Showing papers in "History and Philosophy of Logic in 1991"
TL;DR: Presburger's essay on the completeness and decidability of arithmetic with integer addition but without multiplication is a milestone in the history of mathematical logic and formal metatheory.
Abstract: Presburger's essay on the completeness and decidability of arithmetic with integer addition but without multiplication is a milestone in the history of mathematical logic and formal metatheory. The proof is constructive, using Tarski-style quantifier elimination and a four-part recursive comprehension principle for axiomatic consequence characterization. Presburger's proof for the completeness of first order arithmetic with identity and addition but without multiplication, in light of the restrictive formal metatheorems of Godel, Church, and Rosser, takes the foundations of arithmetic in mathematical logic to the limits of completeness and decidability.
102 citations
TL;DR: The existence of correspondence between George Boole (1815-1864) and William Stanley Jevons (1835-1882) has been known for a long time and part was even published in 1913 as mentioned in this paper.
Abstract: Although the existence of correspondence between George Boole (1815–1864) and William Stanley Jevons (1835–1882) has been known for a long time and part was even published in 1913, it has never been fully noted; in particular, it is not in the recent edition of Jevons's letters and papers. The texts are transcribed here, with indication of their significance. Jevons proposed certain quite radical changes to Boole's system, which Boole did not accept; nevertheless, they were to become well established.
22 citations
TL;DR: In this article, an interpretation of Russell's multiple-relation theory of judgment is presented, which characterizes it as direct application of the 1905 theory of definite descriptions, which was by regarding propositional symbols (when occurring as subordinate clauses) as disguised descriptions of complexes.
Abstract: This paper offers an interpretation of Russell's multiple-relation theory of judgment which characterizes it as direct application of the 1905 theory of definite descriptions The paper maintains that it was by regarding propositional symbols (when occurring as subordinate clauses) as disguised descriptions of complexes, that Russell generated the philosophical explanation of the hierarchy of orders and the ramified theory of types of Principia mathematica (1910) The interpretation provides a new understanding of Russell's abandoned book Theory of knowledge (1913), the ‘direction problems’ and Wittgenstein's criticisms
15 citations
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TL;DR: This paper examined three recent discussions of Aristotle's system of syllogisms with apodeictic and assertoric premisses and showed that a more satisfactory interpretation can be formulated by combining and developing elements drawn from all three.
Abstract: This paper examines three recent discussions of Aristotle's system of syllogisms with apodeictic and assertoric premisses. Though they contain no cross-references, and though they arrive at disparate interpretations, all three pieces share a common aim. That aim is to construct an intuitively graspable interpretation of Aristotle's modal syllogistic which is based on metaphysical considerations. I argue that none of these authors has succeeded in this; nevertheless, I share their broad aim, and attempt to show that a more satisfactory interpretation can be formulated by combining and developing elements drawn from all three.
13 citations
TL;DR: The life and work of Mojzesz Presburger (1904-43?) are summarised in this article and the surviving records of his student time at Warsaw University provide information on a little-studied topic.
Abstract: The life and work of Mojzesz Presburger (1904–1943?) are summarised in this article. Although his production in logic was small, it had considerable impact, both his own researches and his editions of lecture notes of Adjukiewicz and Łukasiewicz. In addition, the surviving records of his student time at Warsaw University provide information on a little-studied topic.
12 citations
11 citations
TL;DR: The relation between Wittgenstein's philosophy of mathematics and mathematical intuitionism has raised a considerable debate as discussed by the authors, and there is an attempt to analyse if there is a commitment to themes characteristic of the intuitionist movement in Mathematics and if that commitment is one important strain running through his Remarks on the foundations of mathematics.
Abstract: The relation between Wittgenstein's philosophy of mathematics and mathematical Intuitionism has raised a considerable debate. My attempt is to analyse if there is a commitment in Wittgenstein to themes characteristic of the intuitionist movement in Mathematics and if that commitment is one important strain that runs through his Remarks on the foundations of mathematics. The intuitionistic themes to analyse in his philosophy of mathematics are: firstly, his attacks on the unrestricted use of the Law of Excluded Middle; secondly, his distrust of non-constructive proofs; and thirdly, his impatience with the idea that mathematics stands in need of a foundation. These elements are Fogelin's starting point for the systematic reconstruction of Wittgenstein's conception of mathematics.
9 citations
TL;DR: In this paper, Dummett's and Gregory Currie's interpretations of Frege's views on the analysis of thoughts are discussed and an alternative interpretation which answers the above problem is proposed.
Abstract: In both his earlier and later writings, Frege claims that distinct sentences can express the same thought, and that there is a correspondence between the parts of a thought and the parts of the sentence expressing it. The joint assertion of these claims gives rise to a problem: how can there be a correspondence between the parts of one thought and the parts of distinct sentences? This paper discusses Michael Dummett's and Gregory Currie's interpretations of Frege's views on the analysis of thoughts and proposes an alternative interpretation which answers the above problem and provides some insight into Frege's epistemology of thoughts and his theory of human communication.
8 citations
TL;DR: In this article, a variety of contributions relating to the connection of probability with logic are described, some are grand attempts at providing a logical foundation for probability and inductive inference, others are concerned with probabilistic inference or, more generally, with the transmittance of probability through the structure (logical syntax) of language.
Abstract: This essay describes a variety of contributions which relate to the connection of probability with logic. Some are grand attempts at providing a logical foundation for probability and inductive inference. Others are concerned with probabilistic inference or, more generally, with the transmittance of probability through the structure (logical syntax) of language. In this latter context probability is considered as a semantic notion playing the same role as does truth value in conventional logic. At the conclusion of the essay two fully elaborated semantically based constructions of probability logic are presented.
7 citations
TL;DR: In this article, the authors reconstruct how Frege thought to reconcile the cognitive value of arithmetic with its analytical nature, and show that the epistemological formulation of the context principle plays a decisive role; it provides a way of obtaining concepts which are truly fruitful and whose contents cannot be grasped beforehand.
Abstract: I try to reconstruct how Frege thought to reconcile the cognitive value of arithmetic with its analytical nature. There is evidence in Frege's texts that the epistemological formulation of the context principle plays a decisive role; it provides a way of obtaining concepts which are truly fruitful and whose contents cannot be grasped beforehand. Taking the definitions presented in the Begriffsschrift,I shall illustrate how this schema is intended to work.
5 citations
TL;DR: The need for paraphrase in formal logic runs deeper than the odd recalcitrant inference, and thus, I shall argue, commits logicians to some interesting consequences as discussed by the authors, namely, the thesis that arguments are valid in virtue of their form must be severely qualified, and it is misleading to view a formal logical theory as a standard for justifying and criticizing inference.
Abstract: In formal logic there is a premium on clever paraphrase, for it subsumes troublesome inferences under a familiar theory. (A paradigm is Davidson's analysis 1967 of inferences like ‘He buttered his toast with a knife; so, he buttered his toast’.) But the need for paraphrase in formal logic runs deeper than the odd recalcitrant inference, and thus, I shall argue, commits logicians to some interesting consequences. First, the thesis that arguments are valid in virtue of their form must be severely qualified (§4). And second, it is misleading to view a formal logical theory as a standard for justifying and criticizing inference (§7). The latter point depends on the nature and role of paraphrase, which permit a range of conflicting logical theories. Conflicting logical theories arise from the conflicting goals of logical theorists and the promiscuous nature of paraphrase makes reconciliation impossible.
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TL;DR: A joint examination of these claims enables one to appreciate the wide scope of Leibniz's interests for mechanical procedures, concerning a variety of philosophical themes further developed both in later logical investigations and in methodological contributions to cognitive psychology.
Abstract: Leibniz's overall view of the relationship between reasoning and computation is discussed on the basis of two broad claims that one finds in his writings, concerning respectively the nature of huma...