How do you prove that given propositions are inconsistent?

Question

I understand that to prove they are consistent, simply find values for each propositions that makes their truth value all true.

However, to prove they are inconsistent, is the only way to test all values in all propositions and show that they are always false?

Answer 1

Using a truth table, as you say, we must check they do not have a single line where they are all true.

However, with a long truth table it would be impractical and it is not going to give us much insight. Consider this definition:

The sentences $\mathcal{A_1, A_2, ..., A_n}$ are provably inconsistent iff a contradiction can be proved from them.

So, if we take $\mathcal A_1, A_2, ..., A_n$ as premises and reach a contradiction using valid rules of inferences, we proved those sentences are provably inconsistent.

For example, if we want to show $A \land B, \neg A$ are provably inconsistent, we must provide a proof of $A \land B, \neg A \vdash \bot$. In words, $A \land B, \neg A$ entail $\bot$ (contradiction).

$ \def\fitch#1#2{\quad\begin{array}{|l}#1\\\hline#2\end{array}} \def\Ae#1{\qquad\mathbf{\forall E} \: #1 \\} \def\Ai#1{\qquad\mathbf{\forall I} \: #1 \\} \def\Ee#1{\qquad\mathbf{\exists E} \: #1 \\} \def\Ei#1{\qquad\mathbf{\exists I} \: #1 \\} \def\R#1{\qquad\mathbf{R} \: #1 \\} \def\ci#1{\qquad\mathbf{\land I} \: #1 \\} \def\ce#1{\qquad\mathbf{\land E} \: #1 \\} \def\oi#1{\qquad\mathbf{\lor I} \: #1 \\} \def\oe#1{\qquad\mathbf{\lor E} \: #1 \\} \def\ii#1{\qquad\mathbf{\to I} \: #1 \\} \def\ie#1{\qquad\mathbf{\to E} \: #1 \\} \def\be#1{\qquad\mathbf{\leftrightarrow E} \: #1 \\} \def\bi#1{\qquad\mathbf{\leftrightarrow I} \: #1 \\} \def\qi#1{\qquad\mathbf{=I}\\} \def\qe#1{\qquad\mathbf{=E} \: #1 \\} \def\ne#1{\qquad\mathbf{\neg E} \: #1 \\} \def\ni#1{\qquad\mathbf{\neg I} \: #1 \\} \def\IP#1{\qquad\mathbf{IP} \: #1 \\} \def\x#1{\qquad\mathbf{X} \: #1 \\} \def\DNE#1{\qquad\mathbf{DNE} \: #1 \\} $

$ \fitch{1.\, A \land B\\2\, \neg A}{ 3.\,A \ce{1} 4.\, \bot \ne{2,3} } $

Resource: "forall x: Calgary. An Introduction to Formal Logic".

Answer 2

If the set contains two sentences that are direct negations each other, i.e. two statements of the form $A$ and $\neg A$, then this is an immediate inconsistency.

If there are no immediate negations, you can try some equivalence transformations. For example the set $\{p \land q, \neg p \lor \neg q\}$ is inconsistent because $\neg p \lor \neg q \equiv \neg(p \land q)$ by DeMorgan's laws.

Other contradictions may be less obvious, e.g. the set $\{p \to q, p, \neg q\}$ is inconsistent because $\{p \to q, p\}$ logically entails $q$, in contradiction to $\neg q$. To find these sorts of inconsistencies, you either have to "think through", draw a truth table, or perform a syntactic derivation. The tableau calculus and resolution are particularly suited for finding contradictions; look these up if you're interested.

Note that as soon as you found one contradiction, this is proof enough and you no longer have to check through the other sentences in the set.