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Nonstandard theories of quantification and identity

Published online by Cambridge University Press:  12 March 2014

A. Trew
A. Trew*
Affiliation:
University of East Anglia
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Extract

In this paper a number of nonstandard systems of predicate logic with or without identity, are translated with subsystems of applied standard system of predicate logic with identity. There are nonstandard theories of quantification which, following [16], are described as inclusive systems; their theorems are valid in all domains, including the empty domain. Theories of quantification which allow for the substitution of denotationless terms for free variables, are described, following [21], as systems of free logic; they are said to be free of the requirement that all singular terms must have denotations. Free logics and inclusive logics may each be of the other type. A nonstandard theory of identity, described, following [12] as a theory of nonreflexive identity, may be combined with a standard or with a nonstandard theory of quantification. Another kind of nonstandard system of predicate logic examined is a nonstandard version of a system of monadic predicate logic in which a distinction is made between sentence and predicate negation, and which is nonstandard in the sense that the laws relating sentence and predicate negation diverge from the standard ones. In the systems examined, this is combined with an inclusive quantification theory.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 35 , Issue 2 , June 1970 , pp. 267 - 294
Copyright
Copyright © Association for Symbolic Logic 1970

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References

[1]Church, A., Review of K. Lambert: Existential import revisited, Notre Dame journal of formal logic, vol. 4 (1963), pp. 268–292, this Journal, vol. 30 (1965), pp. 103–104.Google Scholar
[2]Cochiarella, N., A logic of actual and possible objects, this Journal, vol. 31 (1966), pp. 688–689.Google Scholar
[3]Hailperin, T., Quantification and the empty domain, this Journal, vol. 18 (1953), pp. 197–200.Google Scholar
[4]Hailperin, T., Systems of restricted quantification. I, this Journal, vol. 22 (1957), pp. 19–35.Google Scholar
[5]Hughes, G. and Londey, D., The elements of formal logic, Methuen, London, 1965.Google Scholar
[6]Jaskowski, S., On the rules of supposition informal logic, Studia logica, vol. 1 (1934), pp. 5–32.Google Scholar
[7]Kalish, D. and Montague, R., On Tarski's formalisation of predicate logic with identity, Archiv für matematische Logik und Grundlagenforshung, vol. 7 (1965), pp. 81–101.CrossRefGoogle Scholar
[8]Kleene, S., Introduction to metamathematics, North-Holland, Amsterdam, 1952.Google Scholar
[9]Kripke, S., Semantical considerations on modal logic, Acta philosophica fennica, Fasc. XVI (1963), pp. 83–94.Google Scholar
[10]Lambert, K. and Scharle, T., A translation theorem for two systems of free logic, Logique et analyse, no. 39–40 (1967), pp. 328–341.Google Scholar
[11]Leblanc, H. and Hailperin, T., Non-designating singular terms, Philosophical review, vol. 68 (1959), pp. 239–243.CrossRefGoogle Scholar
[12]Lejewski, C., A theory of non-reflexive identity, Proceedings of the sixth Forschungsgespräch; Institut für Wissenschaftstheorie, Salzburg, 09, 1965.Google Scholar
[13]Mates, B., Leibniz on possible worlds, Logic, methodology and philosophy of science. III, edited by Rootselaar, van and Staal, , North-Holland, Amsterdam, 1968.Google Scholar
[14]Meyer, R. and Lambert, K., Universally free logic, this Journal, vol. 33 (1968), pp. 8–26.Google Scholar
[15]Mostowski, A., On the rules of proof in the pure functional calculus, this Journal, vol. 16 (1951), pp. 107–111.Google Scholar
[16]Quine, W., Quantification and the empty domain, this Journal, vol. 19 (1954), pp. 177–179.Google Scholar
[17]Rescher, N., On the logic of chronological propositions, Mind, vol. 75 (1966), pp. 75–96.CrossRefGoogle Scholar
[18]Routley, R., Some things do not exist, Notre Dame journal of formal logic, vol. 7 (1966), pp. 251–274.CrossRefGoogle Scholar
[19]Scott, D., Existence and description in formal logic, Bertrand Russell, philosopher of the century, edited by Schoenman, R., George Allen & Unwin, London, 1967, pp. 181–200.Google Scholar
[20]Trew, A., Incompleteness of a logic of Routley's, Notre Dame journal of formal logic, vol. 9 (1968), pp. 385–387.CrossRefGoogle Scholar
[21]van Fraassen, B., The completeness of free logic, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 12 (1966), pp. 219–234.CrossRefGoogle Scholar
[22]van Fraassen, B. and Lambert, K., On free description theory, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 13 (1967), pp. 225–240.CrossRefGoogle Scholar