7.5.3. Generalization

Next let us look at the role of an unrestricted universal as a conclusion. Here we have the law for conjunction as a conclusion to use as a model.

We have to expect changes, though, because that law gives separate consideration to each of the two components of the conjunction and we cannot expect to do this for the instances of a universal. Still, the law for conjunction points us in the right direction: we should look for some way of connecting the validity of a universal conclusion with the validity of arguments having its instances as conclusions.

A connection like this is used in geometric proofs when we begin by saying, for example, Let ABC be a triangle, and then go on to use our conclusions concerning ABC to justify general conclusions about all triangles. That is, we sometimes establish universal claims by generalizing from particular instances of them.

Clearly not every generalization from a particular instance will be legitimate. Certain premises may entail The Empire State Building is tall without entailing Every building is tall. In a geometric argument concerning a triangle ABC, we limit the information that we may use about the instance that we are considering to what we may establish concerning any triangle. For example, we ignore the fact we are using a diagram that shows ABC as acute or obtuse, and we probably avoid drawing it as a right triangle or an isosceles triangle to begin with. These restrictions are sometimes expressed by saying that we are arguing about an arbitrary or an arbitrarily chosen triangle. The idea is that what you say about the triangle ABC should hold for a triangle chosen at random or even one chosen by your worst enemy. Let us call an argument like this a general argument since it argues for an instance in a way that will hold generally for values in the domain of a universal.

The law we are looking for should say that an unrestricted universal is a valid conclusion from given premises if we can establish an instance of it by a general argument. But we need to make this more precise. In particular, we need to say how we can recognize a general argument just by looking at the logical forms of the sentences it involves.

If we were to give instructions for making a general argument about a triangle ABC, one thing we might say is that we should not use any special assumptions about ABC. If we are going to generalize about triangles, we may assume that ABC is a triangle but we should not assume that it is acute or obtuse. This is just another way of saying that we should not use special information about this triangle, but it suggests an idea we can apply to arguments when we know only their logical forms. Since we are considering arguments for unrestricted universals, we must be able to generalize not just about triangles, or some other limited class, but about everything; and that means we should use no assumptions at all about the term from which we wish to generalize. To insure that we do not use an special assumptions about a term τ when we generalize from an instance θτ to a universal ∀x θx, we will require that τ not appear in our assumptions. For reasons we will consider in 7.6.1, we will require also that τ not appear in the predicate θ and that it be unanalyzed so that it not only does not appear in the assumptions or θ but in fact shares no vocabulary with them.

Even setting aside these further requirements, you may have noticed a couple of jumps here. Saying we have an assumption containing τ is different from saying we have used that assumption, and saying that τ appears in an assumption is different from saying that the assumption provides special information about τ. For example, The weather is fine and dandy and so is everything else mentions the weather without constituting a special assumption about it (since the same assumption is made about everything). Still, the requirement that the term from which we generalize not appear in the assumptions is easy to check and using it will not limit the entailments we can establish, only the terms we can use to establish them.

These restrictions appear in our law stating the conditions under which a universal can be validly concluded.

Let us say that an unanalyzed term appearing in neither the premises or conclusion of an argument is independent with respect to that argument. In this vocabulary, the law says that an argument with an unrestricted universal conclusion is valid if and only if the premises entail an instance of the universal for an independent term. When arguments are stated in English, phrases like let α be arbitrary or let us choose α arbitrarily function as commitments to use the term α as an independent term.

The crucial part of this law is the if claim since the only-if part says only that a universal cannot be a valid conclusion unless any instance for an independent term is also valid, something that follows from the principle of universal instantiation. The key idea behind the truth of the if part is that, because the independent term α is unanalyzed and does not appear in either Γ or θ, it could be made to refer to anything without affecting the premises Γ or the predicate θ. And this means that, if the premises suffice to entail θα, they suffice to show that θ is true of everything—i.e., that the universal ∀x θx is true. Indeed, given a proof of θα from the premises Γ, we could construct a proof of θτ for any term τ simply by replacing every occurrence of α by τ, our restrictions on α insuring that the premises Γ and θ remain unchanged and that α had no ties to them that are not shared by τ.

This argument recalls the comparison of the universal with conjunction. Since a conjunction can have any components, we must argue for each component individually and, since a conjunction has only two components, there is nothing to keep us from doing this. On the other hand, there would be no hope of providing a separate argument for each instance of a universal since, in general, there is no way of setting a limit on the number of instances it has. However, there is no need to consider each of these instances individually since they all have the same form, so an argument establishing an instance for one independent term can set the pattern for all of the rest.

A principle for the restricted universal as a conclusion follows from this law and the law for the conditional as a conclusion:

That is, we can establish a restricted generalization by showing that an arbitrarily chosen object has the attribute when we assume that it is in the domain.