Modus tollens
Probability calculus

Modus tollens represents an instance of the law of total probability combined with Bayes' theorem expressed as:

where the conditionals Pr (P ∣ Q) {\displaystyle \Pr(P\mid Q)} and Pr (P ∣ ¬ Q) {\displaystyle \Pr(P\mid \lnot Q)} are obtained with (the extended form of) Bayes' theorem expressed as:

and

In the equations above Pr (Q) {\displaystyle \Pr(Q)} denotes the probability of Q {\displaystyle Q}, and a (P) {\displaystyle a(P)} denotes the base rate (aka. prior probability) of P {\displaystyle P}. The conditional probability Pr (Q ∣ P) {\displaystyle \Pr(Q\mid P)} generalizes the logical statement P → Q {\displaystyle P\to Q}, i.e. in addition to assigning TRUE or FALSE we can also assign any probability to the statement. Assume that Pr (Q) = 1 {\displaystyle \Pr(Q)=1} is equivalent to Q {\displaystyle Q} being TRUE, and that Pr (Q) = 0 {\displaystyle \Pr(Q)=0} is equivalent to Q {\displaystyle Q} being FALSE. It is then easy to see that Pr (P) = 0 {\displaystyle \Pr(P)=0} when Pr (Q ∣ P) = 1 {\displaystyle \Pr(Q\mid P)=1} and Pr (Q) = 0 {\displaystyle \Pr(Q)=0}. This is because Pr (¬ Q ∣ P) = 1 − Pr (Q ∣ P) = 0 {\displaystyle \Pr(\lnot Q\mid P)=1-\Pr(Q\mid P)=0} so that Pr (P ∣ ¬ Q) = 0 {\displaystyle \Pr(P\mid \lnot Q)=0} in the last equation. Therefore, the product terms in the first equation always have a zero factor so that Pr (P) = 0 {\displaystyle \Pr(P)=0} which is equivalent to P {\displaystyle P} being FALSE. Hence, the law of total probability combined with Bayes' theorem represents a generalization of modus tollens.
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