David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
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Studia Logica 95 (3):355-378 (2010)
We briefly overview some of the historical landmarks on the path leading to the reduction of the number of logical connectives in classical logic. Relying on the duality inherent in Boolean algebras, we introduce a new operator ( Nallor ) that is the dual of Schönfinkel’s operator. We outline the proof that this operator by itself is sufficient to define all the connectives and operators of classical first-order logic ( Fol ). Having scrutinized the proof, we pinpoint the theorems of Fol that are needed in the proof. Using the insights gained from the proof, we show that there are four binary operators that each can serve as the only undefined logical constant for Fol . Finally, we show that from every n -ary connective that yields a functionally complete singleton set of connectives two Schönfinkel-type operators are definable, and all the latter are so definable.
|Keywords||Philosophy Computational Linguistics Mathematical Logic and Foundations Logic|
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References found in this work BETA
Katalin Bimbó, Combinatory Logic. Stanford Encyclopedia of Philosophy.
Charles S. Peirce (1931). Collected Papers of Charles Sanders Peirce. Cambridge, Harvard University Press.
Francis Jeffry Pelletier & Norman M. Martin (1990). Post's Functional Completeness Theorem. Notre Dame Journal of Formal Logic 31 (3):462-475.
Emil Post (1921). Introduction to a General Theory of Elementary Propositions. American Journal of Mathematics 43 (1):163--185.
Emil Leon Post (1941). The Two-Valued Iterative Systems of Mathematical Logic. London, H. Milford, Oxford University Press.
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