Book Introduction to Logic. Patrick Suppes-Section 2.1-Excercise 4

Question

Anybody can help me with the solution of this exercise?

4. Construct a (non valid) rule of inference which by itself will satisfy Criterion II but violates Criterion I?

Thank you in advance. Diego.

Answer 1

Patrick Suppes provides an example of an inference rule in section 2.1 that satisfies the soundness Criterion I but does not satisfy the completeness Criterion II:

From any sentence P we may infer P.

If the antecedent of the conditional, P, is true, then the consequent, P, is also true and so the rule is sound. However, it does not allow us by itself to derive every valid conclusion. For example, it does not allow us to derive P v Q. We would also need another rule for disjunction introduction.

The exercise asks us to do the something similar:

Construct a (non valid) rule of inference which by itself will satisfy Criterion II but violates Criterion I?

We might try this:

From any sentence P we may infer Q.

This would derive all conclusions which logically follow, but it would also derive those which do not logically follow based on a truth table. So this rule would be complete but not sound.

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Patrick Suppes. Introduction to Logic. (1957) Van Nostrand Rheinhold.