nForum - Discussion Feed (inconsistency) 2022-01-28T23:56:16-05:00 https://nforum.ncatlab.org/ Lussumo Vanilla & Feed Publisher TobyBartels comments on "inconsistency" (35060) https://nforum.ncatlab.org/discussion/4271/?Focus=35060#Comment_35060 2012-09-21T17:13:52-04:00 2022-01-28T23:56:16-05:00 TobyBartels https://nforum.ncatlab.org/account/7/ 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: ϕ\phi is ...

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.

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Mike Shulman comments on "inconsistency" (35012) https://nforum.ncatlab.org/discussion/4271/?Focus=35012#Comment_35012 2012-09-20T14:42:42-04:00 2022-01-28T23:56:16-05:00 Mike Shulman https://nforum.ncatlab.org/account/3/ Thanks; I added some remarks about paraconsistent logic.