Modus tollens

Relation to modus ponens

Every use of modus tollens can be converted to a use of modus ponens and one use of contraposition to the premise which is a material implication. For example:

1. If P, then Q. (premise – material implication)
2. If not Q, then not P. (derived by contraposition)
3. Not Q. (premise)
4. Therefore, not P. (derived by modus ponens)

Likewise, every use of modus ponens can be converted to a use of modus tollens and contraposition.

Formal notation

The modus tollens rule can be stated formally as:

$${\displaystyle \frac{P\to Q,\mathrm{\neg }Q}{\therefore \mathrm{\neg }P}}$$

where ${\displaystyle P\to Q}$ stands for the statement "P implies Q". ${\displaystyle \mathrm{\neg }Q}$ stands for "it is not the case that Q" (or in brief "not Q"). Then, whenever "${\displaystyle P\to Q}$" and "${\displaystyle \mathrm{\neg }Q}$" each appear by themselves as a line of a proof, then "${\displaystyle \mathrm{\neg }P}$" can validly be placed on a subsequent line.

The modus tollens rule may be written in sequent notation:

$${\displaystyle P\to Q,\mathrm{\neg }Q⊢\mathrm{\neg }P}$$

where ${\displaystyle ⊢}$ is a metalogical symbol meaning that ${\displaystyle \mathrm{\neg }P}$ is a syntactic consequence of ${\displaystyle P\to Q}$ and ${\displaystyle \mathrm{\neg }Q}$ in some logical system;

or as the statement of a functional tautology or theorem of propositional logic:

$${\displaystyle ((P\to Q)\wedge \mathrm{\neg }Q)\to \mathrm{\neg }P}$$

where ${\displaystyle P}$ and ${\displaystyle Q}$ are propositions expressed in some formal system;

or including assumptions:

$${\displaystyle \frac{\mathrm{\Gamma }⊢P\to Q\mathrm{\Gamma }⊢\mathrm{\neg }Q}{\mathrm{\Gamma }⊢\mathrm{\neg }P}}$$

though since the rule does not change the set of assumptions, this is not strictly necessary.

More complex rewritings involving modus tollens are often seen, for instance in set theory: $${\displaystyle \begin{array}{cc}P& \subseteq Q\\ x& \notin Q\\ \therefore x& \notin P\end{array}}$$

("P is a subset of Q. x is not in Q. Therefore, x is not in P.")

Also in first-order predicate logic:

$${\displaystyle \mathrm{\forall }x:P(x)\to Q(x)}$$ $${\displaystyle \mathrm{\neg }Q(y)}$$ $${\displaystyle \therefore \mathrm{\neg }P(y)}$$

("For all x if x is P then x is Q. y is not Q. Therefore, y is not P.")

Strictly speaking these are not instances of modus tollens, but they may be derived from modus tollens using a few extra steps.

Justification via truth table

The validity of modus tollens can be clearly demonstrated through a truth table.

In instances of modus tollens we assume as premises that p → q is true and q is false. There is only one line of the truth table—the fourth line—which satisfies these two conditions. In this line, p is false. Therefore, in every instance in which p → q is true and q is false, p must also be false.

Formal proof

Via disjunctive syllogism

Via reductio ad absurdum

Via contraposition

Correspondence to other mathematical frameworks

Probability calculus

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

$${\displaystyle Pr(P)=Pr(P\mid Q)Pr(Q)+Pr(P\mid \mathrm{\neg }Q)Pr(\mathrm{\neg }Q),}$$

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

$${\displaystyle Pr(P\mid Q)=\frac{Pr(Q\mid P)a(P)}{Pr(Q\mid P)a(P)+Pr(Q\mid \mathrm{\neg }P)a(\mathrm{\neg }P)}}$$ and $${\displaystyle Pr(P\mid \mathrm{\neg }Q)=\frac{Pr(\mathrm{\neg }Q\mid P)a(P)}{Pr(\mathrm{\neg }Q\mid P)a(P)+Pr(\mathrm{\neg }Q\mid \mathrm{\neg }P)a(\mathrm{\neg }P)}.}$$

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

Subjective logic

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

$${\displaystyle {\omega }_{P\tilde{\Vert }Q}^A=({\omega }_{Q|P}^A,{\omega }_{Q|\mathrm{\neg }P}^A)\widetilde{⊚}(a_P,{\omega }_Q^A),}$$

where ${\displaystyle {\omega }_Q^A}$ denotes the subjective opinion about ${\displaystyle Q}$, and ${\displaystyle ({\omega }_{Q|P}^A,{\omega }_{Q|\mathrm{\neg }P}^A)}$ denotes a pair of binomial conditional opinions, as expressed by source ${\displaystyle A}$. The parameter ${\displaystyle a_P}$ denotes the base rate (a.k.a. the prior probability) of ${\displaystyle P}$. The abduced marginal opinion on ${\displaystyle P}$ is denoted ${\displaystyle {\omega }_{P\tilde{\Vert }Q}^A}$. The conditional opinion ${\displaystyle {\omega }_{Q|P}^A}$ generalizes the logical statement ${\displaystyle P\to Q}$, i.e. in addition to assigning TRUE or FALSE the source ${\displaystyle A}$ can assign any subjective opinion to the statement. The case where ${\displaystyle {\omega }_Q^A}$ is an absolute TRUE opinion is equivalent to source ${\displaystyle A}$ saying that ${\displaystyle Q}$ is TRUE, and the case where ${\displaystyle {\omega }_Q^A}$ is an absolute FALSE opinion is equivalent to source ${\displaystyle A}$ saying that ${\displaystyle Q}$ is FALSE. The abduction operator ${\displaystyle \widetilde{⊚}}$ of subjective logic produces an absolute FALSE abduced opinion ${\displaystyle {\omega }_{P\tilde{\Vert }Q}^A}$ when the conditional opinion ${\displaystyle {\omega }_{Q|P}^A}$ is absolute TRUE and the consequent opinion ${\displaystyle {\omega }_Q^A}$ 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.