Predicate Logic

Universal Instantiation

- universal instantiation 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. |

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.