##### Predicate Logic

Universal Instantiation

### Subjects to be Learned

• universal instantiation rule

### Contents

Rule: x P(x)
-------
P(c)

where c is some arbitrary element of the universe.

Restrictions:

If c is a variable, then it must not already be quantified somewhere in P(x) --- see Explanation and Note below.

Explanation:

What this rule says is that from x P(x) one can infer P(c) for any object in the universe represented by the variable c, thus stripping off the universal quantifier. It should be noted that P holds for any specifc object in the universe in this case. This rule follows because x P(x) says that P(x) holds true for all objects in the universe. Note that P(x), in general, may be a compound wff invloving a number of predicates, variables, and quantifiers. Thus c in P(c) must not be a variable which is quantified within P(x). See Note below for an example of what can happen c is not properly chosen.

Example:

For example, the following argument can be proven correct using the Universal Instantiation:"No humans can fly. John Doe is human. Therefore John Doe can not fly."

First let us express this using the following notation:
F(x) means "x can fly."
H(x) means "x is human."
d is a symbol representing John Doe.

Then the argument is
[ x [H(x)  F(x)] H(d) ]  F(d).

The proof is
 1. x [H(x)  F(x)] Hypothesis 2. H(d) Hypothesis 3. H(d)  F(d) Universal instantiation on 1. 4. F(d) Modus ponens on 2 and 3.

Note:

If the restriction on the Universal Instantiation does not exist, then y x P(x,y) may become x P(x,x), that is x has been substituted for y by the Universal Instantiation, which is not correct.