論理哲学ワークショップ / Workshop on Philosophy of Logic
After the publication of his Begriffsschrift, Frege distanced himself from the Booleans by insistently opposing the content that his logic was able to express to the abstraction of Boolean logic. Frege’s commitment to the notion of content would last until the end of his work, becoming in particular the source of his famous distinction between sense and denotation. But what did Frege exactly mean by this opposition between “contentual” (inhaltlich) and abstract, as a properties of logical systems? A semiotic perspective on the constitution of Boolean logic shows that this distinction can be associated with the one between Arithmetic and Algebra, considered as different mathematical practices on signs respectively underpinning the constitution of these two logical systems. Moreover, a particular attention paid to Boole’s own voluntary deviations from what would soon become the standard Boolean system permits to identify different figures of what can be seen as a content dimension arising as a logical effect of the semiotic properties of Arithmetic.
Gerhald Gentzen gave three consistency proofs for number theory. These consistency proofs have a common aim that originates from Hilbert’s Program. Hilbert, in his program, aimed to justify the use of ideal propositions in mathematics, by showing that no contradiction can be derived in a formal system of the ideal parts of mathematics. Gentzen aimed to justify the use of ideal propositions in number theory, and this aim is prominent especially in Gentzen’s 1938 consistency proof. In Gentzen’s 1935/36 consistency proofs, there is another aim that is not found in Hilbert’s Program. The aim is to formulate a “finitary” interpretation that gives a meaning to every ideal proposition of number theory and makes the theory sound.
In this talk, first we argue that what motivated Gentzen to give such an interpretation is an intuitionists’ objection against the significance of consistency proofs. Second, we show that his way of the interpretation appealed to a notion being very close to the notion of spreads, which was introduced in intuitionistic mathematics. As a consequence, we claim that intuitionism was deeply related to both Gentzen’s motivation and method for the interpretation.
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