structural:scholar_edition, linguistic:voice_change
Subjective logic

Contraposition represents an instance of the subjective Bayes' theorem in subjective logic expressed as:

where (ω Q | P A, ω Q | ¬ P A) {\displaystyle (\omega _{Q|P}^{A},\omega _{Q|\lnot P}^{A})} denotes a pair of binomial conditional opinions given by source A {\displaystyle A}. The parameter a P {\displaystyle a_{P}} denotes the base rate (aka. the prior probability) of P {\displaystyle P}. The pair of derivative inverted conditional opinions is denoted (ω P | ~ Q A, ω P | ~ ¬ Q A) {\displaystyle (\omega _{P{\tilde {|}}Q}^{A},\omega _{P{\tilde {|}}\lnot Q}^{A})}. The conditional opinion ω Q | P A {\displaystyle \omega _{Q|P}^{A}} generalizes the logical statement P → Q {\displaystyle P\to Q}, i.e. in addition to assigning TRUE or FALSE the source A {\displaystyle A} can assign any subjective opinion to the statement. The case where ω Q ∣ P A {\displaystyle \omega _{Q\mid P}^{A}} is an absolute TRUE opinion is equivalent to source A {\displaystyle A} saying that P → Q {\displaystyle P\to Q} is TRUE, and the case where ω Q ∣ P A {\displaystyle \omega _{Q\mid P}^{A}} is an absolute FALSE opinion is equivalent to source A {\displaystyle A} saying that P → Q {\displaystyle P\to Q} is FALSE. In the case when the conditional opinion ω Q | P A {\displaystyle \omega _{Q|P}^{A}} is absolute TRUE the subjective Bayes' theorem operator ϕ ~ {\displaystyle {\widetilde {\phi \,}}} of subjective logic produces an absolute FALSE derivative conditional opinion ω P | ~ ¬ Q A {\displaystyle \omega _{P{\widetilde {|}}\lnot Q}^{A}} and thereby an absolute TRUE derivative conditional opinion ω ¬ P | ~ ¬ Q A {\displaystyle \omega _{\lnot P{\widetilde {|}}\lnot Q}^{A}} which is equivalent to ¬ Q → ¬ P {\displaystyle \lnot Q\to \lnot P} being TRUE. Hence, the subjective Bayes' theorem represents a generalization of both contraposition and Bayes' theorem.