3.3: Correct Definitions of Substitution Instance and Truth in an Interpretation

In chapter 2 I gave an incorrect definition of 'substitution instance.' I said that we get the substitution instance of (Vu) ( . . . u . . .) with s substituted for u by simply dropping the initial (u) and writing in s wherever we find u in (. . . u . . .). This is correct as long as neither a second (Vu) nor a (Ǝu) occurs within the scope of the initial (Vu), that is, within the sentence ( . . . u . . .). Since I used only this kind of simple sentence in chapter 2, there we could get away with the simple but incorrect defintion. But now we must correct our definition so that it will apply to any sentence. Before reading on, can you see how multiple quantification can make trouble for the simple definition of substitution instance, and can you see how to state the definition correctly?

To correct the definition of substitution instance, all we have to do is to add the qualification that the substituted occurrences of the variable be free:

For any universally quantified sentence (Vu) ( . . . u . . .), the Substitution Instance of the sentence, with the name s substituted for the variable u, is (. . . s . . .), the sentence formed by dropping the initial universal quantifier and writing s for all free occurrences of u in (. . . u . . .).

For any existentially quantified sentence (Ǝu) (. . . u . . .), the Substitution Instance of the sentence, with the name s substituted for the variable u, is (. . . s . . .), the sentence formed by dropping the initial existential quantifier and writing s for all free occurrences of u in (. . . u . . .).

For example, look back at (13). Its substitution instance with 'c' substituted for 'x' is

(16) (Ǝx)(Bx v Lxa) ⊃ (Bc & Lcb)
2 3 4 5 6

The occurrences of 'x' at 3 and 4 are not free in the sentence which results from (13) by dropping the initial quantifier. So we don't substitute 'c' for 'x' at 3 and 4. We substitute 'c' only at the free occurrences, which were at 5 and 6.

Can you see why, when we form substitution instances, we pay attention only to the occurrences which are free after dropping the outermost quantifier? The occurrences at 3 and 4, bound by the '(Ǝx)' quantifier at 2, have nothing to do with the outermost quantification. When forming substitution instances of a quantified sentence, we are concerned only with the outermost quantifier and the occurrences which it binds.

To help make this clear, once again consider (15), which is equivalent to (13). In (15), we have no temptation to substitute 'c' for '2' when forming the 'c'-substitution instance for the sentence at a whole. (15) says that there is some x such that so on and so forth about x. In making this true . for some specific x, say c, we do not touch the occurrences of '2'. The internal 'z'-quantified sentence is just part of the so on and so forth which is asserted about x in the quantified form of the sentence, that is, in (15). So the internal 'z'-quantified sentence is just part of the so on and so forth which is asserted about c in the substitution instance of the sentence. Finally, (13) says exactly what (15) says. So we treat (13) in the same way.

Now let's straighten out the definition of truth of a sentence in an interpretation. Can you guess what the problem is with our old definition? I'll give you a clue. Try to determine the truth value of 'Lxe' in the interpretation of figure 2-1. You can't do it! Nothing in our definition of an interpretation gives us a truth value for an atomic sentence with a free variable. An interpretation only gives truth values for atomic sentences which use no variables. You will have just as much trouble trying to determine the truth value of '(Vx)Lxy' in any interpretation. A substitution instance of '(Vx)Lxy' will still have the free variable 'y', and no interpretation will assign such a substitution instance a truth value.

Two technical terms (mentioned in passing in chapter 1) will help us in talking about our new problem:

A sentence with one or more free variables is called an Open Sentence.

A sentence which is not open (i.e., a sentence with no free variables) is called a Closed Sentence

In a nutshell, our problem is that our definitions of truth in an interpretation do not specify truth values for open sentences. Some logicians deal with this problem by treating all free variables in an open sentence as if they were universally quantified. Others do what I will do here: We simply say that open sentences have no truth value.

If you think about it, this is really very natural What, anyway, is the truth value of the English "sentence" 'He is blond.', when nothing has been said or done to give you even a clue as to who 'he' refers to? In such a situation you can't assign any truth value to 'He is blond.' 'He is blond.' functions syntactically as a sentence-it has the form of a sentence. But there is still something very problematic about it. In predicate logic we allow such open sentences to function syntactically as sentences. Doing this is very useful in making clear how longer sentences get built up from shorter ones. But open sentences never get assigned a truth value, and in this way they fail to be full-fledged sentences of predicate logic.

It may seem that I am dealing with the problem of no truth value for open sentences by simply ignoring the problem. In fact, as long as we acknowledge up-front that this is what we are doing, saying that open sentences have no truth value is a completely adequate way to proceed.

We have only one small detail to take care of. As I stated the definitions of truth of quantified sentences in an interpretation, the definitions were ,said to apply to any quantified sentences. But they apply only to closed sentences. So we must write in this restriction:

A universally quantified closed sentence is true in an interpretation just in case all of the sentence's substitution instances, formed with names in the interpretation, are true in the interpretation.

An existentially quantified closed sentence is true in an interpretation just in case some (i.e., one or more) of the sentence's substitution instances, formed with names in the interpretation, are true in the interpretation.

These two definitions, together with the rules of valuation given in chapters 1 and 4 of volume I for the sentence logic connectives, specify a truth value for any closed sentence in any of our interpretations.

You may remember that in chapter 1 in volume I we agreed that sentences of logic would always be true or false. Sticking by that agreement now means stipulating that only the closed sentences of predicate logic are real sentences. As I mentioned in chapter 1 in this volume, some logicians use the phrase Formulas, or Propositional Functions for predicate logic open sentences, to make the distinction clear. I prefer to stick with the word 'sentence' for both open and closed sentences, both to keep terminology to a minimum and to help us keep in mind how longer (open and closed) sentences get built up from shorter (open and closed) sentences. But you must keep in mind that only the closed sentences are fullfledged sentences with truth values.

Exercise

3-2. Write a substitution instance using 'a' for each of the following sentences:

a) (Vy)(Ǝx)Lxy b) (Ǝz)[(Vx)Bx v Bz]
c) (Ǝx)[Bx ≡ (Vx)(Lax v Bx)]
d) (Vy){(Ǝx)(Bx v [(Ǝu)By ⊃ Bx)} & (Vx)(By ⊃ Bx)]
e) (Vy){(Ǝx)Bx v [(Ǝy)By ⊃ Lyy]}
f) (Vy)(Ǝx)[(Rxy ⊃ Dy) ⊃ Ryx]
g) Wx)(Vy)Wz){[Sxy v (Hz ⊃ Lxz)] ≡ (Scx & Hy))
h) (Ǝx)(Vz){(Pxa ⊃ Kz) & (Ǝy)[(Pxy v Kc) & Pxx])
i) (Ǝz)(Vy){[(Ǝx)Mzx v (Ǝx)(Mxy ⊃ Myz)] & (Ǝx)Mzx)
j) (Vx){[(Vx)Rxa ⊃ Rxb] v [(Ǝx)(Rcx v Rxa) ⊃ Rxx])

3-3. If u does not occur free in X, the quantifiers (Vu) and (Ǝu) are said to occur Vacuously in (Vu)X and (Ǝu)X. Vacuous quantifiers have no effect. Let's restrict our attention to the special case in which X is closed, so that it has a truth value in any of its interpretations. The problem I want to raise is how to apply the definitions for interpreting quantifiers to vacuously occurring quantifiers. Because truth of a quantified sentence is defined in terms of substitution instances of (Vu)X and (Ǝu)X, when u does not occur free in X, we most naturally treat this vacuous case by saying that X counts as a substitution instance of (Vu)(X) and (Ǝu)(X). (If you look back at my definitions of 'substitution instance', you will see that they really say this if by 'for all free occurrences of u' you understand 'for no occurrences of u' when u does not occur free in X at all. In any case, this is the way you should understand these definitions when u does not occur free in X.) With this understanding, show that (Vu)X, (Ǝu)X, and X all have the same truth value in any interpretation of X.

3-4. a) As I have defined interpretation, every object in an interpretation has a name. Explain why this chapter's definitions of truth of existentially and universally quantified sentences would not work as intended if interpretations were allowed to have unnamed objects.

b) Explain why one might want to consider interpretations with unnamed objects. In part I1 we will consider interpretations with unnamed objects and revise the definitions of truth of quantified sentences accordingly.