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# consistent

If $T$ is a theory of $\mathcal{L}$ then it is consistent iff there is some model $\mathcal{M}$ of $\mathcal{L}$ such that $\mathcal{M}\vDash T$. If a theory is not consistent then it is inconsistent.

A slightly different definition is sometimes used, that $T$ is consistent iff $T\not\vdash\bot$ (that is, as long as it does not prove a contradiction). As long as the proof calculus used is sound and complete, these two definitions are equivalent.

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inconsistent

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## Mathematics Subject Classification

03B99*no label found*

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## Comments

## Consistent definitions?

This definition actually disagrees with the definition in the First Order Logics entry, which defines a set of formulae to be consistent if they don't prove a contradiction.

This is one of the many examples of the lack of standardized terminology within logic. Maybe a note should be added to this entry to the effect that there is more than one definition.