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Let $\phi(x_1,...,x_n)$ be a statement about an equality of two expressions having $x_1,...,x_n$ respectively.

If there is no $(x_1,...,x_n)$ such that $\phi(x_1,...,x_n)$ is true, we call this equation $\phi$ inconsistent. I don't know why specially 'consistency and inconsistency' is named for equations. Literally, 'An equation has no solution' makes more sense than 'an equation is inconsistent' to me. If $\phi$ is inconsistent, does this mean something about $Con(\phi)$?

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Logical inconsistency means something more like "you can derive a contradiction from the statement." For example, if I require that $x = 3$ and that $x = 5$, that's logically inconsistent because it implies that $3 = 5$; I'm making inconsistent demands on the value of $x$.

In suitable contexts, it is a theorem, but a nontrivial theorem, that logical inconsistency is equivalent to not having a solution. For example, in first-order logic, the completeness theorem guarantees that a collection of first-order axioms is logically inconsistent iff it doesn't have a model. The Nullstellensatz in algebraic geometry can also be viewed in this light, as can a simpler statement in linear algebra.

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One has to be a little bit careful about what one means by ‘logical inconsistency’ in the context of algebraic theories. For example, when a system of polynomials has no solution, the system is inconsistent – in the sense that every other equation (including $0 = 1$) can be derived from it. (Of course, you could say that $0 \ne 1$ is part of the logical axioms and so derive $\bot$, but I prefer to work in the positive fragment.) – Zhen Lin Mar 13 '13 at 18:41

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