When can we treat an occurrence of a designator, "D", as a designator for the purpose of applying the tableau rules? Or, which comes to the same thing, when can we translate an occurrence of a designator by means of an individual constant: "a", for example?
In the first place, as we have seen, the rules for the language of the predicate calculus dictate that, if one uses individual constants, they must stand for something. If this were not so the "xj rule would not be sound.
For, thanks to that rule, "xFx |- Fa is a correct sequent. But if there did not have to be such a thing as a, there could be an interpretation in which "xFx was true, but Fa wasn't. Indeed an individual constant must stand for just one thing. (Otherwise one of the things might have a property which the other one lacked - so Fa would not necessarily be inconsistent with ¬Fa.) So, if we are to translate an occurrence of "D" with "a", there must be such a thing as D, and just one thing.
In the second place "Fa|-$xFx" is a correct sequent. Does it follow, then, that the occurrence of "D" must pass test 2? (Remember that test 2 says: suppose one replaces "D" in the sentence with "it" and adds "there is or was or will be something such that" at the beginning; does the truth of the new sentence follow from the truth of the original?) In fact there could be two views about this.
According to the first view, our aim should be merely to ensure that the truth-value of our translation is bound to match that of the original sentence given that there is such a thing as D. This is effectively the view which Hodges takes, given what he means by "purely referential". (Section 28)
According to this view, if there were such a thing as Hugh Rice's cow, nothing would go wrong if we translated "Hugh Rice's cow" in (E7) ("Buttercup is not Hugh Rice's cow.") as "a", even though it fails test 2.
According to the second view, our aim should be to ensure that the logical properties of our translation match (as far as possible) the logical properties of the original.
According to this view, then, since "Fa|-$xFx" is a correct sequent, we should translate "D" with "a" only if the occurrence of "D" passes test 2, without the aid of the additional fact that there is such a thing as D. It would be in keeping with the second view also to require that the occurrence of "D" should also pass test 1 - remember that test 1 says: if the sentence is true, must there be such thing as D, and just one such thing?
The idea would be that it is not enough that there should in fact be such a thing as "D", if we are to translate "D" with "a"; we should require also that this particular use of "D" should commit us to the existence of such a thing, just as the use of "a" commits us to the existence of something that "a" stands for. (Note, by the way, that even on the second view, when translating an occurrence of "D" with "a", we shall not require it to possess one property which "a" possesses. In the case of "a", "|-$xx=a" is a correct sequent; we shall not, however, require it to be a necessary truth that there is such a thing as D.)
In the third place the tableau rule for identity which allows us to replace "D" with "E" given that D=E means that the occurrence must pass test 3. (Test 3: (a) (If there is such a thing as D) would it make no difference to the truth‑value of the sentence if "D" were replaced with some other designator with the same reference? (b) (If there is no such thing as D) if there had been such a thing as D, would it have made no difference to the truth value of the sentence, if "D" had been replaced with some other designator with the same reference?)
View 1: it is permissible to translate an occurrence of a designator "D" with an individual constant if and only if:
So, as long as there is such a thing as D, we can translate the occurrence with "a" if it passes our three test for being a p.r.o., but it does not need to pass the tests, apart from test 3.
View 2: it is permissible to translate an occurrence of a designator "D" with an individual constant if and only if:
(i) there is such a thing as D, and just one thing;
(ii) it passes our three tests for being a p.r.o. (whether or not everything that passes our tests is in fact a p.r.o.)