Type

$P$ implies $Q$ . $Q$ is false. Therefore, $P$ must also be false.

$P\rightarrow Q,\neg Q$ $\therefore \neg P$ ^[1]

Correspondence to other mathematical frameworks[edit]

Probability calculus[edit]

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

$\Pr(P)=\Pr(P\mid Q)\Pr(Q)+\Pr(P\mid \lnot Q)\Pr(\lnot Q)\,,$

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

$\Pr(P\mid Q)={\frac {\Pr(Q\mid P)\,a(P)}{\Pr(Q\mid P)\,a(P)+\Pr(Q\mid \lnot P)\,a(\lnot P)}}\;\;\;$ and $\Pr(P\mid \lnot Q)={\frac {\Pr(\lnot Q\mid P)\,a(P)}{\Pr(\lnot Q\mid P)\,a(P)+\Pr(\lnot Q\mid \lnot P)\,a(\lnot P)}}.$

In the equations above $\Pr(Q)$ denotes the probability of $Q$ , and $a(P)$ denotes the base rate (aka. prior probability) of $P$ . The conditional probability $\Pr(Q\mid P)$ generalizes the logical statement $P\to Q$ , i.e. in addition to assigning TRUE or FALSE we can also assign any probability to the statement. Assume that $\Pr(Q)=1$ is equivalent to $Q$ being TRUE, and that $\Pr(Q)=0$ is equivalent to $Q$ being FALSE. It is then easy to see that $\Pr(P)=0$ when $\Pr(Q\mid P)=1$ and $\Pr(Q)=0$ . This is because $\Pr(\lnot Q\mid P)=1-\Pr(Q\mid P)=0$ so that $\Pr(P\mid \lnot Q)=0$ in the last equation. Therefore, the product terms in the first equation always have a zero factor so that $\Pr(P)=0$ which is equivalent to $P$ being FALSE. Hence, the law of total probability combined with Bayes' theorem represents a generalization of modus tollens.^[6]

Subjective logic[edit]

Modus tollens represents an instance of the abduction operator in subjective logic expressed as:

$\omega _{P{\tilde {\|}}Q}^{A}=(\omega _{Q|P}^{A},\omega _{Q|\lnot P}^{A}){\widetilde {\circledcirc }}(a_{P},\,\omega _{Q}^{A})\,,$

where $\omega _{Q}^{A}$ denotes the subjective opinion about $Q$ , and $(\omega _{Q|P}^{A},\omega _{Q|\lnot P}^{A})$ denotes a pair of binomial conditional opinions, as expressed by source $A$ . The parameter $a_{P}$ denotes the base rate (aka. the prior probability) of $P$ . The abduced marginal opinion on $P$ is denoted $\omega _{P{\tilde {\|}}Q}^{A}$ . The conditional opinion $\omega _{Q|P}^{A}$ generalizes the logical statement $P\to Q$ , i.e. in addition to assigning TRUE or FALSE the source $A$ can assign any subjective opinion to the statement. The case where $\omega _{Q}^{A}$ is an absolute TRUE opinion is equivalent to source $A$ saying that $Q$ is TRUE, and the case where $\omega _{Q}^{A}$ is an absolute FALSE opinion is equivalent to source $A$ saying that $Q$ is FALSE. The abduction operator ${\widetilde {\circledcirc }}$ of subjective logic produces an absolute FALSE abduced opinion $\omega _{P{\widetilde {\|}}Q}^{A}$ when the conditional opinion $\omega _{Q|P}^{A}$ is absolute TRUE and the consequent opinion $\omega _{Q}^{A}$ is absolute FALSE. Hence, subjective logic abduction represents a generalization of both modus tollens and of the Law of total probability combined with Bayes' theorem.^[7]

– Relevance fallacy

Evidence of absence

Latin phrases

– Habits of working

Modus operandi

– Rule of logical inference

Modus ponens

– Arrangement that allows conflicting parties to coexist in peace

Modus vivendi

– Faulty deductive reasoning due to a logical flaw

Non sequitur

– Proof by showing that the negation is impossible

Proof by contradiction

– Mathematical logic concept

Proof by contrapositive

– System of propositional logic developed by the Stoic philosophers

Stoic logic

Audun Jøsang, 2016, Springer, Cham, ISBN 978-3-319-42337-1

Subjective Logic; A formalism for Reasoning Under Uncertainty

at Wolfram MathWorld

Type

Type

Field

Statement

Symbolic statement

Correspondence to other mathematical frameworks[edit]

Probability calculus[edit]

Subjective logic[edit]

Evidence of absence

Latin phrases

Modus operandi

Modus ponens

Modus vivendi

Non sequitur

Proof by contradiction

Proof by contrapositive

Stoic logic

Subjective Logic; A formalism for Reasoning Under Uncertainty

Modus Tollens