In sentence logic, we said that an argument is valid if and only if, for all possible cases in which all the premises are true, the conclusion is true also. In predicate logic, the intuitive notion of validity remains the same. We change things only by generalizing the notion of possible case. Where before we meant that all lines in the truth table which made all premises true also make the conclusion true, now we mean that all interpretations which make all the premises true also make the conclusion true:
An argument expressed with sentences in predicate logic is valid if and only if the conclusion is true in every interpretation in which all the premises are true.
You may remember that we got started on predicate logic at the beginning of chapter 1 because we had two arguments which seemed valid but which sentence logic characterized as invalid. To test whether predicate logic is doing the job it is supposed to do, let us see whether predicate logic gives us the right answer for these arguments;
Everyone loves Eve. (
Adam loves Eve. Lae
Suppose we have an interpretation in which '(
Vx)Lxe' is true. Will 'Lae' have to be true in this interpretation also? Notice that 'Lae' is a substitution instance of '( Vx)Lxe'. A universally quantified sentence is true in an interpretation just in case all its substitution instances are true in the interpretation. So in any interpretation in which '( Vx)Lxe' is true, the instance 'Lae' will be true also. And this is just what we mean by the argument being valid.
Let's examine the other argument:
Adam loves Eve. Lae
Someone loves Eve. (Ǝx)Lxe
Suppose we have an interpretation in which 'Lae' is true. Does '(Ǝx)Lxe' have to be true in this interpretation? Notice that 'Lae' is an instance of '(Ǝx)Lxe'. We know that '(Ǝx)Lxe' is true in an interpretation if even one of its instances is true in the interpretation. Thus, if 'Lae' is true in an interpretation, '(Ǝx)Lxe' will also be true in that interpretation. Once again, the argument is valid.
Along with validity, all our ideas about counterexamples carry over from sentence logic. When we talked about the validity of a sentence logic argument, we first defined it in this way: An argument is valid just in case any line of the truth table which makes all the premises true makes the conclusion true also. Then we reexpressed this by saying: An argument is valid just in case it has no counterexamples; that is, no lines of the truth table make all the premises true and the conclusion false. For predicate logic, all the ideas are the same. The only thing that has changed is that we now talk about interpretations where before we talked about lines of the truth table:
A Counterexample to a predicate logic argument is an interpretation in which the premises are all true and the conclusion is false.
A predicate logic argument is Valid if and only if it has no counterexamples.
Let's illustrate the idea of counterexamples in examining the validity of
Is there a counterexample to this argument? A counterexample would be an interpretation with 'Lae' true and '(Ǝx)Lxe' false. But there can be no such interpretation. 'he' is an instance of '(Ǝx)Lxe', and '(Ǝx)Lxe' is true in an interpretation if even one of its instances is true in the interpretation. Thus, if 'he' is true in an interpretation, '(Ǝx)Lxe' will also be true in that interpretation. In other words, there can be no interpretation in ' which 'Lae' is true and '(3x)Lxe' is false, which is to say that the argument has no counterexamples. And that is just another way of saying that the argument is valid.
For comparison, contrast the last case with the argument
It's easy to construct a counterexample to this argument. Any case in which someone other than Adam is blond and Adam is not blond will do the trick. So an interpretation with Adam and Eve in the domain and in which Eve is blond and Adam is not blond gives us a counterexample, showing the argument to be invalid.
This chapter has been hard work.. But your sweat will be repaid. The concepts of interpretation, substitution instance, and truth in an interpretation provide the essential concepts you need to understand quantification. In particular, once you understand these concepts, you will find proof techniques for predicate logic to be relatively easy.
2-6. For each of the following arguments, determine whether the argument is valid or invalid. If it is invalid, show this by giving a counterexample. If it is valid, explain your reasoning which shows it to be valid. Use the kind of informal reasoning which I used in discussing the arguments in this section.
You may find it hard to do these problems because I haven't given you any very specific strategies for figuring out whether an argument is valid. But don't give up! If you can't do one argument, try another first. Try to think of some specific, simple interpretation of the sentences in an argument, and ask yourseIf-"Are the premise and conclusion both true in that interpretation?' Can I change the interpretation so as to make the premise true and the conclusion false? If you succeed in doing that, you will have worked the problem because you will have constructed a counterexample and shown the argument to be invalid. If you can't seem to be able to construct a counterexample, try to understand why you can't. If you can see why you can't and put this into words, you will have succeeded in showing that the argument is valid. Even if you might not succeed in working many of these problems, playing around in this way with interpretations, truth in interpretations, and counterexamples will strengthen your grasp of these concepts and make the next chapter easier.
Vx)Lxe b) Lae c) (Ǝx)Lxe
Vx)(Bx & Lxe) e) ( Vx)(Bx ⊃ Lxe)
f) (Ǝx)Bx & (Ǝx)Lxa g) (
Vx)(Bx ⊃ LxeL) & ( Vx)(~Bx ⊃ Lxa)
(Ǝx)(Bx & Lxa) (
Vx)[(Bx ⊃ Lxe) & (~Bx ⊃ Lxa)]
Chapter SUmmary Exercise
Provide short explanations for each of the following, checking against the text to make sure you understand each term clearly and saving your answers in your notebook for reference and review.
b) Interpretation of a Sentence
c) Substitution Instance
d) Truth in an Interpretation
e) Validity of a Predicate Logic Argument