ABSTRACT: A comprehensive introduction to ancient (western) logic from earliest times to the 6th century CE, with an emphasis on topics which may be of interest to contemporary logicians. Content: 1. Pre-Aristotelian Logic 1.1 Syntax and Semantics 1.2 Argument Patterns and Valid Inference 2. Aristotle 2.1 Dialectics 2.2 Sub-sentential Classifications 2.3 Syntax and Semantics of Sentences 2.4 Non-modal Syllogistic 2.5 Modal Logic 3. The early Peripatetics: Theophrastus and Eudemus 3.1 Improvements and Modifications of Aristotle's Logic 3.2 Prosleptic Syllogisms 3.3 Forerunners (...) of Modus Ponens and Modus Tollens 3.4 Wholly Hypothetical Syllogisms 4. Diodorus Cronus and Philo the Logician 5. The Stoics 5.1 Logical Achievements Besides Propositional Logic 5.2 Syntax and Semantics of Complex Propositions 5.3 Arguments 5.4 Stoic Syllogistic 5.5 Logical Paradoxes 6. Epicurus and the Epicureans 7. Later Antiquity. (shrink)

In De interpretatione 9, Aristotle argues against the fatalist view that if statements about future contingent singular events (e.g. ‘There will be a sea battle tomorrow,’ ‘There will not be a sea battle tomorrow’) are already true or false, then the events to which those statements refer will necessarily occur or necessarily not occur. Scholars have generally held that, to refute this argument, Aristotle allows that future contingent statements are exempt from either the principle of bivalence, or the law of (...) excluded middle. This paper offers a new interpretation of Aristotle’s refutation of fatalism. According to this interpretation, each member of a pair of contradictory future contingent statements, in virtue of expressing modal necessity, is simply false. (shrink)

In the last book of The Consolation of Philosophy, Boethius develops his solution to the problem of divine foreknowledge and free choice. Interpreters standardly hold that this problem and his solution to it presuppose causal indeterminism. In this paper, I argue that Boethius, following a Neoplatonist view found in Proclus, is a causal determinist and compatibilist and maintains that God’s providential knowledge ensures the occurrence of all the events he knows. This alternative interpretation offers a better fit with Boethius’s text (...) and its historical antecedents and provides resources to address the main criticisms of his solution. (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)

Is a mathematical theorem proved because provable, or provable because proved? If Brouwer’s intuitionism is accepted, we’re committed, it seems, to the latter, which is highly problematic. Or so I will argue. This and other consequences of Brouwer’s attempt to found mathematics on the intuition of a move of time have heretofore been insufficiently appreciated. Whereas the mathematical anomalies of intuitionism have received enormous attention, too little time, I’ll try to show, has been devoted to some of the temporal anomalies (...) that Brouwer has invited us to introduce into mathematics. (shrink)

The goal of this paper is to figure out whether Aristotle's response to the argument for fatalism in De Interpretatione 9 is a success. By "response" it is meant not simply the reasons Aristotle offers to highlight why fatalism does not accord with how we conduct our lives, but also the solution he devises to block the argument for fatalism. This paper finds that a) Aristotle's argument for fatalism is essentially bivalence plus that the truth of a proposition implies necessity, (...) b) that Aristotle's solution is to restrict bivalence, c) that this solution is coherent, and d) that while this solution does not rule out the possibility of fatalism, it does succeed in blocking the argument for fatalism offered within chapter 9. El objetivo de este trabajo es esclarecer si la respuesta de Aristóteles al argumento a favor del fatalismo en De Interpretatione 9 resulta exitosa. Por "respuesta" se entienden no solo las razones que da Aristóteles para destacar porqué el fatalismo no concuerda con la manera en que llevamos nuestras vidas, sino también la solución que ofrece para bloquear el argumento a favor del fatalismo. El artículo plantea a) que el argumento de Aristóteles respecto del fatalismo consiste esencialmente en la bivalencia más el hecho de que la verdad de una proposición implica necesidad; b) que la solución de Aristóteles es restringir la bivalencia; c) que esta solución es coherente, y d) que si bien esta solución no elimina la posibilidad del fatalismo, sí logra bloquear el argumento a favor del fatalismo ofrecido en el capítulo 9. O objetivo deste trabalho é esclarecer se a resposta de Aristóteles ao argumento a favor do fatalismo em De Interpretatione 9 é bem-sucedida. Por "resposta" se entendem não somente as razões que Aristóteles dá para destacar por que o fatalismo não está de acordo com a maneira em que levamos nossas vidas, mas também a solução que oferece para bloquear o argumento a favor do fatalismo. Este artigo propõe que: a) o argumento de Aristóteles a respeito do fatalismo consiste essencialmente na bivalência mais o fato de que a verdade de uma proposição implica necessidade; b) a solução de Aristóteles é restringir a bivalência; c) essa solução é coerente, e d) embora essa solução não elimine a possiblidade do fatalismo, consegue bloquear o argumento a favor dele oferecido no capítulo 9. (shrink)

Ancient commentators like Ammonius and Boethius tried to solve Aristotle's “sea battle argument” in On Interpretation 9 by saying that statements about future contingents are “indefinitely” true or false. They were followed by al-Fārābī in his commentary on On Interpretation. The article sets out two possible interpretations of what “indefinitely” means here, and shows that al-Fārābī actually has both conceptions: one applied in his interpretation of Aristotle, and another that he is forced into by the problem of divine foreknowledge. It (...) also explains the relevance of al-Fārābī's remarks as a link between the non-statistical modal theories of Philo and Avicenna. (shrink)

The problem of future contingents is regarded as an important philosophical problem in connection with determinism and it should be treated by tense logic. Prior’s early work focused on the problem, and later Prior studied branching-time tense logic which was invented by Kripke. However, Prior’s idea to use three-valued logic for the problem seems to be still alive. In this paper, we consider partial and paraconsistent approaches to the problem of future contingents. These approaches theoretically meet Aristotle’s interpretation of future (...) contingents. (shrink)

In this article, I will argue that the argument for fatalism based on the relativity of simultaneity fails. The original proponents of the argument called the thesis in terms of ‘determinism’, but Levin refers to it as ‘relativistic fatalism’. Relativistic fatalism is a view supported by the alleged dependence of the property of being future on an arbitrary choice of some coordinate system. First I will try to explain the classic argument, attributed to it a dialectic that justified to call (...) it in the same terms as Levin did. Subsequently, I will refuse the relativistic fatalism using many strategies to deal with it. (shrink)