Predicate Logic
Universal Instantiation
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. |