Predicate Logic

Universal Instantiation

Subjects to be Learned



x P(x)

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
2. H(d) Hypothesis
3. H(d) F(d) Universal instantiation on 1.
4. F(d) Modus ponens on 2 and 3.


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.

Next --- Universal Generalization
Existential Instantiation
Existential Generalization
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