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This paper stresses the importance of identifying the nature of an author's conception of logic when using terms from modern logic in order to avoid, as far as possible, injecting our own conception of logic in the author's texts. Sundholm (2012. “‘Inference versus consequence” revisited: Inference, conditional, implication’, Synthese, 187, 943–956) points out that inferences are staged at the epistemic level and are made out of judgments, not propositions. Since it is now standard to read Aristotelian sullogismoi as inferences, I have taken Alexander of Aphrodisias's commentaries to Aristotle's logical treatises as a basis for arguing that the premises and conclusions should be read as judgments rather than as propositions. Under this reading, when Alexander speaks of protaseis, we should not read the modern notion of proposition, but rather what we now call judgments. The point is not just a matter of terminology, it is about the conception of logic this terminology conveys. In this regard, insisting on judgments rather than on propositions helps bring to light Alexander's epistemic conception of logic. [ABSTRACT FROM AUTHOR]

Łukasiewicz introduced a new methodological approach to the history of logic. It consists of the use of modern formal logic in the research of the history of logic. Although he was not the first to use formal logic in his historical research, Łukasiewicz was the first who used it consistently and formulated it as a requirement for a historian of logic. The aim of this paper is to present Łukasiewicz's contribution and the history of its formulation. In addition, the paper will present its reception. There are two main themes in Łukasiewicz's work on the history of logic. He discussed Stoic logic and Aristotle's syllogistic. The second part of the paper focuses on current reception of Łukasiewicz's contribution and on the question of whether there is a difference between knowledge about his contribution to Stoic logic and to Aristotle's syllogistic, given that it has been hypothesised that Łukasiewicz's contribution is currently more known among Aristotelian scholars than among those who focus on Stoic logic. Based on the papers that have appeared in the last five years at Web of Science and Scopus this hypothesis has not been confirmed. [ABSTRACT FROM AUTHOR]

We take into account two areas of the logical research of the Lvov-Warsaw School. First, we consider a new approach to research in the history of logic introduced and practiced by Łukasiewicz and some of his followers. In this style of doing history of logic, the knowledge of original philosophical and logical texts was combined with competence in modern logic. This method resulted in many important discoveries both in history and in logic and philosophy. At the same time, we pay attention to contemporary historical research devoted to the logic created by the School itself. In this context, we present an overview of the six papers included in the special issue 'Logic and its History in the Lvov-Warsaw School'. These papers were presented at an international symposium, 'History in/of the Lvov-Warsaw School', held in October 2022 in Warsaw, and they cover various aspects of historical-logical and logical achievements of the School. They concern topics from history of ancient logic, history of modern proof theory, and the issues of applying logic to philosophy and theology in the same way as the one implemented at the Lvov-Warsaw School. [ABSTRACT FROM AUTHOR]

'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 $ (\textsf{L}_{\textsf{Plum}}) $ (L Plum ). 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. [ABSTRACT FROM AUTHOR]

The paper is about the connection between logic and discrimination, with special focus on Plumwood's ideas in her groundbreaking article 'The Politics of Reason. Towards a Feminist Logic' (1993). Although Plumwood's paper is not focused on the notion of discrimination, what she writes is useful for illuminating some basic mechanisms of thought that are at the basis of discriminatory practices. After an introductory section about the concepts of logic and discrimination and their possible interconnections, I present Plumwood's ideas in 1993 with a special focus on their relevance for understanding the nature of discrimination. More specifically, I use examples of discriminatory practices that make the connection between logical operations and oppression envisaged by Plumwood clear. I focus especially on two questions: Can logic produce discrimination? Can logic contribute to the fight against discrimination? If so, how? [ABSTRACT FROM AUTHOR]

The article illustrates the different meanings of the term "logic" in Weber's work and then proceeds to discuss his approach to the explanation of historical events and in particular to counterfactual analysis. Weber's epistemology is first situated within the neo-Kantian debates of his time as well as legal positivism and historical jurisprudence. The article then focuses on this author's conception of science as a value sphere, on the aims and methods of explanation in the social and historical sciences and on the key concepts of possibility judgements and adequate causation. The central role played by counterfactuals and modal logic is illustrated through the example of the battle of Marathon, which Weber himself saw as the starting point of a causal chain leading to the rise of Western rationalism. The article concludes by highlighting the pioneering role of Weber's counterfactualism for the subsequent development of possible worlds theory and its use in the causal analysis of singular historical events. [ABSTRACT FROM AUTHOR]

The aim of this special issue is to explore the intersections of logic, intended as the formal science of valid consequence, and politics, the practical science of associated life. The analyses here provided deal with the theme in two ways: by considering what logic can do for politics, so proposing possible formalizations of political arguments, or using logical notions and tools to interpret political facts and problems; by considering what politics can do for logic, so examining how logical choices are and have been affected or determined by extra-logical facts of political relevance. The two perspectives outline a territory of inquiry that is not so much frequented, in political and logical literature, but is becoming more and more interesting nowadays. [ABSTRACT FROM AUTHOR]

This paper advances a version of the theory whereby logic had deep origins in democracy, by re-reading Gorgias' Encomium of Helen. Democracy, 'the government by debate', called political (and scientific) attention to the inferential abilities of citizens and to politicians' ability of taking advantage of them. Sophists, in particular, discovered that people's inferences follow constant repeatable forms, that these forms have impact on choices and decisions concerning public good, and then by dominating them you dominate politics in democracy. With the Encomium Gorgias launched these ideas, so laying the ground for Aristotle's science of deduction. [ABSTRACT FROM AUTHOR]

Arabic logicians in the thirteenth century discussed a set of arguments raised by the theologian Fakhr al-Dīn al-Rāzī (d. 1210) that in some respects closely resembles Carroll's paradox. Roughly, the paradox states that we can never reach a conclusion from a set of premises without incurring an infinite regress. The present article presents and discusses Rāzī's formulation of the problem with syllogistic deduction, his own solutions to the problem, and the contributions of Afḍal al-Dīn al-Khūnajī (d. 1248) and Najm al-Dīn al-Kātibī (d. 1276). It is argued that the proposed solutions developed by these authors are best understood as tending in the direction of Gilbert Ryle's distinction between knowing-that and knowing-how. [ABSTRACT FROM AUTHOR]

Aversion towards logic is a characteristic feature of the Islamic traditionalists. There is in fact a history of opposition to logic in Islam. As any other areas of history, here also the correct picture will not be achieved unless all of the pieces are put together. In what follows, I am going to shed light on a chapter written by Zayn al-Dīn al-ʿĀmilī (d. 966/1558), the Twelver Shīʿī Scholar better known as al-Shahīd al-Thānī. The chapter not only shows al-Shahīd al-Thānī's negative stance towards logic, but also is important because it is a part of less studied Shīʿite traditionalists' tendencies towards logic; those who are considered as the most influential figures in Iranian seminaries from the Safavid period up until today. [ABSTRACT FROM AUTHOR]

At APr 2.4 57a36–13, Aristotle presents a notorious reductio argument in which he derives the claim 'If B is not large, B is large' and calls that result impossible. Aristotle is thus committed to some form of connexivity and this paper argues that his commitment is to a strong form of connexivity which excludes even cases in which 'B is large' is necessary. It is further argued that Aristotle's view of connexivity is best understood as arising from his analysis of affirmation and negation in terms of combination and separation: a proposition that separates two terms cannot entail a proposition that combines the same two terms. In order to motivate this account of connexivity, this paper interprets Aristotle as emphasising the predicative structure, especially the copulae, of the argument's component propositions. The arguments for this are based on a consideration of the larger context of APr 2.2–4. A reconstruction of the argument using propositional variables does not fully capture Aristotle's intentions. [ABSTRACT FROM AUTHOR]

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

Gotthard Günther's early work is inspired by Hegel's logic. Both share the view that the forms of 'classical logic' are insufficient to capture essential aspects of thought. However, while Hegel rules out formalization for his dialectical logic, Günther attributes this to the fact that Hegel's thought itself is still bound to the forms of a classical logic. Günther thus develops a trans-classical logic, with which he believes he can formally capture some of Hegel's insights. The paper presents Günther's critical engagement with Hegel and outlines some basic assumptions arising from his critique. [ABSTRACT FROM AUTHOR]

In 1672 John Eliot, English Puritan educator and missionary to New England, published The Logick Primer: Some Logical Notions to initiate the INDIANS in the knowledge of the Rule of Reason; and to know how to make use thereof (Eliot 1672) The Logick Primer: Some Logical Notions to Initiate the INDIANS in the Knowledge of the Rule of Reason; and to Know How to Make Use Thereof , Cambridge, MA: Marmaduke Johnson]. This roughly 80 page pamphlet introduces syllogistic vocabulary and reasoning so that syllogisms can be created from Biblical texts. The use of logic for proselytizing purposes is not distinctive: What is distinctive about Eliot's book is that it is bilingual, written in both English and Massachusett (Wôpanâak), an Algonquian language spoken in eastern coastal and southeastern Massachusetts. It is one of the earliest bilingual logic textbooks and it is the first, and perhaps only, textbook in an indigenous American language. In this paper, we (1) introduce John Eliot and the linguistic context he was working in; (2) introduce the contents of the Logick Primer – vocabulary, inference patterns, and applications; (3) discuss notions of 'Puritan' logic that inform this primer; and (4) address the importance of his work in documenting and expanding the Massachusett language and the problems that accompany his colonial approach to this work. [ABSTRACT FROM AUTHOR]

This essay addresses the general systematic question underlying any project of 'formalization' of Hegel's dialectic-speculative logic, namely, the question concerning the peculiar concept of logical 'form' and the connected type of 'formalism' at stake in a logic that is, programmatically and innovatively within the historical tradition, a 'dialectic-speculative' logic. [ABSTRACT FROM AUTHOR]

The Yogācāra school of logic developed a theory of sign-inferences that has many features of the Stoic and Epicurean logical teachings with small inclusions of Aristotelian ideas. In the Nyāyabindu of Dharmakīrti, we can find the following schemes of formal reasoning: modus Barbara (Figure I) and modus Camenes (Figure IV) of the Aristotelian syllogistic, and all the inference rules of the Stoic logic: modus ponens, modus tollens, modus ponendo tollens, modus tollendo ponens I, modus tollendo ponens II. The three premises of Yogācāra inference (if p, then q; there is an example that p and q occur together; p; then q) corresponds to the Epicurean doctrine of three premises with a necessary exemplification. Thus, the paper is a comparative study of Greek and Buddhist theory of sign-inferences, showing a deep similarity between them. [ABSTRACT FROM AUTHOR]

Hegel's Doctrine of the Concept advances a theory of conceptual determinacy. As I will demonstrate, Hegel's theory of conceptual determinacy leads him to endorse self-predication and existential implication as features endemic to conceptual content. I first demonstrate some features of this logic, and some of its entailments. Following the reconstruction of Hegel's logic of self-predication, I apply this logic to illuminate Hegel's critique of formal logic. Finally, the self-predicative feature of Hegel's logics offers a measure by which to determine the potential and the limitations of any formal account of Hegel's logic. [ABSTRACT FROM AUTHOR]

Hegel interpreters commonly reject attempts to situate Hegel's logic in relation to modern movements. Appealing to his criticisms of the logic of Verstand or mere understanding with its fixed logical structure, Hegel's logic, it is pointed out, was a logic of Vernunft or reason—a logic more at home in the thought of Plato and Aristotle than in modern mathematical forms. Contesting this implied dichotomy, it is here argued that the ancient roots of Hegel's logic, especially as transmitted by late Neopythagorean/Neoplatonic thinkers such as Proclus, gave it many features similar to ones later found in the type of algebraic transformation of Aristotle, started first by Leibniz, reanimated by Boole in the mid-nineteenth century and then developed by others such as C. S. Peirce and Arend Heyting. In particular, the ancient mathematics upon which Hegel had drawn allowed him to anticipate an answer to the criticism that Frege would later aim at Boole, concerning his inability to unite opposed class and propositional calculi. Hegel's logic would be a hybrid, incorporating features found later in intuitionist and classical logic, but it could be so because of the way he had called upon the mathematics of the ancient Platonist tradition. [ABSTRACT FROM AUTHOR]

In this paper, I examine Ruth Barcan Marcus's early formal work on modal systems and the deduction theorem, both for the material and the strict conditional. Marcus proved that the deduction theorem for the material conditional does not hold for system S2 but holds for S4. This last result is at odds with the recent claim that without proper restrictions the deduction theorem fails also for S4. I explain where the contrast stems from. For the strict conditional, Marcus proved the deduction theorem for S4 though restricted to arguments with necessary premises. I discuss Marcus's result and analyze her philosophical position on the significance of the deduction theorem for modal systems designed to express the notion of deducibility. [ABSTRACT FROM AUTHOR]

This paper aims at disentangling two distinct problems in present philosophy of logic: the a priori/a posteriori divide and the theory choice problem. A confusion of these problems is present in the heart of current anti-exceptionalism about logic, as the use of a posteriori methods is identified with theory choice. We illustrate how the division may be preserved in a version of anti-exceptionalism by discussing Carnap's approach, which had both an a priori epistemology and a pragmatic account of logical theory choice. This contributes to illuminate the claim that the current understanding is mistaken in considering Carnap as an exceptionalist, and it also helps us to address one of the difficulties of current anti-exceptionalism, to know, that, by equating a posteriori with theory choice, it requires appeal to framework-independent facts. We suggest that the pragmatic account of both Quine and Carnap to theory choice was advanced precisely to avoid that kind of difficulty. [ABSTRACT FROM AUTHOR]

The present text discusses whether there is a tension between aphorisms 6.1-6.13 of the Tractatus and the Church-Turing theorem about the decidability of predicate logic. We attempt to establish the following points: (i) Aphorisms 6.1-6.13 are not consistent with the Church-Turing theorem. (ii) The logical symbolism of the Tractatus, built from the N-operator, can (and should) be interpreted as expressively complete with respect to first-order formulas. (iii) Wittgenstein's reasons for believing that Logic is decidable were purely philosophical and the undecidability result shows that there are aspects of his criticisms of Frege and Russell that become unjustified in light of these results. [ABSTRACT FROM AUTHOR]

The paper examines the interplays between logic and politics in the Polish School of Logic starting from 1914. The Polish School of Logic flourished between 1920 and 1939. Philosophically, it was influenced by Kazimierz Twardowski (1866–1938). For Twardowski logic is fundamental for every kind of human activity, professional and private and this means that every argument should be formulated and proceed by correct inferential rules. These rules involve semiotics, formal logic and methodology of science. The paper shows how this position was shared by Twardowski's students, including Jan Łukasiewicz, Stanisław Leśniewski, Kazimierz Ajdukiewicz and Tadeusz Kotrabiński, as well as by the next generation of logicians and philosophers and especially by Alfred Tarski. These authors considered logic, philosophy and science as completely neutral with respect to politics and ideology but they also considered logical skills as indispensable in political activities. [ABSTRACT FROM AUTHOR]

George Boole published the pamphlet The Mathematical Analysis of Logic in 1847. He believed that logic should belong to a universal mathematics that would cover both quantitative and nonquantitative research. With his pamphlet, Boole signalled an important change in symbolic logic: in contrast with his predecessors, his thinking was exclusively extensional. Notwithstanding the innovations introduced he accepted all traditional Aristotelean syllogisms. Nevertheless, some criticisms have been raised concerning Boole’s view of Aristotelean logic as the solution of algebraic equations. In order to circumvent such criticisms, we show here how Boole’s conclusions may be deduced from the axioms and inference rules proposed in his pamphlet, from which Aristotle’s deductions in Prior Analytics (including those that are completed through an impossibility) can be stated as theorems. We clarify his method for dealing with both universal and particular premises by means of his symbol v and demystify some criticisms concerning the way he expressed particular premises. Boole’s conclusions for some of those premises for which according to Aristotle there is no deduction are also discussed. The deductive system presented here has the potential to provide a new perspective on Boole’s approach to logic in The Mathematical Analysis of Logic. [ABSTRACT FROM AUTHOR]

In this paper, the connection between logicism and the principle of tolerance in Carnap's philosophy of logic and mathematics is to be presented in terms of the history of its development. Such development is conditioned by two lines of criticism to Carnap's attempt to combine Logicism and Conventionalism, the first of which comes from Gödel, the second from Alfred Tarski. The presentation will take place in three steps. First, the Logicism of Carnap before the publication of The Logical Syntax of Language will be analyzed in its main features. Second, the relationship between Logicism and the Principle of Tolerance at the time of the publication of Logical Syntax, and thirdly, in the so-called semantical phase of Carnap's thought will be examined. On the basis of the two preceding points, we will then critically analyze two interpretations of the possibility to maintain a philosophical position like that of Carnap nowadays. [ABSTRACT FROM AUTHOR]

We propose a new modal logic endowed with a simple deductive system to interpret Aristotle's theory of the modal syllogism. While being inspired by standard propositional modal logic, it is also a logic of terms that admits a (sound) extensional semantics involving possible states-of-affairs in a given world. Applied to the analysis of Aristotle's modal syllogistic as found in the Prior Analytics A8-22, it sheds light on various fine-grained distinctions which when made allow us to clarify some ambiguities and obtain a completely consistent system and prove all of the modal syllogisms considered valid by Aristotle. This logic allows us also to make a connection with the axioms of modern propositional modal logic and to perceive to what extent these are implicit in Aristotle's reasoning. Further work will involve addressing the question of the completeness of this logic (or variants thereof) together with an extension of the logic to include a calculus of relations (for instance the relational syllogistic treated in Galen's Introduction to Logic) which Slomkowsky has argued is already found in the Topics. [ABSTRACT FROM AUTHOR]

Formalizing categorical propositions of traditional logic in the language of quantifiers and propositional functions is no straightforward matter, especially when modalities get involved. Starting with the formulas for non-modal categoricals, we consider various ways of modalizing the formulas and semantic criteria of their evaluation that can be derived from Buridan. In addition to the logical relations included in the octagon of divided modal propositions, three interrelated aspects are taken into account—existential import, sensitivity to ampliation of terms in modal contexts, and quantification over possibilia. We end by suggesting a representation of Buridan's divided modal propositions that relies on the use of actualist quantification over variable domains. The formulas adequately capture the truth conditions given by Buridan, and they preserve all relations of the octagon, as well as permissible conversions in modal S5. [ABSTRACT FROM AUTHOR]

The twentieth-century logical mainstream, derived from works by Łukasiewicz and Scholz, pictures the history of logic for the most part as the prehistory of Boolean–Fregean mathematical logic. Particularly, with respect to classical propositional calculus, the Stoic logic has been pictured as an early stage of it and Aristotle's or the Peripatetics' logic as a theory that assumes it. Although it was not emphasised, it follows that the ancient logics contain the principle of explosion. In the endmost quarter of the twentieth century, a competitive view began to spread to the effect that the ancient systems of logic were paraconsistent or relevantistic. In the twenty-first century, the latter view prevails and has every chance of becoming a new orthodoxy. It is claimed that although in Łukasiewicz's argument for the classicality of ancient logics, there are gaps, it may be demonstrated that the ancient logics contain the principle of explosion. [ABSTRACT FROM AUTHOR]

Buridan endorses the basic idea that q follows from p iff it is impossible that p is true but q is false. Since he also accepts the law that, if p is impossible, the conjunction (p ∧ q) must be impossible, he comes to regard the principle ‘Ex impossibili quodlibet’ (EIQ) as basically correct. However, his logic is based on a ‘nominalist’ view according to which propositions are tokens of spoken, written or thought language existing in space of time, and the truth of such a proposition presupposes that it exists. This conception prompted Buridan to modify the definition of ‘consequence’, but none of his attempts was fully successful. Yet, a large part of his theory of consequences is based on the usual account of necessary truth-conservation. As regards principle EIQ, Buridan distinguishes two variants of impossibility. Proposition p is possible iff the state of affairs as described by p is possible; this doesn’t guarantee, however, that p is ‘possibly true’. E.g. ‘No proposition is negative’ is considered by Buridan as describing a possible state of affairs, but this proposition is not possibly true, because in order to be true, it would have to exist, and as soon as it comes into existence, it becomes false. Accordingly, EIQ requires that the antecedent p is really impossible and not only impossibly true. [ABSTRACT FROM AUTHOR]

Mathematical logic was first introduced to China in early 1920s. Although, the process of introduction was facilitated by the lectures of Bertrand Russel at Peking University in 1921 and continued by China's most passionate adherents of Russell's philosophy, the establishment of mathematical logic as an academic discipline occurred only in late 1920s, in the framework of a recently reorganised Qinghua University in Peking. The main aim of this paper is to shed some light on the process of establishment of mathematical logic at the department of philosophy at Qinghua University, between the years 1926 and 1945. In its main line of discussion, the article highlights the curricular developments at the department, connecting them with concrete theoretical endeavours by the leading members of the department. Furthermore, in its later parts, the article summarises the main advances made in the context of studies of modern logic at Qinghua University, from the expositions on the quintessential work Principia Mathematica, to the subsequent integration of criticisms coming from circles of logicians at Harvard University and ground-breaking contributions of Kurt Gödel in the mid-1930s. [ABSTRACT FROM AUTHOR]

The aim of this paper is to look at the arguments advanced by three Parisian arts masters about how to understand Prior Analytics II 4 and the more general discussion that medieval authors situate this in, revolving around the validity of various rules of inference. In particular, this paper argues that all three arts masters read Aristotle's remarks as committing him to a weak form of connexive principles where the antecedent needs to be either true or at least formally possible; this is known in the contemporary literature as a weak connexive thesis. [ABSTRACT FROM AUTHOR]

The focus of the article is the notion of normativity in logic and politics and their possible intersections. The twentieth-century divide between the analytical and the continental idea of logic is explored, by noting that they both – with significant differences – can be seen as proposing a 'bottom-up normativity', which may have immediate political effects. Logical normativity postulates universality, and a connection between reality and reason able to orient actions universalistically. For a bottom-up conception (as specifically advanced by Deleuze's logic of sense), the universalistic approach does not descend from above on an elusive reality but is a self-organization of multiple processes occurring in a horizon which is typically affected by political facts and choices. [ABSTRACT FROM AUTHOR]

This paper presents the link between Arthur N. Prior and logicians that belonged to the Lvov-Warsaw School. Although certain members of the Lvov-Warsaw School influenced Prior's views, the amount and the form of the impact are still under discussion. Prior also cooperated with some of them in the development of his systems of logic. This paper focuses on four main areas in which Prior admitted adopting ideas from the Lvov-Warsaw School: systems of propositional logic, the history of logic, modal and temporal logic and ontology. Prior's published works as well as his correspondence, which is stored in Prior's Nachlass, are used to describe the influence. [ABSTRACT FROM AUTHOR]

We present a memorial summary of the professional life and contributions to logic of John Corcoran. We also provide a full list of his many publications. Courtesy of Lynn Corcoran [ABSTRACT FROM AUTHOR]

In 1954, A. N. Prior (1914–1969) discovered a way to formalize tense-logic—as such, there is no doubt that he is the father of modern tense-logic. Despite this, he considered his early teacher in philosophy and logic, J. N. Findlay, to be, in a sense, the father of modern tense-logic. This title has been disputed, and it has been argued that Findlay merely had the luck of inspiring Prior's invention through an important footnote in an article on time from 1941. However, others have accepted the view that Findlay is the father of tense-logic merely based on the same footnote and Prior's appraisal. This article will argue, based upon a reading of Findlay's early authorship, that both views are wrong. Furthermore, an evaluation of Findlay's early work makes it evident that Prior was able to provide the formal foundation that was lacking in Findlay's theory of tenses. Finally, the article argues that Prior's tense-logic and Findlay theory of tenses was part of a different view on the relationship between symbols and metaphysics than the one espoused by Bertrand Russell. [ABSTRACT FROM AUTHOR]

In the Laws of Thought, Boole establishes a theory of secondary propositions based upon the notion of time. This temporal interpretation of secondary propositions has historically been met with wide disapproval and is usually dismissed in the modern literature as a philosophical non-starter. What was Boole thinking? This paper attempts to give an answer to this question. Specifically, it provides an account according to which Boole's temporal interpretation follows from his psychologistic conception of logic, in addition to certain background assumptions regarding the psychological necessity of the concept of time. Once Boole's psychological premises are laid bare, it becomes clearer how he might have viewed the temporal interpretation to be an essential feature of his theory of secondary propositions. [ABSTRACT FROM AUTHOR]

This is a companion article to the translation of 'Zasada sprzeczności a logika symboliczna', the appendix on symbolic logic of Jan Łukasiewicz's 1910 book O zasadzie sprzeczności u Arytotelesa (On the Principle of Contradiction in Aristotle). While the appendix closely follows Couturat's 1905 book L'algebra de la logique (The Algebra of Logic), footnotes show that Łukasiewicz was aware of the work of Peirce, Huntington and Russell (before Principia Mathematica). This appendix was influential in the development of the Polish school of logic, directly inspiring Stanisław Leśniewski and Leon Chwistek and more widely by serving as a text of the new symbolic logic. This appendix was an important source of the dominant algebraic logic in Poland, but also indicates that Łukasiewicz appreciated Russell's axiomatic approach to logic. [ABSTRACT FROM AUTHOR]

The abstract status of Kant's account of his 'general logic' is explained in comparison with Gödel's general definition of a formal logical system and reflections on 'abstract' ('absolute') concepts. Thereafter, an informal reconstruction of Kant's general logic is given from the aspect of the principles of contradiction, of sufficient reason, and of excluded middle. It is shown that Kant's composition of logic consists in a gradual strengthening of logical principles, starting from a weak principle of contradiction that tolerates a sort of contradictions in predication, and then proceeding to the (constructive) principle of sufficient reason, and to a classical-like logic, which includes the principle of excluded middle. A first-order formalisation is applied to this reconstruction, which reveals implicit modalities in Kant's account of logic, and confirms the implementability of Kant's logic into a sound and complete formal system. [ABSTRACT FROM AUTHOR]

In this paper Venn's account of probability inference and induction is examined, tracing their differences as well as how they ‘co-operate’ in inferences from particulars to particulars. We discuss the role of mathematical idealizations in making probability inferences, the celebrated rule of succession and we delve into the nature of the reference class problem arguing that for Venn it is a common problem for both induction and inference in probability. Our approach is both historical and philosophical attempting to sketch Venn's position both in the philosophy of probability and induction of his time and in relation to the twentieth-century frequentism. [ABSTRACT FROM AUTHOR]

This is the first English translation directly based on the original Polish 'Zasada sprzeczności a logika symboliczna', the appendix on symbolic logic of Jan Łukasiewicz's 1910 book O zasadzie sprzeczności u Arytotelesa (On the Principle of Contradiction in Aristotle). [ABSTRACT FROM AUTHOR]

The rather unrestrained use of second-order logic in the neo-logicist program is critically examined. It is argued in some detail that it brings with it genuine set-theoretical existence assumptions and that the mathematical power that Hume's Principle seems to provide, in the derivation of Frege's Theorem, comes largely from the 'logic' assumed rather than from Hume's Principle. It is shown that Hume's Principle is in reality not stronger than the very weak Robinson Arithmetic Q. Consequently, only a few rudimentary facts of arithmetic are logically derivable from Hume's Principle. And that hardly counts as a vindication of logicism. [ABSTRACT FROM AUTHOR]

A new theoretical approach to Aristotelian Logic (AL) based on three axioms has been recently introduced. This formalization of the theory allowed for the unification of its uncommunicated traditional branches, thus restoring the theoretical unity of AL. In this brief paper, the applicability of the three AL axioms to Propositional Logic (PL) is explored. First, it is shown how the AL axioms can be applied to some simple PL arguments in a straightforward manner. Second, the development of a proof method for PL inspired by the AL axioms is presented. This method mimics the underlying mechanics of the proof method from AL, and offers a complementary alternative to proof methods such as truth trees. [ABSTRACT FROM AUTHOR]

In the years following Bertrand Russell's visit in China, fragments from his work on mathematical logic and the foundations of mathematics started to enter the Chinese intellectual world. While up until 1925 Chinese intellectuals like Zhang Shenfu, Zhang Dongsun and others mainly contributed to the dissemination of general notions from Russell's logicism, epistemology and mathematical philosophy, two aspiring Chinese mathematicians, Fu Zhongsun and Zhang Bangming, introduced to Chinese readers the first Chinese translation of Russell's Introduction to Mathematical Philosophy. One year after the second edition of the book was published in 1924, Chinese intellectuals started to respond to Russell's 'mathematical logic' and philosophy of mathematics, as contained in the work. Consequently, in 1925 the first public debate on this topic broke out in Chinese periodicals. This article aims at providing a summary of these 1925 discussions, from criticisms of Russell's work to a general debate on 'infinitesimals'. Through close examination of the arguments and discursive strategies used in the debate, the article tries to highlight the state of Chinese discourse on mathematical logic during the time in question, while also shedding some light on the mechanisms and strategies in the undergoing process of the adoption of Western scientific ideas. [ABSTRACT FROM AUTHOR]

Wittgenstein's conception of the general form of a truth function given in thesis 6 can be presented as a sort of a trade-off: the author of the Tractatus is unable to reconcile the simplicity of his original idea of a series of forms with the simplicity of his generalisation of Sheffer's stroke; therefore, he is forced to sacrifice one of them. As we argue in this paper, the choice he makes – to weaken the logical constraints put on the concept of a series of forms, thus effectively metaphorising that concept for the sake of upholding the N-operation's role of generating the series – is unfortunate. An actual expansion of a series of truth functions as defined in 6 would require either making decisions at each step (Anscombe) or outwardly rejecting the concept of a series (Sundholm). However, neither is faithful to Wittgenstein's own fundamental intuitions regarding the nature of logic. For this reason, a different trade-off that prioritises upholding the basic features of a series of forms over the simplicity of the operation that generates that series seems to be much more reasonable. We offer such a trade-off by developing the schema already present in the Tractatus (5.101). The key element of our alternative solution is the construction of an operation that can perform the task of producing all consecutive truth functions of a given collection of atomic propositions as an invariant difference between any base and its successor throughout the series. We define it as a composition of three functions (in an algebraic sense): the first turns a symbol of a given truth function into a binary number, the second increments that number, the third turns the result back into a symbol of another truth function. We also show how the operation can be defined by means of a different notation without invoking binary arithmetic. [ABSTRACT FROM AUTHOR]

Recently, in 'An inquiry on Fakhr al-Dīn Rāzī's authorship of Al-manṭiq al-kab $ \bar{\iota} $ ι ¯ r (MS Aḥmad iii, no. 3401)', we refuted the quite common view among historians of Arabic logic as to attribute Al-manṭiq al-kab $ \bar{\iota} $ ι ¯ r (Major [book on] logic) to Fakhr al-dīn Rāzī; however, in that paper, we could not identify the real author of the book. In this paper, we try to show that Al-manṭiq al-kab $ \bar{\iota} $ ι ¯ r was written by Burhān al-dīn Nasafī (1203–1288), who authored another book, Sharḥ Asās al-kiyāsa whose logical part is very similar to Al-manṭiq al-kab $ \bar{\iota} $ ι ¯ r in the following significant features: the titles of chapters and sections, the style of writing, some technical as well as some non-technical terms, many sentences word by word. None of these features we find in any other Arabic logical work. So, Al-manṭiq al-kab $ \bar{\iota} $ ι ¯ r and Sharḥ Asās al-kiyāsa are so close to each other that we might most possibly identify the authors of the books as the same scholar, i.e. Burhān al-dīn Nasafī. Fortunately, this has been verified recently by two manuscripts of a medieval Arabic logical book. [ABSTRACT FROM AUTHOR]

Leśniewskian Ontology (LO) is a system in which the basic subject-predicate formula takes the form of a ϵ b and express one-argument predication, e.g. John is a student. In LO's language, there is no many-argument form of predication given that would allow for the structural expression of, for example, the sentence John is Anne's son. In this article, a simple and natural extension of LO is suggested to encompass many-argument predication. The system thus obtained corresponds to polyadic second-order logic. [ABSTRACT FROM AUTHOR]

The goal of this paper is to present a new reconstruction of Aristotle's assertoric logic as he develops it in Prior Analytics, A1-7. This reconstruction will be much closer to Aristotle's original text than other such reconstructions brought forward up to now. To accomplish this, we will not use classical logic, but a novel system developed by Ben-Yami [2014. ‘The quantified argument calculus’, The Review of Symbolic Logic, 7, 120-46] called ‘QUARC’. This system is apt for a more adequate reconstruction since it does not need first-order variables (‘x’, ‘y’, …) on which the usual quantifiers act—a feature also not to be found in Aristotle. Further, in the classical reconstruction, there is also need for binary connectives (‘∧’, ‘→’) that don't have a counterpart in Aristotle. QUARC, again, does not need them either to represent the Aristotelian sentence types. However, the full QUARC is also not called for so that I develop a subsystem thereof (‘QUARC’) which closely resembles Aristotle's way of developing his logic. I show that we can prove all of Aristotle's claims within this systems and, lastly, how it relates to classical logic. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate the problem of existential import in Abelard's modal logic, and to ask whether the system of logical relationships that he proposes for modal propositions maintains its validity when some of the terms included in these propositions are empty. In the following, I first argue that, just as in the case of non-modal propositions, Abelard interprets modal propositions as having existential import, so that it is a necessary condition for the truth of propositions like ‘It is possible for my son to be alive’ or ‘it is necessary that all men are animals’ that their subjects’ referents exist. Then, I present the schemata of inferences that Abelard proposes to describe the logical behaviour of de rebus modal propositions. I argue that these systems of relations are valid only as long as all the terms contained in the formulas have an existing referent. I also claim that Abelard was aware of this difficulty (at least in the Logica Ingredientibus), and, accordingly, he explicitly decided to restrict the validity of his modal system to propositions that do not contain empty terms. [ABSTRACT FROM AUTHOR]

This paper sketches the antilogism of Christine Ladd-Franklin and historical advancement about antilogism, mainly constructs an axiomatic system Atl based on first-order logic with equality and the wholly-exclusion and not-wholly-exclusion relations abstracted from the algebra of Ladd-Franklin, with soundness and completeness of Atl proved, providing a simple and convenient tool on syllogistic reasoning. Atl depicts the empty class and the whole class differently from normal set theories, e.g. ZFC, revealing another perspective on sets and set theories. Two series of Dotterer and the theorem proved by Russinoff are re-characterized in antilogistic formulas corresponding to the square of opposition and twenty-four moods of syllogism. Then the restricted equivalence between two forms of antilogism and minimal inconsistency is built up, implying the internal connection between antilogisms and minimal inconsistency and leaving two related conjectures not proved. Besides, this paper provides a new explanation for the reason why contrariety, subcontrariety, and subalternation and the nine special moods require the non-empty assumption. [ABSTRACT FROM AUTHOR]