Section 2: Interpretations and models in QL

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In SL, an interpretation or symbolization key specifies what each of the sentence letters means. The interpretation of a sentence letter along with the state of the world determines whether the sentence letter is true or false. Since the basic units are sentence letters, an interpretation only matters insofar as it makes sentence letters true or false. Formally, the semantics for SL is strictly in terms of truth value assignments. Two interpretations are the same, formally, if they make for the same truth value assignment.

What is an interpretation in QL? Like a symbolization key for QL, an interpretation requires a UD, a schematic meaning for each of the predicates, and an object that is picked out by each constant. For example:

UD: comic book characters
Fx: \(x\) fights crime.
b: the Batman
w: Bruce Wayne

Consider the sentence \(Fb\). The sentence is true on this interpretation, but— just as in SL— the sentence is not true just because of the interpretation. Most people in our culture know that Batman fights crime, but this requires a modicum of knowledge about comic books. The sentence \(Fb\) is true because of the interpretation plus some facts about comic books. This is especially obvious when we consider \(Fw\). Bruce Wayne is the secret identity of the Batman in the comic books— the identity claim \(b\) = \(w\) is true— so \(Fw\) is true. Since it is a secret identity, however, other characters do not know that \(Fw\) is true even though they know that \(Fb\) is true.

We could try to characterize this as a truth value assignment, as we did for SL. The truth value assignment would assign 0 or 1 to each atomic wff: \(\), \(Fb\)Fw, and so on. If we were to do that, however, we might just as well translate the sentences from QL to SL by replacing \(Fb\) and \(Fw\) with sentence letters. We could then rely on the definition of truth for SL, but at the cost of ignoring all the logical structure of predicates and terms. In writing a symbolization key for QL, we do not give separate definitions for \(Fb\) and \(Fw\). Instead, we give meanings to \(F\), \(b\), and \(w\). This is essential because we want to be able to use quantifiers. There is no adequate way to translate ∀\(xFx\) into SL.

So we want a formal counterpart to an interpretation for predicates and constants, not just for sentences. We cannot use a truth value assignment for this, because a predicate is neither true nor false. In the interpretation given above, \(F\) is true of the Batman (i.e., \(Fb\) is true), but it makes no sense at all to ask whether \(F\) on its own is true. It would be like asking whether the English language fragment ‘...fights crime’ is true.

What does an interpretation do for a predicate, if it does not make it true or false? An interpretation helps to pick out the objects to which the predicate applies. Interpreting \(Fx\) to mean ‘\(x\) fights crime’ picks out Batman, Superman, Spiderman, and other heroes as the things that are \(Fs\). Formally, this is a set of members of the UD to which the predicate applies; this set is called the extension of the predicate.

Many predicates have indefinitely large extensions. It would be impractical to try and write down all of the comic book crime fighters individually, so instead we use an English language expression to interpret the predicate. This is somewhat imprecise, because the interpretation alone does not tell you which members of the UD are in the extension of the predicate. In order to figure out whether a particular member of the UD is in the extension of the predicate (to figure out whether Black Lightning fights crime, for instance), you need to know about comic books. In general, the extension of a predicate is the result of an interpretation along with some facts.

Sometimes it is possible to list all of the things that are in the extension of a predicate. Instead of writing a schematic English sentence, we can write down the extension as a set of things. Suppose we wanted to add a one-place predicate \(M\) to the key above. We want \(Mx\) to mean ‘\(x\) lives in Wayne Manor’, so we write the extension as a set of characters:
extension(\(M\)) = {Bruce Wayne, Alfred the butler, Dick Grayson}

You do not need to know anything about comic books to be able to determine that, on this interpretation, \(Mw\) is true: Bruce Wayne is just specified to be one of the things that is \(M\). Similarly,∃\(xMx\) is obviously true on this interpretation: There is at least one member of the UD that is an \(M\)— in fact, there are three of them.

What about the sentence ∀\(xMx\)? The sentence is false, because it is not true that all members of the UD are \(M\). It requires the barest minimum of knowledge about comic books to know that there are other characters besides just these three. Although we specified the extension of \(M\) in a formally precise way, we still specified the UD with an English language description. Formally speaking, a UD is just a set of members.

The formal significance of a predicate is determined by its extension, but what should we say about constants like \(b\) and \(w\)? The meaning of a constant determines which member of the UD is picked out by the constant. The individual that the constant picks out is called the referent of the constant. Both \(b\) and \(w\) have the same referent, since they both refer to the same comic book character. You can think of a constant letter as a name and the referent as the thing named. In English, we can use the different names ‘Batman’ and ‘Bruce Wayne’ to refer to the same comic book character. In this interpretation, we can use the different constants ‘\(b\)’ and ‘\(w\)’ to refer to the same member of the UD.

Sets

We use curly brackets ‘{’ and ‘}’ to denote sets. The members of the set can be listed in any order, separated by commas. The fact that sets can be in any order is important, because it means that {foo, bar} and {bar, foo} are the same set.

It is possible to have a set with no members in it. This is called the empty set. The empty set is sometimes written as {}, but usually it is written as the single symbol ∅.

Models

As we have seen, an interpretation in QL is only formally significant insofar as it determines a UD, an extension for each predicate, and a referent for each constant. We call this formal structure a model for QL.

To see how this works, consider this symbolization key:

UD: People who played as part of the Three Stooges
Hx: \(x\) had head hair.
f: Mister Fine

If you do not know anything about the Three Stooges, you will not be able to say which sentences of QL are true on this interpretation. Perhaps you just remember Larry, Curly, and Moe. Is the sentence \(Hf\) true or false? It depends on which of the stooges is Mister Fine.

What is the model that corresponds to this interpretation? There were six people who played as part of the Three Stooges over the years, so the UD will have six members: Larry Fine, Moe Howard, Curly Howard, Shemp Howard, Joe Besser, and Curly Joe DeRita. Curly, Joe, and Curly Joe were the only completely bald stooges. The result is this model:

UD = {Larry, Curly, Moe, Shemp, Joe, Curly Joe}
extension(\(H\)) = {Larry, Moe, Shemp}
referent(\(f\)) = Larry

You do not need to know anything about the Three Stooges in order to evaluate whether sentences are true or false in this model. \(Hf\) is true, since the referent of \(f\) (Larry) is in the extension of \(H\). Both ∃\(xHx\) and ∃\(x\)¬\(Hx\) are true, since there is at least one member of the UD that is in the extension of \(H\) and at least one member that is not in the extension of \(H\). In this way, the model captures all of the formal significance of the interpretation.

Now consider this interpretation:

UD: whole numbers less than 10
Ex: \(x\) is even.
Nx: \(x\) is negative.
Lxy: \(x\) is less than \(y\).
Txyz: \(x\) times \(y\) equals \(z\).

What is the model that goes with this interpretation? The UD is the set {1,2,3,4,5,6,7,8,9}.

The extension of a one-place predicate like \(E\) or \(N\) is just the subset of the UD of which the predicate is true. Roughly speaking, the extension of the predicate \(E\) is the set of \(Es\) in the UD. The extension of \(E\) is the subset {2,4,6,8}. There are many even numbers besides these four, but these are the only members of the UD that are even. There are no negative numbers in the UD, so \(N\) has an empty extension; i.e. extension(\(N\)) = ∅.

The extension of a two-place predicate like \(L\) is somewhat vexing. It seems as if the extension of \(L\) ought to contain 1, since 1 is less than all the other numbers; it ought to contain 2, since 2 is less than all of the other numbers besides 1; and so on. Every member of the UD besides 9 is less than some member of the UD. What would happen if we just wrote extension(\(L\)) = {1,2,3,4,5,6,7,8}?

The problem is that sets can be written in any order, so this would be the same as writing extension(\(L\)) = {8,7,6,5,4,3,2,1}. This does not tell us which of the members of the set are less than which other members.

We need some way of showing that 1 is less than 8 but that 8 is not less than 1. The solution is to have the extension of \(L\) consist of pairs of numbers. An ordered pair is like a set with two members, except that the order does matter. We write ordered pairs with angle brackets ‘<’ and ‘>’. The ordered pair <foo, bar> is different than the ordered pair <bar, foo>. The extension of \(\)L is a collection of ordered pairs, all of the pairs of numbers in the UD such that the first number is less than the second. Writing this out completely:

extension(L) = {<1,2>, <1,3>, <1,4>, <1,5>, <1,6>, <1,7>,
<1,8>, <1,9>, <2,3>, <2,4>, <2,5>, <2,6>, <2,7>, <2,8>, <2,9>,
<3,4>, <3,5>, <3,6>, <3,7>, <3,8>, <3,9>, <4,5>, <4,6>, <4,7>,
<4,8>, <4,9>, <5,6>, <5,7>, <5,8>, <5,9>, <6,7>, <6,8>, <6,9>,
<7,8>, <7,9>, <8,9>}

Three-place predicates will work similarly; the extension of a three-place predicate is a set of ordered triples where the predicate is true of those three things in that order. So the extension of \(T\) in this model will contain ordered triples like <2,4,8>, because 2×4 = 8.

Generally, the extension of an \(n\)-place predicate is a set of all ordered \(n\)-tuples <\(a\)₁,\(a\)₂,...,\(a\)_n> such that \(a\)₁–\(a\)_n are members of the UD and the predicate is true of \(a\)₁–\(a\)_n in that order.