Universal generalization: Difference between revisions
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==Generalization with hypotheses== |
==Generalization with hypotheses== |
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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 \phi(y)</math> has been derived. The generalization rule states that <math>\Gamma \vdash \forall x \phi(x)</math> can be derived if '' |
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 \phi(y)</math> has been derived. The generalization rule states that <math>\Gamma \vdash \forall x \phi(x)</math> can be derived if ''y'' is not mentioned in Γ and ''x'' does not occur in φ. |
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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: |
Revision as of 17:39, 29 January 2011
Template:Rules of inference In mathematical logic, generalization (also universal generalization, GEN) is an inference rule of predicate calculus. 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
Prove: .
Proof:
Number | Formula | Justification |
---|---|---|
1 | Hypothesis | |
2 | Hypothesis | |
3 | Axiom PRED-1 | |
4 | From (1) and (3) by Modus Ponens | |
5 | Axiom PRED-1 | |
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, the deduction theorem was applicable in steps 10 and 11 because the formulas being moved have no free variables.
See also
- Categorical imperative redirected from "Generalization in Ethics"
- Derivative (generalizations)
- First-order logic
- Hasty generalization
- Generalization error
- Generalizations of Fibonacci numbers
- Generalizations of Pauli matrices
- Universal instantiation