The Word “Is”
The standard view is that “is” is ambiguous between the “is” of predication and the “is” of identity (we might also add the “is” of composition, as in “this statue is bronze”). Thus we have, “the cup is red” and “Hesperus is Phosphorus”, where the two occurrences of “is” have different meanings. To claim that “is” has the same meaning in both occurrences would produce absurd consequences. If the “is” in “the cup is red” expressed identity, then the sentence would mean that the cup is identical to redness, which is false and absurd. If the “is” of “Hesperus is Phosphorus” expressed predication, then the sentence would mean that Hesperus has the property of Phosphorus, which verges on the meaningless and is certainly not true—“Phosphorus” is not a predicate but a singular term. So “is” must be ambiguous between the two cases, sometimes meaning identity and sometimes meaning predication. That is a serious failing in natural language, requiring linguistic reform: our language systematically confuses two very different concepts.
But this conclusion is too hasty; there is no need to adopt the ambiguity thesis in order to account for the meaning of “is”. For first, it is not difficult to construe the “is” in identity statements as simply the predicative “is”, by expanding such statements in the obvious way, viz. “Hesperus is identical to Phosphorus”. Here we have a predicate expression, “identical to Phosphorus”, coupled with the “is” of predication, so that the sentence is saying, “Hesperus has the property of being identical to Phosphorus”. We don’t need a separate meaning for “is” to account for its use in identity statements; we just need to fill out the predicate in the obvious way. Clearly “is” cannot express identity in the expanded version, or else the sentence would be saying that Hesperus is identical to identity with Hesperus, which is nonsense. The point is even clearer if we add a sortal term to statements of identity, as in “Hesperus is the same planet as Phosphorus”: here “same planet” carries the attribution of identity, with “is” just acting as the predicative copula. When we use “is” alone in an identity statement this is just a shorter version of the explicit expansion that employs the identity concept (“same”) directly. There is no distinctive “is” of identity.
Can we enforce uniformity of meaning from the other direction? That is, can we claim that “is” always expresses identity? It would certainly be difficult to do that if we read the sentences in question naively, as saying (for example) that the cup is identical to redness; but a simple paraphrase can resolve this problem. What if we rephrase “the cup is red” as “the color of the cup is (identical to) red”? That is a straightforward identity statement, and it is straightforwardly true. The same trick can be applied to all predicative uses of “is”, as in “the species of Felix is cat” or “the job of John Smith is philosopher”. Put in stilted philosopher’s language, we are paraphrasing “ais F” as “among the attributes of a is F-ness”, where “is” expresses simple numerical identity. We can take this as a quantified statement along the following lines: “there are attributes G that a instantiates and one of these G’s is identical to F-ness”. Thus: “there is a (unique) color C such that the cup has C and C is identical to redness”. This sounds rather ponderous, no doubt, but it corresponds quite well with the intuitive meaning of the original statement, more colloquially expressed as, “the color of the cup is red”.
So there is nothing compulsory about finding ambiguity in “is”; in fact, it is quite easy to provide paraphrases that employ “is” in one meaning for all sentences that contain “is”. And surely that is the preferable position, since it is hard to believe that natural language could harbor such a disreputable ambiguity—why not simply have two words for such very different concepts? There is the question which of the two theories we should prefer, given that both appear adequate. I incline to a mixed position, combining both types of paraphrase. The second type offers a convincing expansionary analysis, spelling out the underlying meaning of the sentence; but the first type makes it clear that the so-called “is” of identity is really short for “is identical to” or “is the same as”, which contains the “is” of predication. Thus “the cup is red” has the same meaning as, “the color of the cup is identical to red”. We turn the original sentence into a statement of identity, but that statement itself contains in its expansion a predicative use of “is”, with identity conveyed by the attached predicate “identical to red”. Predicative sentences turn out to be identity sentences, but identity sentences turn out to contain the “is” of predication. So in the final analysis “is” is always predicative, though ordinary predicative sentences are equivalent to identity sentences.
How then should we analyze “Hesperus is Phosphorus”—what is its underlying logical form? It turns out to mean the same as, “Among the attributes of Hesperus one of them is that of being identical to Phosphorus”. We quantify over attributes and declare one of them to be identical to identity with Phosphorus—where “is” occurs in its predicative meaning. Thus: “There are attributes that Hesperus has and one of them is identical to identity with Phosphorus”. This sentence expresses an identity proposition concerning the attribute of identity with a given object, but in order to state that identity we need to use “is” predicatively. Given that the “is” in an identity statement so clearly means, “is identical with”, this is just what we would expect on the assumption that identity is at the root of all predication. All propositions are really identity propositions, on this view, formed by quantifying over attributes or properties. The recipe for constructing the underlying identity proposition is simply to refer to a property and declare it one of the properties an object has, as in “the color of the cup is (identical to) (the color) red”. Second-order quantification plus identity therefore enter even into ordinary subject–predicate sentences—which is not what we have been taught to expect. But, as we know from Russell’s theory of descriptions, language can be more complex than it seems on the surface when it is properly analyzed.  First-order logic really embeds second-order logic (with identity) in underlying logical form. Still, “is” remains uniformly a device of predication, even as it occurs in second-order identity sentences. The impression that “is” is ambiguous disappears once we carry out the requisite analysis.
 Any believer in conceptual or logical analysis fully expects ordinary language sentence forms to have expansionary analyses, even complex and taxing ones. See my, Truth by Analysis (Oxford University Press, 2012).