8.5.1. Proof by choice

As has been the case elsewhere in this chapter, our discussion of principles of entailment for existentials can build on our discussion of universals in the last chapter. The differences between the principles governing universal and existential quantifiers will, in most cases, be analogous to differences between the principles for conjunction and disjunction. The laws of entailment for the universal quantifiers were modifications of laws for conjunction, and the rules for the existential quantifiers will nearly all be based in a similar way on rules for disjunction. Our planning rule for existential sentences is the one exception to this, and even it is analogous to a rule that could have been used for disjunction.

These analogies with the universal quantifier on the one hand and with disjunction on the other derive from the truth conditions for the unrestricted existential, which follow the conditions for disjunction in precisely the way the conditions for the universal follow those for conjunction. A sentence ∃x θx is true in a structure if and only if it has at least one true instance in a language expanded by the range R of that structure. In other words, an existential claim behaves like a disjunction of its instances when these instances are taken from a language that has a name for each reference value in the structure. However, as was the case with the universal, the set of instances can vary from one structure to another, so general laws of entailment cannot employ any definite information about what the instances of an existential sentence are.

We will begin our discussion of principles of entailment with the role of an unrestricted existential as a premise. First, recall the corresponding principle for disjunction. A disjunctive premise may be used to draw a conclusion by way of a proof by cases. In such a proof, we suppose in turn that each of the disjuncts is true and argue for the conclusion in each case. A comparable way of arguing from an existential would be to establish many case arguments, each one considering an instance of the existential as one case. Since we cannot associate the existential with any definite set of instances, there is no way to delimit the range of case arguments we would need to consider, so we must use adapt a device from our treatment of the universal: we need to set out the indefinitely many arguments by offering a general pattern. That is, to use an existential premise to draw a conclusion, we draw the conclusion from one instance of the existential in a way that sets a pattern for all other instances.

This sort of argument may be called a proof by choice, a name which reflects another way of looking at the principle behind it. Consider the two arguments below.

The validity of the argument on the left can be traced to the validity of the one on the right. In the latter, we use the premise X broke down in place of the existential Something broke down, so we argue for the conclusion from an instance of the existential. When we replace an existential by an instance of it, we are choosing an name for the example the existential claims to exist, so this is an argument to proceeds by way of the choice of a name.

Of course, we cannot assume that the something claimed to exist by an existential premise is some thing that we have other information about. That is, choosing a name really means choosing a new name. For an unrestricted existential tells us nothing about the example it claims to exist except for the property it is said to exemplify. So the name we choose must be one that could apply to anything that has this property. And that returns us to the first way of looking at proofs by choice: they must argue from one instance of an existential in a way that sets a pattern for all such instances.

Recalling the test we used for the generality of arguments in the case of the universal quantifiers, we can expect our analysis of the role of an existential as a premise to make reference to a term that is independent in an appropriate sense. We will want a term α that has no connection to the premises and conclusion of the argument—including the existential ∃x θx—apart from the assumption θα. So suppose the term α is unanalyzed term and does not appear in the set Γ, the sentence φ, or the existential ∃x θx, and consider the two arguments

We can argue that each is valid if and only if the other is if we can show that each is divided by a structure if and only if the other is. If a structure S divides the premises and conclusion of the first, it will assign θ a non-empty extension, and we can form a structure S′ that divides the second argument by assigning a value in this extension to the term α. For this assignment will not change the extension of θ or the truth values of φ and the members Γ since α does not appear in these expressions, so θα will be true and the conclusion and the other premises will keep the same truth values. On the other hand, any structure dividing the second argument will give θ a non-empty extension (because the value of the term α will be in it) so this structure will make ∃x θx true and also divide the first argument. Thus we will have a structure dividing one argument if and only if we have a structure dividing the other, and each argument is valid if and only if the other is.

This gives us our principle describing the role of the unrestricted existential as a premise.

The corresponding principle for the restricted existential combines these ideas with the properties of conjunction:

That is, having a restricted existential as an assumption comes to the same thing as assuming that an independent term refers to something that is in the domain and has the attribute.