This book provides a detailed exposition of one of the most practical and popular methods of proving theorems in logic, called Natural Deduction. It is presented both historically and systematically. Also some combinations with other known proof methods are explored. The initial part of the book deals with Classical Logic, whereas the rest is concerned with systems for several forms of Modal Logics, one of the most important branches of modern logic, which has wide applicability.

La lógica de la Edad Media se presenta a los lógicos contemporáneos, filósofos medievalistas, historiadores y filósofos de la lógica, como un campo tan fascinante como de difícil acceso. Parece demasiado intrincado para casi cualquier investigador de estas áreas encontrar la punta del ovillo que lo conduzca a transitar una presentación ordenada e inteligible de la lógica medieval. Este libro pretende solucionar este problema. Para ello, presenta de manera ordenada y autocontenida los desarrollos lógicos de la parte técnicamente más evolucionada (...) de la lógica de la Edad Media: el siglo XIV. A diferencia de otras presentaciones no evade los hechos históricos y culturales que conforman el panorama de la época ni las cuestiones filosóficas detrás de los desarrollos lógicos, a fin de entender cabalmente las cuestiones técnicas y no desnaturalizar la propuesta medieval. Los primeros tres capítulos se abocan a cuestiones introductorias, históricas y a la genealogía de los tópicos más importantes. El cuarto capítulo está dedicado a la presentación sintáctica del lenguaje de la lógica medieval. El quinto y el sexto a la metateoría de esta lógica. El séptimo explora y presenta la semántica. Los dos últimos nos brindan una reconstrucción racional, en términos del lenguaje simbólico contemporáneo, de la lógica de finales del siglo XIV. (shrink)

At Varro LL VI.56 and SE M 8.275-276, we find reports of the Stoic view that children and articulate non-rational animals such as parrots cannot genuinely speak. Absent from these testimonia is the peculiar case of the superficiality of the actor’s speech, which appears in one edition of the unstable text of PHerc 307.9 containing fragments of Chrysippus’ Logical Investigations. Commentators who include this edition of the text in their discussions of the Stoic theory of speech do not offer a (...) univocal account of the superficiality of the parrot’s, the child’s, and the actor’s speech. In this paper, I offer a reconstruction of the Stoic account of genuine and superficial speech and show that not only is there an account of superficial speech that univocally explains the superficiality of the speech of parrots, children, and actors, but that this account challenges traditional assumptions about the entities at the heart of the Stoic theory of language—lekta. It will turn out that genuine speech is the expression of a lekton by way of performing a speech act, and that this account of superficial speech can be used to explain other linguistic phenomena that are of interest to the Stoics, such as sentences in insoluble sophisms and sentences containing demonstratives that do not refer to anything in the subject term. Importantly, my reconstruction shows, against the near consensus view of lekta, that lekta do not primarily explain what makes an utterance meaningful. Rather, they primarily explain what makes an utterance an instance of genuine speech. (shrink)

We argue that the extant evidence for Stoic logic provides all the elements required for a variable-free theory of multiple generality, including a number of remarkably modern features that straddle logic and semantics, such as the understanding of one- and two-place predicates as functions, the canonical formulation of universals as quantified conditionals, a straightforward relation between elements of propositional and first-order logic, and the roles of anaphora and rigid order in the regimented sentences that express multiply general propositions. We consider (...) and reinterpret some ancient texts that have been neglected in the context of Stoic universal and existential propositions and offer new explanations of some puzzling features in Stoic logic. Our results confirm that Stoic logic surpasses Aristotle’s with regard to multiple generality, and are a reminder that focusing on multiple generality through the lens of Frege-inspired variable-binding quantifier theory may hamper our understanding and appreciation of pre-Fregean theories of multiple generality. (shrink)

An overview of the ancient philosophers and their philosophical system (divided into the fields of logic, physics, and ethics) comprising the living, organic, enduring, and evolving body of interrelated ideas identifiable as the Stoic perspective.

Since Mates’ seminal Stoic Logic there has been uncertainty and debate about how to treat the term anapodeiktos when used of Stoic syllogisms. This paper argues that the customary translation of anapodeiktos by ‘indemonstrable’ is accurate, and it explains why this is so. At the heart of the explanation is an argument that, contrary to what is commonly assumed, indemonstrability is rooted in the generic account of the Stoic epistemic notion of demonstration. Some minor insights into Stoic logic ensue.

'Reason, Causation and Compatibility with the Phenomena' strives to give answers to the philosophical problem of the interplay between realism, explanation and experience. This book is a compilation of essays that recollect significant conceptions of rival terms such as determinism and freedom, reason and appearance, power and knowledge. This title discusses the progress made in epistemology and natural philosophy, especially the steps that led from the ancient theory of atomism to the modern quantum theory, and from mathematization to analytic philosophy. (...) Moreover, it provides possible gateways from modern deadlocks of theory either through approaches to consciousness or through historical critique of intellectual authorities. This work will be of interest to those either researching or studying in colleges and universities, especially in the departments of philosophy, history of science, philosophy of science, philosophy of physics and quantum mechanics, history of ideas and culture. Greek and Latin Literature students and instructors may also find this book to be both a fascinating and valuable point of reference. (shrink)

In his semiotic system Peirce distinguished between two interpretants and two objects of a sign: an immediate and a dynamical interpretant, and an immediate and a dynamical object. It is argued that Peirce’s immediate object can be interpreted a qua-object which has the dynamical object as its basis, and the dynamical interpretant consists of an interpreter’s conception of the object of the sign. Peirce semiotic system is compared with the accounts given by Frege, Husserl, Meinong, and the Stoics.

Logical form has always been a prime concern for philosophers belonging to the analytic tradition. For at least one century, the study of logical form has been widely adopted as a method of investigation, relying on its capacity to reveal the structure of thoughts or the constitution of facts. This book focuses on the very idea of logical form, which is directly relevant to any principled reflection on that method. Its central thesis is that there is no such thing as (...) a correct answer to the question of what is logical form: two significantly different notions of logical form are needed to fulfil two major theoretical roles that pertain respectively to logic and to semantics. This thesis has a negative and a positive side. The negative side is that a deeply rooted presumption about logical form turns out to be overly optimistic: there is no unique notion of logical form that can play both roles. The positive side is that the distinction between two notions of logical form, once properly spelled out, sheds light on some fundamental issues concerning the relation between logic and language. (shrink)

Conditional logics were originally developed for the purpose of modeling intuitively correct modes of reasoning involving conditional—especially counterfactual—expressions in natural language. While the debate over the logic of conditionals is as old as propositional logic, it was the development of worlds semantics for modal logic in the past century that catalyzed the rapid maturation of the field. Moreover, like modal logic, conditional logic has subsequently found a wide array of uses, from the traditional (e.g. counterfactuals) to the exotic (e.g. conditional (...) obligation). Despite the close connections between conditional and modal logic, both the technical development and philosophical exploitation of the latter has outstripped that of the former, with the result that noticeable lacunae exist in the literature on conditional logic. My dissertation addresses a number of these underdeveloped frontiers, producing new technical insights and philosophical applications. -/- I contribute to the solution of a problem posed by Priest of finding sound and complete labeled tableaux for systems of conditional logic from Lewis' V-family. To develop these tableaux, I draw on previous work on labeled tableaux for modal and conditional logic; errors and shortcomings in recent work on this problem are identified and corrected. While modal logic has by now been thoroughly studied in non-classical contexts, e.g. intuitionistic and relevant logic, the literature on conditional logic is still overwhelmingly classical. Another contribution of my dissertation is a thorough analysis of intuitionistic conditional logic, in which I utilize both algebraic and worlds semantics, and investigate how several novel embedding results might shed light on the philosophical interpretation of both intuitionistic logic and conditional logic extensions thereof. -/- My dissertation examines deontic and connexive conditional logic as well as the underappreciated history of connexive notions in the analysis of conditional obligation. The possibility of interpreting deontic modal logics in such systems (via embedding results) serves as an important theoretical guide. A philosophically motivated proscription on impossible obligations is shown to correspond to, and justify, certain (weak) connexive theses. Finally, I contribute to the intensifying debate over counterpossibles, counterfactuals with impossible antecedents, and take—in contrast to Lewis and Williamson—a non-vacuous line. Thus, in my view, a counterpossible like "If there had been a counterexample to the law of the excluded middle, Brouwer would not have been vindicated" is false, not (vacuously) true, although it has an impossible antecedent. I exploit impossible (non-normal) worlds—originally developed to model non-normal modal logics—to provide non-vacuous semantics for counterpossibles. I buttress the case for non-vacuous semantics by making recourse to both novel technical results and theoretical considerations. (shrink)

Šiuo straipsniu mėginama parodyti, jog Indijos ir stoikų logika yra panašesnės viena į kitą nei į standartinę logiką. Šiuo tikslu analizuojama ištrauka iš Kathāvatthu, kurioje pateikiamas sąlyginio teiginio apibrėžimas dažnai interpretuojamas kaip atitinkantis modernioje teiginių logikoje vartojamą materialiosios implikacijos sampratą. Teigiama, jog tokia interpretacija nėra visiškai pagrįsta. Todėl daroma išvada, jog tai, kas iš tiesų sakoma aptariamoje ištraukoje, nuosekliai susiję su Chrisipo Soliečio pasiūlytu sąlyginio teiginio teisingumo kriterijumi.

This paper discusses the reports in Diogenes Laertius and in Sextus Empiricus concerning the classification of propositions. It is argued that the material in Sextus uses a source going back to the Dialectical school whose most prominent members were Diodorus Cronus and Philo of Megara. The material preserved in Diogenes Laertius, on the other hand, goes back to Chrysippus.

This paper is a critical discussion of A.N. Prior’s contribution to the modern understanding of indeterminism and human freedom of choice. Prior suggested that these ideas should be conceived in terms of his tense logic. It can be demonstrated that his approach provides an attractive formalization that makes it possible to discuss indeterminism and human freedom of choice in a very precise manner and in a broader metaphysical context. It is also argued that Prior’s development of this approach was closely (...) linked to his very personal struggles with fundamental religious and metaphysical questions. In his opinion, holding the doctrine of divine foreknowledge together with the doctrine of human freewill gives rise to difficult logical and philosophical problems. It appears that Prior, rather early on, decided to support what was later known as the Peircean solution, although he also analysed what he considered to be the most important alternative solution—the Ockhamist solution. This paper offers a discussion of some of Prior’s published books and papers as well as some of the papers in his Nachlass. (shrink)

This paper seeks to elucidate the distinctive nature of the rational impression on its own terms, asking precisely what it means for the Stoics to define logikē phantasia as an impression whose content is expressible in language. I argue first that impression, generically, is direct and reflexive awareness of the world, the way animals get information about their surroundings. Then, that the rational impression, specifically, is inherently conceptual, inferential, and linguistic, i.e. thick with propositional content, the way humans receive incoming (...) information from the world. When we suspend certain contemporary assumptions about propositional content, the textual evidence can be taken at face value to reveal why, for the Stoics, rational impressions are called thoughts (noēseis) and how the Stoics’ novel semantic entities called lekta (roughly, the meanings of our words) depend on rational impressions for their subsistence. (shrink)

What characterizes indexical thinking is the fact that the modes of presentation through which one thinks of objects are context-bound and perspectival. Such modes of presentation, I claim, are mental files presupposing that we stand in certain relations to the reference : the role of the file is to store information one can gain in virtue of standing in that relation to the object. This raises the communication problem, first raised by Frege : if indexical thoughts are context-bound and relation-based, (...) how is it possible to communicate them to those who are not in the same context and do not stand in the right relations to the object? Following Frege, I argue that the solution comes from an important distinction between linguistic and psychological modes of presentation. Psychological modes of presentation are mental files. They are perspectival and context-bound. But linguistic modes of presentation are fixed by the conventions of the language and they are shared by the language users. They are public and serve to coordinate mental files in communication by constraining them to contain the piece of information they encode. In this way communication takes place even though the indexical thoughts entertained by the speaker are, in some sense, private and cannot be shared by the audience. (shrink)

Commentators have not said much regarding Berkeley and Stoicism. Even when they do, they generally limit their remarks to Berkeley’s Siris (1744) where he invokes characteristically Stoic themes about the World Soul, “seminal reasons,” and the animating fire of the universe. The Stoic heritage of other Berkeleian doctrines (e.g., about mind or the semiotic character of nature) is seldom recognized, and when it is, little is made of it in explaining his other doctrines (e.g., immaterialism). None of this is surprising, (...) considering how Stoics are considered arch-materialists and determinists. My aim is to suggest that our understanding of Berkeley’s philosophy is improved significantly by acknowledging its underlying Stoic character. I argue that Berkeley proposes not only a semantic ontology based on assumptions of Stoic logic but also a doctrine in which perceptions or ideas are intelligible precisely because they are always embedded in the propositions of a discourse or language. (shrink)

Void is at the heart of Stoic metaphysics. As the incorporeal par excellence, being defined purely in terms of lacking body, it brings into sharp focus the Stoic commitment to non-existent Somethings. This article argues that Stoic void, far from rendering the Stoic system incoherent or merely ad hoc, in fact reflects a principled and coherent physicalism that sets the Stoics apart from their materialist predecessors and atomist neighbors.

In Roman rhetoric, contrarium was variably considered either a figure of speech or an argument. The paper examines the logical pattern of this type of argument, which according to Cicero is based on a third Stoic indemonstrable syllogism: $$ \neg ({\hbox{p}} \wedge {\hbox{q}});<$> <$>{\hbox{p}} \to \neg {\hbox{q}}{\hbox{.}} $$ The persuasiveness of this type of argument, however, vitally depends on the validity of the alleged ‹incompatibility’ forming its major premiss. Yet this appears to be the argument’s weak point, as the ‹incompatibilities’ (...) employed generally hold for the most part only, and are reducible to topical argument schemes. This is why in practical usage such arguments are most often phrased as rhetorical questions, the persuasive force of which, enhanced by certain strategical maneuverings and fallacies, makes the audience swallow the argument. (shrink)

Husserl’s philosophy, by the usual account, evolved through three stages: 1. development of an anti-psychologistic, objective foundation of logic and mathematics, rooted in Brentanian descriptive psychology; 2. development of a new discipline of "phenomenology" founded on a metaphysical position dubbed "transcendental idealism"; transformation of phenomenology from a form of methodological solipsism into a phenomenology of intersubjectivity and ultimately (in his Crisis of 1936) into an ontology of the life-world, embracing the social worlds of culture and history. We show that this (...) story of three revolutions can provide at best a preliminary orientation, and that Husserl was constantly expanding and revising his philosophical system, integrating views in phenomenology, ontology, epistemology and logic with views on the nature and tasks of philosophy and science as well as on the nature of culture and the world in ways that reveal more common elements than violent shifts of direction. We argue further that Husserl is a seminal figure in the evolution from traditional philosophy to the characteristic philosophical concerns of the late twentieth century: concerns with representation and intentionality and with problems at the borderlines of the philosophy of mind, ontology, and cognitive science. (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)

The paper deals with the possible readings of The Reaper Argument premisses. Some conjectures related to the Stoics’ alleged proof of the argument are discussed.

It has been noted before in the history of logic that some of Frege's logical and semantic views were anticipated in Stoicism. In particular, there seems to be a parallel between Frege's Gedanke (thought) and Stoic lekton; and the distinction between complete and incomplete lekta has an equivalent in Frege's logic. However, nobody has so far claimed that Frege was actually influenced by Stoic logic; and there has until now been no indication of such a causal connection. In this essay, (...) we attempt, for the first time, to provide detailed evidence for the existence of this connection. In the course of our argumentation, further analogies between the positions of Frege and the Stoics will be revealed. The classical philologist Rudolf Hirzel will be brought into play as the one who links Frege with Stoicism. The renowned expert on Stoic philosophy was Frege's tenant and lived in the same house as the logician for many years. In der Geschichte der Logik ist häufig bemerkt worden, dass einige der logischen und semantischen Auffassungen Freges in der Stoa antizipiert worden sind. Genannt wurden insbesondere die Parallelen zwischen dem Fregeschen Gedanken und dem stoischen Lekton sowie die Unterscheidung zwischen vollständigen und unvollständigen Lekta, die bei Frege ihre Entsprechung hat. Ein Wirkungszusammenhang ist allerdings nicht behauptet worden. Dazu gab es bislang auch keinen Anlass. Der vorliegende Beitrag versucht erstmalig, einen detaillierten Indizienbeweis für das Bestehen eines solchen Zusammenhangs vorzulegen. Dabei werden weitere charakteristische Übereinstimmungen zwischen Frege und der Stoa aufgewiesen. Als Mittelsmann wird der Altphilologe Rudolf Hirzel vorgestellt. Er wohnte lange Jahre als Mieter zusammen mit Frege im selben Haus und war ein anerkannter Experte der stoischen Philosophie. (shrink)

The relation between ex falso and disjunctive syllogism, or even the justification of ex falso based on disjunctive syllogism, is an old topic in the history of logic. This old topic reappears in contemporary logic since the introduction of minimal logic by Johansson. The disjunctive syllogism seems to be part of our general non-problematic inferential practices and superficially it does not seem to be related to or to depend on our acceptance of the frequently disputable ex falso rule. We know (...) that the acceptance of the ex falso is a sufficient condition for the acceptance of the disjunctive syllogism, but the interesting question is: is the ex falso a necessary condition for the acceptance of the disjunctive syllogism? The aim of the present paper is to discuss some possible ways to define systems that combines the preservation of the disjunctive syllogism with the rejection of the ex falso. (shrink)

This open access book is a superb collection of some fifteen chapters inspired by Schroeder-Heister's groundbreaking work, written by leading experts in the field, plus an extensive autobiography and comments on the various contributions by Schroeder-Heister himself. For several decades, Peter Schroeder-Heister has been a central figure in proof-theoretic semantics, a field of study situated at the interface of logic, theoretical computer science, natural-language semantics, and the philosophy of language. -/- The chapters of which this book is composed discuss the (...) subject from a rich variety of angles, including the history of logic, the proper interpretation of logical validity, natural deduction rules, the notions of harmony and of synonymy, the structure of proofs, the logical status of equality, intentional phenomena, and the proof theory of second-order arithmetic. All chapters relate directly to questions that have driven Schroeder-Heister's own research agenda and to which he has made seminal contributions. The extensive autobiographical chapter not only provides a fascinating overview of Schroeder-Heister's career and the evolution of his academic interests but also constitutes a contribution to the recent history of logic in its own right, painting an intriguing picture of the philosophical, logical, and mathematical institutional landscape in Germany and elsewhere since the early 1970s. The papers collected in this book are illuminatingly put into a unified perspective by Schroeder-Heister's comments at the end of the book. Both graduate students and established researchers in the field will find this book an excellent resource for future work in proof-theoretic semantics and related areas. (shrink)

In The Uses of Argument, Stephen Toulmin proposed a model for the layout of arguments: claim, data, warrant, qualifier, rebuttal, backing. Since then, Toulmin’s model has been appropriated, adapted and extended by researchers in speech communications, philosophy and artificial intelligence. This book assembles the best contemporary reflection in these fields, extending or challenging Toulmin’s ideas in ways that make fresh contributions to the theory of analysing and evaluating arguments.

Controversy has surrounded the interpretation of the so-called 'Diodorean' and 'Chrysippean' conditionals of the Stoics. I critically evaluate and reject two interpretations of each of them: as expressing natural laws, and as strict conditionals. In doing so I engage with the work of authors such as Frede, Gould, Hurst, the Kneales, Mates, and Prior. I conclude by offering my own proposal for where these Stoic conditionals should be located on a 'ladder' of logical strength.

En este artículo se analiza el debate sobre la forma adecuada de entender el condicional producido en el siglo IV a. C. El análisis se lleva a cabo mediante el método proporcionado por Rudolf Carnap para estudiar el significado de las expresiones, es decir, mediante el método de extensión e intensión. Los resultados de ese análisis parecen mostrar que, aunque, según Sexto Empirico, el debate fue sobre cuatro criterios diferentes para comprender el condicional, en realidad tres de esos criterios parecen (...) ser iguales y tienen una forma lógica común muy clara en el marco de Carnap. (shrink)

The main issue of the present Article is a discussion about the conditional proposition in Stoic- Megarian Logic. There were, in Stoic-Megarian school, many theories of meaning to interpret these propositions. Since conditional propositions made the structure’s core of their Logic, so for the understanding of the key concept of their thought, we need searching, prior to anything, in the semantics of these propositions. We will, in the first place and much survey, compare between Stoic-Megarian Logic and Aristotelian Logic and (...) present where is the state of conditional propositions in the Stoic-Megarian Logic; then, in the second place, we explore all kinds of conditional proposition and their conditions of truth values in Stoic-Megarian Logic and compare them with the concept of implication in modern logic and reinterpret them with new instruments in modern logic. Most logicians, in this field, evaluate and compare Stoic-Megarian conditional propositions with definitions of Material implication, strict implication, and Natural laws and we, in this Article, evaluate and analyze these propositions with this perspective and ideas. (shrink)

Resumen Atendiendo a la lógica estándar, solo uno de los cinco indemostrables propuestos por Crisipo de Solos es realmente indemostrable. Sus otros cuatro esquemas son demostrables en tal lógica. La pregunta, por tanto, es: si cuatro de ellos no son verdaderamente indemostrables, por qué Crisipo consideró que sí lo eran. López-Astorga mostró que si ignoramos el cálculo proposicional estándar y asumimos que una teoría cognitiva contemporánea, la teoría de la lógica mental, describe correctamente el razonamiento humano, se puede entender por (...) qué Crisipo pensó que todos sus indemostrables eran tan básicos. No obstante, en este trabajo trato de argumentar que la teoría de la lógica mental no es el único marco que puede explicar esto. En concreto, sostengo que otra importante teoría sobre el razonamiento en el presente, la teoría de los modelos mentales, también puede ofrecer una explicación al respecto.According to standard logic, only one of the five indemonstrables proposed by Chrysippus of Soli is actually indemonstrable. The other four schemata are demonstra-ble in that logic. The question hence is, if four of them are not really indemonstrable, why Chrysippus considered them to be so. López-Astorga showed that, if we ignore standard propositional calculus and assume that a current cognitive theory, the mental logic theory, truly describes human reasoning, it can be explained why Chrysippus thought that all of his indemonstrables were so basic. However, in this paper, I try to argue that the mental logic theory is not the only framework that can account for that. In particular, I hold that another important reasoning theory at present, the mental models theory, can offer an explanation in that regard as well. (shrink)

In modal logic, propositions are evaluated relative to possible worlds. A proposition may be true relative to a world w, and false relative to another world w'. Relativism is the view that the relativization idea extends beyond possible worlds and modalities. Thus, in tense logic, propositions are evaluated relative to times. A proposition (e.g. the proposition that Socrates is sitting) may be true relative to a time t, and false relative to another time t'. In this paper I discuss, and (...) attempt to rebut, two classical objections to Relativism. The first objection, due to Frege, is the objection from incompleteness. I distinguish two possible relativist responses to that objection, one of which corresponds to the view I actually defend : Moderate Relativism. The second objection is due to Mark Richard, who argued that the objects of belief cannot be relativistic. I show that that objection can be met within the Moderate Relativist framework. In the last section, I deal with special forms of disagreement that have loomed large in recent discussions of Relativism. (shrink)

ArgumentThis paper argues that ancient Babylonian signs (omens) reflect a mode of inferential reasoning as a function of their syntactic and logical structure as conditionals. Taking into account the institutional context that produced a systematic written body of omens, the paper is principally interested in the cognitive disposition of such texts. Investigating what constitutes system in these works, formal aspects of the material are examined in terms of the nature of conditionals and the logic of conditional statements. It is argued (...) that conditional statements about ominous phenomena speak as much to a notion of conceivability and possibility as to empirical physical actuality. It is claimed that the epistemological character of the Babylonian compendia of signs cannot be understood without due attention to epistemic possibility and further, that the cuneiform omen texts comprise a different kind of scientific knowledge from that limited to a knowledge of physical phenomena and their laws. (shrink)

We show how to construct certain " $[Unrepresented Character]_{M,T}$ -type" interpreted languages, with each such language containing meaningfulness and truth predicates which apply to itself. These languages are comparable in expressive power to the $[Unrepresented Character]_{T}$ -type, truth-theoretic languages first considered by. Kripke, yet each of our $[Unrepresented Character]_{M,T}$ -type languages possesses the additional advantage that, within it, the meaninglessness of any given meaningless expression can itself be meaningfully expressed. One therefore has, for example, the object level truth (and meaningfulness) (...) of the claim that the strengthened Liar is meaningless. (shrink)

Mediaeval logicians inherited from Boethius an account of conditional propositions and the syllogisms which may be constructed using them. In the following paper it is shown that there are considerable difficulties with Boethius' account which arise from his failure to understand the nature of compound propositions and in particular to provide for their negation. Boethius suggests that there are two different conditions which may be imposed for the truth of a conditional proposition but he really gives no adequate account of (...) how such propositions may be obtained. The true greatness of Peter Abaelard as a philosophical logician is revealed in what he is able to do with the material which he found in Boethius. It is shown that he developed a precise theory of conditionals giving an account of how true conditionals may be obtained and principles which may be used to reject others as false. Unlike Boethius Abaelard properly appreciates that conjunctions must be treated as logical units. Even he, however, falls victim to difficulties which arise when this connective is brought into contact with negation and the conditions which he lays down for the truth of a conditional. (shrink)

1 Logic in philosophy The century that was Logic has played an important role in modern philosophy, especially, in alliances with philosophical schools such as the Vienna Circle, neopositivism, or formal language variants of analytical philosophy. The original impact was via the work of Frege, Russell, and other pioneers, backed up by the prestige of research into the foundations of mathematics, which was fast bringing to light those amazing insights that still impress us to-day. The Golden Age of the 1930s (...) deeply affected philosophy, and heartened the minority of philosophers with a formalanalytical bent. As Brand Blanshard writes in Reason and Analysis (1964) – I quote from memory here, to avoid the usual disappointment when re-reading an original text. (shrink)

This article presents a detailed exploration of what Sextus and Pyrrhonists regarded as mnemonic signs, where one experience reminds us of another, such as seeing smoke reminds us of a fire that is not yet evident to our present observations. For the skeptic the use of mnemonic signs obviates the need for reasoned, theoretical interpretations or elaborated belief formation. It allows the skeptic or the theory-free physician, for that matter, to live a life or practice symptomatic medicine without the need (...) for philosophizing. (shrink)

This is a thorough treatment of first-order modal logic. The book covers such issues as quantification, equality (including a treatment of Frege's morning star/evening star puzzle), the notion of existence, non-rigid constants and function symbols, predicate abstraction, the distinction between nonexistence and nondesignation, and definite descriptions, borrowing from both Fregean and Russellian paradigms.

Chrysippus claims that some propositions perish. including some true conditionals whose consequent is impossible and antecedent is possible, to which he appeals against Diodorus?s Master Argument. On the standard interpretation. perished propositions lack truth values. and these conditionals are true at the same time as their antecedents arc possible and consequents impossible. But perished propositions are false, and Chrysippus?s conditionals are true when their antecedent and consequent arc possible, and false when their antecedent is possible and consequent impossible. The claim (...) of the Master Argument that Chrysippus rejects, then, is stronger ihan usually supposed. (shrink)

This entry introduces the reader to the main ideas in Frege's philosophy of logic, mathematics, and language.