Showing papers in "History and Philosophy of Logic in 2016"
TL;DR: This paper provided an interpretation of the universal quantifier in the Organon in terms of Paul Lorenzen's dialogical logic, and provided textual evidence of the use of that rule in Plato's dialogues.
Abstract: In this paper we provide an interpretation of Aristotle's rule for the universal quantifier in Topics Θ 157a34–37 and 160b1–6 in terms of Paul Lorenzen's dialogical logic. This is meant as a contribution to the rehabilitation of the role of dialectic within the Organon. After a review of earlier views of Aristotle on quantification, we argue that this rule is related to the dictum de omni in Prior Analytics A 24b28–29. This would be an indication of the dictum’s origin in the context of dialectical games. One consequence of our approach is a novel explanation of the doctrine of the existential import of the quantifiers in dialectical terms. After a brief survey of Lorenzen's dialogical logic, we offer a set of rules for dialectical games based on previous work by Castelnerac and Marion, to which we add here the rule for the universal quantifier, as interpreted in terms of its counterpart in dialogical logic. We then give textual evidence of the use of that rule in Plato's dialogues, thus showing that Aris...
29 citations
TL;DR: While Pāṇini's Sanskrit grammar is computationally limited, the metalanguage through which his formalism is defined can be directly used to define any rule-based system by mimicking standard formal language definitions as an extension of the grammatical system proper.
Abstract: Pāṇini's fourth (?) century BC Sanskrit grammar uses rewrite rules utilizing an explicit formal language defined through a semi-formal metalanguage. The grammar is generative, meaning that it is capable of expressing a potential infinity of well-formed Sanskrit sentences starting from a finite symbolic inventory. The grammar's operational rules involve extensive use of auxiliary markers, in the form of Sanskrit phonemes, to control grammatical derivations. Pāṇini's rules often utilize a generic context-sensitive format to identify terms used in replacement, modification or deletion operations. The context-sensitive rule format is itself defined using Pāṇini's more general method of auxiliary markers, the latter used to define many dozens of linguistic categories and rules controlling derivations of Sanskrit sentences through the manipulation of ‘non-terminal’ and ‘terminal’ symbols. This technique for controlling formal derivations was rediscovered by Emil Post in the 1920s and later shown by him to be ca...
12 citations
TL;DR: In this article, the authors argue that to dispute well logic's chiefest end provides no greater miracle than the creation of the art of logic itself, which is itself a miracle.
Abstract: Sweet Analytics, ‘tis thou hast ravish'd me,Bene disserere est finis logices.Is to dispute well logic's chiefest end?Affords this art no greater miracle?(Christopher Marlow, Doctor Faustus, Act 1, ...
11 citations
TL;DR: The authors showed that Peirce studied the scholastic theory of consequentiae as early as 1866-67, and that that theory constituted a source of logical doctrine for a lifetime of creative and original work.
Abstract: In 1898 C. S. Peirce declares that the medieval doctrine of consequences had been the starting point of his logical investigations in the 1860s. This paper shows that Peirce studied the scholastic theory of consequentiae as early as 1866–67, that he adopted the scholastics’ terminology, and that that theory constituted a source of logical doctrine that sustained Peirce for a lifetime of creative and original work.
10 citations
TL;DR: Arnauld and his collaborators in the Port-Royal Grammar and Logic as discussed by the authors argued that these mental objects are really distinct from the external objects they represent, and therefore called "ideas".
Abstract: In his dispute with Malebranche about the nature of ideas, Arnauld endorses a form of direct realism. This appears to conflict with views put forward by Arnauld and his collaborators in the Port-Royal Grammar and Logic where ideas are treated as objects in the mind. This tension can be resolved by a careful examination of Arnauld's remarks on the semantics of ‘perception’ and ‘idea’ in light of the Port-Royal theory of language. This examination leads to the conclusion that Arnauld's ideas really are objects in the mind, and not perceptual acts as many commentators hold. What Arnauld denies is that these mental objects are really distinct from the external objects they represent. Instead, Arnauld holds that, by the act of conception, the external objects themselves—not copies—come to be present in the mind and are therefore called ‘ideas’.
9 citations
TL;DR: In this paper, 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 (syllogism) and reveals that Aristotle's syllogismhood is non-monotonic, and thus narrower than classical validity as consisting in, or coextensive with, necessary truth-preservation.
Abstract: 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 (NCF). 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 (‘syllogism’). 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...
9 citations
TL;DR: In an appendix to his influential article On Second Order Logic, George Boolos offered an ingenious argument that seems to diminish Quine's account of logical truth on a deeper level as discussed by the authors.
Abstract: The paper is concerned with Quine's substitutional account of logical truth. The critique of Quine's definition tends to focus on miscellaneous odds and ends, such as problems with identity. However, in an appendix to his influential article On Second Order Logic, George Boolos offered an ingenious argument that seems to diminish Quine's account of logical truth on a deeper level. In the article he shows that Quine's substitutional account of logical truth cannot be generalized properly to the general concept of logical consequence. The purpose of this paper is threefold: first, to introduce the reader to the metamathematics of Quine's substitutional definition of logical truth; second, to make Boolos' result accessible to a broader audience by giving a detailed and self-contained presentation of his proof; and, finally, to discuss some of the possible implications and how a defender of the Quinean concepts might react to the challenge posed by Boolos' result.
6 citations
TL;DR: For a recent book about Alfred Tarski, a reviewer needed to describe briefly and edit carefully some translations of Polish works on mathematical philosophy that lay outside his background in mat... as discussed by the authors.
Abstract: For a recent book about Alfred Tarski this reviewer needed to describe briefly and edit carefully some translations of Polish works on mathematical philosophy that lay outside his background in mat...
6 citations
TL;DR: The fact that medieval logic has become an object of increased interest in the last 20 years, for many reasons, one of them is, possibly, the fact that Medievals dealt with logical questions considering their in...
Abstract: Medieval logic has become an object of increased interest in the last 20 years, for many reasons. One of them is, possibly, the fact that Medievals dealt with logical questions considering their in...
5 citations
TL;DR: The Wolfson Chair in Philosophy at Brandeis University, where Sommers taught from 1963 to 1993 as mentioned in this paper, was named after the author of this article. But it was not a chair he used in this paper.
Abstract: Fred Sommers passed away in October of 2014 in his 92nd year. Having begun his teaching at Columbia University, he eventually became the Harry A. Wolfson Chair in Philosophy at Brandeis University, where he taught from 1963 to 1993. During his long and productive career, Sommers authored or co-authored over 50 books, articles, reviews, etc., presenting his ideas on numerous occasions throughout North America and Europe. His work was characterized by a commitment (often implicit) to the preservation and application of historical insights and to the value of a well-articulated, coherent logical system. He was recognized for his independence and refusal to accept any view on the basis of authority alone. This made him a formidable critic but accounted in part for his many innovative and original ideas. In spite of his general contrariness in logic, Sommers earned the respect of the majority of his contemporaries, including Russell, Quine, van Benthem, Hacking, Suppes, and Strawson. In 2005, he was the subjec...
5 citations
TL;DR: In this article, it was shown that the techniques for finding arguments described in Topics I-VII are, by and large, locations that Aristotle thought of as appropriate for use in philosophical inquiry.
Abstract: I claim that, in the Topics, Aristotle advises dialectical questioners to intentionally argue fallaciously in order to escape from some dialectically awkward positions, and I work through the consequences of that claim. It will turn out that, although there are important exceptions, the techniques for finding arguments described in Topics I–VII are, by and large, locations that Aristotle thought of as appropriate for use in philosophical inquiry. The text that grounds this claim, however, raises a further problem: it highlights the solitary nature of philosophical inquiry, which puts into question the philosophical relevance of Topics VIII. I find that the Topics provides inadequate grounds for thinking that Aristotle saw Topics VIII as describing standards or techniques of argument that were appropriate for philosophy, and so these texts cannot be used by contemporary commentators to shed light on Aristotle's philosophical practice. Finally, although Aristotle saw philosophy as a solitary activity, he th...
TL;DR: In this paper, it was shown that the five criticisms are not independent but rather amount to five successive filters through which an argument may be passed before its non-genuineness (as a syllogism or refutation) is exposed, which sheds light on Aristotle's curious insistence, in the face of the heterogeneity of his examples of BTQ in Topics 8.13, that every question-begging refutation contains a premise that is ‘the same as the conclusion.
Abstract: At Topics 8.11 161b19–33 Aristotle lists five criticisms () which may be leveled against a dialectical argument ‘in itself’ (). The five criticisms correspond in many respects to the familiar conditions Aristotle places on syllogism and refutation. However, begging the question (BTQ)—the violation of the condition that the conclusion of a syllogism be something different () from the premises—seems not to appear on the list of five criticisms . That this omission is only apparent becomes clear once it is seen that the five criticisms are not independent but rather amount to five successive filters through which an argument may be passed before its non-genuineness (as a syllogism or refutation) is exposed. This result sheds light on Aristotle's curious insistence, in the face of the heterogeneity of his examples of BTQ in Topics 8.13, that every question-begging refutation contains a premise that is ‘the same’ as the conclusion. The fact that the five criticisms expose BTQ as a defect in syllogistic reasoni...
TL;DR: A philosophical interpretation of the ancient Mohist treatment of the parallel inference concerning its semantic-syntactic structure is presented and a modern logical treatment via an expanded predicate logic account is suggested.
Abstract: The purpose of this paper is to explore the issue of how the validity of the parallel inference (as a type of deductive reasoning) is possible in view of its deep semantic-syntactic structure. I first present a philosophical interpretation of the ancient Mohist treatment of the parallel inference concerning its semantic-syntactic structure. Then, to formally and accurately capture the later Mohist point in this connection for the sake of giving a general condition for the validity of the parallel inference, I suggest a modern logical treatment via an expanded predicate logic account.
TL;DR: Whitehead's bad emendations of the cardinal number in the Principia Mathematica are discussed in this paper, where it is claimed that it is a fallacy to apply Modus Ponens and infer Nc'α from Θα and Θ'α.
Abstract: There are many wonderful puzzles concerning Principia Mathematica, but none are more striking than those arising from the crisis that befell Whitehead in November of 1910. Volume 1 appeared in December of 1910. Volume 2 on cardinal numbers and Russell's relation arithmetic might have appeared in 1911 but for Whitehead's having halted the printing. He discovered that inferences involving the typically ambiguous notation ‘Nc‘α’ for the cardinal number of α (i.e. the class of all classes similar with α) might generate fallacies. When the volume appeared in 1912, it was extensively emended by Whitehead and accompanied by a Prefatory Statement of Symbolic Conventions. This paper endeavors to recover from Whitehead's bad emendations—including his bewildering thesis that since ‘‘α’ is ‘true whenever significant,’ ‘α is to be accepted. It is supposedly a fallacy to apply Modus Ponens and infer Nc‘α from ‘α and‘‘α.
TL;DR: This article made some observations on the general interrelationship between the logical writings of Albert and Buridan and gave an account of a "semantic logical model" (S-L) for analyzing complex subject terms in some basic categorical propositions.
Abstract: The essay first makes some observations on the general interrelationship between the logical writings of Albert and Buridan. Second, it gives an account of a ‘semantic logical model’ (S-L) for analyzing complex subject terms in some basic categorical propositions which is defended by Albert of Saxony, and briefly recounts Buridan's criticisms of that model. Finally, the essay maintains that the Albertian (S-L) model is typically compatible with, and a further development of, what is called by a late-fourteenth century anonymous scholar ‘the English-Rule’ but the ‘determinable/determinate’ grammatical model defended by Buridan is not.
TL;DR: In this paper, an account of the theory of disputation given by Robert Kilwardby (d. 1279) in his commentary, composed in Paris during the 1240s, on Aristotle's Prior Analytics is presented.
Abstract: The article is concerned with the account of Aristotle's theory of disputation given by Robert Kilwardby (d. 1279) in his commentary, composed in Paris during the 1240s, on Aristotle's Prior Analytics. Specifically, I show that Kilwardby covers demonstrative as well as dialectical disputations, and gives an elementary account of the rules governing such disputations, in their adversarial forms as well as in an idealized form where the interlocutors engage in a cooperative activity. I describe the resemblances and the differences between disputations as theorized by Kilwardby and the game of obligationes as understood by some of his contemporaries.
TL;DR: In this article, French and Standefer present a detailed discussion of the book and discuss the view according to logical pluralism, which they call Logical Pluralism (LPL).
Abstract: 11We thank Rohan French for a detailed discussion of this review. We also wish to reciprocally thank Shawn Standefer for detailed discussions about the book.Logical pluralism is the view according ...
TL;DR: A note correcting some technical terminology from linguistics found in "Distributive Terms, Truth, and The Port Royal Logic" as mentioned in this paper is found in this journal, Jan. 17, 2013, 133-54.
Abstract: A note correcting some technical terminology from linguistics found in ‘Distributive Terms, Truth, and The Port Royal Logic’, this journal, Jan. 17, 2013, 133–54.
TL;DR: In this paper, the Crocodile Paradox is reformulated and analyzed as a game whose Nash equilibrium is shown to trigger a cyclic choice and to invite a rationality gap, and it is shown that choosing the Nash equilibrium of CP conforms to the principles Wolfgang Spohn and Haim Gaifman introduced to, allegedly, guarantee acyclicity but, in fact, does not prevent self-referential/cyclic choice.
Abstract: This article discusses rationality gaps triggered by self-referential/cyclic choice, the latter being understood as choosing according to a norm that refers to the choosing itself. The Crocodile Paradox is reformulated and analyzed as a game—named CP—whose Nash equilibrium is shown to trigger a cyclic choice and to invite a rationality gap. It is shown that choosing the Nash equilibrium of CP conforms to the principles Wolfgang Spohn and Haim Gaifman introduced to, allegedly, guarantee acyclicity but, in fact, does not prevent self-referential/cyclic choice and rationality gaps. It is shown that CP is a counter-example to Gaifman's solution of the rationality gaps problem.