Summary of John McDowell's article 'Identity mistakes: Plato and the logical atomists'

NOTE: McDowell's article is a densely packed one that does not admit of ready summary, because he is doing many things and analyzing things so precisely and concisely that a summary which attempts to include all the important points would be impossibly dense. On the other hand, a 'summary' that explained McDowell's argument in everyday language that we all would understand would be easily twice as long as the article itself. In what follows, I have slimmed down the article and omitted much detail. I only hope that I have included enough detail to follow the arguments I have presented.

Socrates: Now isn't it true about all things, together or individually, that we must either know them or not know them? I am ignoring for the moment the intermediate conditions of learning and forgetting, as they don't affect the argument here.
Theaetetus: Of course, Socrates, in that case there is no alternative. With each thing we either know it or we do not.
Socrates: Then when a man judges, the objects of his judgments are necessarily either things which he knows or things which he doesn't know?
Theaetetus: Yes, that must be so.
Socrates: Yes, if he knows a thing, it is impossible that he should not know it; or if he does not know it, he cannot know it.
Theaetetus: Yes, of course.
Socrates: Now take the man who judges what is false. Is he thinking that things which he knows are not those things but some other things which he knows--so that knowing both he is ignorant of both?
Theaetetus: But that would be impossible, Socrates.
Socrates: Then is he imagining that things which he doesn't know are other things which he doesn't know? Is it possible that a man who knows neither Theaetetus nor Socrates should take it into his head that Socrates is Theaetetus or Theaetetus Socrates?
Theaetetus: I don't see how that could happen.
Socrates: But a man certainly doesn't think that things he knows are things he does not know, or again that things he doesn't know are things he knows.
Theaetetus: No, that would be a very odd thing.
Socrates: Then in what way is false judgement still possible? There is evidently no possibility of judgement outside the cases we have mentioned, since everything is either a thing we know or a thing we don't know; and within these limits there appears to be no place for false judgement to be possible.
Theaetetus: That's perfectly true.

The problem is as follows: "With any given thing (e.g. Theaetetus) one either knows it or not. This applies, in particular, to things which figure in one's judgements. With two things (e.g. Theaetetus and Theodorus) there are four combinations:

Plato assumes that one must know a thing if it figures in one's judgements in any way. In combinations b, c, and d above, at least one of the two things is not known. Thus in b, c, and d at least one of the two things cannot figure into the judgement that the one thing is the other. Thus it is clear that it is impossible to form a judgement at all involving these two things if one is in the positions described by b, c, or d.

The only possibility left in which a judgement, false or true, can be made is a, that one knows both. So if we are to make a false judgement, we must falsely judge that one thing we know is another thing we know.

Plato claims at 188b3-5 that one cannot know both things and make a false judgement about them, namely that the one thing is the other. One cannot make that false judgement because that false judgement would surely mean that one does not really know both things. But one knows ex hypothesi both things, and knowing both things precludes judging that one is the other.

Furthermore, ex hypothesi, it looks like there are definitely two things, not one, and one knows both, so one must know they are two things. McDowell, however, does not say that.

Russell held that all true sentences of the form "x is y" are "tautological." A tautology is, in its most basic form, a claim that "A is A" or "You are you." With tautologies all you need to know they are true is understanding of the terms. In order to accept that claim, we need some further assumptions: I) one cannot judge to be true what one is in a position to know to be false, and II) one cannot correctly express a judgement with a sentence containing terms one does not understand. Russell says that when you assert that "x is y," x and y refer to the bearer of the names 'x' and 'y,' rather than the names themselves. To understand x is to understand its bearer, and so to assert that 'x is y' is to say that there is only one bearer there. We can say that Russell says that what a sentence says depends on the meanings of the terms, not the terms themselves.

Frege, however, held that the true sentence 'x is y' is not quite the same as the true sentence 'x is x,' because the sense of x is different from that of y. Their reference (i.e. their bearer) may, however, be the same.

Leibniz had a principle that reference-preserving substitutions in a sentence preserve its truth value.

But there are referentially opaque contexts and referentially transparent contexts. For example, "Jacques is a professor" remains true if you substitute "Dr. Bailly" or "Isidora's father" for "Jacques" in it. Thus the context is referentially transparent. In the sentence, "Nine is necessarily greater than seven," you simply cannot substitute "the number of planets" for "nine." That context is referentially opaque, even though 'the number of planets' has the same referent as 'nine.' Likewise, "Isidora smiles at her mother" is not the same as "Isidora smiles at a beautiful lady" even though 'her mother' is 'a beautiful lady.' What is happening is that two terms can have the same referent but different senses.

McDowell proposes that when we think about (1) above (it is impossible to judge that one thing is the other, whichever of these combinations obtains), we need to think about opacity and transparency. (1) only holds true in transparent contexts.

Russell's assertion that all true sentences 'x is y' are tautological can yield the same paradox about assertions as Plato's paradox does about judgements. If all true positive identity statements 'x is y' are obviously true, or 'self-intimatingly' true, as McDowell says, then how can one make false ones?

It is simple: one makes them when the context is opaque. One may know Jacques and one may know Isidora's father, but one may not know that Jacques is Isidora's father (for any number of reasons: one may only have contact with Isidora's father after a run, in a certain context (the daycare center), with jeans and a baseball cap, in the dark, etc.). So there is a paradox only if you confuse transparent with opaque contexts.

Another suggestion that will get rid of the paradox is to say that by 'know' in the initial paradox, one means 'know all about.' McDowell rejects that idea, because it would prove the impossibility of any false judgements whatsoever, not just those in the form of an identity, and Plato takes himself to be talking about identity judgements.