A 'self-refutation argument' is any argument which aims at showing that a certain thesis is self-refuting. This study was the first book-length treatment of ancient self-refutation and provides a unified account of what is distinctive in the ancient approach to the self-refutation argument, on the basis of close philological, logical and historical analysis of a variety of sources. It examines the logic, force and prospects of this original style of argumentation within the context of ancient philosophical debates, dispelling various misconceptions (...) concerning its nature and purpose and elucidating some important differences which exist both within the ancient approach to self-refutation and between that approach, as a whole, and some modern counterparts of it. In providing a comprehensive account of ancient self-refutation, the book advances our understanding of influential and debated texts and arguments from philosophers like Democritus, Plato, Aristotle, Epicurus, the Stoics, the Academic sceptics, the Pyrrhonists and Augustine. (shrink)

When in classical formal logic the notions of deduction, valid inference and logical consequence are defined, causal language plays no role. The founder of western logic, Aristotle, identified ‘non-cause’, or ‘positing as cause what is not a cause’, as a logical fallacy. I argue that a systematic re-examination of Aristotle's analysis of NCF, and the related language of logical causality, in the Sophistical Refutations, Topics, Analytics and Rhetoric, helps us to understand his conception of. It reveals that Aristotle's syllogismhood is (...) non-monotonic, and thus narrower than classical validity as consisting in, or coextensive with, necessary truth-preservation. It also supports the conclusion that a syllogism is a form of relevant consequence: the premisses must be causes of the conclusion in the sense that they are not only sufficient but also necessary for the conclusion to follow. Close analysis of NCF also suggests that syllogistic relevance is not a semantic matter, but a syntactic.. (shrink)

This Companion provides a comprehensive guide to ancient logic. The first part charts its chronological development, focussing especially on the Greek tradition, and discusses its two main systems: Aristotle's logic of terms and the Stoic logic of propositions. The second part explores the key concepts at the heart of the ancient logical systems: truth, definition, terms, propositions, syllogisms, demonstrations, modality and fallacy. The systematic discussion of these concepts allows the reader to engage with some specific logical and exegetical issues and (...) to appreciate their transformations across different philosophical traditions. The intersections between logic, mathematics and rhetoric are also explored. The third part of the volume discusses the reception and influence of ancient logic in the history of philosophy and its significance for philosophy in our own times. Comprehensive coverage, chapters by leading international scholars and a critical overview of the recent literature in the field will make this volume essential for students and scholars of ancient logic. (shrink)

The usual representation of quantum algorithms, limited to the process of solving the problem, is physically incomplete. We complete it in three steps: extending the representation to the process of setting the problem, relativizing the extended representation to the problem solver to whom the problem setting must be concealed, and symmetrizing the relativized representation for time reversal to represent the reversibility of the underlying physical process. The third steps projects the input state of the representation, where the problem solver is (...) completely ignorant of the setting and thus the solution of the problem, on one where she knows half solution. Completing the physical representation shows that the number of computation steps required to solve any oracle problem in an optimal quantum way should be that of a classical algorithm endowed with the advanced knowledge of half solution. (shrink)

The article examines Aristotle’s seminal discussion of the fallacy of begging the question, reconstructing its complex articulation within a variety of different, but related, contexts. I suggest that close analysis of Aristotle’s understanding of the fallacy should prompt critical reconsideration of the scope and articulation of the fallacy in modern discussions and usages, suggesting how begging the question should be distinguished from a number of only partially related argumentative faults.

Bob hides a ball in one of four drawers. Alice is to locate it. Classically she has to open up to three drawers, quantally just one. The fundamental reason for this quantum speedup is not known. The usual representation of the quantum algorithm is limited to the process of solving the problem. We extend it to the process of setting the problem. The number of the drawer with the ball becomes a unitary transformation of the random outcome of the preparation (...) measurement. This extended, time-symmetric, representation brings in relational quantum mechanics. It is with respect to Bob and any external observer and cannot be with respect to Alice. It would tell her the number of the drawer with the ball before she opens any drawer. To Alice, the projection of the quantum state due to the preparation measurement should be retarded at the end of her search; in the input state of the search, the drawer number is determined to Bob and undetermined to Alice. We show that, mathematically, one can ascribe any part of the selection of the random outcome of the preparation measurement to the final Alice’s measurement. Ascribing half of it explains the speedup of the present algorithm. This leaves the input state to Bob unaltered and projects that to Alice on a state of lower entropy where she knows half of the number of the drawer with the ball in advance. The quantum algorithm turns out to be a sum over histories in each of which Alice knows in advance that the ball is in a pair of drawers and locates it by opening one of the two. In the sample of quantum algorithms examined, the part of the random outcome of the initial measurement selected by the final measurement is one half or slightly above it. Conversely, given an oracle problem, the assumption it is one half always corresponds to an existing quantum algorithm and gives the order of magnitude of the number of oracle queries required by the optimal one. (shrink)

This article aims at reconstructing the logic and assessing the force of Socrates' argument against Protagoras' 'Measure Doctrine' at Theaetetus 171a–c. I examine and criticise some influential interpretations of the passage, according to which, e.g., Socrates is guilty of ignoratio elenchi by dropping the essential Protagorean qualifiers or successfully proves that md is self-refuting provided the missing qualifiers are restored by the attentive reader. Having clarified the meaning of MD, I analyse in detail the broader section 170a–171d and argue, against (...) an extensive scholarly consensus, that it contains two slightly different formulations of the same argument, and not two distinct arguments, that Socrates does not highlight his own strategy at 171a–c as especially clever, and that his argument successfully shows that md turns out to be untenable for Protagoras himself when submitted to scrutiny in dialectical contexts, without aiming at proving its absolute falsehood. Finally, I clarify the philosophical import of the final image of Protagoras' momentary return from the underworld. (shrink)

Through a close reading of Plutarch’s Against Colotes 3-9, the paper reconstructs and interprets the original Epicurean criticism against Democritean epistemology and ontology, and in particular against Democritus’ theory of sensible qualities, and Plutarch’s twofold criticism of Epicurean epistemology, on similar grounds, and of the questionable exegetical and argumentative manoeuvres used by the Epicurean Colotes. In the process of interpreting Plutarch’s text, the paper reflects on the nature, motivation and plausibility of Plutarch’s own exegetical and argumentative strategies. Finally, the paper (...) suggests that a careful reading of Against Colotes will help us to put into focus the important question of whether, independently of their apparently opposite claims about the “reality” of secondary qualities, Democritus and Epicurus shared either a form of radical relativism or a form of commonsense “intersubjectivism” in their evaluations of ascriptions of such qualities to macroscopic objects. (shrink)

This is a comprehensive introduction to the two great logical systems of antiquity, Aristotelian logic and Stoic logic. Although advanced undergraduate and graduate students are the intended primary readership, classical and philosophy scholars with non-specialist interests in the subject will find the fresh approach useful and stimulating. A conscious effort has been made to create a fruitful dialogue between the Aristotelian and Stoic systems both in terms of theoretical content and in terms of methodological issues and approaches. In all sections (...) emphasis is placed on broader questions concerning the distinctive nature of ancient logic, its relation with modern logic and the study of the history of logic, and the methodological difficulties posed by the reconstruction, analysis and assessment of ancient views, both in themselves and in the light of modern logic. (shrink)

The usual representation of quantum algorithms is limited to the process of solving the problem. We extend it to the process of setting the problem. Bob, the problem setter, selects a problem-setting by the initial measurement. Alice, the problem solver, unitarily computes the corresponding solution and reads it by the final measurement. This simple extension creates a new perspective from which to see the quantum algorithm. First, it highlights the relevance of time-symmetric quantum mechanics to quantum computation: the problem-setting and (...) problem solution, in their quantum version, constitute pre- and post-selection, hence the process as a whole is bound to be affected by both boundary conditions. Second, it forces us to enter into relational quantum mechanics. There must be a representation of the quantum algorithm with respect to Bob, and another one with respect to Alice, from whom the outcome of the initial measurement, specifying the setting and thus the solution of the problem, must be concealed. Time-symmetrizing the quantum algorithm to take into account both boundary conditions leaves the representation to Bob unaltered. It shows that the representation to Alice is a sum over histories in each of which she remains shielded from the information coming to her from the initial measurement, not from that coming to her backwards in time from the final measurement. In retrospect, all is as if she knew in advance, before performing her problem-solving action, half of the information that specifies the solution of the problem she will read in the future and could use this information to reach the solution with fewer computation steps. This elucidates the quantum computational speedup in all the quantum algorithms examined. (shrink)

Greek Memories aims to identify and examine the central concepts underlying the theories and practices of memory in the Greek world, from the archaic period to Late Antiquity, across all the main literary genres, and to trace some fundamental changes in these theories and practices. It explores the interaction and development of different 'disciplinary' approaches to memory in Ancient Greece, which will enable a fuller and deeper understanding of the whole phenomenon, and of its specific manifestations. This collection of papers (...) contributes to enriching the current scholarly discussion by refocusing it on the question of how various theories and practices of memory, recollection, and forgetting play themselves out in specific texts and authors from Ancient Greece, within a wide chronological span, and across a broad range of genres and disciplines. (shrink)