Rules of inference for quantified statements

3.1

?xP(x)

Universal instantiation: ---------

P(c)

P(c)

Existential generalization: --------

?xP(x)

Hypotheses (axioms):

• All Canny integers are non-Slotty

• 3 is Canny

• All Bumpy integers are Slotty

Prove that

• There is an integer that is not Bumpy

?x(C(x) ? ? S(x))

C(3)

?x(B(x) ? S(x))

?x ? B(x)

1. ?x(C(x) ? ? S(x)) Axiom

2. C(3) ? ? S(3) From 1 by Universal instantiation

3. C(3) Axiom

4. ? S(3) From 2,3 by Modus ponens

5. ?x(B(x) ? S(x)) Axiom

6. B(3) ? S(3) From 5 by Universal instantiation

7. ? B(3) From 4,6 by Modus tollens

8. ?x ? B(x) From 7 by Existential generalization

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