Introduction to Relations
 

In the last section we introduced the idea that a propositional function can have more than one argument. We refer to polyadic propositional functions as relations, and relations can have two or more relata.  For now we'll look at relations that take two arguments. Common examples are "taller than", "smaller than", "smarter than" but also "loves", "is the father of", "eats", and "talks to". Notice that "eats" as a verb in English can function without a relatum, for example in the sentence "Fred eats." but it can also function as a relation in sentences like "Fred eats fish."

Singular propositions expressing relations work just like singular propositions we've already seen. The only difference is that relations require more than one name.

When expressing relations, we need to use more than one quantifier to bind all the variables in our propositional functions. For each individual variable, we will need one quantifier.

Consider the sentence:

This is a relation. We can write the propositional function as Txy. But one of the relata is Martha, and so we have:

Now we need to bind the variable. The sentence asserts that someone is taller than Martha, so we use the existential quantifier:

Here's an Easy Polyadic Translation Exercise to begin with.