To this point our primary interest has been in one-place predicates.We will now expand our focus and turn to some multi-place predicates. There are English predicates with more than two places (for example, "_ is between . and ,"), but we shall not concern ourselves with them. We shall confine ourselves to two-place predicates.

In the discussion below we shall use the following symbolization key:

Let us now symbolize:

These are, respectively:

Note that these two do not say the same thing. Now let us consider:

The first of these comes out as:

We may read this as the statement that there is someone x such that Steven is taller than x. This does indeed say what we wanted to say. The second comes out as:

We may read this as the statement that there is someone x such that x is taller than Steven. Again, this says what we want it to say.

As usual, we must be very careful with negations. Consider:

The first is an existential sentence:

And you must watch out for the following as well:

The first is an existential sentence:

The second is a disjunction:

Mastering all of this is, of course, a matter of practice and thought. It will be difficult unless you pay careful attention to the kinds of sentences you are working with.

Let us now turn to some sentences in which we have no singular terms. Here we will often need more than one variable. And as we will see, the order of quantifiers is often of considerable importance.

The sentence:

We will now use "T_ ." for the predicate "_ is at least as tall as .". (I am at least as tall as someone if I am the same height or taller.) Our domain will remain the same. The sentence "(x)($ y)Txy" says that for every person x there is some person y such that x is at least as tall as y. Consider the sentence:

It should be clear that the sentences

and

do not say the same thing. Let us use "T_." for t." for tcate "_was born on day.". Consider the claim that every person was born on some day. As symbolized by the first sentence, this claim is the true one that everyone has a a day on which they were born. However, the second sentence yields the false claim that there is some one day everyone was born on. In order to make sure that we see this let us look at two worlds.

It is not always easy, if one hears or reads claims like these, to tell exactly what was meant. Consider the claim that everyone is loved by someone. This might mean that everyone is loved by someone or another. This is the first type of symbolization. But it could also mean there is some being such that everyone is loved by that being. This is the interpretation captured by the second type of symbolization. Typically we appeal to the context and our background knowledge in order to decide precisely what is meant. The second symbolization might, for example, be the most plausible interpretation if someone argued that since God exists everyone is loved by someone.

One of our constant themes has been that we must be very careful with sentences involving negations. That obtains here as well. Consider the statement:

Here we should see that the statement is a negation. But we still have two ways of construing the sentence being negated. The statement might be taken as either:

or:

Here, the first seems to be the most natural interpretation, but the second interpretation is also possible. Or consider the claim that it is false that everyone in your logic classroom speaks some language. This sentence seems to admit of both interpretations. Certainly if we said explicitly that it is false that there is some language that everyone in that room speaks, we would have an example of a statement with a structure like that of the second sentence.

Let us now look at a few statements that have a more complex structure. We shall use the following symbolization key.

Notice that it is quite legitimate to take "father" as a two-place as well as a one-place predicate: a father is always a father of someone. We will, then, symbolize the statement that Steven is a father as:

How should we symbolize the following?

This is, in effect, an I sentence. It would be symbolized as:

We can read this as the statement that for everyone x, if x is a male then there is some y or other such that x is the parent of y. Trickier yet is the statement:

Recall that we already know how to symbolize A statements, and already know how to symbolize both "x is a father" and "x is a parent". So our statement is symbolized in the following way:

Here we said that everything x is such that, if x is a father, then x is a parent. Finally, let us symbolize:

These come out as, respectively:

What we have done is utilized "(Mx & ($ y)Pxy" as our means of symbolizing "x is a male parent".