# Modus tollens

In classical logic, modus tollens (or modus tollendo tollens)[1] (Latin for "the way that denies by denying")[2] has the following argument form:

If P, then Q.
¬Q
Therefore, ¬P.[3]

It can also be referred to as denying the consequent, and is a valid form of argument (unlike similarly-named but invalid arguments such as affirming the consequent or denying the antecedent). (See also modus ponens or "affirming the antecedent".)

Modus tollens is sometimes confused with indirect proof (assuming the negation of the proposition to be proved and showing that this leads to a contradiction) or proof by contrapositive (proving If P, then Q by a proof of the equivalent contrapositive If not-Q, then not-P).

## Formal notation

The modus tollens rule may be written in logical operator notation:

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

where ${} \vdash {}$ represents the logical assertion.

It can also be written as:

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

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:

$P\subseteq Q$
$x\notin Q$
$\therefore x\notin P$

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

Also in first-order predicate logic:

$\forall x.~P(x) \to Q(x)$
$\exists x.~\neg Q(x)$
$\therefore \exists x.~\neg P(x)$

("All P's are Q's. There's an x that's not a Q. Therefore, there's an x that's not a 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 an intruder is detected, the alarm goes off.
The alarm does not go off.
Therefore, no intruder is detected.

Supposing that the premises are both true (the intrusion alarm is functioning properly, and does indeed not go off), 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 the alarm is malfunctioning and does not go off even though its sensors have detected an intrusion, but that does not invalidate the argument; it just means that the first premise is not satisfied.)

Another example:

If Lizzie is the murderer, then she owns an axe.
Lizzie does not own an axe.
Therefore, Lizzie is not the murderer.

Modus tollens became well known when it was used by Karl Popper in his proposed response to the problem of induction, falsificationism. However, here the use of modus tollens is much more controversial, as "truth" or "falsity" are inappropriate concepts to apply to theories (which are generally approximations to reality) and experimental findings (whose interpretation is often contingent on other theories). Thus (to take a historical example)

If special relativity is true, then the mass of the electron has a specific dependence on velocity.
Experimentally, the mass of the electron does not have this dependence (Kaufmann (1906)[4]).
Therefore, special relativity is false.

Einstein rejected this argument on the grounds that the alternative theories that appeared to be validated by the experiment were inherently less plausible than his own.

## 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.

## Notes

1. ^ Sanford, David Hawley. 2003. If P, Then Q: Conditionals and the Foundations of Reasoning. London, UK: Routledge: 39 "[Modus] tollens is always an abbreviation for modus tollendo tollens, the mood that by denying denies."
2. ^ Stone, Jon R. 1996. Latin for the Illiterati: Exorcizing the Ghosts of a Dead Language. London, UK: Routledge: 60.
3. ^ University of North Carolina, Philosophy Department, Logic Glossary. Accessdate on .
4. ^ Kaufmann, W. (1906), "Über die Konstitution des Elektrons", Annalen der Physik 19: 487–553