Modus tollens

The form of a modus tollens argument is a mixed hypothetical syllogism, with two premises and a conclusion:

:If P, then Q.

:Not Q.

:Therefore, not P.

The first premise is a conditional ("if-then") claim, such as P implies Q. The second premise is an assertion that Q, the consequent of the conditional claim, is not the case. From these two premises it can be logically concluded that P, the antecedent of the conditional claim, is also not the case.

For example:

:If the dog detects an intruder, the dog will bark.

:The dog did not bark.

:Therefore, no intruder was detected by the dog.

Supposing that the premises are both true (the dog will bark if it detects an intruder, and does indeed not bark), it logically follows that no intruder has been detected. This is a valid argument since it is not possible for the conclusion to be false if the premises are true. (It is conceivable that there may have been an intruder that the dog did not detect, but that does not invalidate the argument; the first premise is "if the dog detects an intruder". The thing of importance is that the dog detects or does not detect an intruder, not whether there is one.)

Example 1:

:If I am the burglar, then I can crack a safe.

:I cannot crack a safe.

:Therefore, I am not the burglar.

Example 2:

:If Rex is a chicken, then he is a bird.

:Rex is not a bird.

:Therefore, Rex is not a chicken.

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

:If P, then Q. (premise – material implication)

:If not Q, then not P. (derived by transposition)

:Not Q . (premise)

:Therefore, not P. (derived by modus ponens)

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

The modus tollens rule can be stated formally as:

:$\backslash frac\{P\; \backslash to\; Q,\; \backslash neg\; Q\}\{\backslash therefore\; \backslash neg\; P\}$

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

The modus tollens rule may be written in sequent notation:

:$P\backslash to\; Q,\; \backslash neg\; Q\; \backslash vdash\; \backslash neg\; P$

where $\backslash vdash$ is a metalogical symbol meaning that $\backslash neg\; P$ is a syntactic consequence of $P\; \backslash to\; Q$ and $\backslash neg\; Q$ in some logical system;

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

:$((P\; \backslash to\; Q)\; \backslash land\; \backslash neg\; Q)\; \backslash to\; \backslash neg\; P$

where $P$ and $Q$ are propositions expressed in some formal system;

or including assumptions:

:$\backslash frac\{\backslash Gamma\; \backslash vdash\; P\backslash to\; Q\; ~~~\; \backslash Gamma\; \backslash vdash\; \backslash neg\; Q\}\{\backslash Gamma\; \backslash vdash\; \backslash neg\; P\}$

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:

:$P\backslash subseteq\; Q$

:$x\backslash notin\; Q$

:$\backslash therefore\; x\backslash notin\; P$

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

Also in first-order predicate logic:

:$\backslash forall\; x:~P(x)\; \backslash to\; Q(x)$

:$\backslash neg\; Q(y)$

:$\backslash therefore\; ~\backslash neg\; P(y)$

("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.

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.

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

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

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

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

In the equations above $\backslash Pr(Q)$ denotes the probability of $Q$, and $a(P)$ denotes the base rate (aka. prior probability) of $P$. The conditional probability $\backslash Pr(Q\backslash mid\; P)$ generalizes the logical statement $P\; \backslash to\; Q$, i.e. in addition to assigning TRUE or FALSE we can also assign any probability to the statement. Assume that $\backslash Pr(Q)\; =\; 1$ is equivalent to $Q$ being TRUE, and that $\backslash Pr(Q)\; =\; 0$ is equivalent to $Q$ being FALSE. It is then easy to see that $\backslash Pr(P)\; =\; 0$ when $\backslash Pr(Q\backslash mid\; P)\; =\; 1$ and $\backslash Pr(Q)\; =\; 0$. This is because $\backslash Pr(\backslash lnot\; Q\backslash mid\; P)\; =\; 1\; -\; \backslash Pr(Q\backslash mid\; P)\; =\; 0$ so that $\backslash Pr(P\backslash mid\; \backslash lnot\; Q)\; =\; 0$ in the last equation. Therefore, the product terms in the first equation always have a zero factor so that $\backslash Pr(P)\; =\; 0$ which is equivalent to $P$ being FALSE. Hence, the law of total probability combined with Bayes' theorem represents a generalization of modus tollens.Audun Jøsang 2016:p.2

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

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

where $\backslash omega^\{A\}\_\{Q\}$ denotes the subjective opinion about $Q$, and $(\backslash omega^\{A\}\_\{Q|P\},\backslash omega^\{A\}\_\{Q|\backslash lnot\; P\})$ 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 $\backslash omega^\{A\}\_\{P\backslash tilde\{\backslash |\}Q\}$. The conditional opinion $\backslash omega^\{A\}\_\{Q|P\}$ generalizes the logical statement $P\; \backslash to\; Q$, i.e. in addition to assigning TRUE or FALSE the source $A$ can assign any subjective opinion to the statement. The case where $\backslash omega^\{A\}\_\{Q\}$ is an absolute TRUE opinion is equivalent to source $A$ saying that $Q$ is TRUE, and the case where $\backslash omega^\{A\}\_\{Q\}$ is an absolute FALSE opinion is equivalent to source $A$ saying that $Q$ is FALSE. The abduction operator $\backslash widetilde\{\backslash circledcirc\}$ of subjective logic produces an absolute FALSE abduced opinion $\backslash omega^\{A\}\_\{P\backslash widetilde\{\backslash |\}Q\}$ when the conditional opinion $\backslash omega^\{A\}\_\{Q|P\}$ is absolute TRUE and the consequent opinion $\backslash omega^\{A\}\_\{Q\}$ 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.Audun Jøsang 2016:p.92

• {{annotated link|Evidence of absence}}
• {{annotated link|Latin phrases}}
• {{annotated link|Modus operandi|Modus operandi}}
• {{annotated link|Modus ponens|Modus ponens}}
• {{annotated link|Modus vivendi|Modus vivendi}}
• {{annotated link|Non sequitur (logic)|Non sequitur}}
• {{annotated link|Performative contradiction}}
• {{annotated link|Proof by contradiction}}
• {{annotated link|Proof by contrapositive}}
• {{annotated link|Stoic logic}}
• Law of excluded middle

{{Reflist}}

• Audun Jøsang, 2016, [https://books.google.com/books?id=nqRlDQAAQBAJ&q=%22Modus+tollens%22 Subjective Logic; A formalism for Reasoning Under Uncertainty] Springer, Cham, {{ISBN|978-3-319-42337-1}}

• [http://mathworld.wolfram.com/ModusTollens.html Modus Tollens] at Wolfram MathWorld

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Category:Classical logic

Category:Rules of inference

Category:Latin logical phrases

Category:Theorems in propositional logic