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

Background: The schema P→ Q,\ P\ ∴\ Q is not modus ponens restated for negated propositions. Its premise, P→ 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 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 XP\!·\!XP→ Q=XPXQ. 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.