and norms that govern practical, creative, and religious dimensions of human life. Just as the exploration of reality beyond human life is an important philosophical endeavor, the (...) study of human life itself has equal significance. Yet, such activity is itself a human one. The fact that philosophizing constitutes a part of human life implies that an inquiry into the nature of philosophical activity is also the philosopher’s duty. Given that philosophical activity often manifests in the form of debates concerning various problems, we are thus prompted to investigate the nature of philosophical debates. This raises the following question: What is the nature of philosophical debates? The present study represents a partial response to this question. Specifically, I aim to address a question that is often overlooked by philosophers: What is the logical structure of ontological debates? In other words, this study aims to explore the philosophical logic underlying ontology-talks. For this purpose, I focus on the logical structure of essential expressions for ontological debates—that is, their logical syntax and semantics. This investigation is significant not only as an exploration of a philosophical task which is not explored, but also as a meta-philosophical undertaking that critically reflects on philosophical activity itself. In Chapter 2, I develop a theory of \(\mathcal{H}^{S\times O}\), a formal language for ontology-describing expressions, which includes counterontological conditionals and ontology-quoting sentences. I argue that a language capable of accommodating ontology-describing expressions must incorporate impossible worlds within its logical space and adopt a two-dimensional structure partitioned along the axes of metaphysical possibilities and ontologies. After defining \(\mathcal{H}^{S\times O}\) within this framework, I outline a translation procedure for ontology-describing expressions into \(\mathcal{H}^{S\times O}\)-sentence forms. I further demonstrate that this translation procedure preserves the logical and semantic properties of the original expressions. Chapter 3 shifts focus to ontology-evaluating expressions. I first analyze their logical properties and subsequently extend \(\mathcal{H}^{S\times O}\) to include these expressions, introducing a set of pure axioms to characterize their formal features. To illustrate the practical applicability of ontological evaluative expressions in actual ontological debates, I introduce and justify three impure axioms and employ them to construct model cases for resolving ontological debates. In Chapter 4, I propose a philosophical interpretation of the second dimension of the logical space articulated in \(\mathcal{H}^{S\times O}\). Here, I advocate a conventionalist approach, arguing that this second dimension represents alternative conventions governing the term “exist.” To justify my approach, first, I critically evaluate three alternative interpretations, demonstrating their limitations. I then argue that to introduce a conventional dimension into the logical space is both natural and necessary. Finally, I provide examples that suggest the view that discourse on ontology can be viewed as discourse concerning the linguistic conventions associated with the term “exist.” The present study seeks to elucidate the logical properties of essential expressions that constitute ontological debates and to theorize these properties through a robust framework of logical syntax and semantics. Furthermore, this study aims to show that the ontological alternatives in question are not merely contradictory situations, epistemically possible scenarios, or fiction. Rather, they represent alternative world-descriptions grounded in alternative linguistic conventions. If successful, this study will contribute to a broader and more profound understanding of the human activity of engaging in ontological debate. (shrink)

Āptamīmāṃsā – Deep Reflection On The Omniscient Lord – known also as Devāgama or Devāgamastotra, by Ācārya Samantabhadra (2nd century CE) is the cornerstone of Philosophy and Nyāya. It starts with a discussion, in a philosophical-cum-logical manner, on the concept of omniscience and the attributes of the Omniscient. Ācārya Samantabhadra questions the validity of the attributes that are traditionally associated with a praiseworthy deity and goes on to establish the logic of accepting the Omniscient as the most trustworthy and praiseworthy (...) Supreme Being. Employing the doctrine of conditional predication (syādvāda) – the logical expression of reality in light of the foundational principle of non-absolutism (anekāntavāda) – he faults certain conceptions based on absolutism: existence (bhāvaikānta) and non-existence (abhāvaikānta), non-dualism (advaita-ekānta) and separateness (pŗthaktva-ekānta), and permanence (nityatva-ekānta) and momentariness (kṣaṇika-ekānta). He elucidates the important concept of the ‘seven-nuance system’ (saptabhaṅgī). It embraces the seven limbs (saptabhaṅga) of assertion, the one-sided but relative method of comprehension called the standpoint (naya), and also the acceptance and rejection of the assertion. Ācārya Samantabhadra finally elucidates correct perspective on issues including fate (daiva) and human-deed (pauruṣa), and the bases of merit (puṇya) and demerit (pāpa). The book includes a comprehensive Index of Sanskrit terms. (shrink)

Logical rationalism asserts that we can acquire immediate, non-inferential justification for beliefs in basic logical principles. The intuitions that arise when we consider particular cases of validity can offer justification for our foundational logical beliefs about rules of inference. I motivate rationalism through an argument from the indispensability of intuitions. This argument shows that rationalism is the theory best equipped to solve the problem of background logic. This is the challenge of explaining how we gain justified beliefs in rules of (...) inference without using those rules in a viciously circular way to arrive at the beliefs. As rationalism is the best epistemology which solves this important challenge, it is our best account of logical knowledge. In addition, I argue that a second major motivation for logical rationalism arises from work on logical practice. Recently, it has been claimed that we should attend to the practices of logicians when selecting an epistemology of logic. I argue that these considerations from logical practice also favour the adoption of logical rationalism. (shrink)

This book presents a philosophical conception of logic -- "logical expressivism"-- according to which the role of logic is to make explicit reason relations, which are often neither monotonic nor transitive. It reveals new perspectives on inferential roles, sequent calculi, representation, truthmakers, and many extant logical theories.

This paper raises some questions about the formalization of sentences containing ‘if’ or similar expressions. In particular, it focuses on three kinds of sentences that resemble conditionals in some respects but exhibit distinctive logical features that deserve separate consideration: whether-or-not sentences, biscuit conditionals, and concessive conditionals. As will be suggested, the examples discussed show in different ways that an adequate formalization of a sentence must take into account the content expressed by the sentence. This upshot is arguably what one should (...) expect on the view that logical form is determined by truth conditions. (shrink)

Arthur Schopenhauer was one of the first logicians of the 19th century to develop a visual representation of the sorites with which it is possible to depict multiple terms in one diagram. With the help of an example from Seneca, he explains how the sorites diagram works and provides two reading interpretations for the diagram. Since Schopenhauer’s sorites diagram in particular or sorites diagrams in general have never been dealt with in research before, this paper will introduce and analyse Schopenhauer’s (...) sorites diagram and its different reading possibilities. Thereby, it will not only be shown how the sorites diagram can help to dissolve the sorites back into individual syllogisms, but also other topics such as the reading directions, the extension and intension of the sorites diagram will be discussed in more detail. It turns out that Schopenhauer’s sorites was just the foundation for further works in his philosophical system. (shrink)

We give a new and elementary construction of primitive positive decomposition of higher arity relations into binary relations on finite domains. Such decompositions come up in applications to constraint satisfaction problems, clone theory and relational databases. The construction exploits functional completeness of 2-input functions in many-valued logic by interpreting relations as graphs of partially defined multivalued ‘functions’. The ‘functions’ are then composed from ordinary functions in the usual sense. The construction is computationally effective and relies on well-developed methods of functional (...) decomposition, but reduces relations only to ternary relations. An additional construction then decomposes ternary into binary relations, also effectively, by converting certain disjunctions into existential quantifications. The result gives a uniform proof of Peirce’s reduction thesis on finite domains, and shows that the graph of any Sheffer function composes all relations there. (shrink)

This book provides a first introduction to two key systems that play a role in any variety or application of logic: classical propositional logic and classical predicate logic. It also contains pointers to variations and alternatives, to logics that deal with a wider range of concepts. This book provides examples of the application of logic to philosophical problems, and illustrates some of the philosophical problems that are raised by logic itself. This book aims to give the reader a first glimpse (...) of the wide variety of tastes, smells, and textures that await them once they have trained their palate to appreciate the standard dishes of the logic cuisine. (shrink)

Many tasks in statistical and causal inference can be construed as problems of entailment in a suitable formal language. We ask whether those problems are more difficult, from a computational perspective, for causal probabilistic languages than for pure probabilistic (or “associational”) languages. Despite several senses in which causal reasoning is indeed more complex—both expressively and inferentially—we show that causal entailment (or satisfiability) problems can be systematically and robustly reduced to purely probabilistic problems. Thus there is no jump in computational complexity. (...) Along the way we answer several open problems concerning the complexity of well-known probability logics, in particular demonstrating the ${\exists \mathbb {R}}$ -completeness of a polynomial probability calculus, as well as a seemingly much simpler system, the logic of comparative conditional probability. (shrink)

The truth table method, natural deduction, and the truth tree method, the three validity proving methods standardly taught in an introductory logic course, are too clumsy for the battlefield of real-life. The “short truth table” test is handy at times, but it stumbles at many other times. In this paper, we set up a general method that can beat all the methods mentioned above in a contest of speed. Furthermore, the procedure can be step-by-step paraphrased in a natural language, so (...) that, unlike the other methods, a real-life logical problem can be analyzed and explained in a real-life language too. (shrink)

The semantic analysis of conditional sentences does not entirely align with their morphosyntactic structure. I substantiate this hypothesis with instances from both Czech and English that extend beyond conventional textbook examples. I also highlight that logicians and philosophers often make terminological errors when they disregard the insights from linguistic disciplines. Despite the early analytic philosophy’s emphasis on terminological precision, the practical application falls significantly short of this ideal. I firmly believe that a proper understanding of the morphosyntax and semantics of (...) conditional sentences is a prerequisite for their classification and analysis. In our pursuit of a comprehensive exploration of conditional clauses, we must also grapple with methodological challenges that delve into the very core of the philosophy of language. The primary issue revolves around the fact that two out of the three a priori methods in philosophy rely heavily on the analysis of conditional sentences: thought experiments and modal metaphysics. (shrink)

The purpose of this paper is to analyse and compare two concepts which tend to be treated as synonymous, and to show the difference between them: these are critical thinking and logical culture. Firstly, we try to show that these cannot be considered identical or strictly equivalent: i.e. that the concept of logical culture includes more than just critical thinking skills. Secondly, we try to show that Christian philosophers, when arguing about philosophical matters and teaching philosophy to students, should not (...) focus only on critical thinking skills, but rather also consider logical culture. This, as we argue, may help to improve debate both within and outside of Christian philosophy. (shrink)

My first aim in this paper is to use time diagrams in the style of Brentano to analyze constructions in Brouwer's separable mathematics more precisely. I argue that constructions must involve not only pairing and projecting as basic operations guaranteed by the intuition of twoity, as sometimes assumed in the literature, but also a recalling operation. My second aim is to argue that Brouwer's views on the intuition of twoity and arithmetic lead to an ontological explosion. Redeveloping the constructions of (...) natural numbers and systems sketched in an appendix to Brouwer's Cambridge lectures, I observe that the only plausible way he can make some elementary arithmetic in his separable mathematics is by allowing for the same canonical number to be determined by multiple separable entities, resulting in an overabundant mathematical ontology. (shrink)

In this short paper we consider Linnebo's thin/thick dichotomy: first, we show that it does not overlap with the very common one between abstract/concrete objects; second, on the basis of some difficulties with the distinction, we propose, as a possible way out, to move from thin/thick objects to thin/thick concepts.

Chapter 1 constitutes an introduction to Gentzen calculi from two perspectives, logical and philosophical. It introduces the notion of generalisations of Gentzen sequent calculus and the discussion on properties that characterize good inferential systems. Among the variety of Gentzen-style sequent calculi, I divide them in two groups: syntactic and semantic generalisations. In the context of such a discussion, the inferentialist philosophy of the meaning of logical constants is introduced, and some potential objections – mainly concerning the choice of working with (...) semantic generalizations – are addressed. Finally, I’ll introduce the case studies that I’ll be dealing with in part II. Chapter 2 is concerned with the origins and development of Jaśkowski’s discussive logic. The main idea of this chapter is to systematize the various stages of the development of discussive logic related researches from two different angles, i.e., its connections to modal logics and its proof theory, by highlighting virtues and vices. Chapter 3 focuses on the Gentzen-style proof theory of discussive logic, by providing a characterization of it in terms of labelled sequent calculi. Chapter 4 deals with the Gentzen-style proof theory of relevant logics, again by employing the methodology of labelled sequent calculi. This time, instead of working with a single logic, I’ll deal with a whole family of them. More precisely, I’ll study in terms of proof systems those relevant logics that can be characterised, at the semantic level, by reduced Routley-Meyer models, i.e., relational structures with a ternary relation between states and a unique base element. Chapter 5 investigates the proof theory of a modal expansion of intuitionistic propositional logic obtained by adding an ‘actuality’ operator to the connectives. This logic was introduced also using Gentzen sequents. Unfortunately, the original proof system is not cut-free. This chapter shows how to solve this problem by moving to hypersequents. Chapter 6 concludes the investigations and discusses the future of the research presented throughout the dissertation. (shrink)

In the context of nonstandard analysis, the somewhat vague equality relation of near-equality allows us to relate objects that are indistinguishable but not necessarily equal. This relation appears to enable us to better understand certain paradoxes, such as the paradox of Theseus’s ship, by identifying identity at a time with identity over a short period of time. With this view in mind, I propose and discuss two mathematical models for this paradox.

Qualifiers such as “insofar as” and “in itself” have always been important ingredients in key philosophical claims. Descartes, for instance, famously argues that insofar as he is a thinker, he is not made of matter, and Kant equally famously argues that we cannot know things in themselves. Neither of these claims is meant to be true without qualification. Descartes is not simply denying that humans consist of matter, and Kant is not simply denying that we know things. Therefore, we cannot (...) even begin to understand such claims without knowing how qualifiers work. Unlike the logic of quantification, however, the logic of qualification is rather underexplored. In this paper, I examine several instances of philosophical uses of qualifiers, taken from Aristotle, Avicenna, Descartes, Kant, and 20th c. action theory. In the light of these examples, I discuss several accounts of how such qualifiers work. (shrink)

Logicism is the thesis that all or, at least parts, of mathematics is reducible to deductive logic in at least two senses: (A) that mathematical lexis can be defined by sole recourse to logical constants [a definition thesis]; and, (B) that mathematical theorems are derivable from solely logical axioms [a derivation thesis]. The principal proponents of this thesis are: Frege, Dedekind, and Russell. The central question that I raise in this paper is the following: ‘How did Russell construe the philosophical (...) worth of logicism?’ The argument that I build in response to this is that Russell perceived an inverse proportion between a logical reduction of mathematics and the certitude of non-novel mathematical theorems—such that the more we reduce mathematics to logic, the more certain we become of our mathematical theorems; this was portrayed through a presentation of mathematical knowledge as coherent. Therefore, I set out to sketch Russell’s coherence theory and appraise it in relation to the presence discourse: i.e., in relation to logicism and mathematical certainty. (shrink)

This paper has been presented at the Charles S. Peirce Society’s 10-Minute Thesis Initiative: “His Glassy Essence in Relation” on February 18, 2023, where papers were also presented by Professor Doctor António Manuel Martins and Professor Doctor Mohammad Shafiei, respectively affiliated to the Coimbra Institute for Philosophical Studies and Shahid Beheshti University. -/- The edition “His Glassy Essence in Relation” of the Charles S. Peirce Society’s 10-Minute Thesis Initiative has been jointly organized by Aaron Wilson, António Manuel Martins, Mohammad Shafiei, (...) Richard Kenneth Atkins, and Robert Junqueira. The second, the third, and the last all presented papers, while the first one presided over the proceedings. The second to last was responsible for creating the bridge that enabled it all to take place. -/- The 10-Minute Thesis Initiative is an ongoing cycle of the Charles S. Peirce Society. This edition in particular benefited from the collaboration of the Coimbra Institute for Philosophical Studies and the Shahid Beheshti University. This text has been published in PhicaRe (Philosophy & Care Repository) of the Institute for Philosophical Studies of the Faculty of Arts and Humanities of the University of Coimbra. -/- How to cite (Chicago Manual of Style 17th Edition): Junqueira, Robert. “Peirce and the Coimbra Jesuit Course: A Bond Far More Pervasive Than Commonly Believed.” In PhicaRe (Philosophy & Care Repository), 1–12. Coimbra: Institute for Philosophical Studies, 2023. DOI: 10.5281/zenodo.7607786. (shrink)

The original interest of this article lies in existential import. It provides a broader view on the problem by reference to modern, symbolic logic (ch.1). Gradually, however, our interest will change into the amalgamated expressions often used in logic; that is, why are such expressions as “x is a round triangle” applied in logic? We critically discuss this question from an extensional viewpoint, namely model theoretic semantics (ch.2). We also touch on Church’s λ-calculus in the appendix (app.2).

Trata-se de capítulo de um livro sobre a noção de "substância" na história da filosofia e contém uma discussão sobre a noção ordinária de "corpo" do ponto de vista da filosofia analítica.

According to the analysis of concessive conditionals suggested by Crupi and Iacona, a concessive conditional $$p{{\,\mathrm{\hookrightarrow }\,}}q$$ p ↪ q is adequately formalized as a conjunction of conditionals. This paper presents a sound and complete axiomatic system for concessive conditionals so understood. The soundness and completeness proofs that will be provided rely on a method that has been employed by Raidl, Iacona, and Crupi to prove the soundness and completeness of an analogous system for evidential conditionals.

It is a metaphysical orthodoxy that interesting non-symmetric relations cannot be reduced to symmetric ones. This orthodoxy is wrong. I show this by exploring the expressive power of symmetric theories, i.e. theories which use only symmetric predicates. Such theories are powerful enough to raise the possibility of Pythagrapheanism, i.e. the possibility that the world is just a vast, unlabelled, undirected graph.

ABSTRACT Theories of sets such as Zermelo Fraenkel set theory are usually presented as the combination of two distinct kinds of principles: logical and set-theoretic principles. The set-theoretic principles are imposed ‘on top’ of first-order logic. This is in agreement with a traditional view of logic as universally applicable and topic neutral. Such a view of logic has been rejected by the intuitionists, on the ground that quantification over infinite domains requires the use of intuitionistic rather than classical logic. In (...) the following, I consider constructive set theories, which use intuitionistic rather than classical logic, and argue that they manifest a distinctive interdependence or an entanglement between sets and logic. In fact, Martin-Löf type theory identifies fundamental logical and set-theoretic notions. Remarkably, one of the motivations for this identification is the thought that classical quantification over infinite domains is problematic, while intuitionistic quantification is not. The approach to quantification adopted in Martin-Löf’s type theory is subtly interconnected with its predicativity. I conclude by recalling key aspects of an approach to predicativity inspired by Poincaré, which focuses on the issue of correct quantification over infinite domains and relate it back to Martin-Löf type theory. (shrink)

In support of logical nihilism, according to which there are no logical laws, Gillian Russell offers purported counterexamples to two laws of logic. Russell’s examples rely on cleverly constructed predicates not found in ordinary English. I show that similar apparent counterexamples to the same logical laws can be constructed without exotic predicates but using only what ordinary language provides. We correctly analyze such arguments so that they do not actually constitute counterexamples to any logic laws. I claim that we can (...) and should do the same for Russell’s arguments. (shrink)

We argue that traditional formulations of the reduction thesis that tie it to privileged relational operations do not suffice for Peirce’s justification of the categories and invite the charge of gerrymandering to make it come out as true. We then develop a more robust invariant formulation of the thesis, one that is immune to that charge, by explicating the use of triads in any relational operations. The explication also allows us to track how Thirdness enters the structure of higher order (...) relations and even to propose a numerical measure of it. Our analysis reveals new conceptual phenomena when negation or disjunction are used to compound relations. (shrink)

This paper formulates a bilateral account of harmony that is an alternative to one proposed by Francez. It builds on an account of harmony for unilateral logic proposed by Kürbis and the observation that reading the rules for the connectives of bilateral logic bottom up gives the grounds and consequences of formulas with the opposite speech act. I formulate a process I call 'inversion' which allows the determination of assertive elimination rules from assertive introduction rules, and rejective elimination rules from (...) rejective introduction rules, and conversely. It corresponds to Francez's notion of vertical harmony. I also formulate a process I call 'conversion', which allows the determination of rejective introduction rules from assertive elimination rules and conversely, and the determination of assertive introduction rules from rejective elimination rules and conversely. It corresponds to Francez's notion of horizontal harmony. The account has a number of features that distinguish it from Francez's. (shrink)

How is it that false statements, such as “horses have eight legs”, can be just as meaningful as true statements, such as “horses have four legs”? Where does logical structure come from? We can describe what the world would be like if the laws of physics were different – could we do the same for the laws of logic? Are there facts about ethics? If we ever managed to answer the philosophical questions that humans have pondered for thousands of years, (...) what would life look like on the other side? -/- All of these questions and more are addressed in Ludwig Wittgenstein’s early work, the Tractatus Logico-Philosophicus…which was first published 100 years ago. (shrink)

Book Review of Impossible Worlds, by Francesco Berto and Mark Jago. Oxford: Oxford University Press, 2019.

This book constitutes the refereed proceedings of the 12th International Conference on the Theory and Application of Diagrams, Diagrams 2021, held virtually in September 2021. The 16 full papers and 25 short papers presented together with 16 posters were carefully reviewed and selected from 94 submissions. The papers are organized in the following topical sections: design of concrete diagrams; theory of diagrams; diagrams and mathematics; diagrams and logic; new representation systems; analysis of diagrams; diagrams and computation; cognitive analysis; diagrams as (...) structural tools; formal diagrams; and understanding thought processes. 10 chapters are available open access under a Creative Commons Attribution 4.0 International License via link.springer.com. (shrink)

In his paper, “On paradox without self-reference”, Neil Tennant proposed the conjecture for self-referential paradoxes that any derivation formalizing self-referential paradoxes only generates a looping reduction sequence. According to him, the derivation of the Liar paradox in natural deduction initiates a looping reduction sequence and the derivation of the Yablo's paradox generates a spiral reduction. The present paper proposes the counterexample to Tennant's conjecture for self-referential paradoxes. We shall show that there is a derivation of the Liar paradox which generates (...) a spiraling reduction procedure. Since the Liar paradox is a self-referential paradox, the result is a counterexample to his conjecture. Tennant has believed that classical reductio has no essential role to formalize paradoxes. As our counterexample applies the rule of classical reductio, he may reject the counterexample. In this sense, it will be briefly argued that classical reductio and his rules for the liar sentence share some inferential role. If classical reductio should not be used in paradoxical reasoning, neither should be his rules for the liar sentence. (shrink)

We examine a contribution L. Susan Stebbing made to the understanding of critical thinking and its relation to formal logic. Stebbing took expertise in formal logic to authorise logical intervention in public debate, specifically in assessing of the validity of everyday reasoning. She held, however, that formal logic is purely the study of logical form. Given the problems of ascertaining logical form in any particular instance, and that logical form does not always track informal validity, it is difficult to see (...) how she could justify her belief in logical interventionism. Her answer to this problem is the contribution we explore here. It involves the view that although the logician’s expertise is not sufficient to assess arguments made in everyday contexts on its own, it nevertheless plays a unique role in giving systematicity and direction to the critique of such arguments, in particular, in public debate. (shrink)

Attempts to articulate the real meaning or ultimate significance of a famous theorem comprise a major vein of philosophical writing about mathematics. The subfield of mathematical logic has supplie...

According to conciliatory views on the significance of disagreement, it’s rational for you to become less confident in your take on an issue in case your epistemic peer’s take on it is different. These views are intuitively appealing, but they also face a powerful objection: in scenarios that involve disagreements over their own correctness, conciliatory views appear to self-defeat and, thereby, issue inconsistent recommendations. This paper provides a response to this objection. Drawing on the work from defeasible logics paradigm and (...) abstract argumentation, it develops a formal model of conciliatory reasoning and explores its behavior in the troubling scenarios. The model suggests that the recommendations that conciliatory views issue in such scenarios are perfectly reasonable—even if outwardly they may look odd. (shrink)

The structure "home - world - ideals" is presented as the structure of "philosophical striving" (F. Marković). It could be formally described as a model consisting of a domain, relations and a valuation. On that basis, the identity, openness, and the significance of Croatian philosophy is investigated. The programme of the renewal of Croatian philosophy (as proposed 1882 by Franjo Marković) is re-examined, and some unsolved historical-cultural discontinuities within the programme are described. The written beginnings of Croatian philosophical thought are (...) detected in the Charter of Duke Trpimir (852). (shrink)

In his Lecture on Ethics, Wittgenstein mentions the difficulty a recipient of his philosophy has in ‘seeing both the road he is led and the goal which it leads to’. Oskari Kuusela's b...