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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

Brutius
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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.

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Mauro ALLEGRANZA
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  • thanks for the quick response. Yes I do know what a tautology is. i) and ii) seem easy, even trivial in many cases. iii) Is where I am having the most trouble I guess. I can easily prove this system is sound, but am having problems with the completeness part. I also have other problems where I must prove that the system is complete which seems equally as difficult. – Brutius Mar 06 '14 at 16:12
  • @user114620 - I think that a tautology that you cannot prove is $\lnot \lnot p \rightarrow p$. The problem is to find a way to prove that you cannot prove it. The usual technique to show independence result is through many-valued truth-tables. See Elliott Mendelson, Introduction to Mathematical Logic (4th ed - 1997), page 43. – Mauro ALLEGRANZA Mar 06 '14 at 16:19