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  1. William M. Farmer & Joshua D. Guttman (2000). A Set Theory with Support for Partial Functions. Studia Logica 66 (1):59-78.
    Partial functions can be easily represented in set theory as certain sets of ordered pairs. However, classical set theory provides no special machinery for reasoning about partial functions. For instance, there is no direct way of handling the application of a function to an argument outside its domain as in partial logic. There is also no utilization of lambda-notation and sorts or types as in type theory. This paper introduces a version of von-Neumann-Bernays-Gödel set theory for reasoning about sets, proper (...)
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  2. William M. Farmer (1995). Reasoning About Partial Functions with the Aid of a Computer. Erkenntnis 43 (3):279 - 294.
    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 (...)
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  3. William M. Farmer (1993). A Simple Type Theory with Partial Functions and Subtypes. Annals of Pure and Applied Logic 64 (3):211-240.
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  4. William M. Farmer (1992). Theory Interpretations in Computerized Mathematics. Journal of Symbolic Logic 57 (1):356.
     
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  5. William M. Farmer (1991). A Unification-Theoretic Method for Investigating the K-Provability Problem. Annals of Pure and Applied Logic 51 (3):173-214.
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  6. William M. Farmer (1991). Review: Jan Krajicek, On the Number of Steps in Proofs. [REVIEW] Journal of Symbolic Logic 56 (1):334-335.
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  7. William M. Farmer (1990). A Partial Functions Version of Church's Simple Theory of Types. Journal of Symbolic Logic 55 (3):1269-1291.
    Church's simple theory of types is a system of higher-order logic in which functions are assumed to be total. We present in this paper a version of Church's system called PF in which functions may be partial. The semantics of PF, which is based on Henkin's general-models semantics, allows terms to be nondenoting but requires formulas to always denote a standard truth value. We prove that PF is complete with respect to its semantics. The reasoning mechanism in PF for partial (...)
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  8. William M. Farmer (1989). Krajíček Jan and Pudlák Pavel. The Number of Proof Lines and the Size of Proofs in First Order Logic. Archive for Mathematical Logic, Vol. 27 (1988), Pp. 69–84. [REVIEW] Journal of Symbolic Logic 54 (3):1107-1108.
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  9. William M. Farmer (1989). Review: Jan Krajicek, Pavel Pudlak, The Number of Proof Lines and the Size of Proofs in First Order Logic. [REVIEW] Journal of Symbolic Logic 54 (3):1107-1108.
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  10. William M. Farmer (1988). A Unification Algorithm for Second-Order Monadic Terms. Annals of Pure and Applied Logic 39 (2):131-174.
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  11. William M. Farmer (1988). Review: Larry Wos, Automated Reasoning: 33 Basic Research Problems. [REVIEW] Journal of Symbolic Logic 53 (4):1258-1259.
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