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Logical Pluralism
 To appear, Special Logic issue of the Australasian Journal of Philosophy
, 2000
"... Abstract: A widespread assumption in contemporary philosophy of logic is that there is one true logic, that there is one and only one correct answer as to whether a given argument is deductively valid. In this paper we propose an alternative view, logical pluralism. According to logical pluralism th ..."
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Abstract: A widespread assumption in contemporary philosophy of logic is that there is one true logic, that there is one and only one correct answer as to whether a given argument is deductively valid. In this paper we propose an alternative view, logical pluralism. According to logical pluralism there is not one true logic; there are many. There is not always a single answer to the question “is this argument valid?” 1 Logic, Logics and Consequence Anyone acquainted with contemporary Logic knows that there are many socalled logics. 1 But are these logics rightly socalled? Are any of the menagerie of nonclassical logics, such as relevant logics, intuitionistic logic, paraconsistent logics or quantum logics, as deserving of the title ‘logic ’ as classical logic? On the other hand, is classical logic really as deserving of the title ‘logic ’ as relevant logic (or any of the other nonclassical logics)? If so, why so? If not, why not? Logic has a chief subject matter: Logical Consequence. The chief aim of
Carnap on the foundations of logic and mathematics
, 2009
"... Throughout most of his philosophical career Carnap upheld and defended three distinctive philosophical positions: (1) The thesis that the truths of logic and mathematics are analytic and hence without content and purely formal. (2) The thesis that radical pluralism holds in pure mathematics in that ..."
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Throughout most of his philosophical career Carnap upheld and defended three distinctive philosophical positions: (1) The thesis that the truths of logic and mathematics are analytic and hence without content and purely formal. (2) The thesis that radical pluralism holds in pure mathematics in that any consistent system of postulates is equally legitimate and that there is no question of justification in mathematics but only the question of which system is most expedient for the purposes of empirical science. (3) A minimalist conception of philosophy in which most traditional questions are rejected as pseudoquestions and the task of philosophy is identified with the metatheoretic study of the sciences. In this paper, I will undertake a detailed analysis of Carnap’s defense of the first and second thesis. This will involve an examination of his most technical work The Logical Syntax of Language (1934), along with the monograph “Foundations of Logic and Mathematics ” (1939). These are the main works in which Carnap defends his views concerning the nature of truth and radical
Beyond the axioms: The question of objectivity in mathematics
"... I will be discussing the axiomatic conception of mathematics, the modern version of which is clearly due to Hilbert... ..."
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I will be discussing the axiomatic conception of mathematics, the modern version of which is clearly due to Hilbert...
Reflections on a categorical foundations of mathematics
, 2008
"... We examine Gödel’s completeness and incompleteness theorems for higher order arithmetic from a categorical point of view. The former says that a proposition is provable if and only if it is true in all models, which we take to be local toposes, i.e. Lawvere’s elementary toposes in which the terminal ..."
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We examine Gödel’s completeness and incompleteness theorems for higher order arithmetic from a categorical point of view. The former says that a proposition is provable if and only if it is true in all models, which we take to be local toposes, i.e. Lawvere’s elementary toposes in which the terminal object is a nontrivial indecomposable projective. The incompleteness theorem showed that, in the classical case, it is not enough to look only at those local toposes in which all the numerals are standard. Thus, for a classical mathematician, Hilbert’s formalist program is not compatible with the belief in a Platonic standard model. However, for pure intuitionistic type theory, a single model suffices, the linguistically constructed free topos, which is the initial object in the category of all elementary toposes and logical functors. Hence, for a moderate intuitionist, formalism and Platonism can be reconciled after all. The completeness theorem can be sharpened to represent any topos by continuous sections of a sheaf of local toposes.
Gödel on Intuition and on Hilbert’s finitism
"... There are some puzzles about Gödel’s published and unpublished remarks concerning finitism that have led some commentators to believe that his conception of it was unstable, that he oscillated back and forth between different accounts of it. I want to discuss these puzzles and argue that, on the con ..."
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There are some puzzles about Gödel’s published and unpublished remarks concerning finitism that have led some commentators to believe that his conception of it was unstable, that he oscillated back and forth between different accounts of it. I want to discuss these puzzles and argue that, on the contrary, Gödel’s writings represent a smooth evolution, with just one rather small doublereversal, of his view of finitism. He used the term “finit ” (in German) or “finitary ” or “finitistic ” primarily to refer to Hilbert’s conception of finitary mathematics. On two occasions (only, as far as I know), the lecture notes for his lecture at Zilsel’s [Gödel, 1938a] and the lecture notes for a lecture at Yale [Gödel, *1941], he used it in a way that he knew—in the second case, explicitly—went beyond what Hilbert meant. Early in his career, he believed that finitism (in Hilbert’s sense) is openended, in the sense that no correct formal system can be known to formalize all finitist proofs and, in particular, all possible finitist proofs of consistency of firstorder number theory, P A; but starting in the Dialectica paper
Russell’s Absolutism vs.(?)
"... Along with Frege, Russell maintained an absolutist stance regarding the subject matter of mathematics, revealed rather than imposed, or proposed, by logical analysis. The Fregean definition of cardinal number, for example, is viewed as (essentially) correct, not merely adequate for mathematics. And ..."
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Along with Frege, Russell maintained an absolutist stance regarding the subject matter of mathematics, revealed rather than imposed, or proposed, by logical analysis. The Fregean definition of cardinal number, for example, is viewed as (essentially) correct, not merely adequate for mathematics. And Dedekind’s “structuralist” views come in for criticism in the Principles. But, on reflection, Russell also flirted with views very close to a (different) version of structuralism. Main varieties of modern structuralism and their challenges are reviewed, taking account of Russell’s insights. Problems of absolutism plague some versions, and, interestingly, Russell’s critique of Dedekind can be extended to one of them, ante rem structuralism. This leaves modalstructuralism and a category theoretic approach as remaining nonabsolutist
—Carnap, The Logical Syntax of Language
"... “... before us lies the boundless ocean of unlimited possibilities.” ..."