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Is Objectual Identity Really Dispensable?

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Abstract

Kai Wehmeier’s Wittgensteinian Predicate Logic (W-logic) is a formulation of first-order logic under the exclusive interpretation of the quantifiers. W-logic has a distinguished relation constant for co-reference but no sign for objectual identity. Wehmeier denies that objectual identity exists on the grounds that it cannot be a genuine binary relation. Fortunately W-logic is equi-expressive with standard first-order logic with identity and it appears that objectual identity is dispensable across the broader logical enterprise. This paper challenges the latter claim as objectual identity seems to be needed in the exclusive interpretation of quantified modal logic, specifically, for implementing certain kinds of de re quantification.

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Notes

  1. Interpreting ‘=’ as objectual identity seemed necessary in order to adequately account for truths like ‘∀x x = x’ in FOL=. Under this interpretation the truth of each of the instances generated by ‘x = x’ is grounded in the appropriate relational fact.

  2. Hintikka (1956) anticipates W-logic but does not work out a number of technical details supplied by Wehmeier. Hintikka speculated that the impredicative comprehension/abstraction principle may be taken as an additional axiom in an extension of first-order logic without identity so long as the quantifiers are interpreted exclusively. As evidence he shows that many of the traditional paradoxes are no longer virulent under this interpretation. Unfortunately a counterexample to the consistency of the abstraction schema, when the quantifiers are interpreted exclusively, was discovered by Hintikka (1957).

  3. In the Tractatus (5.5303) Wittgenstein (1922) similarly attacks the coherence of objectual identity as does Russell (1903, p. 63) by noting that one particular cannot do the work of two. Russell (1903, p. 64) only admits the relation given its indispensability in mathematics but without offering a solution accounting for its philosophically puzzling features.

  4. Wehmeier (2012, p. 765, fn. 10) makes the same point wherein he claims that “Since the translation [into W-logic] easily extends to second- and higher-order logics, there is no reason to think that W-logic might not be able to accommodate any part of mathematics”.

  5. One might also work in a W-logic version of second-order logic. Though identity is definable in second-order logic under the inclusive interpretation the attempted definition fails when interpreted exclusively, as shown by Wehmeier (2012, p. 774, fn. 21).

  6. The translation works since in W-logic the sentence ‘Σx\(\Box \)Πy¬Taut(y)’ says that there is some individual x in the actual world such that in all accessible possible worlds any individual other than x fails to satisfy some tautology (there can be no objects like that), hence, for all accessible worlds the only individual which can be in their respective domains is x itself. As some readers might countenance worlds with no individuals the additional condition Taut(x) guarantees that x is in the domain of all worlds accessible from the world of evaluation.

  7. If the language is restricted to just QML (without identity) the W-logician has no problem handling de re quantification. For example, the W-logic translation of ‘∀x\(\Box \)∃yPxy’ is Πx\(\Box \)(Pxx ∨ ΣyPxy). But this won’t work with the translation of the necessity of being into quantified modal W-logic since on the offered procedure we would get Πx\(\Box \)(x = x ∨ Σy x = y) as its translation. The latter is equivalent to Πx\(\Box \)x = x (since for any x ‘Σy x = y’ is false in W-logic). But Πx\(\Box \)x = x is equivalent to ΠxTaut(x) for Taut(x) any tautology, and the necessity of being becomes a necessary truth. What was supposed to be a routine translation procedure instead yields a substantive truth of metaphysics.

  8. There are many competing formal descriptions of QML= built, for example, over different kinds of normal modal propositional logics. The phrase “the theories QMWL and QML= ought to be equi-expressive” means that for any set of axioms and rules in which quantifiers are interpreted inclusively and in which the necessity of being can be translated there is a W-logic analogue (a version of QMWL) which is equi-expressive.

  9. For example, SQML, the simplest quantified modal logic, entails the necessity of being since the existence predicate is definable via objectual identity. This follows since ∀x\(\Box \)x=x, and consequently ∀x\(\Box \)∃y y=x, is a thesis of SQML. But then∀x\(\Box \)Ex holds under the assumption that ‘E’ is interpreted as the predicate λx. ∃y y=x.

  10. Of course not all controversy would be eliminated for there are philosophers who claim that de re quantification is not even intelligible e.g. Quine (1953).

  11. Let me give a simple example of a violation of the desideratum. There is a solution to our problem which is couched in a language containing infinitary operations, for example, those necessary for forming Scott sentences (infinite disjunctions). Let D be the union of all domains (of all worlds) and suppose that each member has a name in the list c0, c1,…,cα … , for α some cardinal number (possibly transfinite). Let ‘Ref(ci, x, w)’ mean that cirefers to x at world w. With the new metalinguistic vocabulary it is possible to express the necessity of being in the extended language as for every object x and in all possible worlds w eitherRef(c0, x, w) orRef(c1, x, w) or …. The sentence has the intended meaning in an extension of W-logic to one with infinitary linguistic resources. But now the commitment to infinite sets and to sentences with an infinite number of disjuncts is part of the formal apparatus for translating the necessity of being and this clearly goes well beyond the merely formal conventions of the exclusive interpretation. What is wanted is not just a technical solution but one grounded on philosophical considerations as well.

  12. Alternatively one could countenance impossible worlds at which some instances of tautologies fail.

  13. The just given argument depends on the claim that ‘\(\Box \)’ and ‘◇’ are interdefinable. One could work in Prior’s (1957) system Q in which the alethic operators are no longer duals, in particular, simply because one can prove that ¬Σx◇¬Taut(x) does not license the move to Πx\(\Box \)Taut(x) in a system like Q. Rather, we may only infer that Πx¬◇¬Taut(x) which does not imply the necessity of being (it states that for any object x we cannot find a possible world in which ¬Taut(x) holds). Prior’s Q is formally and philosophically problematic (e.g. Menzel 1991, pp. 346–348; Williamson 2013, p. 71) and the W-logician would likely not want to take Prior’s system as the right way to build QMWL.

  14. I thank Kai Wehmeier for pointing out that adding the existence predicate is a solution route completely open to the W-logician.

  15. For example, when one is first introduced to free logic the existence predicate is often defined in this way as if it were a perfectly ordinary part of logical technology.

  16. Thanks to Kai Wehmeier for noting that the meaning of the existence predicate may at least be partially specified as a constraint rather than the identification of a real-world correlate (like a property).

  17. The claim that the expression ‘λx.Taut(x)’ does not denote a property (given its universality) implies that ‘λx.E(x)’ also fails to denote a property (as existence applies to everything).

  18. As noted by an anonymous reviewer.

  19. I thank an anonymous reviewer for raising this point.

  20. Thanks to an anonymous reviewer for this observation.

  21. This is not to say that the converse Barcan is without controversy in some quarters even without the existence predicate.

  22. I thank Kai Wehmeier for raising this objection via email communication.

  23. In the lectures “Philosophy of Logical Atomism” Russell claimed in the discussion after Lecture 5 that “… it is perfectly clear that, if there were such a thing as this existence of individuals that we talk of, it would be absolutely impossible for it not to apply, and that is the characteristic of a mistake” (quoted from Kripke 2013, p. 7).

  24. For example, the normal modal propositional logic S5 is sound and complete with respect to several descriptions of its frames. It is sound and complete with respect to the class of all reflexive and Euclidean frames and also with respect to all frames in which R is an equivalence. But it is also sound and complete with respect to the class of all frames in which R is empty (that is, these frames are non-empty sets of worlds).

  25. For Wehmeier identity is not completely eliminated from W-logic as it persists as the property of self-identity (Wehmeier 2012, p. 768). It then seems that for Wehmeier NB3 is the most plausible (second-order) W-logic analogue to the necessity of being.

  26. It is important to see how ΠxΣF(Fx ∧ ¬ΣuΣv(Fu ∧ Fv)) expresses that for every object a there is a singleton property applying only to a itself. Fix any object a. The sentence says that there is some property F such that Fa, hence, F applies to at least one thing. Since ‘u’ occurs free in ‘Σv(Fu ∧ Fv)’ the values of ‘u’ and ‘v’ cannot coincide. Hence, ‘¬ΣuΣv(Fu ∧ Fv)’ states that F cannot apply to two or more things and since ‘Fa’ holds it follows that F must be a singleton property.

  27. There are several other solutions in the same vicinity as NB1–NB3, for example, Skolem functions may be used to eliminate the second-order quantifiers. Another option is to admit primitive pairing (π) and projection (π1, π2) functions such that the necessity of being is formalized along the lines of Πx\(\Box \)Σz (π1(z) := x), where ‘:=’ is the sign for functional evaluation. These solutions again require the postulation of entities other than individuals and both solutions implicitly assume the being-constraint.

  28. Hintikka (1956) distinguishes between weak and strong interpretations of the exclusive reading. Wehmeier works with the weak interpretation and it is the one described here.

  29. Cf. Wehmeier (2012, p. 763).

  30. Hintikka (1956, pp. 225–226) offers several examples in which the exclusive reading fits the logical form of some English sentences.

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Acknowledgements

Thanks to Kai Wehmeier for several very helpful email exchanges on earlier drafts of this paper. These discussions improved the paper in innumerable ways and his intellectual generosity is most appreciated. I am also sincerely grateful for the extensive comments, both technical and philosophical, provided by the anonymous referees.

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  1. Department of Philosophy and Religious Studies, Glendale Community College, 6000 West Olive Ave, Glendale, AZ, 85302, USA

    Eric T. Updike

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Appendix: A Brief Introduction to W-Logic

Appendix: A Brief Introduction to W-Logic

The main feature of W-logic is the exclusive interpretation of the quantifiers, an interpretation committed to the Tractarian ideal that distinct variables take distinct values. The standard convention for interpreting a sentence like ‘∀x∃yRxy’ is that the range of the displayed quantifiers is the same (the choice for ‘y’ includes the value assigned to ‘x’ relative to some background interpretation). This is the inclusive reading of the quantifiers. On the exclusive reading the choice for ‘y’ excludes the value assigned to ‘x’; an acceptable W-logic translation of ‘∀x∃yRxy’ is ‘Πx(ΣyRxy ∨ Rxx)’ (where ‘Π’ and ‘Σ’ are the universal and existential quantifiers, respectively, under their exclusive readings).

Formally, the exclusive interpretation of the quantifiers introduces the following convention for handling quantifiers with overlapping scopes: the range of a bound variable ‘x’ excludes all values of variables which occur free within the scope of the quantifier which binds that occurrence of ‘x’. More generally, for a sentence having the general form Qxφ(x1,x2,…,xn, c1,…,cm), with the displayed free variables ‘xi’, individual constants ‘cj’ and for ‘Qx’ some quantifier expression, the exclusive reading requires that the variable ‘x’ must exclude from its range the values assigned to each free variable ‘xi’ and to each ‘cj’ with respect to some fixed interpretation.Footnote 28 For example, the exclusive interpretation of the formula Πy(Rxy ∨ ΣzSzy) excludes from the range of ‘y’ the value assigned to ‘x’ and excludes from the range of ‘z’ the value assigned to ‘y’ (the range of ‘z’ includes the value assigned to ‘x’ as this variable does not occur free within the scope of the displayed occurrence of ‘Σz’).

In W-logic functions are treated as single-valued relations and rather than an expression like ‘f(x) = y’ one may introduce an application operator APP whereby APP(f, x, y) is interpreted as ‘the value of f at x is y’.Footnote 29 In lieu of identity W-logic includes a distinguished relation constant ‘≡’ for co-denotation or co-reference, a relation which can only obtain between individual constants (as such, it is not an objectual relation). Indeed, no symbol of the base language is interpreted as an objectual relation (though free relation variables may be so interpreted). Partisan supporters of objectual identity may always interpret a free relation variable as their favored reading of identity, suffering the corresponding metaphysical commitments.

To help ease the transition into interpreting first-order logic in the light of the lately given conventions for W-logic let [φ]w be the W-logic translation of a sentence φ of FOL= where again ‘Π’ and ‘Σ’ are the universal and existential quantifiers of W-logic. Their respective interpretations are the same as their inclusive mates except in the case of overlapping quantifier scopes, as discussed above. The following translations from FOL= into W-logic are readily forthcoming (Taut(x) is an arbitrary tautology and a, b are individual constants):

$$ \begin{array}{*{20}l} {\left[ {a = b} \right]^{w} } \hfill & = \hfill & {a \equiv b} \hfill \\ {[\forall xx = x]^{w} } \hfill & = \hfill & {Taut\left( x \right)} \hfill \\ {[\forall x(\neg x = a \leftrightarrow Px)]^{w} } \hfill & = \hfill & {\Pi x(\neg Pa \wedge Px)} \hfill \\ {[\exists x\exists yx \ne y]^{w} } \hfill & = \hfill & {\Sigma x\Sigma y(Taut\left( x \right) \wedge Taut\left( y \right))} \hfill \\ {[\exists x\exists y(Fx \wedge Fy)]^{w} } \hfill & = \hfill & {\Sigma x\Sigma y(Fx \wedge Fy) \vee \exists xFx} \hfill \\ {[\exists x\forall yRxy]^{w} } \hfill & = \hfill & {\Sigma x(\Pi yRxy \wedge Rxx)} \hfill \\ \end{array} $$

Wehmeier (2004), building on the work of Hintikka (1956), showed that W-logic is equi-expressive with FOL= without individual constants while Wehmeier (2008) showed that it is impossible for W-logic to express that two constants co-denote. To fix this lacuna Wehmeier added the sign ‘≡’ to W-logic. Given the additional expressive power W-logic becomes equi-expressive with FOL=. In this respect one can live without identity as any thesis of FOL= can be translated into a W-logic correlate which has exactly the same models. The exclusive reading of the quantifiers often yields complicated sentences, but, in its favor, there are English sentences that are best understood in the exclusive sense.Footnote 30

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Updike, E.T. Is Objectual Identity Really Dispensable?. Erkenn 84, 761–782 (2019). https://doi.org/10.1007/s10670-018-0100-y

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