A NOTE ON WIGGINS' PROOF OF THE NECESSITY OF IDENTITYⁱ

David Wiggins (Wiggins, 1980) has offered a proof that if x is identical with y, x is necessarily identical with y. The proof relies on Leibniz' Law, and makes ingenious use of £-bindingⁱⁱ.

'(£x)[Fx]' may be understood as 'the property that a thing has if it is F' ('F' may, of course, be complex). To say that an individual a has the property represented by '(£x)[fx]' we may write '(£x)[Fx],<a>'. Since this is equivalent, one assumes, for cases not involving opacity, to 'Fa', one would want a rule warranting passage from '(£x)[...x...],<a>' to '...a...' (£-conversion).

Wiggins then introduces an operator 'NEC' on such 'abstracts': [NEC(£x)[Fx]]', e.g., is the property of being necessarily F. It is important to note that in '[NEC(£x)[Fx]],<a>' 'a' is not in the scope of 'NEC': the context is perfectly extensional; all the moves of ordinary quantificational logic would be in order.

The proof now proceeds thus: Grant that there is a truth about H (Hesperus) expressed as follows:

[NEC[(£x)(£y)(x=y)]],<H,H>

This is an abstractable property, so we have:

(£z)[[NEC[(£x)(£y)(x=y)]],<H,z>]

But H has this property, and since H=P, P has it also:

(£z)[[NEC[(£x)(£y)(x=y)]],<H,z>],<P>

Then £-conversion gives us:

[NEC[(£x)(£y)(x=y)]],<H,P>

We may then conditionalize on the assumption that H=P, and universally quantify the result.

We may reasonably ask why we should accept '[NEC[(£x)(£y)(x=y)]],<H,H>', and what it means. Offhand, it would most naturally be taken as 'H is necessarily identical with itself'. Perhaps it might be read 'H has the property: being necessarily identical with H'. One might think it a matter of indifference whether one read 'Faa' as 'a has F to itself' or 'a has F to a'. But, as Geach (Geach, 1962, pp. 136-143) has cogently noted, they are different.

Suppose we have a dyadic predicate 'Fxy'. For a given finite universe 'F' can be represented by a Boolean matrix; in general, an N-adic predicate would be represented by an N-dimensional Boolean array. Now consider 'Fxx': for the given universe, it would be represented as a one dimensional Boolean string, the leading diagonal of the two dimensional array for 'Fxy'. Being represented by a one dimensional string is, of course, the mark of a monadic predicate. And so we should consider 'Fxx'. This is the conclusion Geach comes to, by different (and more persuasive) arguments.

To reflect the monadic character of 'Fxx' Geach suggests a notation: '(x;u,v)Fuv'. The variables 'u' and 'v' here are, of course, bound; the only free variable is 'x'. The entire complex could be replaced by 'Gx', if one were not concerned to reveal the fact that the monadic predicate is derived from a dyadic one. Rather than simply write '(£x)Sx' for 'everything is self-identical', for example, we should write '(£x)(x;u,v)u=v'.

So Geach does not claim that we should eschew the practice of writing 'Fxx' when doing logic. The rules of any given system will guard against any fallacies that might result from dyadic/monadic confusion. And, on the whole, things are simpler without his suggested notation, since it, too, would have to come accompanied by rules which told us, e.g., how to get from 'a has F to itself' to 'something has F to something'. The notation 'Fxx' may be misleading, but, as Geach puts it, 'What the signs conceal, their use reveals' (p.142).

But in new contexts it may be well to demand complete perspicuity of signs. Using a notation according to the ordinary rules may lead us into fallacy if new bits of notation are added to the system.

Suppose, then, that we insist on Geach's notation for reflexive cases. 'x is self identical' will come out '(x;u,v)u=v'. The £-abstract on that will be '(£x)(x;u,v)u=v' and the 'NEC' counterpart will be '[NEC[(£x)(x;u,v)u=v]]'. But now notice that the move which took us from the first to the second line of the proof can no longer be made.

From

[NEC[(£x)(x;u,v)u=v]],<H>

we can only get the innocuous

[NEC[(£x)(x;u,v)u=v]],<P>

It is obvious that the proof requires that '[NEC[(£x)(£y)(x=y)]]' have two argument places, so that, so to speak, Phosphorus can be smuggled into one and not the other. And sometimes we will want to treat something of the form 'Fxx' as if it really had two argument places, as when we allow existential generalization with different variables in each place. But we have to decide antecedently that such existential generalization is truth preserving. That is what is at issue here: should '[NEC[(£x)(£y)(x=y)]],<H,H>' be read as saying that Hesperus is necessarily self-identical, or as saying that Hesperus has the detachable property of being necessarily identical with Hesperus? It might be easy to get someone to accept the premise, if given the first reading, but then the proof wouldn't go. On the second the proof will go, but skeptics (like me) are likely to dig in their heels at the first mention of properties like: being necessarily identical with Hesperus. Persuade me that there is a property like that and it won't take much to get me to agree that Phosphorus has it.

I have not, of course, given any reason to reject Wiggins' conclusion. I have only claimed that the passage from premise to conclusion is by no means the harmless exercise in quantificational logic that it superficially appears. The necessity of identity is something logic might be brought to reflect, if metaphysics established it. But the metaphysics wears the stripes.

Nollaig MacKenzie, York University. Nollaig@YorkU.CA

References

Wiggins, D., Sameness and Substance, Basil Blackwell, Oxford, 1980.

Geach, P. T., Reference and Generality, Cornell University Press, Ithaca, 1962.