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Is there a difference between 'inconsistent' and 'contrary'? As far as I understand two statements are inconsistent when they can not both be true. Does 'contrary' have the same definition?

As far as I understand, neither term is perfectly synonymous with the other nor with 'contradictory' (where two variables cannot bear the same truth-value, be it true or false)

So do 'inconsistent', 'contrary', and 'contradictory' each have their own meaning, or are two of those terms mutually synonymous?

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Two statements are said to be contradictory if for both statements, the truth of one implies the falsity of the other.

Two statements are said to be contrary if they can both be false, but they cannot both be true.

(I know you used the word "inconsistent" instead of "contradictory", but the latter seems to be the more conventional choice. They're synonymous here though, so take your pick.)

The point you seem to have overlooked about contradictory pairs is that they cannot both be false either.

  • Hi David. I should have more clearly stated my question. I meant to ask whether there is a distinct meaning for each of the terms 'contradictory', 'contrary', and 'inconsistent'. I believe that lexical redundancy is wasteful, so I had hoped that each had its own denotation. – Hal May 14 '13 at 13:59
  • Yes, they all all have distinct meanings, in some instances a few distinct meanings each! (I know I say in my post that inconsistent and contradiction are synonymous but apparently I'm only quasi-correct about that.) I presume you might like me to elaborate a little bit more here... But first, I noticed you tagged Aristotle in your question. Before I continue with an explanation in 1st-Order Logic, I suppose I should ask you if you're asking this question from the perspective of Aristotle's Logic because then I'll have to give a different answer. – David H May 14 '13 at 16:00
  • I think it's best to stick with what's in style. So I think the contemporary definitions would be more helpful. Thank you. – Hal May 14 '13 at 16:43
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Inconsistent, in formal logic, is a property of sets of propositions. For instance, the set {A, ¬A} is an inconsistent set.

On the other hand, contrary is a relationship between two propositions, that is to say the proposition A stands in the "contrary" relation to the proposition ¬A.

A set of propositions is inconsistent just in case, for some two propositions in the set, they are contrary to each other.

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