Universal generalization: Difference between revisions
No edit summary |
No edit summary |
||
Line 7: | Line 7: | ||
These restrictions are necessary for soundness. Without the first restriction, one could conclude <math>\forall x P(x)</math> from the hypothesis <math>P(y)</math>. Without the second restriction, one could make the following deduction: |
These restrictions are necessary for soundness. Without the first restriction, one could conclude <math>\forall x P(x)</math> from the hypothesis <math>P(y)</math>. Without the second restriction, one could make the following deduction: |
||
#<math>\exists z \, \exists w ( z \not = w) </math> (Hypothesis) |
#<math>\exists z \, \exists w \, ( z \not = w) </math> (Hypothesis) |
||
#<math>\exists w (y \not = w) </math> (Existential instantiation) |
#<math>\exists w \, (y \not = w) </math> (Existential instantiation) |
||
#<math>y \not = x</math> (Existential instantiation) |
#<math>y \not = x</math> (Existential instantiation) |
||
#<math>\forall x (x \not = x)</math> (Faulty universal generalization) |
#<math>\forall x \, (x \not = x)</math> (Faulty universal generalization) |
||
This purports to show that <math>\exists z \, \exists w ( z \not = w) \vdash \forall x (x \not = x),</math> which is an unsound deduction. |
This purports to show that <math>\exists z \, \exists w \, ( z \not = w) \vdash \forall x \, (x \not = x),</math> which is an unsound deduction. |
||
==Example of a proof== |
==Example of a proof== |
Revision as of 21:39, 20 October 2018
In predicate logic, generalization (also universal generalization oder universal introduction,[1][2][3] GEN) is a valid inference rule. It states that if has been derived, then can be derived.
Generalization with hypotheses
The full generalization rule allows for hypotheses to the left of the turnstile, but with restrictions. Assume Γ is a set of formulas, a formula, and has been derived. The generalization rule states that can be derived if y is not mentioned in Γ and x 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:
- (Hypothesis)
- (Existential instantiation)
- (Existential instantiation)
- (Faulty universal generalization)
This purports to show that which is an unsound deduction.
Example of a proof
Prove: is derivable from and .
Proof:
Number | 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.