Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

We have some sound deduction system (every provable sentence is true), which has property of principle of explosion and some theory T described in that system. Lets assume that theory T is inconsistent($P$ and $\neg P$ both belongs to the theory). Because of principle of explosion you can prove some contradictions $Y$ and $\lnot Y$ which both will be true thanks to soundness. That I suppose shows that there is no possibility that you could create inconsistent theory but it is not true because you can always create theory such as $\{P, \neg P\}$ and prove some other contradiction.

What is wrong with this reasoning?

share|improve this question
6  
I have been to many concerts where the sound system was inconsistent. –  Gerry Myerson Feb 19 '12 at 11:52

1 Answer 1

up vote 7 down vote accepted

Nono! If the system is sound and you have a theory $T$, then everything (structure, or whatever the context is that you are talking about) that satisfies $T$ will also satisfy everything that the system can deduce from $T$. But if $T$ itself is inconsistent, there is no structure (or whatever your context is) satisfying $T$.
So it is no problem that there is no structure satisfying the things you can deduce from $T$ in your system.

This is like with the truth table of the implication: If the premise $\phi$ is always false, then the implication $\phi\Rightarrow\psi$ is true.

share|improve this answer
    
Does it mean that I can not create model(which I think you refered as structure) for inconsistent set of axioms? –  Trismegistos Feb 19 '12 at 12:15
    
@Trismegistos: This is what soundness says: Every satisfiable theory is consistent. –  Apostolos Feb 19 '12 at 12:24
    
@Trismegistos: When I said structure, I was talking about first order logic. You are right, though, model is the better and more general term in this context. –  Stefan Geschke Feb 19 '12 at 12:33

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.