where P(c) holds for every element c of the universe of discourse.
x must not appear as a free variable in P(c).
What this rule says is that if P(c) holds for any arbitrary element c of the universe, then we can conclude that x P(x).
If, however, c is supposed to represent some specific element of the universe that has the property P, then one can not generalize it to all the elements. For example, if P(x) means "x is fast", then all it means is that an unspecified element represented by x is fast. It does not necessarily mean that everything in the universe is fast.
This rule is something we can use when we want to prove that a certain property holds for every element of the universe. That is when we want to prove x P(x), we take an abrbitrary element x in the universe and prove P(x). Then by this Universal Generalization we can conclude x P(x).
For example, consider the following argument: For every number x if x > 1, then x - 1 > 0. Also for every number x, x > 1. (Here we are making a hypothetical argument. We know, of course, every number is not greater than 1.)
Hence we conclude that for every number x, x - 1 >0.
This argument can be justified as follows using this Universal Generalization.
Let P(x) represent x > 1, and Q(x) represent x - 1 > 0.
Then the argument above is represented by
[x [P(x) Q(x)] x P(x)] x Q(x).
To prove it we proceed as follows:
|1. x [P(x) Q(x)]||Hypothesis|
|2. x P(x)||Hypothesis|
|3. [P(x) Q(x)]||Universal Instantiation on 1.|
|4. P(x) for the same x as in 3.||Universal Instantiation on 2.|
|5. Q(x)||Modus ponens on 3 and 4.|
|6. x Q(x)||Universal Generalization on 6.|