• Source: Universal generalization

In predicate logic, generalization (also universal generalization, universal introduction, GEN, UG) is a valid inference rule. It states that if



⊢

P
(
x
)


{\displaystyle \vdash \!P(x)}

has been derived, then



⊢

∀
x

P
(
x
)


{\displaystyle \vdash \!\forall x\,P(x)}

can be derived.




Generalization with hypotheses


The full generalization rule allows for hypotheses to the left of the turnstile, but with restrictions. Assume



Γ


{\displaystyle \Gamma }

is a set of formulas,



φ


{\displaystyle \varphi }

a formula, and



Γ
⊢
φ
(
y
)


{\displaystyle \Gamma \vdash \varphi (y)}

has been derived. The generalization rule states that



Γ
⊢
∀
x

φ
(
x
)


{\displaystyle \Gamma \vdash \forall x\,\varphi (x)}

can be derived if



y


{\displaystyle y}

is not mentioned in



Γ


{\displaystyle \Gamma }

and



x


{\displaystyle x}

does not occur in



φ


{\displaystyle \varphi }

.
These restrictions are necessary for soundness. Without the first restriction, one could conclude



∀
x
P
(
x
)


{\displaystyle \forall xP(x)}

from the hypothesis



P
(
y
)


{\displaystyle P(y)}

. Without the second restriction, one could make the following deduction:




∃
z

∃
w

(
z
≠
w
)


{\displaystyle \exists z\,\exists w\,(z\not =w)}

(Hypothesis)




∃
w

(
y
≠
w
)


{\displaystyle \exists w\,(y\not =w)}

(Existential instantiation)




y
≠
x


{\displaystyle y\not =x}

(Existential instantiation)




∀
x

(
x
≠
x
)


{\displaystyle \forall x\,(x\not =x)}

(Faulty universal generalization)
This purports to show that



∃
z

∃
w

(
z
≠
w
)
⊢
∀
x

(
x
≠
x
)
,


{\displaystyle \exists z\,\exists w\,(z\not =w)\vdash \forall x\,(x\not =x),}

which is an unsound deduction. Note that



Γ
⊢
∀
y

φ
(
y
)


{\displaystyle \Gamma \vdash \forall y\,\varphi (y)}

is permissible if



y


{\displaystyle y}

is not mentioned in



Γ


{\displaystyle \Gamma }

(the second restriction need not apply, as the semantic structure of



φ
(
y
)


{\displaystyle \varphi (y)}

is not being changed by the substitution of any variables).




Example of a proof


Prove:



∀
x

(
P
(
x
)
→
Q
(
x
)
)
→
(
∀
x

P
(
x
)
→
∀
x

Q
(
x
)
)


{\displaystyle \forall x\,(P(x)\rightarrow Q(x))\rightarrow (\forall x\,P(x)\rightarrow \forall x\,Q(x))}

is derivable from



∀
x

(
P
(
x
)
→
Q
(
x
)
)


{\displaystyle \forall x\,(P(x)\rightarrow Q(x))}

and



∀
x

P
(
x
)


{\displaystyle \forall x\,P(x)}

.
Proof:

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

Kata Kunci Pencarian:

• Kuantifikasi semesta
• Generalisasi
• Fungsi Kempner
• Teori representasi
• Domestikasi hewan
• Asal-usul anjing domestik
• Universal generalization
• Universal instantiation
• Universal quantification
• Generalization (disambiguation)
• Modus tollens
• List of rules of inference
• Universal law of generalization
• Subset
• High school in the United States
• Rule of inference