3.2.2: Truth in an Interpretation

Just like a line of a truth table, an interpretation tells us whether each atomic sentence formed from predicates and names is true or false. What about compound sentences? If the main connective of a compound sentence does not involve a quantifier, we simply use the old rules for the connectives of sentence logic. We have only one more piece of work to complete: We must make more exact our informal description of the conditions under which a quantified sentence is true or is false in an interpretation.

Intuitively, a universally quantified sentence is going to be true in an interpretation if it is true in the interpretation for everything to which the variable could refer in the interpretation. (Logicians say, "For every value of the universally quantified variable.") An existentially quantified sentence will be true in an interpretation if it is true for something to which the variable could refer in the interpretation (that is, "for some value of the existentially quantified variable.") What we still need to do is to make precise what it is for a quantified sentence to be true for a value of a variable. Let's illustrate with the same example we have been using, the interpretation given in figure 2-1.

Consider the sentence '(Vx)Bx'. In the interpretation we are considering, there are exactly two objects, a, and e. '(Vx)Bx' will be true in the interpretation just in case, roughly speaking, it is true both for the case of 'x' referring to a and the case of 'x' referring to e. But when 'x' refers to a, we have the sentence 'Ba'. And when 'x' refers to 'e', we have the sentence 'Be'. Thus '(Vx)Bx' is true in this interpretation just in case both 'Ba' and 'Be' are true. We call 'Ba' the Substitution Instance of '(Vx)Bx' formed by substituting 'a' for 'x'. Likewise, we call 'Be' the substitution instance of '(Vx)Bxl formed by substituting 'e' for 'x'. Our strategy is to explain the meaning of universal quantification by defining this notion of substitution - instance and then specifying that a universally quantified sentence is true in an interpretation just in case it is true for all substitution instances in the interpretation:

Definition (Incomplete): the Substitution Instance

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 wherever u had occurred.

A word of warning: This definition is not yet quite right. It works only as long as we don't have multiple quantification, that is, as long as we don't have sentences which stack one quantifier on top of other quantifiers. But until chapter 3 we are going to keep things simple and consider only simple sentences which do not have one quantifier applying to a sentence with another quantifier inside. When we have the basic concepts we will come back and give a definition which is completely general. Now we can easily use this definition of substitution instance to characterize truth of a universally quantified sentence in an interpretation:

Definition: (Incomplete Definition)

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

Another word of warning: As with the definition of substitution instance, this definition is not quite right. Again, Chapter 3 will straighten out the details.

To practice, let's see whether '(Vx)(Bx ⊃ Lxe)' is true in the interpretation of figure 2-1. First we form the substitution instances with the names of the interpretation, 'a', and 'e'. We get the first substitution instance by dropping the quantifier and writing in 'a' everywhere we see 'x'. This gives

\[Ba ⊃ Lae. \]

Note that because 'Ba' and 'Lae' are both true in the interpretation, this first substitution instance is true in the interpretation. Next we form the second substitution instance by dropping the quantifier and writing in 'e' wherever we see 'x':

\[Be ⊃ Lee.\]

Because 'Be' is false and 'Lee' is true in the interpretation, the conditional 'Be ⊃ Lee' is true in the interpretation. We see that all the substitution instances of '(Vx)(Bx ⊃ Lxe)' are true in the interpretation. So this universally quantified sentence is true in the interpretation.

To illustrate further our condition for truth of a universally quantified sentence, consider the sentence '(Vx)(Bx ⊃ Lxa)'. This has the substitution instance 'Ba ⊃ Laa'. In this interpretation 'Ba' is true and 'Laa' is false, so 'Ba 3 Laa' is false in the interpretation. Because '(Vx)(Bx ⊃ Lxa)' has a false substitution instance in the interpretation, it is false in the interpretation.

You may have noticed the following fact about the truth of a universally quantified sentence and the truth of its substitution instances. By definition '(Vx)(Bx ⊃ Lxe)' is true in the interpretation just in case all of its instances are true in the interpretation. But its instances are all true just in case the conjunction of the instances is true. That is, '(Vx)(Bx ⊃ Lxe)' is true in the interpretation just in case the conjunction

\[(Ba\, ⊃\, Lae)\, \& \,(Be\, ⊃ \,Lee)\]

is true in the interpretation. If you think about it, you will see that this will hold in general. In the interpretation we have been discussing (or any interpretation with two objects named 'a' and 'e'), any universally quantified sentence, '(Vx)(. . . x . . .)', will be true just in case the conjunction of its substitution instance, '(. . . a . . .)&(. . . e . . .)', is true in the interpretation.

It's looking like we can make conjunctions do the same work that the universal quantifier does. A universally quantified sentence is true in an interpretation just in case the conjunction of all its substitution instances is true in the interpretation. Why, then, do we need the universal quantifier at all?

To answer this question, ask yourself what happens when we shift to a new interpretation with fewer or more things in its domain. In the new interpretation, what conjunction will have the same truth value as a given universally quantified sentence? If the new interpretation has a larger domain, our conjunction will have more conjuncts. If the new interpretation has a smaller domain, our conjunction will have fewer conjuncts. In other words, when we are looking for a conjunction of instances to give us the truth value of a universally quantified sentence, the conjunction will change from interpretation to interpretation. You can see in this way that the universal quantifier really does add something new. It acts rather like a variable conjunction sign. It has the effect of forming a long conjunction, with one conjunct for each of the objects in an interpretation's domain. If an interpretation's domain has infinitely many objects, a universally quantified sentence has the effect of an infinitely long conjunction!

What about existentially quantified sentences? All the work is really done. We repeat everything we said for universal quantification, replacing the word 'all' with 'some':

(Incomplete) Definition: Substitution Instance

For any existentially quantified sentence (Ǝ)(. . . 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 wherever u had occurred.

(incomplete) Definition: existentially quantified sentence

An existentially quantified 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.

As with the parallel definitions for universally quantified sentences, these definitions will have to be refined when we get to chapter 3.

To illustrate, let's see whether the sentence '(Ǝx)(Bx & Lxe)' is true in the interpretation of figure 2-1. We will need the sentence's substitution instances. We drop the quantifier and write in 'a' wherever we see 'x', giving 'Ba & Lae', the instance with 'a' substituted for 'x'. In the same way, we form the instance with 'e' substituted for 'x', namely, 'Be & Lee'. '(Ǝx)(Bx & Lxe)' is true in the interpretation just in case one or more of its substitution instances are true in the interpretation. Because 'Ba' and 'Lae' are true in the interpretation, the first instance, 'Ba & Lae', is true, and so '(Ǝx)(Bx & Lxe)' is true.

Have you noticed that, just as we have a connection between universal quantification and conjunction, we have the same connection between existential quantification and disjunction: '(Ǝx)(Bx & Lxe)' is true in our interpretation just in case one or more of its instances are true. But one or more of its instances are true just in case their disjunction

\[(Ba & Lae) v (Be & Lee)\]

is true. In a longer or shorter interpretation we will have the same thing with a longer or shorter disjunction. Ask yourself, when is an existentially quantified sentence true in an interpretation? It is true just in case the disjunction of all its substitution instances in that interpretation is true in the interpretation. Just as the universal quantifier acted like a variable conjunction sign, the existential quantifier acts like a variable disjunction sign. In an interpretation with an infinite domain, an existentially quantified sentence even has the effect of an infinite disjunction.

I hope that by now you have a pretty good idea of how to determine whether a quantified sentence is true or false in an interpretation. In understanding this you also come to understand everything there is to know about the meaning of the quantifiers. Remember that we explained the meaning of the sentence logic connectives '~', '&', 'v', '⊃', and '≡' by giving their truth table definitions. For example, explaining how to determine whether or not a conjunction is true in a line of a truth table tells you everything there is to know about the meaning of '&'. In the same way, our characterization of truth of a quantified sentence in an interpretation does the same kind of work in explaining the meaning of the quantifiers.

This point about the meaning of the quantifiers illustrates a more general fact. By a "case" in sentence logic we mean a line of a truth table, that is, an assignment of truth values to sentence letters. The interpretations of predicate logic generalize this idea of a case. Keep in mind that interpretations do the same kind of work in predicate logic that assignments of truth values to sentence letters do in sentence logic, and you will easily extend what you already know to understand validity, logical truth, contradictions, and other concepts in predicate logic.

By now you have also seen how to determine the truth value which an interpretation gives to any sentence, not just to quantified sentences. An interpretation itself tells you which atomic sentences are true and which are false. You can then use the rules of valuation for sentence logic connectives together with our two new rules for the truth of universally and existentially quantified sentences to determine the truth of any compound sentence in terms of the truth of shorter sentences. Multiple quantification still calls for some refinements, but in outline you have the basic ideas.

Exercise

2-2. Consider the interpretation.

D ≡ {a,b}; ~Ba & Bb & Laa & ~Lab & Lba & ~Lbb.

For each of the following sentences, give all of the sentence's substitution instances in this interpretation, and for each substitution instance say whether the instance is true or false in the interpretation. For example, for the sentence '(Vx)Bx', your answer should look like this:

GIVEN SENTENCE SUBSTITUTION INSTANCES
(Vx)Bx Ba, false in the interpretation
Bb, true

a) (Ǝx)Bx b) (Ǝx)~Lxa c) (Vx)Lxa

d) (Ǝx)Lbx e) (V)(Bx v Lax) f) (Ǝx)(Lxa & Lbx)

g) (Vx)(Bx ⊃ Lbx) h) (Ǝx)[(Lbx &Bb) vBx] i) (Ǝx)[Bx ⊃ (Lxx ⊃ Lxa)]

j) (Vx)[(Bx v Lax) ⊃ (Lxb v~Bx)] k) (Ǝx)[(Lax & Lxa) ≡ (Bx v Lxb)]

2-3. For each of the sentences in exercise 2-2, say whether the sentence is true or false in the interpretation of exercise 2-2.

2-4. For each of the following sentences, determine whether the sentence is true or false in the interpretation of exercise 2-2. In this exercise, you must carefully determine the main connective of a sentence before applying the rules to determine its truth in an interpretation. Remember that a quantifier is a connective which applies to the shortest full sentence which follows it. Remember that the main connective of a sentence is the last connective that gets used in building the sentence up from its parts. To determine whether a sentence is true in an interpretation, first determine the sentence's main connective. If the connective is '&', 'v', '~', '⊃', or '≡', you must first determine the truth value of the components, and then apply the rules for the main connective (a conjunction is true just in case both conjuncts are true, and so on). If the main connective is a quantifier, you have to determine the truth value of the substitution instances and then apply the rule for the quantifier, just as you did in the last exercise.

a) (Ǝx)Lxx ⊃ (Vx)(Bx v Lbx)
b) ~(Ǝx)(Lxx ⊃ Bx) & (Vx)(Bx ⊃ Lxx)
c) (Ǝx)[Bx ≡ (Lax v Lxb)]
d) (Ǝx)(Lxb v Bx) ⊃ (Lab v ~Ba)
e) ~(Vx)(~Lxx v Lxb) ⊃ (Lab v ~Lba)
f) (Ǝx)[(Lbx v Bx) ⊃ (Lxb & -Bx)]
g) (Vx)~[(~Lxx ≡ Bx) ⊃ (Lax ≡ Lxa)]
h) (Vx)(Lax v Lxb) v (3x)(Lax v Lxb)
i) (Ǝx)[Lxx & (Bx Ǝ Laa)] & (Ǝx)~(Lab ≡ Lxx)
j) (Vx){[Bx v (Lax & ~Lxb)] ⊃ (Bx ⊃ Lxx)}

2-5. In the past exercises we have given interpretations by explicitly listing objects in the domain and explicitly saying which predicates apply to which things. We can also describe an interpretation in more general terms. For example, consider the interpretation given by

i) Domain: All U.S. citizens over the age of 21.
ii) Names: Each person in the domain is named by 'a' subscripted by his or her social security number.
iii) Predicates: Mx: x is a millionaire.
Hx: x is happy.

(That is, a one place predicate 'Mx' which holds of someone just in case that person is a millionaire and a one place predicate 'Hx' which holds of someone just in case that person is happy.)

a) Determine the truth value of the following sentences in this interpretation. In each case explain your answer. Since you can't write out all the substitution instances, you will have to give an informal general statement to explain your answer, using general facts you know about being a millionaire, being happy, and the connection (or lack of connection) between these.

a1) (Ǝx)Mx a2) (Vx)Hx a3) (Vx)(Hx Mx) a4) (Ǝx)(Mx & ~Hx)
a5) (Vx)[(Mx ⊃ Hx) & (~Mx ⊃ ~Hx)]
a6) (Ǝx)[(Hx & Mx) v (~Hx & ~Mx)]
a7) (Ǝx)(Mx & Hx) & (Ǝx)(Mx & ~Hx)
a8) (Vx)(Hx ⊃ Mx) ⊃ ~(Ǝx)Mx

Here is another example:

i) Domain: All integers, 1, 2, 3, 4, . . .
ii) Names: Each integer is named by 'a' subscripted by that integer's numeral. For example, 17 is named by 'a₁₇'.
iii) Predicates: Ox: x is odd.
Kxy: x is equal to or greater than y.

b) Again, figure out the truth value of the following sentences, and explain informally how you arrived at your answer.

b1) (Ǝx)Ox b2) (Vx)~Ox b3) (Ǝx)(Ox & Kxx)
b4) (Vx)Kxa₁₇ b5) (Vx)(Ox v ~Ox)
b6) (Ǝx)(Ox & Kxa₁₇)
b7) (Vx)[Ox ≡ (~Kxa₁₈ & Kxa₁₇)]
b8) (Ǝx)(Kxa₁₇ ⊃ Kxa₁₈) & (Vx)(~Kxa₁₇ v Kxa₁₈)
b9) (Vx)(Ox ⊃ Kxa₁₇) & (Vx)(~Ox ⊃ ~ Kxa₁₇)