Showing a Set is Inconsistent In Logic

Question

Let $\Phi = \left \{ \alpha, \beta, \gamma \right \}$ be a set of three well-formed formulas. To show $\Phi$ is inconsistent, should I use deduction to show that $\Phi \vdash \phi$ for all $\phi \in \Phi$? If so, how would that look? Do I have to use all elements in $\Phi$ to deduce every $\phi \in \Phi$? If so, do I use axioms that include all three formulas or what? I apologize for my elementary understanding!

Answer 1

Actual answer

First of all, no, there's no obligation to use everything in $\Phi$. The set $\Phi$ is the set of sentences you're allowed to use in a proof. Adding stuff to $\Phi$ only makes it easier to prove things.

You're also mixing up $\Phi$ and the set of all sentences in the language. Obviously $\Phi$ proves every sentence in $\Phi$. What you want to talk about is the sentences in general which $\Phi$ proves.

Now, on to the issue of inconsistency. There are two notions of inconsistency of a set of sentences $\Phi$:

• $\Phi$ proves everything.

• For some $p$, $\Phi$ proves both $p$ and $\neg p$.

Conveniently, in classical propositional logic these are equivalent: clearly if $\Phi$ is inconsistent in the first sense, it's inconsistent in the second sense, and conversely if $\Phi$ proves both $p$ and $\neg p$ we can use proof by contradiction to prove whatever we want from $\Phi$: "Suppose $\neg q$. Then ... we conclude $p$ and $\neg p$. So since we got a contradiction from $\neg q$, we have $q$."

So all you need to do is find a single sentence such that $\Phi$ proves both the sentence and its negation.

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Irrelevant but interesting note

Now note that I said "classical logic" above. There are other logics out there, and in some of them it is not the case that proving a contradiction means proving everything. For such logics we do need to distinguish between the notions of inconsistency given above: generally, the second (weaker) one is called "inconsistent" while the first is called "trivial." In these logics, and unlike classical logic, inconsistent theories may be interesting; see e.g. here.