First- vs. Second-Order Logic

There is a lot confusion, terminological and other, about the distinction between first- and second-order logic. Everyone agrees that second-order logic is first-order logic supplemented by a new kind of quantification and that the new quantification is over "second-order entities," and that second-order entities are over and above first-order entities, but that's about all people agree on (oh, and that second-order variables are capital letters and function letters, typically .

So, if you see a formula with or , it is a second-order formula. So what. For all I've said so far, that is completely uninteresting, so far, second-order logic is a mere notational variant of first-order logic, one with more than one sort of variable.

Indeed, if all there were to second-order logic was what I have said so far, it is just a special case of sorted logic, which is a mere notational variant of first-order logic, and all the problems of first-order logic would be inherited by second-order logic. It would be a mere notational variant.

Given a domain of "individuals" (that is, picking a domain, and deciding to call its members individuals), the properties and functions on that domain, thought of as objects, are second-order objects with respect to that domain.

Since the notation of second-order logic is just that of sorted first-order logic, and since the distinction between first- and second-order objects is thoroughly relative, how can this make any difference? (Example: a line may be thought of as the set of points on it or the property of being a point on it, but a point may be thought of as the set of lines that go through it.)

Suppose we have a system of second-order logic, and we take it to be just a sorted first-order logic. Then we have two domains, the domain of individuals and the domain of sets of individuals. Not all objects need be in the domain of individuals. Similarly, not all sets of individuals need be in the domain of sets of individuals. This sort of second-order logic is known as Henkin logic, and it is just (paraphrasing Quine) first-order logic in wolf's clothing.

However, and I am finally getting to the point, we can interpret statements in second-order logic as being entirely about a first-order domain and its properties, which are determined by the first-order domain and need no separate specification. Second-order logic interpreted this way is often called full second-order logic, though it is too often called simply second-order logic.

There is nothing wrong with second-order logic, just as there is nothing wrong with the claim that points have no extension. However, if our purpose is saying, defining, specifying what they are, or of being informative to someone who doesn't already know, such methods are useless. -- ProfessorShaughanLavine? - 02 Mar 2005