Modal Objection to Naive Leibnizian Identity

Abstract

This essay examines an argument of perennial importance against naive Leibnizian absolute identity theory, originating with Ruth Barcan in 1947 (Barcan, R. 1947. ‘The identity of individuals in a strict functional 3 calculus of second order’, Journal of Symbolic Logic, 12, 12–15), and developed by Arthur Prior in 1962 (Prior, A.N. 1962. Formal Logic. Oxford: The Clarendon Press), presented here in the form offered by Nicholas Griffin in his 1977 book, Relative Identity (Griffin, N. 1977. Relative Identity. Oxford: The Clarendon Press). The objection considers the property of being necessarily identical to a specific object a as a counterexample to Leibnizian identity conditions, and more particularly to the indiscernibility of identicals, when it is only contingently true that a=b. The inquiry eliminates necessity and reference to a specifically designated object as responsible for the counterexample, leaving only identity. The requirements for an exact reinterpretation of Leibniz's Law in light of counterexamples involving converse intentional properties and the family of properties suggested by the property of being necessarily identical to a where a=b (or an equivalent definite descriptor variation) is only logically contingently true, are formalized in a more powerful, counterexample-resistant version of Leibniz's Law that does not succumb to relative identity as a necessary alternative motivated by desperation over the failure of the absolute identity principle.

Acknowledgements

I am grateful to the students in my Identität seminar at Universität Bern, Fall Semester 2009, to whom I presented several versions of this essay as we worked through the classic conflict of relative versus absolute identity in the two seminal studies by Griffin and Wiggins. Thanks are also due to three anonymous referees who offered useful technical and historical recommendations for improving a previous version of the essay.

Notes

²Leibniz's historical views on identity are undoubtedly immune to the criticism that the principle of the indiscernibility of identicals so-designated in contemporary analytic metaphysics and philosophy of logic is subject to counterexamples involving converse intentional properties, and to counterexamples involving the kinds of properties freely defined by Griffin. The historical issues that I canvass begin only with Barcan (1947); it is this relatively recent span of the history and philosophy of logic on which I am focused throughout. For more discussions of the again so-called identity of indiscernibles, see Adams 1979, pp. 5–26. Adams makes it clear that, at least with respect to the identity of indiscernibles — he does not discuss the indiscernibility of identicals — Leibniz distinguishes between three degrees of the principle's strength, concerning all of an individual's properties, purely qualitative properties, or non-relational qualitative properties. Griffin's modal objection to the indiscernibility of identicals and my efforts to respond to it evidently target specifically that version of weakest degree of strength in which quantification is made over all properties without qualification, allowing Leibniz an escape route that he seems actually to favor in the more strengthened formulations of either or both parts of his identity thesis. See also Mates (1972, pp. 81–118).

³Griffin attributes the proof originally to Barcan (1947, p. 15, proposition 2.31), and in the more recent form he discusses owing to Prior (1962, pp. 205–206).

⁴Note that (LL) or the indiscernibility of identicals in particular cannot avoid the objection concerning converse intentional properties by holding that there is no difficulty for Leibniz's principle provided only that such properties are construed de re rather than de dicto, as properties belonging to individual things falling under the identity thesis independently of the way these things and their properties are thought of or linguistically described, as when the same thing is considered under alternative descriptions. The reason that such a strategy cannot avail is obvious when we reflect that converse intentional properties by definition are extrinsic properties that an object has entirely in virtue of a particular propositional attitude that a thinking intending subject takes up toward it in a particular intending episode. If I love Naples, it does not follow that I love Napoli, even though Naples = Napoli, if I do not know, as is the distinguishing mark of a de dicto context, that Naples and Napoli are the same city named, respectively, in English and Italian. The problem is dramatized by Kripke (1979).

⁵See Jacquette 2009, pp. 12–20.

⁶See Griffin 1977, esp., pp. 2–4.

⁷A paraphrastic solution to Quine's argument is developed by Jacquette (1986).