In predicate calculus
a designator is a single word or phrase that is used to refer to one single thing. Designators can be proper name
s, non-count noun
s, singular personal pronoun
s, and definite description
s. Some examples would be:
It is important to say that plurals are never designators. So, "cats" and "dogs" are not designators. Other things that are not designators are "a book" or "some books" as these do not refer to any specific thing. However, we can say "my book" or "that book" as these do refer to something specific (but not "these books" as this is plural).
Purely referential occurrences:
To be a purely referential occurrence, a designator must meet all of the following three criteria:
1) The designator must refer to something that actually exists.
So in the sentence "I want to be the first man on Mars", the designator "the first man on Mars" is not a purely referential occurrence as it does not meet the first criterion - there is no such thing as the first man on Mars.
2) The designator has to refer to one unique thing.
The designator "Bill Clinton" fails the second criterion in the following sentence - "Bill Clinton was the President". Bill Clinton was not unique in being the President - there have been others.
3) The designator can be replaced by any co-referring designator without altering the truth-value of the sentence.
Take the sentence, "Dartmouth is so called because it is situated on the mouth of the River Dart." The designator "Dartmouth" refers to the same thing as "the home of the Naval Academy", but whereas the previous sentence was true, it is not true to say "The home of the Naval Academy is so called because it is situated on the mouth of the River Dart." Therefore this designator fails to meet the third criterion. This third criterion is the one which causes the most problems for designators being purely referential given the context in which the sentence is said, as will be shown in the following section.
Some phrases that are usually designators can sometimes not be within certain contexts. Take the sentence:
"If any boy gets lots of Christmas presents, he will be very happy."
"He" is usually a designator, but in this sentence it is not. It is merely the same variable as the first part of the sentence. A phrase referring to a variable cannot be a designator.
Intensional contexts are when someone believes, wants, hopes, fears, knows, says, and so forth. In such contexts, designators often do not meet the third criterion of being a purely referential occurrence. This is best illustrated through example. Take the following sentence:
"Lois Lane believes Superman can fly."
A co-referring designator to "Superman" is "Clark Kent". Now, although it is true that Lois Lane believes Superman can fly, she does not believe that Clark Kent can fly. The following use of "Superman" is purely referential within an intensional context:
"Superman believes he can fly."
Modal contexts are concerned with the necessary, possible, or impossible. Within such contexts the truth value of what is being said depends on there being, or not being, a relation of some kind between the words being used. Sometimes this means that designators within such contexts do not meet the third criterion of being purely referential as despite referring to the same thing, two designators might mean different things in a given context. Take the sentence:
"It is a necessary truth that anyone who is the Queen of England is a queen."
The designator "the Queen of England" is not purely referential, as it is not necessarily true that anyone who is "the wife of Prince Philip" is a queen. Here, there is a connection between "the Queen of England" and the meaning of the phrase "a queen" that is missing between "the wife of Prince Philip" and "a queen". But sometimes designators in modal contexts are purely referential.
"Descartes is necessarily a thinking thing."
This is just as true as saying:
"The author of Meditations on First Philosophy is necessarily a thinking thing."
De dicto and de re modalities:
Modalities which are concerned with the meanings of words can be split into two categories. De dicto modalities are concerned with the meaning of the words, or, what is said - like in the above "Queen of England" example. De re modalities are concerned with the nature of the thing being said - such as in the above example concerning Descartes. It is important to consider what sort of modality we are dealing with sometimes as it can affect the truth of a sentence. Consider the sentence:
"32 is necessarily 9."
Does the designator "32" meet the third criterion of being purely referential? What if we replace the designator with the co-referring "the number of planets in the solar system"?
"The number of planets in the solar system is necessarily 9."
Looking at this from a de dicto necessity, it is not a necessarily true since there is no real connection in meaning between the number of planets and nine. We cannot say that there are necessarily nine planets - the number of planets is not bound to nine, as there could have been more or less. But we can also look at this from a de re viewpoint. From this angle, the sentence is true since the number of planets is nine in reality, so therefore has the same intrinsic meaning as "nine". As can be seen, whether "32" is a purely referential occurrence of the designator depends entirely on the assertion of whether the sentence is asserted to be a de dicto or de re modality.
Wilfred Hodges, "Logic"