Is Paraconsistent Negation Really Negation?

Question

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?

Answer 1

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.

Answer 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.