Logic Problems

(a) Verify that the propositional formula

(P IMPLIES Q) OR (Q IMPLIES P)

is valid.

(b) The valid formula of part (a) leads to sound proof method: to prove that an implication is true, just prove that its converse is false. For example, from elementary calculus we know that the assertion

If a function is continuous, then it is differentiable

is false. This allows us to reach at the correct conclusion that its converse,

If a function is differentiable, then it is continuous

is true, as indeed it is.

But wait a minute! The implication

If a function is differentiable, then it is not continuous

is completely false. So we could conclude that its converse

If a function is not continuous, then it is differentiable,

should be true, but in fact the converse is also completely false. So something has gone wrong here. Explain what.