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{{Transformation rules}} | |||
In [[predicate logic]], '''generalization''' (also '''universal generalization''',<ref>Copi and Cohen</ref><ref>Hurley</ref><ref>Moore and Parker</ref> '''GEN''') is a [[validity|valid]] [[rule of inference|inference rule]]. It states that if <math> \vdash P(x) </math> has been derived, then <math> \vdash \forall x \, P(x) </math> can be derived. | |||
==Generalization with hypotheses== | |||
The full generalization rule allows for hypotheses to the left of the [[turnstile (symbol)|turnstile]], but with restrictions. Assume Γ is a set of formulas, φ a formula, and <math>\Gamma \vdash \varphi(y)</math> has been derived. The generalization rule states that <math>\Gamma \vdash \forall x \varphi(x)</math> can be derived if ''y'' is not mentioned in Γ and ''x'' does not occur in φ. | |||
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 w (y \not = w) </math> (Existential instantiation) | |||
#<math>y \not = x</math> (Existential instantiation) | |||
#<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. | |||
==Example of a proof== | |||
'''Prove:''' <math> \forall x \, (P(x) \rightarrow Q(x)) \rightarrow (\forall x \, P(x) \rightarrow \forall x \, Q(x)) </math>. | |||
'''Proof:''' | |||
{| border="1" cellpadding="3" | |||
! style="background:#93D7AE;"| Number | |||
! style="background:#93D7AE;"| Formula | |||
! style="background:#93D7AE;"| Justification | |||
|- | |||
| 1 | |||
| <math> \forall x \, (P(x) \rightarrow Q(x)) </math> | |||
| Hypothesis | |||
|- | |||
| 2 | |||
| <math> \forall x \, P(x) </math> | |||
| Hypothesis | |||
|- | |||
| 3 | |||
| <math> (\forall x \, (P(x) \rightarrow Q(x))) \rightarrow (P(y) \rightarrow Q(y))) </math> | |||
| [[Universal instantiation]] | |||
|- | |||
| 4 | |||
| <math> P(y) \rightarrow Q(y) </math> | |||
| From (1) and (3) by [[Modus ponens]] | |||
|- | |||
| 5 | |||
| <math> (\forall x \, P(x)) \rightarrow P(y) </math> | |||
| [[Universal instantiation]] | |||
|- | |||
| 6 | |||
| <math> P(y) \ </math> | |||
| From (2) and (5) by [[Modus ponens]] | |||
|- | |||
| 7 | |||
| <math> Q(y) \ </math> | |||
| From (6) and (4) by [[Modus ponens]] | |||
|- | |||
| 8 | |||
| <math> \forall x \, Q(x) </math> | |||
| From (7) by Generalization | |||
|- | |||
| 9 | |||
| <math> \forall x \, (P(x) \rightarrow Q(x)), \forall x \, P(x) \vdash \forall x \, Q(x) </math> | |||
| Summary of (1) through (8) | |||
|- | |||
| 10 | |||
| <math> \forall x \, (P(x) \rightarrow Q(x)) \vdash \forall x \, P(x) \rightarrow \forall x \, Q(x) </math> | |||
| From (9) by [[Deduction theorem]] | |||
|- | |||
| 11 | |||
| <math> \vdash \forall x \, (P(x) \rightarrow Q(x)) \rightarrow (\forall x \, P(x) \rightarrow \forall x \, Q(x)) </math> | |||
| 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== | |||
*[[First-order logic]] | |||
*[[Hasty generalization]] | |||
*[[Universal instantiation]] | |||
== References == | |||
{{reflist}} | |||
{{DEFAULTSORT:Generalization (Logic)}} | |||
[[Category:Rules of inference]] | |||
[[Category:Predicate logic]] | |||
Revision as of 22:56, 19 January 2014
In predicate logic, generalization (also universal generalization,[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, φ 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 φ.
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
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.
See also
References
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