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# Hyperclassical Logic (A.K.A. If Logic) and its Implications for Logical Theory

Abstract

Let us assume that you are entrusted by UNESCO with an important task. You are asked to devise a universal logical language, a Begriffsschrift in Frege's sense, which is to serve the purposes of science, business and everyday life. What requirements should such a “conceptual notation” satisfy? There are undoubtedly many relevant desiderata, but here I am focusing on one unmistakable one. In order to be a viable lingua universalis, your language must in any case be capable of representing any possible configuration of dependence and independence between different variables. For if such a configuration is possible in principle, there is no guarantee that it might not one day show up among the natural, human or social phenomena we have to study.

But how are dependencies and independencies between variables expressed in our familiar logical notation? Every logician worth his or her truth-table knows the answer. Dependencies between two variables are expressed by dependencies between the quantifiers to which they are bound. For instance, in

the variable y depends on x, while in

z depends on x but not on y, while u depends on both x and y.

But how is the dependence of a quantifier on another one expressed in familiar logical languages? Obviously by occurring in its scope, indicated by the pair of parentheses following it (cf. here Hintikka [1997]). But the nesting of scopes is a transitive and antisymmetrical relation which allows branching only in one direction. Hence other kinds of structures of dependence and independence between variables are not representable in the received logical notation. Such previously inexpressible structures form the subject matter of what has been referred to as independence-friendly (IF) logic.

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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

Jon Barwise [1979], On branching quantifiers in English, Journal of Philosophical Logic, vol. 8, pp. 4780.

H.B. Enderton [1970], Finite partially ordered quantifiers, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 16, pp. 393397.

Jaakko Hintikka [1996], The principles of mathematics revisited, Cambridge University Press.

Jaakko Hintikka [1997], No scope for scope?, Linguistics and Philosophy, vol. 20, pp. 515544.

Wilfrid Hodges [1997 (b)], Some strange quantifiers, Lecture notes in computer science (J. Myclielski et al., editors), vol. 1261, Springer-Verlag, Berlin, pp. 5165.

Neil Tennant [1998], Games some people would have all of us play, Philosophia Mathematica, vol. 6, pp. 90115.

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