Reasoning: An Introduction to Logic, Sets, and Functions

Subsection 5.1.12 Universal Instantiation

Universal Instantiation: ∀x (P(x))

∴ P(c/x)

Note: The substitution of c for x must be for all free occurrences of x and for no other variables.

Thus, from the fact that P is true for all values of x, conclude that it must be true for any particular object c.

This is what we have been calling new-rule-to-come-1.

Uses of Universal Instantiation usually occur early in proofs for which the premises have universally quantified statements.

There are two main ways in which we use this rule. The first is the simplest: we want to prove a claim about some particular individual. We reason that, if P must be true of everyone, it must, in particular be true of Smokey or Fred or 275 or whatever.

For example, we can use Universal Instantiation to make the following argument:

No cats are birds. Lucy is a cat. Therefore, Lucy is not a bird.

Define:

Bird(x): True if x is a bird.

Cat(x): True if x is a cat.

Proof:

[1] x (Cat(x)  Bird(x)) Premise

[2] Cat(Lucy) Premise

[3] Cat(Lucy)  Bird(Lucy) Universal Instantiation [1]

[4] Bird(Lucy) Modus Ponens [2] , [3]

And now for the second use of Universal Instantiation: We can use it to make claims about arbitrary elements (about which we know nothing else). To do this, we must use names about which we have no other premises (unlike what we did in the Lucy case above). If we introduce arbitrary elements, then we can prove new general claims by doing what we’ve already done in our proof of the <ex Breathes> syllogism. We introduce arbitrary elements with Universal Instantiation, reason about them, and then conclude by using what we’ve been calling new-rule-to-come-2 (next) to derive a new universally quantified claim. We’ll hold off giving an example of this until we have new-rule-to-come-2 to work with.