Aristotle’s modal syllogistic is his study of patterns of reasoning about necessity and possibility. Many scholars think the modal syllogistic is incoherent, a ‘realm of darkness’. Others think it is coherent, but devise complicated formal modellings to mimic Aristotle’s results. This volume provides a simple interpretation of Aristotle’s modal syllogistic using standard predicate logic. Rini distinguishes between red terms, such as ‘horse’, ‘plant’ or ‘man’, which name things in virtue of features those things must have, and green terms, such as (...) ‘moving’, which name things in virtue of their non-necessary features. By applying this distinction to the Prior Analytics, Rini shows how traditional interpretive puzzles about the modal syllogistic melt away and the simple structure of Aristotle’s own proofs is revealed. The result is an applied logic which provides needed links between Aristotle’s views of science and logical demonstration. The volume is particularly valuable to researchers and students of the history of logic, Aristotle’s theory of modality, and the philosophy of logic in general. (shrink)

This collection of papers, celebrating the contributions of Swedish logician Dag Prawitz to Proof Theory, has been assembled from those presented at the Natural Deduction conference organized in Rio de Janeiro to honour his seminal research. Dag Prawitz’s work forms the basis of intuitionistic type theory and his inversion principle constitutes the foundation of most modern accounts of proof-theoretic semantics in Logic, Linguistics and Theoretical Computer Science. The range of contributions includes material on the extension of natural deduction with higher-order (...) rules, as opposed to higher-order connectives, and a paper discussing the application of natural deduction rules to dealing with equality in predicate calculus. The volume continues with a key chapter summarizing work on the extension of the Curry-Howard isomorphism, via methods of category theory that have been successfully applied to linear logic, as well as many other contributions from highly regarded authorities. With an illustrious group of contributors addressing a wealth of topics and applications, this volume is a valuable addition to the libraries of academics in the multiple disciplines whose development has been given added scope by the methodologies supplied by natural deduction. The volume is representative of the rich and varied directions that Prawitz work has inspired in the area of natural deduction. (shrink)

Essays on Aristotle's Sea-Battle, Lazy Argument, Argument Reaper, Diodorus' Master Argument -/- The book is devoted to the ancient logical theories, reconstruction of their semantic proprieties and possibilities of their interpretation by modern logical tools. The Ancient arguments are frequently misunderstood in modern interpretations since authors usually have tendency to ignore their historical proprieties and theoretical background what usually leads to a quite inappropriate picture of the argument’s original form and mission. Author’s primary intention was to draw attention to the (...) complexity of some historical arguments and to the theoretical context in which arguments were created, circulated, developed, and finally tuned. Four well-known ancient arguments – with a common central subject related to the future contingencies problem – are reconstructed from available historical sources: “The Sea Battle”, which is drawn from Aristotle’s treatise De Interpretatione; two arguments, usually ascribed to the Stoics, “The Lazy Argument” and “The Reaper”; “The Master Argument” of the Megarian philosopher Diodorus. Arguments are linguistically and semantically detaily analyzed, formally presented by reflecting some relevant corresponding hypotheses based on physical or logical theories of their ancient authors, and finally covered by appropriate logical tools familiar to a modern reader. Two appendices are added at the closing part of the book. One covering some assumptions relevant for understanding of rival streams in ancient theories of meaning related to the nature of names and naming; the other is devoted to the ancient understanding of logical proposition and attempts to find an adequate Latin translation of the Greek delicate philosophical term “ἀξίωμα”. (shrink)

Contemporary accounts of logic and language cannot give proper treatments of plural constructions of natural languages. They assume that plural constructions are redundant devices used to abbreviate singular constructions. This paper and its sequel, "The logic and meaning of plurals, II", aim to develop an account of logic and language that acknowledges limitations of singular constructions and recognizes plural constructions as their peers. To do so, the papers present natural accounts of the logic and meaning of plural constructions that result (...) from the view that plural constructions are, by and large, devices for talking about many things (as such). The account of logic presented in the papers surpasses contemporary Fregean accounts in its scope. This extension of the scope of logic results from extending the range of languages that logic can directly relate to. Underlying the view of language that makes room for this is a perspective on reality that locates in the world what plural constructions can relate to. The papers suggest that reflections on plural constructions point to a broader framework for understanding logic, language, and reality that can replace the contemporary Fregean framework as this has replaced its Aristotelian ancestor. (shrink)

This article examines the traditional and modern doctrines of categorical propositions and argues that both doctrines have serious problems. While the doctrines disagree about existential imports...

A theory of syllogistic reasoning is proposed, derived from the medieval doctrine of 'distribution of terms'. This doctrine may or may not furnish an adequate ground for the logic of the syllogism but does appear to illuminate the psychological processes involved. Syllogistic thinking is shown to have its origins in the approach and avoidance behaviour of pre-verbal organisms and, in verbal (human) organisms, to bridge the gap between the intuitive grasp shown by most of us of the validity of simple (...) logical arguments and the failure of intuition in more complex arguments that require resort to calculation. Some difficulties are considered affecting the use of syllogisms as experimental material. These include failure on the part of the investigator to take account of the fact that a syllogism is always part of a continuing argument in which the topic of the argument is known to all parties and the possibility that subjects may find ways of appearing to solve syllogisms without actually doing so. (shrink)

Older approaches to planning lack the capacity to be responsive to complexity. A new philosophy supporting a new mode of practice could improve significantly the capacity of society to cope with complexity in design, planning, and policy making. The new philosophy and practice must be generic; in other words, it must be divorced in its philosophy and approach from any particular kind of planning activity. It must emphasize the capacity to enhance the work of groups in designing new conceptual structures (...) whether they be plans, policies, or tangible products. (shrink)

In his recent article, titled ‘Royaumont Revisited’, Overgaard challenges Dummett's view that one needs to go as far back as the late nineteenth century in order to discover examples of genuine dialogue between ‘analytic’ and ‘continental’ philosophy. Instead, Overgaard argues that in the 1958 Royaumont colloquium, generally judged as a failed attempt at communication between the two camps, one can find some elements which may be utilized towards re-establishing a dialogue between these two sides. Yet, emphasising this image of Royaumont (...) as a kind of battleground between ‘analytic’ and ‘continental’ philosophy obscures the plurality of philosophical approaches involved. Royaumont was the meeting point of more than two philosophical traditions, as can be shown by the divergent interests of its participants. Thus, though the potential for rapprochement between Oxford ‘linguistic philosophy’ and a certain strand of phenomenological thought may indeed be found among the discussions that took place during the colloquium, one should keep in mind that such rapprochement took place in the context of a meeting between, among others: continental ‘analytics’, Anglophone non-‘analytics’, French historians of philosophy, ‘analytic’ opponents of Oxford philosophy, Franciscan phenomenologists, and Oxonians who called their work ‘phenomenology’. (shrink)

The traditional definition per genus et differentiam is argued to be cognitively grounded in perception and in order to avoid needless argument, definitions are stipulated to assert boundaries. An analysis of the notion of perspective shows that a boundary is a composite of two distinctions: similarity that includes and difference that excludes. The concept is applied to the type-token distinction and percepts are shown to be the result of a comparison between a token as representing some phenomenon and a type (...) as representing the kind of object the phenomenon might be. The Principle of Connection is proposed as the mental architecture that gives rise to percepts. The value of a percept is shown to be one of Identical, Similar, Different or Contrary and its perceptual aspects represent the cognitive schema of Essence, Quality, Quantity and Relation. The two notions of similarity and difference are claimed to constitute the necessary and sufficient conditions for definitions. Accordingly, the qualitative elements of a definition signify the genus as representing the similarity of the defined entity in terms of others of its kind and the quantitative elements signify the differentia as representing the difference it has from everything else. Paying proper attention to both qualitative and quantitative aspects will enable formulation of definitions that are ‘good’. (shrink)

Though Arthur Prior is now best known for his founding of modern temporal logic and hybrid logic, much of his early philosophical career was devoted to history of logic and historical logic. This interest laid the foundations for both of his ground-breaking innovations in the 1950s and 1960s. Because of the important rôle played by Prior's research in ancient and medieval logic in his development of temporal and hybrid logic, any student of Prior, temporal logic, or hybrid logic should be (...) familiar with the medieval logicians and their work. In this article we give an overview of Prior's work in ancient and medieval logic. (shrink)

This article presents Tarski's Address at the Princeton Bicentennial Conference on Problems of Mathematics, together with a separate summary. Two accounts of the discussion which followed are also included. The central topic of the Address and of the discussion is decision problems. The introductory note gives information about the Conference, about the background of the subjects discussed in the Address, and about subsequent developments to these subjects.

This article presents Tarski's Address at the Princeton Bicentennial Conference on Problems of Mathematics, together with a separate summary. Two accounts of the discussion which followed are also included. The central topic of the Address and of the discussion is decision problems. The introductory note gives information about the Conference, about the background of the subjects discussed in the Address, and about subsequent developments to these subjects.

Inference versus consequence , an invited lecture at the LOGICA 1997 conference at Castle Liblice, was part of a series of articles for which I did research during a Stockholm sabbatical in the autumn of 1995. The article seems to have been fairly effective in getting its point across and addresses a topic highly germane to the Uppsala workshop. Owing to its appearance in the LOGICA Yearbook 1997 , Filosofia Publishers, Prague, 1998, it has been rather inaccessible. Accordingly it is (...) republished here with only bibliographical changes and an afterword. (shrink)

An explanation for the uncertain progress of formalist linguistics is sought in an examination of the concept of syntax. The idea of analyzing language formally was made possible by developments in 20th century logic. It has been pointed out by many that the analogy between natural language and a formal system may be imperfect, but the objection made here is that the very concept of syntax, when applied to any non-abstract system of communication, is flawed as it is commonly used. (...) Syntax is properly defined with respect to an individual transformation rule that might be applied to some message. Collections of syntax rules, however, are inevitably due to categories imposed by an observer, and do not correspond to functional features found in non-abstract systems. As such, these categories should not be relied upon as aids to understanding any natural system. (shrink)

Well over a century after its introduction, Frege's two-dimensional Begriffsschrift notation is still considered mainly a curiosity that stands out more for its clumsiness than anything else. This paper focuses mainly on the propositional fragment of the Begriffsschrift, because it embodies the characteristic features that distinguish it from other expressively equivalent notations. In the first part, I argue for the perspicuity and readability of the Begriffsschrift by discussing several idiosyncrasies of the notation, which allow an easy conversion of logically equivalent (...) formulas, and presenting the notation's close connection to syntax trees. In the second part, Frege's considerations regarding the design principles underlying the Begriffsschrift are presented. Frege was quite explicit about these in his replies to early criticisms and unfavorable comparisons with Boole's notation for propositional logic. This discussion reveals that the Begriffsschrift is in fact a well thought-out and carefully crafted notation that intentionally exploits the possibilities afforded by the two-dimensional medium of writing like none other. (shrink)

Published in 1908, C.S. Peirce’s ‘A Neglected Argument for the Reality of God’ is one of his most difficult articles. Presenting a peculiar entanglement of scientific method and theology, it sketches a ‘humble’ argument for the reality—and not the existence—of God for Musers, that is, those who pursue the activity he calls ‘Musement’. In Musement, Peirce claims, we can achieve a kind of perception of the intertwinement of the three universes of experience: of feeling, of brute fact, and of reason. (...) He also somehow relates each universe to a distinct phase of inquiry, which is described by the use of induction, deduction, and abduction or retroduction. The way that he develops his claims allows him to outline God as an abductive vague hypothesis to explain how those three universes make up a single whole. The hypothesis being vague means that the principle of contradiction does not hold for it. In this presentation, we aim at throwing some light at these points, focusing on the concept of Musement and what is understood as the vagueness of the hypothesis. (shrink)

Швидкий технологічний прогрес спричинив зрушення інтересів науковців з усього світу в бік когнітивної науки та інформатики, що, безперечно, вплинуло на формування нового типу мислення. Логіка, як наука про специфічний логічний тип людського мислення, не може залишитися осторонь. У статі проаналізовано методологічні засади когнітивістики та дуальність самої науки логіки, які відкривають перспективи їх співпраці. Для цього автор робить історико-методологічний розгляд обох дисциплін. Доводиться, що, з одного боку, формальна логіка багато вкладає в розвиток штучних формалізованих мов, які активно застосовуються для програмування Штучного (...) Інтелекту, а з іншого – неформальна логіка, як і когнітивістика, займається вивченням природньої мови, що дає їй можливість наблизитися до реального людського мислення. Показано, в який спосіб логіка, застосувавши для вивчення реального процесу мислення людей методологічну базу когнітивістики, може розширити межі своїх досліджень з теоретичного поля до практично-експериментального. (shrink)

Jan Lukasiewicz's treatise on Aristotle's Syllogistic, published in the 1950s, has been very influential in framing contemporary understanding of Aristotle's logical systems. However, Lukasiewicz's interpretation is based on a number of tendentious claims, not least, the claim that the syllogistic was intended to apply only to non-empty terms. I show that this interpretation is not true to Aristotle's text and that a more coherent and faithful interpretation admits empty terms while maintaining all the relations of the traditional square of opposition.

In this essay, I discuss some observations by Peirce which suggest he had some idea of the substantive metalogical differences between logics which permit both quantifiers and relations, and those which do not. Peirce thus seems to have had arguments?which even De Morgan and Frege lacked?that show the superior expressiveness of relational logics.

De acuerdo con los principales enfoques al respecto la lógica de los estoicos es principalmente un sistema deductivo, lo que, en términos actuales, ha sido visto como un sistema de lógica proposicional. La obra de Crísipo acerca de los cinco argumentos indemostrables constituye la principal base de dicho sistema. En este artículo se examina la naturaleza de dichos cinco indemostrables así como el llamando teorema de Antipatro y los esquemas básicos de inferencia, o zemas. Por otra parte y en particular, (...) sostengo que dichos argumentos indemostrables tienen otro importante rol para la filosofía y la historia de la lógica, cual es el de constituirse en una justificación de la deducción, un problema central en el presente. Se analiza, primero, la justificación por los indemostrables y por el principio de condicionalización y, segundo, la justificación cognitiva de la lógica implícita en un sistema estoico. (shrink)

Of the various accounts of negation that have been offered by logicians in the history of Western logic, that of negation as cancellation is a very distinctive one, quite different from the explosive accounts of modern "classical" and intuitionist logics, and from the accounts offered in standard relevant and paraconsistent logics. Despite its ancient origin, however, a precise understanding of the notion is still wanting. The first half of this paper offers one. Both conceptually and historically, the account of negation (...) as cancellation is intimately connected with connexivist principles such as ¬( ¬). Despite this, standard connexivist logics incorporate quite different accounts of negation. The second half of the paper shows how the cancellation account of negation of the first part gives rise to a semantics for a simple connexivist logic. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Whose Aristotle? Which Aristotelianism? A Historical Prolegomenon to Thomas Farrell’s Norms of Rhetorical CultureCarol PosterThe description of various works of logical and rhetorical theory as “Aristotelian,” although far from unusual, is not particularly informative, because it assumes, incorrectly, that there is some ultimate singular Aristotle being imitated by all authors who consider themselves, or who are labeled by others, Aristotelian. In fact, there never has been an interpretation of (...) Aristotle that satisfies the canon of St. Vincent of Lerins, “quod ubique, quod semper, quod ab omnibus creditum est”; instead, there have been almost as many Aristotles as there have been interpreters of Aristotle. Thomas Farrell’s Norms of Rhetorical Culture (1993), therefore, is not so much situated against the background of some single, unique, undisputed Aristotle but, rather, as he himself mentions, within an evolving tradition of Aristotelianism.The historical transformations of Aristotle include two distinct traditions, one oriented toward scientific-technical philosophy andthe other toward rhetorical-poetic and ethical humanism.1 The technical-philosophccical tradition, in which Aristotle was conceived primarily [End Page 375] as metaphysician and natural philosopher, emphasized the logical treatises of the organon, the speculative systems of the Physics and Metaphysics, and the scientific works of the biological treatises. This tradition originated in the Greco-Roman middle and Neoplatonic schools and has had a continuous history stretching through the Middle Ages, Counterreformation, neo-Thomism, and certain contemporary European postmodernisms. The rhetoricalhumanistic tradition, on the other hand, emphasizing the Rhetoric, Poetics, Politics, and ethical treatises, views Aristotle as a practical philosopher, concerned with the productive arts and those matters about which knowledge is probable rather than certain. This rhetorical-humanistic tradition was a relatively negligible aspect of Aristotle’s ancient reception, had a slight presence in the Islamic and Latin traditions, developed in certain strands of Renaissance humanism, and came to fruition in the late eighteenth century, primarily, though not exclusively, within a Protestant tradition. In eighteenth-and early nineteenth-century Britain, the rhetorical-humanistic Aristotle found expression directly in neoclassical poetics and, in a more complicated fashion, in a group of authors (including the Scottish moderates and the slightly later Oxford Noetics) who enlisted Aristotle in a battle against skepticism and materialism. It is the latter of the two strands of the rhetorical-humanistic Aristotle that resurfaces (albeit in an indirect and mediated fashion) in the work of Alastair MacIntyre, Richard McKeon (and other members of the Chicago school), and Thomas Farrell, who used Aristotle’s model of probable deliberation and habituated virtue to construct an alternative to twentieth-century skepticism and materialism.Significant Strata of Aristotelian ReceptionThe history of Aristotelian reception is not monolithic. There has been tremendous variation across periods and national traditions of what works were considered authentically Aristotelian, which of those works were considered important, and how they were interpreted. The phases of Aristotelian reception immediately relevant to Farrell’s work include Greco-Roman Platonism, Islamic commentary, medieval Latin Christianity, Renaissance humanistic textual recovery, neoclassical “Aristotelianism,” early modern Aristotelianism and anti-Aristotelianism, and the nineteenthcentury rehabilitation of Aristotle. This article will summarize briefly pre eighteenth-century interpretations of Aristotle as prolegomena to a close analysis of the works of two eighteenth-century Scotsmen, James Moor [End Page 376] and John Gillies, whose reconfiguration of Aristotle’s Rhetoric, Poetics, and Organon as tools for practical wisdom in modern society anticipated (and, albeit indirectly, influenced) Farrell.2AntiquityTwo distinct strands of Aristotelian reception, the rhetorical-humanistic and technical-philosophical, originated in antiquity. Of these, the rhetorical-humanistic strand was substantially less important. Neither the Poetics nor the Rhetoric was well known in antiquity. The Rhetoric was not the subject of a Greek commentary until the Byzantine period and, although occasionally cited by Greco-Roman rhetoricians including Cicero, Quintilian, and Dionysus of Halicarnassus, was not widely or carefully read.3 The Poetics was also virtually unknown and was never the subject of an ancient commentary.4The Greek commentaries on Aristotle, which are predominantly technical-philosophical in character, include some fifteen thousand pages of extant text. Despite the existence of a distinct ancient Peripatetic school, the majority of commentaries were produced in the Platonic schools, where the works of Aristotle... (shrink)

We present a set-theoretic model of the mental representation of classically quantified sentences (All P are Q, Some P are Q, Some P are not Q, and No P are Q). We take inclusion, exclusion, and their negations to be primitive concepts. We show that although these sentences are known to have a diagrammatic expres- sion (in the form of the Gergonne circles) that constitutes a semantic representation, these concepts can also be expressed syntactically in the form of algebraic formulas. (...) We hypothesized that the quantified sen- tences have an abstract underlying representation common to the formulas and their associated sets of dia- grams (models). We derived 9 predictions (3 semantic, 2 pragmatic, and 4 mixed) regarding people’s as- sessment of how well each of the 5 diagrams expresses the meaning of each of the quantified sentences. We report the results from 3 experiments using Gergonne’s (1817) circles or an adaptation of Leibniz (1903/ 1988) lines as external representations and show them to support the predictions. (shrink)

My first paper on the Is/Ought issue. The young Arthur Prior endorsed the Autonomy of Ethics, in the form of Hume’s No-Ought-From-Is (NOFI) but the later Prior developed a seemingly devastating counter-argument. I defend Prior's earlier logical thesis (albeit in a modified form) against his later self. However it is important to distinguish between three versions of the Autonomy of Ethics: Ontological, Semantic and Ontological. Ontological Autonomy is the thesis that moral judgments, to be true, must answer to a realm (...) of sui generis non-natural PROPERTIES. Semantic autonomy insists on a realm of sui generis non-natural PREDICATES which do not mean the same as any natural counterparts. Logical Autonomy maintains that moral conclusions cannot be derived from non-moral premises.-moral premises with the aid of logic alone. Logical Autonomy does not entail Semantic Autonomy and Semantic Autonomy does not entail Ontological Autonomy. But, given some plausible assumptions Ontological Autonomy entails Semantic Autonomy and given the conservativeness of logic – the idea that in a valid argument you don’t get out what you haven’t put in – Semantic Autonomy entails Logical Autonomy. So if Logical Autonomy is false – as Prior appears to prove – then Semantic and Ontological Autonomy would appear to be false too! I develop a version of Logical Autonomy (or NOFI) and vindicate it against Prior’s counterexamples, which are also counterexamples to the conservativeness of logic as traditionally conceived. The key concept here is an idea derived in part from Quine - that of INFERENCE-RELATIVE VACUITY. I prove that you cannot derive conclusions in which the moral terms appear non-vacuously from premises from which they are absent. But this is because you cannot derive conclusions in which ANY (non-logical) terms appear non-vacuously from premises from which they are absent Thus NOFI or Logical Autonomy comes out as an instance of the conservativeness of logic. This means that the reverse entailment that I have suggested turns out to be a mistake. The falsehood of Logical Autonomy would not entail either the falsehood Semantic Autonomy or the falsehood of Ontological Autonomy, since Semantic Autonomy only entails Logical Autonomy with the aid of the conservativeness of logic of which Logical Autonomy is simply an instance. Thus NOFI or Logical Autonomy is vindicated, but it turns out to be a less world-shattering thesis than some have supposed. It provides no support for either non-cognitivism or non-naturalism. (shrink)

Contextual theories of truth are motivated primarily by the resolution they provide to paradoxical reasoning about truth. The principal argument for contextual theories of truth relies on a key intuition about the truth value of the proposition expressed by a particular utterance made during paradoxical reasoning, which Anil Gupta calls “the Chrysippus intuition.” In this paper, I argue that the principal argument for contextual theories of truth is circular, and that the Chrysippus intuition is false. I conclude that the philosophical (...) motivation for contextual theories of truth fails. (shrink)

Inference is elementary and ubiquitous: Cognition always goes beyond the data. Thinking—including problem solving, decision making, judgement, planning, and argumentation—is here defined as the deliberate application and coordination of one's inferences to serve one's purposes. Reasoning, in turn, is epistemologically self-constrained thinking in which the application and coordination of inferences is guided by a metacognitive commitment to what are deemed to be justifiable inferential norms. The construction of rationality, in this view, involves increasing consciousness and control of logical and other (...) inferences. This metacognitive conception of rationality begins with logic rather than ending with it, and allows for developmental progress without positing a state of maturity. (shrink)

An investigation of Proclus' logic of the syllogistic and of negations in the Elements of Theology, On the Parmenides, and Platonic Theology. It is shown that Proclus employs interpretations over a linear semantic structure with operators for scalar negations (hypemegationlalpha-intensivum and privative negation). A natural deduction system for scalar negations and the classical syllogistic (as reconstructed by Corcoran and Smiley) is shown to be sound and complete for the non-Boolean linear structures. It is explained how Proclus' syllogistic presupposes converting the (...) tree of genera and species from Plato's diairesis into the Neoplatonic linear hierarchy of Being by use of scalar hyper and privative negations. (shrink)

Logic is usually thought to concern itself only with features that sentences and arguments possess in virtue of their logical structures or forms. The logical form of a sentence or argument is determined by its syntactic or semantic structure and by the placement of certain expressions called “logical constants.”[1] Thus, for example, the sentences Every boy loves some girl. and Some boy loves every girl. are thought to differ in logical form, even though they share a common syntactic and semantic (...) structure, because they differ in the placement of the logical constants “every” and “some”. By contrast, the sentences Every girl loves some boy. and Every boy loves some girl. are thought to have the same logical form, because “girl” and “boy” are not logical constants. Thus, in order to settle questions about logical form, and ultimately about which arguments are logically valid and which sentences logically true, we must distinguish the “logical constants” of a language from its nonlogical expressions. (shrink)

It is often assumed that Aristotle, Boethius, Chrysippus, and other ancient logicians advocated a connexive conception of implication according to which no proposition entails, or is entailed by, its own negation. Thus Aristotle claimed that the proposition ‘if B is not great, B itself is great […] is impossible’. Similarly, Boethius maintained that two implications of the type ‘If p then r’ and ‘If p then not-r’ are incompatible. Furthermore, Chrysippus proclaimed a conditional to be ‘sound when the contradictory of (...) its consequent is incompatible with its antecedent’, a view which, in the opinion of S. McCall, entails the aforementioned theses of Aristotle and Boethius. Now a critical examination of the historical sources shows that the ancient logicians most likely meant their theses as applicable only to ‘normal’ conditionals with antecedents which are not self-contradictory. The corresponding restrictions of Aristotle’s and Boethius’ theses to such self-consistent antecedents, however, turn out to be theorems of ordinary modal logic and thus don’t give rise to any non-classical system of genuinely connexive logic,. (shrink)

To believe that logic has no history might at first seem peculiar today. But since the early 20th century, this position has been repeatedly conflated with logical monism of Kantian provenance. This logical monism asserts that only one logic is authoritative, thereby rendering all other research in the field marginal and negating the possibility of acknowledging a history of logic. In this paper, I will show how this and many related issues have developed, and that they are founded on only (...) one prominent statement by Kant. I will argue, however, that this statement takes on a very different meaning in a broader context of the history and philosophy of science, and that Kant and his supporters never advocated the logical monism that they are still said to hold today. (shrink)

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. (shrink)

This paper describes Peirce's systems of logic diagrams, focusing on the so-called ''existential'' graphs, which are equivalent to the first-order predicate calculus. It analyses their implications for the nature of mental representations, particularly mental models with which they have many characteristics in common. The graphs are intended to be iconic, i.e., to have a structure analogous to the structure of what they represent. They have emergent logical consequences and a single graph can capture all the different ways in which a (...) possibility can occur. Mental models share these properties. But, as the graphs show, certain aspects of propositions cannot be represented in an iconic or visualisable way. They include negation, and the representation of possibilities qua possibilities, which both require representations that do not depend on a perceptual modality. Peirce took his graphs to reveal the fundamental operations of reasoning, and the paper concludes with an analysis of different hypotheses about these operations. (shrink)

Logical psychologism is the position that logic is a special branch of psychology, that logical laws are descriptíons of experience to be arrived at through observation, and are a posteriori.The accepted arguments against logical psychologism are effective only when directed against this extreme version. However, the clauses in the above characterization are independent and ambiguous, and may be considered separately. This separation permits a reconsideration of less extreme attempts to tie logic to psychology, such as those defended by Mill and (...) Boole. It also provides the basis for a reexamination of the relationship between logic and psychology, and raises the possibility of a deeper investigation into the nature of logic itself. (shrink)

Despite its basically syllogistic character, Richard Whately's Elements of logic presents the subject in a modern theoretical setting. Whately, for instance, regarded logic as an abstract science, and defined the syllogism as a purely formal device to be used as a means of determining the validity of all arguments. In this paper, I argue that such instances of abstractive ascent place Whately's theory in closer proximity to later 19th-century developments than to the work of his 17th-century predecessors. In addition to (...) discussing the theory itself, other sections deal with its historical setting, and various reactions to it. (shrink)

We consider the history of logic in pre-Petrine. Petrine. and immediate post-Pctrine Russia (from the 15th to the mid-18th centuries) and especially of the Petrine era from the late 17th to early 18th century. Throughout much of this time, the clergy evinced strong hostility towards logic. Nevertheless, a small number of academics and clerics such as Stefan Iavorskii and Fcofan Prokopovich kept Aristotelian logic alive during this period and provided the foundation for its development in the modern era.

In various theories of truth, people have set forth many definitions to clarify in what sense a set of sentences is paradoxical. But what, exactly, is _a_ paradox per se? It has not yet been realized that there is a gap between ‘being paradoxical’ and ‘being a paradox’. This paper proposes that a paradox is a minimally paradoxical set meeting some closure property. Along this line of thought, we give five tentative definitions based upon the folk notion of paradoxicality implied (...) in Tarski’s undefinability theorem of truth and the dependence relation proposed by Leitgeb (J Philos Log 34(2):155–192, 2005). A definition of ‘being a paradox’ is acceptable if its extension must be sufficiently comprehensive to include all known paradoxes on the one hand, and its standard must also be high enough to exclude those that are evidently not paradoxes on the other hand. It turns out that only the last attempt is acceptable: a set of sentences is a paradox if it is paradoxical and self-dependent, and at the same time, it and any of its paradoxical and self-dependent subsets bisimulate each other. In this way, we articulate in what sense a set of sentences is a paradox, and so establish a connection between the notion of a paradox and the notion of paradoxicality. (shrink)