where c is some arbitrary element of the universe.
If c is a variable, then it must not already be quantified somewhere in P(x) --- see Explanation and Note below.
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.
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|
|3. H(d) F(d)||Universal instantiation on 1.|
|4. F(d)||Modus ponens on 2 and 3.|