Diagonalization in Logic and Mathematics

Jacquette, Dale

doi:10.1007/978-94-017-0466-3_2

Cited by 2 publications

(3 citation statements)

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“…I am in fact only too interested in rejecting Tarski's truth schemata on any legitimate formal or philosophical grounds. My philosophical dissatisfaction with Tarskian truth schemata for reasons independent of Barker's complaints can be found already in (Jacquette, 2002;2010). I just do not agree that the particular conclusions Barker tries to support in his arguments can in this case be correctly attributed entirely or even largely to the responsibility of Tarski's truth schemata, and I do not regard Barker's argument as forcing their repudiation.…”

Section: Understanding the Liarmentioning

confidence: 88%

“…This is to say that Gödel numbering expressions like those Barker uses are substitution-inviolable intensional contexts. I explain the intensionality of Gödel contexts in (Jacquette, 1987;2002). ¬FALSE(a), which is evidently not a paradox or an alternative formulation of the liar paradox.…”

Section: Intensionality Of Literal Self-referencementioning

confidence: 90%

“…I find it interesting that in this context Barker now significantly demurs, qualifying his claim: 'The basic technique can be found in (Gödel, 1931), though Gödel does not actually derive an instance of (3) [(L*) in (Gödel, 1931)] there' (Barker, 2009, p. 15). I explain my concept of diagonalization, and its applications in the Gödel, Church and Rosser proofs, in (Jacquette, 2002). but I repeat, contrary to Barker's unsupported claim, that, reverting to the notation in which I define L* as the biconditonal liar sentence, that L*  ¬TRUE(  L*  ) is not, in Barker's words, 'provable' (Barker, 2008, p. 139).…”

Section: Lemmamentioning

confidence: 99%

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Liar Paradox and Substitution into Intensional Contexts

Jacquette¹

2010

Polish Journal of Philosophy

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John Barker, in two recent essays, raises a variety of intriguing criticisms to challenge my interpretation of the liar paradox and the type of solution I propose in 'Denying the Liar' and 'Denying the Liar Reaffirmed.' Barker continues to believe that I have misunderstood the logical structure of the liar sentence and its expression, and that as a result my solution misfires. I shall try to show that on the contrary my analysis is correct, and that Barker does not properly grasp what my solution to the liar paradox involves. Additionally, I argue that Barker makes fundamental errors in the explanation of liar sentence formulations in intensional contexts and in the classical metatheory he invokes to support his criticisms.

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Section: Understanding the Liarmentioning

confidence: 88%

Section: Intensionality Of Literal Self-referencementioning

confidence: 90%

Section: Lemmamentioning

confidence: 99%

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Liar Paradox and Substitution into Intensional Contexts

Jacquette¹

2010

Polish Journal of Philosophy

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Computable Diagonalizations and Turing’s Cardinality Paradox

Jacquette

2014

J Gen Philos Sci

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A. N. Turing's 1936 concept of computability, computing machines, and computable binary digital sequences, is subject to Turing's Cardinality Paradox. The paradox conjoins two opposed but comparably powerful lines of argument, supporting the propositions that the cardinality of dedicated Turing machines outputting all and only the computable binary digital sequences can only be denumerable, and yet must also be nondenumerable. Turing's objections to a similar kind of diagonalization are answered, and the implications of the paradox for the concept of a Turing machine, computability, computable sequences, and Turing's effort to prove the unsolvability of the Entscheidungsproblem, are explained in light of the paradox. A solution to Turing's Cardinality Paradox is proposed, positing a higher geometrical dimensionality of machine symbol-editing information processing and storage media than is available to canonical Turing machine tapes. The suggestion is to add volume to Turing's discrete two-dimensional machine tape squares, considering them instead as similarly ideally connected massive three-dimensional machine information cells. Three-dimensional computing machine symbol-editing information processing cells, as opposed to Turing's twodimensional machine tape squares, can take advantage of a denumerably infinite potential for parallel digital sequence computing, by which to accommodate denumerably infinitely many computable diagonalizations. A three-dimensional model of machine information storage and processing cells is recommended on independent grounds as better representing the biological realities of digital information processing isomorphisms in the threedimensional neural networks of living computers.

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Diagonalization in Logic and Mathematics

Cited by 2 publications

References 44 publications

Liar Paradox and Substitution into Intensional Contexts

Liar Paradox and Substitution into Intensional Contexts

Computable Diagonalizations and Turing’s Cardinality Paradox

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