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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeSep 20th 2012
• (edited Sep 20th 2012)

I have created stubs for inconsistency and contradiction

• CommentRowNumber2.
• CommentAuthorMike Shulman
• CommentTimeSep 20th 2012

Thanks; I added some remarks about paraconsistent logic.

• CommentRowNumber3.
• CommentAuthorTobyBartels
• CommentTimeSep 21st 2012

Although the language is not always used this way, I find it helpful (when considering paraconsistent logics) to strictly distinguish a contradiction from an inconsistency as follows:

• $\phi$ is contradictory if, for some $\psi$, $\phi \vdash \psi$ and $\phi \vdash \neg{\psi}$ (equivalently, $\phi \vdash \psi \wedge \neg{\psi}$ if $\wedge$ obeys the usual rules);
• $\phi$ is inconsistent if, for every $\chi$, $\phi \vdash \chi$ (equivalently, $\phi \vdash \bot$ if $\bot$ obeys the usual rules);
• $\phi$ is paraconsistent if it is contradictory but not inconsistent.

(Since the definition of contradiction depends on the operator $\neg$, one might also say a $\neg$-contradiction in case of multiple candidates for negation, as in linear logic.)

One can generalise this to speak of more general contexts or even entire logics as being contradictory, inconsistent, or paraconsistent; then we recover the usual meaning of when a logic is paraconsistent.