Barwise, J.: 1985, Model-Theoretic Logics: Background and Aims, Chapter 1 of ModelTheoretic Logics, edited by J. Barwise and S. Feferman.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Learner s Dictionary of Current English (by A. S. Hornby, E. V. Gatenby, and H. Wakefield, London, U.K. Oxford University Press, 1958) logic is the science or art of reasoning, proof, and clear thinking. Thus, the commonly accepted equation logic = first order logic is highly suspect. cf. [6] for an extended argument on this. The environment is dubbed BABY SIT because we believe that presently it includes far too many provisional, make shift design decisions. We trust, on the other hand, that as the baby grows up, these haphazard dimensions will be trimmed and a natural, ....

J. Barwise. "Model-Theoretic Logics: Background and Aims," in J. Barwise and S. Feferman, editors, ModelTheoretic Logics, Berlin, Germany: Springer-Verlag, 1985, pp. 3--23.

....of a general phenomenon. Specifically, in the nonfinite case it follows from the Craig Interpolation Theorem that if a class and its complement are both Sigma 1 1 , then the class is first order definable. A logic, such as first order logic, that obeys this property is called Delta closed; cf. [Bar85]. Of course, first order logic is not Delta closed if we restrict our attention to finite structures: for example, evenness is Sigma 1 1 , as is the complement ( oddness ) but, as we have noted, evenness is not first order definable. The final line of research Vardi calls positive; here we ....

....is closed under complement, and the related result by Gurevich and Shelah [GS86] that for finite structures, different natural fixed point logics all have the same expressive power. A recent nice example was 13 This contrasts in an interesting way with results in abstract model theory (cf. [Bar85, Mak85]) where Robinson consistency is a very strong property, and in fact is equivalent to the combination of compactness and Craig interpolation. 24 obtained by Kolaitis [Kol90] As noted earlier, Beth s Theorem (which says that implicit definability is equivalent to explicit definability) fails for ....

J. Barwise. Model-theoretic logics: background and aims. In J. Barwise and S. Feferman, editors, Model-Theoretic Logics, pages 3--23. Springer-Verlag, Berlin/New York, 1985.

....due to Lyndon 1959b. Theorem 4.59 is due to Ressayre 1977. Theorem 4.66 is from Lindstrom 1969. The proof in the text is in the spirit of Friedman s rediscovery of this result; the one indicated in Exercise 165 is closer to the original. A general reference for the abstract notion of a logic is Barwise 1985. The Lyndon Interpolation Theorem from Exercise 158 is from Lyndon 1959a. The result in Exercise 162 is due to Svenonius. 92 A Deduction and Completeness The subject of this appendix, the Completeness Theorem for first order logic (with respect to a system of natural deduction) does not ....

Barwise, J. 1985. Model-theoretic logics: background and aims. In ModelTheoretic Logics, ed. J. Barwise and S. Feferman. 3--23. Springer-Verlag.

....Non well founded sets have been extensively studied through decades, but did not show up in notable applications until Aczel. This is probably due to the fact that the classical well founded universe was a rather satisfying domain for the practicing mathematician ( the mathematician in the street (Barwise 1985)) Aczel s work on non well founded sets evolved from his interest in modeling concurrent processes. He adopted the graph representation for sets to use in his theory. A set like a = fb; fc; dgg can be unambiguously depicted as in Figure 6 in this representation (Aczel 1988) where an arrow from a ....

Barwise, J. (1985). Model-Theoretic Logics: Background and Aims. In Barwise, J. and Feferman, S. (eds.) Model-Theoretic Logics , 3--23, Springer-Verlag, New York.

....sound (truth preserving) whether according to ordinary models or, as with Shoham s theory, according to a restricted class of models. In the latter case, the logic has embedded concepts, and may have important nonstandard characteristics (e.g. proof procedures do not always exist; see especially (Barwise 1985)) Even though the logical conception of manifest and constructive belief is, through the use of nonstandard logics, wide enough to incorporate many interesting theories, it is not free of difficulties. The first problem is whether every interesting representation function f arising in realistic ....

Barwise, J., 1985. Model-theoretic logics: background and aims, Model-Theoretic Logics (J. Barwise and S. Feferman, eds.), New York: Springer-Verlag, 3-23.

.... sets have been extensively studied through decades, but did not show up in notable applications until Aczel (1988) This is probably due to the fact that the classical wellfounded universe was a rather satisfying domain for the practicing mathematician the mathematician in the street (Barwise, 1985). Aczel s work on nonwellfounded sets evolved from his interest in b c,d c d a Figure 4: a = fb; fc; dgg in hyperset notation W Figure 5: The Aczel picture of Omega modeling concurrent processes. He adopted the graph representation for sets to use in his theory. A set a = fb; fc; dgg can be ....

Barwise, J. (1985). Model-Theoretic Logics: Background and Aims. In Barwise, J., and Feferman, S. (eds.) Model-Theoretic Logics, 3--23, Springer-Verlag, New York.

.... the gulf between first order logic and mathematical practice in these words: As logicians, we do our subject a disservice by convincing others that logic is first order, and then convincing them that almost none of the concepts of modern mathematics can really be captured in first order logic [ Barwise, 1985 ] A detailed compendium of even the more important higher order constructions in mathematical practice is well beyond the scope of this survey. Instead, we propose to consider the general issue of second order axioms versus first order schemas for principles such as Induction and Replacement. We ....

J. Barwise. Model-theoretic logics: Background and aims. In J. Barwise and S. Feferman, editors, Model Theoretic Logics. SpringerVerlag, New York, 1985.

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Barwise, J.: 1985, Model-Theoretic Logics: Background and Aims, Chapter 1 of ModelTheoretic Logics, edited by J. Barwise and S. Feferman.

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J. Barwise. 1985. Model theoretic logics: background and aims. In J. Barwise and S. Feferman (eds) Model-Theoretic Logics, Springer, New York, pages 2--23.

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J. Barwise. 1985. Model-theoretic logics: background and aims. In J. Barwise and S. Feferman (eds) Model-Theoretic Logics, Springer, New York, pages 2--23.

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