Three Types of Negation of Triple and its Elements and an Extension of Triple

Summary

The paper argues that the classical triple <subject, predicate, object> can only assert positive facts and that triples and their individual elements exhibit three distinct negations: contradictory, opposite (strong), and intermediary (weak). Building on the author's earlier set system SCOI and logic LCOI+PLCOI, it defines the TCOI triple, in which each of s, p, o and the whole triple can carry any of the three negation operators, plus an optional correlation degree in [0,1]. It distinguishes clear from fuzzy triples, introduces TCOI-entailment to connect triple implication reasoning with LCOI+PLCOI inference, and proves a soundness theorem for a new continuous-valued [0,1] semantics, while conceding that completeness fails for the chosen implication R0. A teacup-temperature example illustrates three fuzzy 'counterfactual' patterns matched to the three negations, supported by a truth-value algorithm built on t-norms and R-implications; a worked calculation assumes d(A→B)=0.9 and d(B)=0.9 and yields 0.1 and lambda-dependent closed forms. Main limitations: the formalization contradicts its own informal story — the 'contradictory negation' of a crisp true element evaluates to 1/2, never 0; the flagship example collapses the contradictory and intermediary conclusions to the same value; the 'counterfactual' patterns are fuzzy modus tollens; and there is no empirical evaluation, no baseline comparison (e.g., ERDF), no implementation, and no guidance for choosing lambda.