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a characterisation of a weak* connexivity in some* abelian logics

december 5th, 2025

in standard logic, we have double-negation elimination, of which one of the formulations is $\overline{\vdash((\alpha\to\bot)\to\bot)\to\alpha}.$ abelian logics are logics that validate the more general axiom of relativity, namely $\overline{\vdash((\alpha\to\beta)\to\beta)\to\alpha}.$ in that vein, one can generate an abelian logic $\textsf L$ with value-functional semantics, where the set of values forms some kind of abelian group $\langle V_{\textsf L},+\rangle$ and in which :

$0_{\textsf L}\in V_{\textsf L}^+$
$(\alpha\to\beta)^{\textsf L}=\beta^{\textsf L}-\alpha^{\textsf L},$
$(\neg\alpha)^{\textsf L}=\textrm-\alpha^{\textsf L}.$

note that $0_{\textsf L}$ is necessarily a glut, as $\textrm-0_{\textsf L}=0_{\textsf L}\in V_{\textsf L}^+.$ given abelian logics tend to generalise negation as well, with $\neg_BA:=A\to B,$ having $\neg A$ be a case where $B$ is a glut, thus of which the negation is (also) designated in particular, like $0_{\textsf L},$ technically makes sense. (estrada-gonzález, 2020)

a connexive logic, on the other hand, is generally defined as a logic which validates the following :

aristotle's theses : $\overline{\vdash\neg(\alpha\to\neg\alpha)},$ and $\overline{\vdash\neg(\neg\alpha\to\alpha)}.$
boethius' theses : $\overline{\vdash(\alpha\to\beta)\to\neg(\alpha\to\neg\beta)},$ and $\overline{\vdash(\alpha\to\neg\beta)\to\neg(\alpha\to\beta)}.$
non-symmetry of implication : for some $a,b$ wff, $\not\vdash(a\to b)\to(b\to a)$.

however, on top of the way we proposed to construct some abelian logics earlier, there's an incompatibility that quickly arises : \begin{equation*}\begin{split} (\neg(\alpha\to\textrm-\alpha))^{\textsf L} &=\textrm-(\textrm-\alpha^{\textsf L}-\alpha^{\textsf L})\\ &=\textrm2\alpha^{\textsf L} \end{split}\end{equation*} therefore, $2V_{\textsf L}\subseteq V_{\textsf L}^+.$ however, \begin{equation*}\begin{split} ((\alpha\to\beta)\to(\beta\to\alpha))^{\textsf L} &=\alpha^{\textsf L}-\beta^{\textsf L}-(\beta^{\textsf L}-\alpha^{\textsf L})\\ &=2\alpha^{\textsf L}-2\beta^{\textsf L}\\ &=2(\alpha^{\textsf L}-\beta^{\textsf L})\\ &\in2V_{\textsf L}\subseteq V_{\textsf L}^+ \end{split}\end{equation*} if we hope to talk about connexive kinds of these kinds of abelian logics, we may need to weaken that last condition as such, thus describing a form of weak* connexive logic :

aristotle's theses : $\overline{\vdash\neg(\alpha\to\neg\alpha)},$ and $\overline{\vdash\neg(\neg\alpha\to\alpha)}.$
boethius' theses : $\overline{\vdash(\alpha\to\beta)\to\neg(\alpha\to\neg\beta)},$ and $\overline{\vdash(\alpha\to\neg\beta)\to\neg(\alpha\to\beta)}.$
weakened non-symmetry of implication : for some $a,b$ wff, $\vdash a\to b$ and $\not\vdash b\to a.$

characterisation theorem :

for all abelian group $\langle V_{\textsf L},+\rangle$ in which :

$0_{\textsf L}\in V_{\textsf L}^+$
$(\alpha\to\beta)^{\textsf L}=\beta^{\textsf L}-\alpha^{\textsf L},$
$(\neg\alpha)^{\textsf L}=\textrm-\alpha^{\textsf L},$
there is an equivalence between the following items :

$\textsf L$ is weakly* connexive, in the sense that it verifies :

aristotle's theses : $\overline{\vdash\neg(\alpha\to\neg\alpha)},$ and $\overline{\vdash\neg(\neg\alpha\to\alpha)}.$
boethius' theses : $\overline{\vdash(\alpha\to\beta)\to\neg(\alpha\to\neg\beta)},$ and $\overline{\vdash(\alpha\to\neg\beta)\to\neg(\alpha\to\beta)}.$
weakened non-symmetry of implication : for some $a,b$ wff, $\vdash a\to b$ and $\not\vdash b\to a.$

$2V_{\textsf L}\subsetneq V_{\textsf L}^+$ and $\textrm-g\not\in V_{\textsf L}^+$ for at least some $g\in V_{\textsf L}^+.$

proof.

$\boxed\Longrightarrow$ assume $\textsf L$ to be weakly* connexive. let's first show that $2V_{\textsf L}\subseteq V_{\textsf L}^+:$ \begin{equation*}\begin{split} (\neg(\alpha\to\neg\alpha))^{\textsf L} &=\textrm-(\textrm-\alpha^{\textsf L}-\alpha^{\textsf L})\\ &=\textrm2\alpha^{\textsf L}\\ &\in2V_{\textsf L} \end{split}\end{equation*} this shows the desired result. now, let's show that $\textrm-g\not\in V_{\textsf L}^+$ for at least some $g\in V_{\textsf L}^+.$ we know there are some $a,b$ wff such that $\vdash a\to b$ and $\not\vdash b\to a.$ thus, $$\begin{split} b^{\textsf L}-a^{\textsf L} &=(a\to b)^{\textsf L}\\ &\in V_{\textsf L}^+\\ \textrm-(b^{\textsf L}-a^{\textsf L}) &=a^{\textsf L}-b^{\textsf L}\\ &=(b\to a)^{\textsf L}\\ &\not\in V_{\textsf L}^+ \end{split} $$ which means $g=b^{\textsf L}-a^{\textsf L}$ works. given $2V_{\textsf L}$ is closed by inversion, $g\in V_{\textsf L}^+\setminus2V_{\textsf L},$ so $2V_{\textsf L}\ne V_{\textsf L}^+.$

$\boxed\Longleftarrow$ assume $\textsf L$ be such, that $2V_{\textsf L}\subsetneq V_{\textsf L}^+$ and $\textrm-g\not\in V_{\textsf L}^+$ for at least some $g\in V_{\textsf L}^+.$ \begin{equation*} \begin{split} (\neg(\alpha\to\neg\alpha))^{\textsf L} &=\textrm-(\textrm-\alpha^{\textsf L}-\alpha^{\textsf L})\\ &=2\alpha^{\textsf L}\\ &\in2V_{\textsf L}\subset V_{\textsf L}^+\\ (\neg(\neg\alpha\to\alpha))^{\textsf L} &=\textrm-(\alpha^{\textsf L}-(\textrm-\alpha^{\textsf L}))\\ &=\textrm-2\alpha^{\textsf L}\\ &\in2V_{\textsf L}\subset V_{\textsf L}^+\\ ((\alpha\to\beta)\to\neg(\alpha\to\neg\beta))^{\textsf L} &=\textrm-(\textrm-\beta^{\textsf L}-\alpha^{\textsf L})-(\beta^{\textsf L}-\alpha^{\textsf L})\\ &=\beta^{\textsf L}+\alpha^{\textsf L}-\beta^{\textsf L}+\alpha^{\textsf L}\\ &=2\alpha^{\textsf L}\\ &\in2V_{\textsf L}\subset V_{\textsf L}^+\\ ((\alpha\to\neg\beta)\to\neg(\alpha\to\beta))^{\textsf L} &=\textrm-(\beta^{\textsf L}-\alpha^{\textsf L})-(\textrm-\beta^{\textsf L}-\alpha^{\textsf L})\\ &=\textrm-\beta^{\textsf L}+\alpha^{\textsf L}+\beta^{\textsf L}+\alpha^{\textsf L}\\ &=2\alpha^{\textsf L}\\ &\in2V_{\textsf L}\subset V_{\textsf L}^+ \end{split} \end{equation*} finally, let $g\in V_{\textsf L}^+$ such that $\textrm-g\not\in V_{\textsf L}^+,$ which exists by assumption. we can take some $\alpha,\beta$ wff, and $f$ a truth function, such that $f(\alpha)=0_{\textsf L}$ and $f(\beta)=g.$ this way, \begin{equation*} \begin{split} f(\alpha\to\beta) &=f(\beta)-f(\alpha)\\ &=g\in V_{\textsf L}^+\\ f(\beta\to\alpha) &=f(\alpha)-f(\beta)\\ &=\textrm-g\not\in V_{\textsf L}^+ \end{split} \end{equation*} thus, $\textsf L$ is indeed weakly* connexive.

such connexive abelian logics described in this characterisation, are also strongly inconsistent, as for any $\textsf L$ such logic, $(\alpha\to\neg\alpha)^{ \textsf L}=\textrm-2\alpha^{\textsf L}\in2V_{\textsf L}\subset V_{\textsf L}^+,$ so we have both $\overline{\vdash\neg(\alpha\to\neg\alpha)}$ and $\overline{\vdash\alpha\to\neg\alpha}.$ well, at least, paraconsistency has become quite cool, but moreover, these logics do not satisfy modus ponens : given $\alpha,\beta$ wff, and $f$ truth fuction such that $f(\alpha)=g$ and $f(\beta)=\textrm-g,$ with $g\in V_{\textsf L}^+$ such that $\textrm-g\in V_{\textsf L}^+,$ we have $f(\alpha\to\beta)=\textrm-2g\in2V_{\textsf L}\subset V_{\textsf L}^+$ and $f(\alpha)=g\in V_{\textsf L}^+,$ yet $f(\beta)=\textrm-g\not\in V^+.$ the fact they don't satisfy modus ponens, in my opinion, legitimise the weakening of the rule of non-symetry of implication here ; but well, not having this fundamental inference can be controversial, though it doesn't shock all that much at this point.
nonetheless, they do satisfy contraposition : $(\alpha\to\beta)^{\textsf L}=\beta^{\textsf L}-\alpha^{\textsf L}=(\textrm-\alpha^{\textsf L}-(\textrm-\beta^{\textsf L}))=(\neg\beta\to\neg\alpha)^{\textsf L}.$ it's also easy to prove they invalidate : affirming the consequent, negating the antecedent $(\textrm{hint}:g-(\textrm-g));$ positive paradox $(\textrm{hint}:(\alpha^{\textsf L}-g)-\alpha^{\textsf L});$ vacuous truth $(\textrm{hint}:(\textrm-g-\alpha^{\textsf L})-(\textrm-\alpha^{\textsf L}))\ldots$

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