Rule of inference in predicate logic
Universal generalizationType | Rule of inference |
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Field | Predicate logic |
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Statement | Suppose is true of any arbitrarily selected , then is true of everything. |
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Symbolic statement | , ![{\displaystyle \vdash \!\forall x\,P(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/15b82a59925ed0531a224874f5fa115c7c1e49dd) |
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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.
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:
(Hypothesis)
(Existential instantiation)
(Existential instantiation)
(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).
Prove:
is derivable from
and
.
Proof:
Step
|
Formula
|
Justification
|
1
|
|
Hypothesis
|
2
|
|
Hypothesis
|
3
|
|
From (1) by Universal instantiation
|
4
|
|
From (1) and (3) by Modus ponens
|
5
|
|
From (2) by 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.
- ^ Copi and Cohen
- ^ Hurley
- ^ Moore and Parker