Logic soundness and completeness

Question

I have to do a number of similar type questions, but I am having trouble grasping the general concepts around soundness and completeness. I have read up on the general definitions and think I understand them, but when asked to show something like the following I come up blank. Thanks for any ideas/help.

Show that the following propositional proof system, PCa, is sound but not complete.

PCa

Connectives (and constants) ∨, ∧, →, ↔, ¬, 0, 1

Axioms All tautologies of the form F→F

Rules of inference Hypothetical Syllogism

Answer 1

See the following post about Sound and complete

About your problem, I assume that you know the definition of tautology.

So you must show that :

(i) your axiom $F \rightarrow F$ is a tautology

and that :

(ii) your rule of inference preserve truth, i.e. if the premise is true also the conclusion is.

These two steps amount at a proof of the soundness of your calculus.

About the completeness side, you must show that

(iii) your calculus is not complete.

This amount to finding a tautology that is not derivable (i.e.provable) from the axiom using only your rule of inference.