W. Farmer and J. Guttman. A Set Theory with Support for Partial Functions. Logica Studia, 66(1):59--78, 2000.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....caching, but one which we do not pursue here, might be some kind of explicit de nition or abbreviation. 3. 1 Preliminaries The standard set theoretic notion of partial function is assumed, together with related set theoretic operations such as union, which will be performed on functions (cf. [9]) For a unary partial function e, we use the notation e(x) # to mean e is de ned at x and e(x) to mean it is not. The domain of de nition of e is denoted Def(e) Expressions are formalized as unary partial functions from the set of positions (De nition 2) to a set of symbols (as in, e.g. ....

W. Farmer and J. Guttman. A Set Theory with Support for Partial Functions. Logica Studia, 66(1):59-78, 2000.

....In First Order Logic, an atomic formula that contains the application of an operator symbol to a sequence of terms is translated into a complex formula because the translation of the application may be undefined in some domains. The variant of Partial First Order Logic used here is taken from [3] where it forms the basis of a set theory with support for partial functions. The paper also describes a sort system used to classify terms. pfol, the set theory, and the sort system are intended to serve as a foundation for mechanized mathematical systems. The set theory, and the sort system may ....

....# i# it is a model of every member of #. 6 2.2.1 Semantics of Full PFOL Structures for First Order Logic associate a total function with each operator symbol, while structures for pfol associate a partial function with each operator symbol. The definition of a model for pfol is given in [3]; this paper will give the semantics of pfol by translating arbitrary formulas into regular formulas over an operator free language. Assume L = C, O, P) and define P = P# p o : o # O and L = C, #, P) where p o # P and p o is (n 1) ary if o is n ary for all o # O. For each ....

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W. M. Farmer and J. D. Guttman. A set theory with support for partial functions. Studia Logica, 65:59--70, 2000. Forthcoming.

....a local context in E at p with respect to (L) Many traditional logics (including propositional logic, rst order logic, and simple type theory) can be formulated as admissible logics. Logics that admit unde nedness, such as lutins [17, 18, 19] the logic of imps, and other related logics (see [23, 24]) can also be formulated as admissible logics. For examples later in the paper, let K stt = L; be an admissible logic formulation of Church s simple type theory [11] 3. Theories In this section we introduce the central notion of a biform theory , which is a generalization of both an ....

Farmer, W. M. and J. D. Guttman: 2000, `A Set Theory with Support for Partial Functions'. Studia Logica 66, 59-78.

....in high school and college. 3 Partial First Order Logic In this section we introduce a variant of first order logic called Partial FirstOrder Logic (pfol) to illustrate how predicate logic can be modified to support the traditional approach. A more detailed presentation of pfol is given in [5]. Several systems similar to pfol have been presented in the literature; see [4] for references. pfol has the usual connectives of first order logic: oe; j;8; 9: In addition, it has a definite description operator I that is used to construct terms of the form I x : I is given a ....

....T , there is an ordinary firstorder logic (fol) theory T and a translation from each formula of T to a formula of T such that T involves no use of function symbols or the I operator and T j= pfol iff T j= fol : Moreover, if contains no function symbols nor I. Proof See [5]. 2 Most of the logical axiom schemas of pfol are exactly the same as those for ordinary first order logic. However, those dealing with instantiation and equality substitution are slightly different. For example, universal instantiation holds only for defined terms: 8 x : t# oe [x 7 t] ....

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W. M. Farmer and J. D. Guttman. A set theory with support for partial functions. Technical report, The mitre Corporation, 1996.

....simple logics which are too inexpressive or on type theories which are too esoteric for most typical mathematics practitioners. To be successful, an iml needs an underlying logic which is highly expressive, familiar to users, and well understood by developers. We have proposed a logic called stmm [8, 9] that is intended to serve as a foundation for mechanized mathematics. stmm is a version of von Neumann BernaysG odel (nbg) set theory [13, 18] with convenient machinery for reasoning with unde nedness and partial functions. 5 Proposal It will take a tremendous e ort to make the idea of an iml ....

W. M. Farmer and J. D. Guttman. A set theory with support for partial functions. Studia Logica, 2000. Forthcoming.

....theory is presented in section 5. Techniques for de ning new sound transformers is the subject of section 6. The papers ends with a short conclusion. 2 STMM The underlying logic of the integrated framework is a version of vonNeumann Bernays G odel (nbg) set theory [10, 11] called stmm [4, 5]. Unlike traditional set theories (such as Zermelo Fraenkel (zf) set theory and nbg) stmm is a well suited foundation for mechanized mathematics. It allows terms to be unde ned, has a de nite description operator, provides 2 a sort system for classifying terms by value, and includes ....

W. M. Farmer and J. D. Guttman. A set theory with support for partial functions. Studia Logica, 2000. Forthcoming. 10

....in high school and college. 3 Partial First Order Logic In this section we introduce a variant of rst order logic called Partial FirstOrder Logic (pfol) to illustrate how predicate logic can be modi ed to support the traditional approach. A more detailed presentation of pfol is given in [5]. Several systems similar to pfol have been presented in the literature; see [4] for references. pfol has the usual connectives of rst order logic: 8; 9: In addition, it has a de nite description operator I that is used to construct terms of the form I x : I is given a ....

....ordinary rstorder logic (fol) theory T and a translation from each formula of T to a formula of T such that T involves no use of function symbols or the I operator and T j= pfol i T j= fol : Moreover, if contains no function symbols nor I. Proof See [5]. 2 Most of the logical axiom schemas of pfol are exactly the same as those for ordinary rst order logic. However, those dealing with instantiation and equality substitution are slightly di erent. For example, universal instantiation holds only for de ned terms: 8 x : t# [x 7 t] ....

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W. M. Farmer and J. D. Guttman. A set theory with support for partial functions. In E. Thysse and F. Lepage, editors, Partial, Epistemic, and Dynamic Logic, Applied Logic Series. Kluwer. Forthcoming.

....logic for formalizing traditional mathematics (e.g. see [17] The following are the major ingredients of stmm. In contrast to zf and nbg, the underlying logic of stmm is Partial First Order Logic (pfol) a version of rst order logic that admits partial functions and unde ned terms (see [14, 15]) stmm contains the usual vocabulary and axioms of nbg. It also contains a sort system for classifying terms by value which is similar to the sort system of lutins. Lastly, stmm has term constructors for function application and function abstraction which provides stmm with lambda notation for ....

....is de ned and what kind of value an application of a function may have when it is de ned. The special machinery in stmm for reasoning with functions is discussed in more detail in section 5. 3.4. Relationship to NBG An alternate version of stmm called nbg is described in [14] and de ned in [15]. nbg is de ned in stages, while stmm is de ned in a direct way. The purpose of nbg is to illustrate precisely how it (and stmm) are related to nbg. It is for study, not use. stmm, on the other hand, is intended to be a logic that can actually be implemented and used as a component of a ....

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W. M. Farmer and J. D. Guttman. A set theory with support for partial functions. Studia Logica, 2000. Forthcoming.

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W. Farmer and J. Guttman. A Set Theory with Support for Partial Functions. Logica Studia, 66(1):59--78, 2000.

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