I want to better explain why I reject the idea that logic refers to something, such as abstractions or Platonic forms.
Words and sentences, of themselves, do not refer to anything. Rather, people can use words and sentences to refer to things. (This should be clear when we remember that the same words and sentences can refer to different things, depending on the context of utterance.) Furthermore, the meaning of a sentence is not always its referent; for we can understand sentences even when a referent is unspecified, and also in cases where the referent is non-existant. (E.g., "The King of France is bald.") From these points it follows, first, that the referent of a sentence depends on how it is used in a particular context; and, second, that sentences can be meaningful even if they have no known referent.
When we look at the meaning of a syllogism, we may easily find referents. For example,
- All men are mortal.
- Socrates is a man.
- Therefore, Socrates is mortal.
Taken by themselves, each of these sentences can be (and normally would be) used to refer to things, namely men, mortality, and Socrates. However, they can also be used to illustrate a certain logical form. The fact that there are referring terms is incidental. We could easily replace "Socrates" with "Alfred," and the logical validity would remain intact, even if nobody had any idea who (or what) Alfred might be.
The use of these sentences as an illustration of a valid syllogism is not their usual referring use. We can say that the meaning of each sentence, in this illustration, is based on their grammatical form, and not on what extra-linguistic entities they might be used to point to. We most clearly present rules of inference when we use symbols which have no referring use in our common language. We thus can say,
- All members of A are members of B.
- x is a member of A.
- Therefore, x is a member of B.
By taking out words like "men" and "Socrates," we help avoid the confusion of thinking that the meaning of our logical demonstration somehow depended on the referring use of our terms.
Of course, some students of logic might ask, "What does 'x' refer to?"
The correct answer is, nothing. The letter 'x' is a place-holder, and anything could be substituted for it. This does not mean that 'x' refers to anything, as though anything were something specific we could point to. And it doesn't mean that 'x' refers to some abstract category of 'logical thingness' or what have you.
Again, the meaning of these sentences, as they are used, is to demonstrate a form of logical deduction. That function does not depend on any references (except perhaps a reference to logic itself; but this reference is not found in any of the three sentences used in the syllogism). The very reason we use arbitrary symbols without referential meaning is to make this clear.
So, when people say that logic must refer to something, such as Platonic forms, not only is it not clear what they are getting at; it is not even clear why they feel the need to get anywhere.