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Let a logic be paraconsistent, if $\phi \wedge \neg \phi \not \models \psi$ for some $\phi, \psi$ (where $\models$ is the logic's consequence relation). There are different ways to prevent a contradiction from entailing everything, the most common being to define a notion of model and a notion of truth in a model so that some contradiction is true in a model, although some formula is not true in that same model.

For instance, let a relational model (for an ordinary propositional language $L$), $r$, be a subset of $At \times \lbrace 0,1\rbrace$, where $At$ is the set of atoms. $r$ can be extended to $L$ as follows: $\neg \phi~ r~ 1 \Leftrightarrow \phi~ r~ 0$; $\neg \phi~ r~ 0 \Leftrightarrow \phi~ r~ 1$; $\phi \wedge \psi ~ r~ 1 \Leftrightarrow \phi~r~1~\text{and}~ \psi ~r~1$; $\phi \wedge \psi ~ r~ 0 \Leftrightarrow \phi~r~0~\text{or}~ \psi ~r~0$. Finally let logical consequence be truth preservation under all $r$. Let $r$ be a model such that $p ~r~ 1, p~ r~ 0, q~ r~ 0$. One easily sees that $p \wedge \neg p ~r~1, q ~r~0$ (this is one version of FDE, a fragment of relevant logic).

Now, here's a problem for this kind of paraconsistent logic. Intuitively a sentence and its negation are contradictories, i.e. in every model at least and at most one of them is true. But the above countermodel shows that $p$ and $\neg p$ are true in one and the same model. But then paraconsistent negation is not negation. This argument seems somehow flawed, but where exactly is the flaw?

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See Paraconsisten Logic. Of course, negation will "change its meaning"; see 3.6 Many-Valued Logics :"Perhaps the simplest way of generating a paraconsistent logic [...], is to use a many-valued logic. Classically, there are exactly two truth values. The many-valued approach is to drop this classical assumption and allow more than two truth values. The simplest strategy is to use three truth values: true (only), false (only) and both (true and false) for the evaluations of formulas. So, the negation is no more "classical". –  Mauro ALLEGRANZA Feb 4 at 16:02
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Quine argued in his Philosophy of Logic that "deviant logics" (his words) represented a change of subject; that things like paraconsistent logic, or even certain forms of intuitionistic logic, aren't actually alethic. I'm not sure how far I go along with Quine there, but you would not be the first person to think that non-classical "negation" isn't really negation. –  Malice Vidrine Feb 4 at 20:22
    
But is hard to think to a "definitive" answer to the question : "what is really negation ?". In classical logic, negation is defined truth-functionally. Since Aristotle, LEM and LNC are basic laws of being and thinking. But since him (and before) there were discussion about this topic... –  Mauro ALLEGRANZA Feb 4 at 20:27
    
@Mauro ALLEGRANZA Of course $\neg$ somehow changes its meaning in FDE and allied systems; otherwise it would be unclear why there are counter models to explosion (contradictions entail everything). The point of my post is: What is it about this new (non-classical) meaning of negation that we may still regard it as a kind negation? Part of this point are the questions: What restrictions must an operator satisfy in order to count as negation? And which of these restrictions are compatible with the failure of explosion? I think there should be a definitive answer to these questions. –  Jon Feb 5 at 1:40

2 Answers 2

The standard answer from the literature would seem to be that only classical negation is a contradictory-forming operator. Paraconsistent negations are at most subcontrary-forming operators (they allow for some formula $A$ to be true together with its negation, while possibly forbidding $A$ and its negation to be both false). Dually, paracomplete negations are at most contrary-forming operators (they allow for some formula $A$ to be false together with its negation, while possibly forbidding $A$ and its negation to be both true) --- one such contrary-forming operator is the negation of intuitionistic logic. Is intuitionistic negation really negation? Is any sub-classical negation really negation?

For a discussion of this topic, you might want to check the paper "Paraconsistent Logics?", by Hartley Slater, and the response "Paraconsistent Logic!", by Béziau.

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The most basic definition of negation is:

A one place connective m is a negative modality if and only if the rule $$ \frac {A \vdash B}{mB \vdash mA} $$ is valid for all formulae A and B. Characteristic examples of negative modalities are the connectives we use when we deny something, or when we express the fact tthat we want to reject something, or when we contradict someone. If A entails B then excluding B entails excluding A

(Restall, "Introduction to substructural logic" (1999), page 59,60 )

In other words(mine): The most basic definition of negation is any operation N so that "If P entails Q then N(Q) entails N(P)"

Sub-minimal logic is the weakest logic with negation $ (P \to Q) \to (\lnot Q \to \lnot P) $ is the only axiom for negation in this logic.

Sub-minimal logic is very weak it has hardly has any theorems with negation, even $ (\lnot P \to \ P) \to P $ (Consequentia mirabilis) and $ ((P \to Q) \to ((P \to \lnot Q) \to \lnot P ) $ are not theorems of this logic.

Minimal logic (Johansson ) adds $ (P \to Q) \to ((P \to \lnot Q) \to \lnot P ) $ as axiom and that makes this logic a bit stronger is minimal logic

Other logics add other axioms but that is another story

Paraconsistent logic is more a catagory of logics than one specific logic, including the 2 logics mentioned above, so I cannot help you any further.

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I don't think this half of contraposition defines negation. Rather it is a restriction on the interpretation of the conditional $\rightarrow$, that a logic might fulfill or not. Indeed there a are conditional logics , such as Stalnaker's, where this half of contraposition fails. But surely this doesn't show that Stalnaker' negation operator isn't really negation. My post focuses on a certain subgroup of paraconsistent logics: Those that invalidate explosion by providing models where both a formula and its negation are true. Their semantic differences are unimportant for my point. –  Jon Feb 5 at 1:25
    
@Jon I am not familiar with Stalnaker' negation, can you give me its axioms or rules? I changed my answer a bit maybe consequently.org/writing/topics/#negation can help you further if has links to lots of papers (but all by the same person) maybe one of them can help you further, not sure about what you mean by semantic differences , I only do syntactics –  Willemien Feb 5 at 11:06
    
Oh, I see, what you mean is a meta-language inference rule, not an object-language formula. This might also hold in Stalnaker's logic. I remember the quoted section from Restall's book, where negation takes center stage; thanks for the reminder. I think he even wrote a paper on negation in Routley's star semantics for relevant logic. I'll have another look at it. Stalnaker's negation is classical negation indexed to possible worlds. By semantic differences I mean the fact that different parac. logics use different notions of model, truth, and consequence. –  Jon Feb 5 at 17:52
    
Saying this is 'the most basic definition of negation' is really arguable. The literature on paraconsistent logics and the literature on many-valued logics both abound with examples of negations failing such global version of contraposition, and with good reason. However, paraconsistent negations with modal semantics (such as the ones that Restall is interested upon), may well satisfy this property. In fact, any normal modal logic may be rewritten in a language that extends the classical language by the addition of a paraconsistent negation satisfying global contraposition. –  J Marcos Feb 8 at 19:43
    
@JMarcos can you give an example (preferably a multi valued one) ps I don't think that Post's cyclic "negation " counts as negation because it is increasing –  Willemien Feb 9 at 9:11

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