Generalization (logic)

Transformation rules
Propositional calculus
Rules of inference Modus ponens Modus tollens Biconditional introduction Biconditional elimination Conjunction introduction Simplification Disjunction introduction Disjunction elimination Disjunctive syllogism Hypothetical syllogism Constructive dilemma Destructive dilemma Absorption Rules of replacement Associativity Commutativity Distributivity Double negation De Morgan's laws Transposition Material implication Exportation Tautology
Predicate logic
Universal generalization Universal instantiation Existential generalization Existential instantiation

In predicate logic, generalization (also universal generalization, GEN) is a valid inference rule. It states that if $\vdash P(x)$ has been derived, then $\vdash \forall x \, P(x)$ can be derived.

Generalization with hypotheses

The full generalization rule allows for hypotheses to the left of the turnstile, but with restrictions. Assume Γ is a set of formulas, φ a formula, and $\Gamma \vdash \varphi(y)$ has been derived. The generalization rule states that $\Gamma \vdash \forall x \varphi(x)$ 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 $\forall x P(x)$ from the hypothesis $P(y)$. Without the second restriction, one could make the following deduction:

1. $\exists z \exists w ( z \not = w)$ (Hypothesis)
2. $\exists w (y \not = w)$ (Existential instantiation)
3. $y \not = x$ (Existential instantiation)
4. $\forall x (x \not = x)$ (Faulty universal generalization)

This purports to show that $\exists z \exists w ( z \not = w) \vdash \forall x (x \not = x),$ which is an unsound deduction.

Example of a proof

Prove: $\forall x \, (P(x) \rightarrow Q(x)) \rightarrow (\forall x \, P(x) \rightarrow \forall x \, Q(x))$.

Proof:

Number	Formula	Justification
1	$\forall x \, (P(x) \rightarrow Q(x))$	Hypothesis
2	$\forall x \, P(x)$	Hypothesis
3	$(\forall x \, (P(x) \rightarrow Q(x))) \rightarrow (P(y) \rightarrow Q(y)))$	Axiom PRED-1
4	$P(y) \rightarrow Q(y)$	From (1) and (3) by Modus Ponens
5	$(\forall x \, P(x)) \rightarrow P(y)$	Axiom PRED-1
6	$P(y) \$	From (2) and (5) by Modus Ponens
7	$Q(y) \$	From (6) and (4) by Modus Ponens
8	$\forall x \, Q(x)$	From (7) by Generalization
9	$\forall x \, (P(x) \rightarrow Q(x)), \forall x \, P(x) \vdash \forall x \, Q(x)$	Summary of (1) through (8)
10	$\forall x \, (P(x) \rightarrow Q(x)) \vdash \forall x \, P(x) \rightarrow \forall x \, Q(x)$	From (9) by Deduction Theorem
11	$\vdash \forall x \, (P(x) \rightarrow Q(x)) \rightarrow (\forall x \, P(x) \rightarrow \forall x \, Q(x))$	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