Modus tollens

In propositional logic, modus tollens (or modus tollendo tollens and also denying the consequent) (Latin for "the way that denies by denying") is a valid argument form and rule of inference.

It is the inference that

1. if $P$ implies $Q$, and the second premise, $Q$, is false,
2. then it can be logically concluded that $P$ must be false.

Modus tollens is closely related to another valid form of argument, modus ponens. There are also similar, but invalid, arguments such as affirming the consequent and denying the antecedent.

The modus tollens rule can be stated formally as:

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

Where $P\to Q$ means "P implies Q", $\neg Q$ means Q is false (not Q). The modus tollens rule, then, is that wherever both "$P\to Q$" and "$\neg Q$" appear by themselves on a line of a proof, then "$\neg P$" can validly be placed on a subsequent line.

Formal notation

The modus tollens rule may be written in sequent notation:

$P\to Q,\neg Q\vdash \neg P$

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

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

$((P\to Q)\land \neg Q)\to \neg P$

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

or including assumptions:

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

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:

$P\subseteq Q$

$x\notin Q$

$\therefore x\notin P$

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

Also in first-order predicate logic:

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

$\exists x:~\neg Q(x)$

$\therefore \exists x:~\neg P(x)$

("For all x if x is P then x is Q. There exists some x that is not Q. Therefore, there exists some x that is not P.")

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

Explanation

The argument has two premises. The first premise is the conditional "if-then" statement, namely that P implies Q. The second premise is that Q is false. From these two premises, it can be logically concluded that P must be false.

Consider an example:

If the watch-dog detects an intruder, the dog will bark.

The dog did not bark

Therefore, no intruder was detected by the watch-dog.

Supposing that the premises are both true (the dog will bark if it detects an intruder, and does indeed not bark), it follows then that no intruder has been detected. This is a valid argument since it is not possible for the premises to be true and the conclusion false. (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 goes "if the watch-dog detects an intruder." The thing of importance is that the dog detects or doesn't detect an intruder, not if there is one.)

Another example:

If I am the axe murderer, then I used an axe.

I cannot use an axe.

Therefore, I am not the axe murderer.

Relation to modus ponens

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 Q is false, then P is false. (derived by transposition)

Q is false. (premise)

Therefore, P is false. (derived by modus ponens)

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

Justification via truth table

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

p	q	p → q
T	T	T
T	F	F
F	T	T
F	F	T

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

Step	Proposition	Derivation
1	$P\rightarrow Q$	Given
2	$\neg Q$	Given
3	$\neg P\lor Q$	Material implication (1)
4	$\neg P$	Disjunctive syllogism (2,3)

Via Reductio ad absurdum

Step	Proposition	Derivation
1	$P\rightarrow Q$	Given
2	$\neg Q$	Given
3	$P$	Assumption
4	$Q$	Modus ponens (1,3)
5	$Q\land \neg Q$	Conjunction introduction (2,4)
6	$\neg P$	Reductio ad absurdum (3,5)

See also

• Evidence of absence
• Non sequitur
• Proof by contradiction
• Proof by contrapositive

External links

• Modus Tollens at Wolfram MathWorld