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'Consistent' and 'inconsistent'

A set of claims is consistent if and only if it is possible for all of the claims in the set to be true together. A set of claims is inconsistent if and only if it is not possible for all of the claims in the set to be true together.

Identifying when sets of claims are consistent and when they’re inconsistent is obviously important in working out what to believe. If we find out that some of the claims we believe are inconsistent, then something’s got to give. It doesn’t make sense to go on believing them all, because there’s no way they could all be true. We at least want the things we believe to be consistent.

We can test for consistency and inconsistency based on propositional logical form (PL consistent/inconsistent) using truth-tables. We construct a truth-table for the sentences in the set and then check to see if there is at least one row on which all of the sentences are true. If there is, the set is PL consistent; if there is not, it is PL inconsistent.

Look at the truth-table for the sentences ‘(P ⋁ ~Q)’ and ‘(~P ⋁ Q)’ (Fig. 1).

Consistency truth-table

Figure 1. Example of a truth-table showing consistency

The sentences are PL consistent. There is at least one row on which they are both true. In fact, there are two: they’re both true when ‘P’ and ‘Q’ are both true (first row), and they’re also both true when ‘P’ and ‘Q’ are both false (final row).

Now look at the truth-table for the sentences ‘~(P ⋁ Q)’, ‘~P’, ‘Q’ (Fig. 2).

Inconsistency truth-table

Figure 2. Example of a truth-table showing inconsistency

We can see from this that these three sentences are (PL) inconsistent. There is no row on which (no kind of possible situation in which) all three sentences are true.

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This article is from the free online course:

Logic: The Language of Truth

University of York