Modus Inversus. On the Algebraic, Probabilistic and Tensorial Foundations of an Independent Inference Figure

Abstract

Background: The schema $\neg P\rightarrow\neg Q,\ \neg P\ \therefore\ \neg Q$ is not modus ponens restated for negated propositions. Its premise, $\neg P\rightarrow\neg Q$, is secured directly and taken as primitive -- nothing more and nothing less. Material and methods:Modus inversus is developed here, on the model of modus tollens, as \emph{modus inversus}, an independent figure of inference standing on its own secured premise, not derived from, reducible to, or dependent upon modus ponens.Results:Representing propositions by idempotent $\{0,1\}$-valued indicator variables, we show that the inference is forced, without further postulate, by the single algebraic identity $X_{\neg P}\!\cdot\!X_{\neg P\rightarrow\neg Q}=X_{\neg P}X_{\neg Q}$. Passing to a probability space and taking expectations, the identity strengthens into a theorem asserting that the conclusion is certain (probability one) whenever the antecedent condition occurs with positive probability. The result is generalised to finite systems of propositions via a necessity tensor, and to jointly necessary multi-factor structures realised in nature by the fire triangle of heat, fuel and oxidiser.Conclusion:Throughout, the secured natural fact that no wax candle burns without gaseous oxygen serves as the natural grounding instance and, for any $P,Q$ whatsoever, as one substitution instance among countless others.