2.1: Interpretations

Recall that we used truth tables to give very precise definitions of the meaning of '&', 'v' '~', '⊃', and '≡'. We would like to do the same for the meaning of quantifiers. But, as you will see very soon, truth tables won't do the job. We need something more complicated.

When we were doing sentence logic, our atomic sentences were just sentence letters. By specifying truth values for all the sentence letters with which we started, we already fixed the truth values of any sentence which we could build up from these smallest pieces. Now that we are doing predicate logic, things are not so easy. Suppose we are thinking about all the sentences which we could build up using the one place predicate 'B', . the two place predicate 'L', the name 'a', and the name 'e'. We can form six atomic sentences from these ingredients: 'Ba', 'Be', 'Laa', 'Lae', 'Lea', and 'Lee'. The truth table formed with these six atomic sentences would have 64 lines. Neither you nor I are going to write out a 64-line truth table, so let's consider just one quite typical line from the truth table:

Figure 2-1

Ba, Be, Laa, Lae, Lea, Lee
t f f t f t á é

Even such an elementary case in predicate logic begins to get quite complicated, so I have introduced a pictorial device to help in thinking about such cases (see Figure 2-1). I have drawn a box with two dots inside, one labeled 'a' and the other labeled 'e'. This box is very different from a Venn diagram. This box is supposed to picture just one way the whole world might be. In this very simple picture of the world, there are just two things, Adam and Eve. The line of the truth table on the left gives you a completed description of what is true and what is false about Adam and Eve in this very simple world: Adam is blond, Eve is not blond, Adam does not love himself, Adam does love Eve, Eve does not love Adam, and Eve does love herself.

You can also think of the box and the description on the left as a very short novel. The box gives you the list of characters, and the truth table line on the left tells you what happens in this novel. Of course, the novel is not true. But if the novel were true, if it described the whole world, we would have a simple world with just Adam and Eve having the properties and relations described on the left.

Now, in writing this novel, I only specified the truth value for atomic sentences formed from the one and two place predicates and from the two names. What about the truth value of more complicated sentences? We can use our old rules for figuring out the truth value of compounds formed from these atomic sentences using '&', 'v', '~', '⊃', and '≡'. For example, in this novel 'Ba & Lae' is true because both the components are true.

What about the truth value of '(Ǝx)Bx'? Intuitively, '(Ǝx)Bx' should be true in the novel because in the novel there is someone, namely Adam, who is blond. As another example, consider '(Ǝx)Lxa'. In this novel '(Ǝx)Lxa' is false because Eve does not love Adam and Adam does not love Adam. And in this novel there isn't anyone (or anything) else. So no one loves Adam. In other words, in this novel it is false that there is someone who loves Adam.

Let's move on and consider the sentence '(Vx)Lxe'. In our novel this sentence is true, because Adam loves Eve, and Eve loves herself, and that's all the people there are in this novel. If this novel were true, it would be true that everyone loves Eve. Finally, '(Vx)Bx' is false in the novel, for in this novel Eve is not blond. So in this novel it is false that everyone is blond.

Remember what we had set out to do: We wanted to give a precise account of the meaning of the quantifiers very like the precise account which truth table definitions gave to '&' and the other sentence logic connectives. In sentence logic we did this by giving precise rules which told us when a compound sentence is true, given the truth value of the compound's components.

We now have really done the same thing for '(Vx)' and '(Ǝx)' in one special case. For a line of a truth table (a "novel") that gives a truth value for all atomic sentences using 'B', 'L', 'a', and 'e', we can say whether a universally quantified or an existentially quantified sentence is true or false. For example, the universally quantified sentence '(Vx)Lxe' is true . just in case 'Lxe' is true for all values of 'x' in the novel. At the moment we are considering a novel in which the only existing things are Adam and Eve. In such a novel '(Vx)Lxe' is true if both 'Lxe' is true when we take 'x' to refer to Adam and 'Lxe' is also true when we take 'x' to refer to Eve. Similarly, '(Ǝx)Bx' is true in such a novel just in case 'Bx' is true for some value of 'x' in the novel. As long as we continue to restrict attention to a novel with only Adam and Eve as characters, '(Ǝx)Bx' is true in the novel if either 'Bx' is true when we take 'x' to refer to Adam or 'Bx' is true if we take 'x' to refer to Eve.

If the example seems a bit complicated, try to focus on this thought: All we are really doing is following the intuitive meaning of "all x" and "some x" in application to our little example. If you got lost in the previous paragraph, go back over it with this thought in mind.

Now comes a new twist, which might not seem very significant, but which will make predicate logic more interesting (and much more complicated) than sentence logic. In sentence logic we always had truth tables with a finite number of lines. Starting with a fixed stock of atomic sentence letters, we could always, at least in principle, write out all possible cases to consider, all possible assignments of truth values to sentence letters. The list might be too long to write out in practice, but we could at least understand everything in terms of such a finite list of cases.

Can we do the same thing when we build up sentences with predicates and names? If, for example, we start with just 'B', 'L', 'a', and 'e', we can form six atomic sentences. We can write out a 64-line truth table which will give us the truth value for any compound built up from these six atomic sentences, for any assignment of truth values to the atomic sentences. But the fact that we are using quantifiers means that we must also consider further possibilities.

Consider the sentence '(Vx)Bx'. We know this is false in the one case we used as an example (in which 'Ba' is true and 'Be' is false). You will immediately think of three alternative cases (three alternative "novels") which must be added to our list of relevant possible cases: the case in which Eve is blond and Adam is not, the case in which Adam and Eve are both blond, and the case in which both are not blond. But there are still more cases which we must include in our list of all possible cases! I can generate more cases by writing new novels with more characters. Suppose 1 write a new novel with Adam, Eve, and Cid. I now have eight possible ways of distributing hair color (blond or not blond) among my characters, which can be combined with 512 different possible combinations of who does or does not love whom! And, of course, this is just the beginning of an unending list of novels describing possible cases in which '(Vx)Bx' will have a truth value. 1 can always expand my list of novels by adding new characters. I can even describe novels with infinitely many characters, although I would not be able to write such a novel down.

How are we going to manage all this? In sentence logic we always had, for a given list of atomic sentence, a finite list of possible cases, the finite number of lines of the corresponding truth table. Now we have infinitely many possible cases. We can't list them all, but we can still say what any one of these possible cases looks like. Logicians call a possible case for a sentence of predicate logic an Interpretation of the sentence. The example with which we started this chapter is an example of an interpretation, so actually you have already seen and understood an example of an interpretation. We need only say more generally what interpretations are.

We give an interpretation, first, by specifying a collection of objects which the interpretation will be about, called the Domain of the interpretation. A domain always has at least one object. Then we give names to the objects in the domain, to help us in talking about them. Next, we must say which predicates will be involved. Finally, we must go through the predicates and objects and say which predicates are true of which objects. If we are concerned with a one place predicate, the interpretation specifies a list of objects of which the object is true. If the predicate is a two place predicate, then the interpretation specifies a list of pairs of objects between which the two place relation is supposed to hold, that is, pairs of objects of which the two place relation is true. Of course, order is important. The pair a-followed-by-b counts as a different pair from the pair b-followed-by-a. Also, we must consider objects paired with themselves. For example, we must specify whether Adam loves himself or does not love himself. The interpretation deals similarly with three and more place predicates.

In practice, we often specify the domain of an interpretation simply by giving the interpretation's names for those objects. I should mention that in a fully developed predicate logic, logicians consider interpretations which have unnamed objects. In more advanced work, interpretations of this kind become very important. But domains with unnamed objects would make it more difficult to introduce basic ideas and would gain us nothing for the work we will do in part I of this volume. So we won't consider interpretations with unnamed objects until part 11.

The following gives a summary and formal definition of an interpretation:

Definition: Interpretation

An Interpretation consists of

1. A collection of objects, called the interpretation's Domain. The domain always has at least one object.
2. A name for each object in the domain. An object may have just one name or more than one name. (In part I1 we will expand the definition to allow domains with unnamed objects.)
3. A list of predicates.
4. A specification of the objects of which each predicate is true and the objects of which each predicate is false-that is, which one place predicates apply to which individual objects, which two place predicates apply to which pairs of objects, and so on. In this way every atomic sentence formed from predicates and names gets a truth value.
5. An interpretation may also include atomic sentence letters. The interpretation specifies a truth value for any included atomic sentence letter.

By an Interpretation of a Sentence, we mean an interpretation which is sure to have enough information to determine whether or not the sentence is true or false in the interpretation.

Definition: Interpretation of a Sentence

An Interpretation of a Sentence is an interpretation which includes all the names and predicates which occur in the sentence and includes truth values for any atomic sentence letters which occur in the sentence.

For example, the interpretation of Figure 2-1 is an interpretation of 'Ba' and of '(Vx)Lxx'. In this interpretation 'Ba' is true and '(Vx)Lxx' is false. Note that for each of these sentences, the interpretation contains more information than is needed to determine whether the sentence is true or false. This same interpretation is not an interpretation of 'Bc' or of '(Ǝx)Txe'. This is because the interpretation does not include the name 'c' or the two place predicate 'T', and so can't tell us whether sentences which use these terms are true or false.

Exercise

2-1. I am going to ask you to give an interpretation for some sentences. You should use the following format. Suppose you are describing an interpretation with a domain of three objects named 'a', 'b', and 'c'. Specify the domain in this way: D ≡ {a,b,c}. That is. specify the domain by giving a list of the names of the objects in the domain. Then specify what is true about the objects in the domain by using a sentence of predicate logic. Simply conjoin all the atomic and negated atomic sentences which say which predicates are true of which objects and which are false. Here is an example. The following is an interpretation of the sentence 'Tb & Kbd':

D ≡ {b,d); Tb & Td & Kbb & Kbd & Kdb & Kdd.

In this interpretation all objects have property T and everything stands in the relation K to itself and to everything else. Here is another interpretation of the same sentence:

D ≡ (b,d); -Tb & Td & Kbb & -Kbd & -Kdb & Kdd.

2-2. Sometimes students have trouble understanding what I want in this exercise. They ask, How am I supposed to decide which interpretation to write down? You can write down any interpretation you want as long as it is an interpretation of the sentence I give you. In every case you have infinitely many interpretations to choose from because you can always get more interpretations by throwing in more objects and then saying what is true for the new objects. Choose any you like. Just make sure you are writing down an interpretation of the sentence I give you.

1. Lab
2. Lab ⊃ Ta
3. Lab v ~Lba
4. (Vx)(Fx≡Rxb)
5. Ga & (Ǝx)(Lxb v Rax)
6. (Kx & (Vx)Rax) ⊃ (Ǝx)(Mx v Rcx)