“Universal generalization”的意思、由来-开放百科全书

词条

Universal generalization

释义

Generalization with hypotheses
Example of a proof
See also
References

In predicate logic, generalization (also universal generalization or 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.

References

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

2 : Rules of inference|Predicate logic

随便看

开放百科全书收录14589846条英语、德语、日语等多语种百科知识，基本涵盖了大多数领域的百科知识，是一部内容自由、开放的电子版国际百科全书。

Generalization with hypotheses

Example of a proof

See also

References