David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Ezio Di Nucci
Jack Alan Reynolds
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Erkenntnis 43 (3):279 - 294 (1995)
Partial functions are ubiquitous in both mathematics and computer science. Therefore, it is imperative that the underlying logical formalism for a general-purpose mechanized mathematics system provide strong support for reasoning about partial functions. Unfortunately, the common logical formalisms — first-order logic, type theory, and set theory — are usually only adequate for reasoning about partial functionsin theory. However, the approach to partial functions traditionally employed by mathematicians is quite adequatein practice. This paper shows how the traditional approach to partial functions can be formalized in a range of formalisms that includes first-order logic, simple type theory, and Von-Neumann—Bernays—Gödel set theory. It argues that these new formalisms allow one to directly reason about partial functions; are based on natural, well-understood, familiar principles; and can be effectively implemented in mechanized mathematics systems.
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References found in this work BETA
Elliott Mendelson (1964). Introduction to Mathematical Logic. Princeton, N.J.,Van Nostrand.
Alonzo Church (1940). A Formulation of the Simple Theory of Types. Journal of Symbolic Logic 5 (2):56-68.
Joseph R. Shoenfield (1954). A Relative Consistency Proof. Journal of Symbolic Logic 19 (1):21-28.
Timothy Smiley (1960). Sense Without Denotation. Analysis 20 (6):125 - 135.
Hugues Leblanc & Theodore Hailperin (1959). Nondesignating Singular Terms. Philosophical Review 68 (2):239-243.
Citations of this work BETA
Raymond D. Gumb (2002). The Lazy Logic of Partial Terms. Journal of Symbolic Logic 67 (3):1065-1077.
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