Predicate

A predicate is an undetermined statement containing one or more variables that becomes a proposition when all of its variables are substituted for values from their domains. Because of this, predicates can be thought of as functions that output propositions, written like \(P(x)\), \(Q(x, y)\), or \(R(x, y, z)\), for example. Alternatively, predicates can also become propositions when they are quantified.

Here is a simple predicate:

\(P(x)\) : \(x\) can fly.

As soon as we input "birds," it becomes the following proposition:

\(P(\)birds\()\) : Birds can fly.

And it is true. However, if we input "fish" instead:

\(P(\)fish\()\) : Fish can fly.

The resulting proposition \(P(\)fish\()\) is false.

As you can see, depending on the input, a predicate can become an entirely different proposition.

Now, here's a predicate that takes in two variables:

\(Q(x, y)\) : \(x\) can fly and \(y\) can swim.

If we fill in just one of the variables:

\(Q(\)birds, \(y)\) : Birds can fly and \(y\) can swim.

The predicate is not yet a proposition. All variables must be given values for it to become a proposition:

\(Q(\)birds, fish\()\) : Birds can fly and fish can swim.

This proposition evaluates as true.

Logic & Proofs
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