# Is there more than one inconsistent theory?

For a given language, is there more than one inconsistent theory? My intuition told me no, but I'm not sure.

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If by theory you mean "a set of sentences closed under logical deduction", then... – Andrés E. Caicedo May 4 '13 at 22:33
I thought that's what a theory was defined as, yes. I'm sorry if this is obvious. Usually we say things like "this theory is inconsistent", not this is "the" inconsistent theory. – Phdetermined May 4 '13 at 22:38
Typically we discuss theories by discussing a set of axioms for the theory (the theory is then the deductive closure of the set of axioms). Of course, the same theory can be described by many different sets of axioms. It is common to identify a theory with a set of axioms, in which case when we say that a theory is inconsistent, we typically mean that the theory generated by a given set of axioms is the inconsistent theory (or, equivalently, that the set of axioms is inconsistent). – Andrés E. Caicedo May 4 '13 at 23:04
@AsafKaragila I would not say that either usage is more typical than the other. I've frequently encountered both. – Andrés E. Caicedo May 4 '13 at 23:05

What do you mean by theory? And what makes for inconsistency?

If by a theory couched in the language L you just mean a set of L-sentences (a common logician's usage), then of course there can be distinct inconsistent theories whatever your definition of inconsistency.

If you require a theory to be a set of sentences closed under consequence, inconsistency is a matter of entailing a contradiction, and the consequence relation allows us to infer anything from a contradiction, then there is only one inconsistent theory couched in the language L, the set of all L sentences.

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I was unaware that was the common logician's usage of the word theory. But your last comment was helpful since I meant a set of sentences closed under consequence. – Phdetermined May 4 '13 at 22:51

It depends on which logical framework you're working in. If you're working in classical logic, then there's only one inconsistent theory (closed under consequence), viz. the theory with every sentence of the language. But there are a number of logical systems which reject the principle of explosion, i.e. the inference $A,\neg A \vdash B$ for any $A$ and $B$. Such logics are called paraconsistent logics, and they allow for many distinct inconsistent theories (closed under consequence), apart from the trivial one (i.e. the set of all sentences of your language).

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Thank you for your answer! I guess I need some non-classical logic experience. – Phdetermined May 4 '13 at 23:11
Graham Priest is perhaps the most famous advocate of paraconsistent logic. He has a book called An Introduction to Non-Classical Logic that surveys a whole host of non-classical logics if you're interested. – Alex Kocurek May 5 '13 at 2:30

One might indeed have plenty of reasons for wanting to distinguish between varieties of inconsistency. As it has already been mentioned by another colleague, a paraconsistent logic $L$, in particular, may help in distinguishing between what one might call simple inconsistency, or contradictoriness ---namely, the presence of a contradiction in a theory of $L$--- and absolute inconsistency, also known as overcompleteness or triviality ---namely, the presence of every $L$-sentence in a theory of $L$.

Another well-known variety of inconsistency (let's call it semi-inconsistency) is one that allows for the derivation of any sentence $B$ from any sentence $A$, while at the same time not allowing for the derivation of an arbitrary $B$ from the empty set of premisses ---that is, for such a semi-inconsistent logic $L$ the empty set is not trivial, but the closure of any other collection of sentences is trivial (a necessary condition for this to be possible, under the usual interpretation of consequence, is that the logic $L$ should contain no sentence that behaves as bottom). This variety of inconsistency may appear, for instance, when the logics turn out to be characterized by incoherent collections of inference rules ---containing for instance some connective such as Prior's tonk.

Finally, one variety of partial inconsistency which is often overlooked is one in which the contradictoriness of your theory allows you to derive all sentences of a certain format, but not necessarily all sentences of $L$. For instance, in minimal intuitionistic logic (Kolmogorov-Johansson's logic), any theory containing $A$ and $\neg A$ must also contain $\neg B$, for arbitrary $B$, while some such inconsistent theories may still be non-trivial, that is, they still do not contain every $B$. In this logic the principle of explosion fails, but partial explosion holds good: $A,\neg A\vdash \neg B$. (Paraconsistent logics that avoid this phenomenon are known as boldly paraconsistent.)

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