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# Modus tollens

In classical logic, modus tollens (or modus tollendo tollens) (Latin for "the way that denies by denying") has the following argument form:
If P, then Q.
¬Q
Therefore, ¬P.

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:
$Pto Q, neg Q vdash neg P$
where $vdash$ represents the logical assertion.

It can also be written as:

$frac\left\{Pto Q ~,~~ neg Q\right\}\left\{neg P\right\}$

or including assumptions:

$frac\left\{Gamma vdash Pto Q ~~~ Gamma vdashneg Q\right\}\left\{Gamma vdash neg P\right\}$
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:

$Psubseteq Q$
$xnotin Q$
$therefore xnotin P$
("P is a subset of Q. x is not in Q. Therefore, x is not in P.")

Also in first-order predicate logic:

$forall x.~P\left(x\right) to Q\left(x\right)$
$exists x.~neg Q\left(x\right)$
$therefore exists x.~neg P\left(x\right)$
("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 there is fire here, then there is oxygen here.
There is no oxygen here.
Therefore, there is no fire here.

Supposing that the premises are both true, if there is a fire here, then there must be oxygen. It is a fact that there is no oxygen here. It follows, then, that there cannot be a fire here. An argument is valid if it is not possible for the premises to be true and the conclusion false. (A counter-example demonstrates that Hydrogen gas burns efficiently with Halogen gases like Chlorine and Fluorine and will combust with Iodine, with no Oxygen present.)

Another example:

If Lizzie were the murderer, then she owns an axe.
Lizzie does not own an axe.
Therefore, Lizzie was 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 (Kauffmann (1906))
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.