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Jaakko Hintikka and Gabriel Sandu a REVOLUTION in LOGIC? Logic Might at First Sight Seem an Unlikely Arena for Revolutions, Even

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Jaakko Hintikka and Gabriel Sandu a REVOLUTION in LOGIC? Logic Might at First Sight Seem an Unlikely Arena for Revolutions, Even Jaakko Hintikka and Gabriel Sandu AREVOLUTIONINLOGIC? Logic might at first sight seem an unlikely arena for revolutions, even for revolutions in Jefferson’s sense rather than Lenin’s. Kant maintained that Aristotelian logic had not changed in two thousand years and could never change.1 Even though Kant’s view of Aristo- tle’s logic has been thoroughly discredited (even by Aristotle’s own standards) both systematically and historically,2 most contemporary philosophers and linguists are adopting the same Kantian attitude to Frege’s logic, or strictly speaking rather to that part of Frege’s logic that has come to be known variously as first-order logic, quantification theory, or lower predicate calculus. When a critic once suggested to a prominent philosopher of language that one of the cornerstones of ordinary first-order logic, Frege’s treatment of verbs for being like is, does not capture the true Sprachlogik, he looked at the speaker with an expression of horror—mock horror, we hope—and said, ‘Nothing is sacred in philosophy any longer!’ Yet Frege’s formulation of first-order logic contains a fundamental error. It is manifested already in his formation rules. And this same virus has infected all the subsequent formulations and versions of first- order logic. In order to see what Frege’s mistake is, we have to go back to the basics and ask what first-order logic is all about. The answer is obvious. First-order logic is about quantifiers. It is not for nothing called quantification theory. But what is often overlooked is that quantification theory is not a study of quantifiers in splendid isolation from each other. If you use quantifiers one by one unrelated to each other, the only logic you end up with is monadic first-order 1Immanuel Kant, Kritik der reinen Vernunft B 7–8. 2Cf. Jaakko Hintikka, “Aristotle’s Incontinent Logician”, Ajatus vol. 37 (1978), pp. 48–65, where Hintikka argues that Aristotle’s system of logic was an unhappy compromise between the different guiding ideas he was trying to implement. Nordic Journal of Philosophical Logic, Vol. 1, No. 2, pp. 169–183. c 1996 Scandinavian University Press. 170 jaakko hintikka and gabriel sandu logic or some other mild generalization of Aristotelian syllogistic. The real power of first-order logic lies in the use of dependent quantifiers, as in (1) (∀x)(∃y)S[x, y] where the truth-making choice of a value of y depends on the value of x. Without dependent quantifiers, one cannot even express the func- tional dependence of a variable on another. Quantifier dependence is therefore the veritable secret of the strength of first-order logic. One can almost say that to understand first-order logic is to understand the notion of quantifier dependence. Several of the typical applications of first-order logic in philosophical analysis, such as uncovering quantifier- switch fallacies, exploit directly the idea of quantifier dependence. But to understand quantifier dependence is ipso facto to under- stand the notion of quantifier independence. The latter is simply the same concept as the former in a different dress. Hence it does not mean transcending in the least the conceptual repertoire of ordinary quantifi- cation theory if we introduce into it an explicit independence indicator by means of which the independence of (Q1x) of another quantifier (Q2x) is expressed by writing it (Q1x/Q2x). The same slash notation can be applied to propositional connec- tives. It is intuitively obvious what is meant by such formulas as (2) (∀x)(A(x)(∨/∀x)B(x)) or (3) (∀x)(∃y)(∀z)(S1[x, y, z](∨/∀x)S2[x, y, z]) It turns out that if we limit our attention to formulas in negation normal form (i.e. in a form where all negation-signs precede immediately atomic formulas or identities), it suffices to consider only two kinds of uses of the slash, viz. (∃x/∀y)and(∨/∀y). The result of adding the slash to the conceptual arsenal of the received first-order logic results in what will be called independence- friendly (IF) first-order logic. What is IF first-order logic like? Does it really go beyond from received first-order logic? In simplest cases, the new notation does not yield anything new. For instance, (4) (∀x)(∃y/∀x)S[x, y] is obviously logically equivalent with (5) (∃y)(∀x)S[x, y] a revolution in logic? 171 Likewise (2) is equivalent with (6) (∀x)A(x) ∨ (∀x)B(x) But in more complex cases something remarkable happens. Consider, for instance, the sentence (7) (∀x)(∀z)(∃y/∀z)(∃u/∀x)S[x, y, z, u] which is equivalent to (8) (∀x)(∃y)(∀z)(∃u/∀x)S[x, y, z, u] Can (7) be expressed without the slash notation? If so, there would be a linear ordering of the four initial quantifiers (∀x), (∃y), (∀z), (∃u) that would result in the same logical force as (7). But what kind of ordering might that be? Since (∃y)dependsin(7)on(∀x) but not on (∀z), the relative order of the three would have to be (9) (∀x)(∃y)(∀z) But since (∃u) depends on (∀z) but not on (∀x), their relative order wouldhavetobe(∀z)(∃u)(∀x), which is incompatible with (9). Likewise, it can be shown that (3) cannot be expressed without the slash notation (or equivalent). Here we are beginning to see the whole horror of Frege’s mistake. The notation he introduced (like the later notation of Russell and Whitehead) arbitrarily rules out certain perfectly possible patterns of dependence and independence between quantifiers or between connec- tives and quantifiers. These patterns are the ones represented by ir- reducible slashed quantifier combination (and similar combinations in- volving quantifiers and connectives). Anyone who understands received first-order logic understands sentences which involve such patterns, for instance understands sentences of the form (7) or (8), in the concrete sense of understanding the precise conditions that their truth imposes on reality. Hence they ought to be expressible in the language of our basic logic. And the only way of doing so is to build our true logic of quantification so as to dispense with the artificial restrictions Frege im- posed on the received first-order logic. The real logic of quantification, in other words the real ground floor of the edifice of logic, is not the ordinary first-order logic. It is IF first-order logic. Terminologically speaking, we are doing ourselves an injustice by calling the received first-order logic “ordinary”. The limitations we have been discussing make Fregean logic systematically speaking quite extraordinary, as is 172 jaakko hintikka and gabriel sandu also reflected by the fact that the properties of the received first-order logic do not reflect at all faithfully what one can expect to happen in logic in general.3 In order to reach IF first-order logic, it is not necessary to introduce any new notation like our slash notation. All that is needed is to for- mulate the rules for parentheses (scope) more liberally than Frege and Russell. If you reflect on the matter (and even if you don’t), there is absolutely no reason why the so-called scope of a quantifier, expressed by the associated pair of parentheses, should comprise a continuous segment of a formula adjacent to the quantifier and following it.4 Ex- pressions violating this requirement (and consequently violating the usual formation rules) can have a perfectly understandable semantical interpretation. Indeed, (8) might as well be written as follows: (10) (∀x)[(∃y)(∀z)](∃u)[S[x, y, z, u]] where the two outer pairs of square brackets indicate the (discontin- uous) scope of (∀x). In more complicated cases, alas, the liberated parentheses notation turns out to be unintuitive to the point of being unreadable. Hence for practical reasons we prefer the slash notation. A fragment of IF first-order logic is known to the cognoscendi as the logic of partially ordered (“branching”) quantifiers.5 This logic is nevertheless not entirely representative of the conceptual situation in general. For one thing, propositional connectives can exhibit the same independence phenomena as quantifiers.6 Moreover, even though quantifier dependencies and independencies can always be dealt with in terms of partial ordering, this is not true of other concepts. For there is in general no reason why the dependence of logical notions on each other should be transitive. 3Cf. Jaakko Hintikka, The Principles of Mathematics Revisited. Cambridge Uni- versity Press, Cambridge, 1996. 4Note that the usual scope notation for propositional connectives already violates some of these requirements. 5Branching quantifiers were introduced by Leon Henkin, “Some Remarks on In- finitely Long Formulas”, in Infinitistic Methods, Warsaw, 1959, pp. 167–183. For their theory, see e.g. W. Walkoe, “Finite Partially Ordered Quantification”, in Journal of Symbolic Logic, vol. 35 (1970), pp. 535–550; Herbert Enderton, “Finite Partially-Ordered Quantifiers”, in Zeitschrift f¨ur Mathematische Logik und Grund- lagen der Mathematik, vol. 16 (1970), pp. 393–397; and Jon Barwise, “Some Ap- plications of Henkin Quantifiers”, in Israel Journal of Mathematics, 25, 1976, pp. 47–80. 6See Gabriel Sandu and Jouko V¨a¨an¨anen, “Partially Ordered Connectives”, Zeitschrift f¨ur Mathematische Logik und Grundlagen der Mathematik, vol. 38 (1992), pp. 361–372. a revolution in logic? 173 Furthermore, the treatment of negation in the context of indepen- dent quantifiers requires more attention than it has been paid in the theory of partially ordered quantifiers.7 By excluding the possibility of genuine independence between quan- tifiers (and by implication between other concepts) Frege excluded from the purview of logic a wealth of important subjects. Many logicians are still apt to consider the logic of branching quantifiers and also IF first-order logic as a marginal curiosity.
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