Universal generalization


In predicate logic, generalization (also universal generalization, universal introduction,[1][2][3] GEN, UG) is a valid inference rule. It states that if has been derived, then can be derived.

Universal generalization
TypeRule of inference
FieldPredicate logic
StatementSuppose is true of any arbitrarily selected , then is true of everything.
Symbolic statement,

Generalization with hypotheses edit

The full generalization rule allows for hypotheses to the left of the turnstile, but with restrictions. Assume   is a set of formulas,   a formula, and   has been derived. The generalization rule states that   can be derived if   is not mentioned in   and   does not occur in  .

These restrictions are necessary for soundness. Without the first restriction, one could conclude   from the hypothesis  . Without the second restriction, one could make the following deduction:

  1.   (Hypothesis)
  2.   (Existential instantiation)
  3.   (Existential instantiation)
  4.   (Faulty universal generalization)

This purports to show that   which is an unsound deduction. Note that   is permissible if   is not mentioned in   (the second restriction need not apply, as the semantic structure of   is not being changed by the substitution of any variables).

Example of a proof edit

Prove:   is derivable from   and  .


Step Formula Justification
1   Hypothesis
2   Hypothesis
3   Universal instantiation
4   From (1) and (3) by Modus ponens
5   Universal instantiation
6   From (2) and (5) by Modus ponens
7   From (6) and (4) by Modus ponens
8   From (7) by Generalization
9   Summary of (1) through (8)
10   From (9) by Deduction theorem
11   From (10) by Deduction theorem

In this proof, universal generalization was used in step 8. The deduction theorem was applicable in steps 10 and 11 because the formulas being moved have no free variables.

See also edit

References edit

  1. ^ Copi and Cohen
  2. ^ Hurley
  3. ^ Moore and Parker