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Modus tollens
 
 
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In logicLogic

Logic, from Classical Greek ?????, originally meaning the word, or what is spoken, is most often said to be the stud...
, modus tollendo tollens is the formal name for indirect proof or proof by contrapositionContraposition Overview

In logic, contraposition is a form of immediate inference in which from a given proposition another is inferred having for i...
 (contrapositive inference), often abbreviated to MT or modus tollens. It can also be referred to as denying the consequent, and is a validValidity

In logic, the form of an argument is valid precisely if it cannot lead from true premises to a false conclusion....
 form of argumentArgument

An argument is a statement or group of statements offered in support of another statement ....
 (unlike similarly-named but invalid arguments such as affirming the consequentAffirming the consequent

Affirming the consequent is a logical fallacy in the form of a hypothetical proposition....
 or denying the antecedentDenying the antecedent

Denying the antecedent, also known as vacuous implication, is a type of logical fallacy....
). It is closely related to another valid form of argument, modus ponensModus ponens

In logic, modus ponens is a validity, simple argument form....
 or "affirming the antecedentAffirming the antecedent

Affirming the antecedent is a valid argument form which proceeds by affirming the truth of the first part of a conditional,...
".

Modus tollens has the following argument formArgument form

In logic, the argument form or test form of an argument results from replacing the different words, or sentences, that m...
:
If P, then Q.
¬Q
Therefore, ¬P.

Formal notation

The modus tollens rule may be written in logical operator notation:
where represents the logical assertionLogical assertion

The logical assertion is a statement that asserts that a certain premise is true, and is useful for statements in proof....
.

Or in set-theoreticSet theory

Set theory is the mathematical theory of sets, which represent collections of abstract objects....
 form:
("P is a subset of Q. x is not in Q. Therefore, x is not in P.")

It can also be written as:

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 validValidity

In logic, the form of an argument is valid precisely if it cannot lead from true premises to a false conclusion....
 if it is not possible for the premises to be true and the conclusion false. (A counter-example demonstrates that HydrogenHydrogen

|-| Triple point || 13.8033 K, 7.042 kPa...
 gas burns efficiently with HalogenHalogen

The halogens are a chemical series....
 gases like ChlorineChlorine

Chlorine , is the chemical element with atomic number 17 and symbol Cl....
 and FluorineFluorine

Fluorine , is the chemical element in the periodic table that has the symbol F and atomic number 9....
 and will combust with IodineIodine Summary

Iodine , is a chemical element in the periodic table that has the symbol I and atomic number 53....
, with no OxygenOxygen

Oxygen is a chemical element with the chemical symbol O and atomic number 8....
 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 PopperKarl Popper

Sir Karl Raimund Popper, CH, MA, Ph.D., D.LITT, FBA, FRS , was an Austrian and British philosopher and a professor at the Lo...
 in his proposed response to the problem of inductionProblem of induction

The problem of induction is the philosophical issue involved in deciding the place of induction in determining empirical tru...
, 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 ponensModus ponens

In logic, modus ponens is a validity, simple argument form....
 and one use of transpositionTransposition (logic) Overview

In the methods of deductive reasoning in classical logic, "transposition is the rule of inference that permits one to infer ...
 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 tableFacts About Truth table

A truth table is a mathematical table used in logic specifically in connection with boolean algebra, boolean functions, and...
.
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.

See also

  • Modus ponensModus ponens

    In logic, modus ponens is a validity, simple argument form....
  • Modus tollendo ponensModus tollendo ponens

    Modus tollendo ponens , or MTP, is a valid, simple argument form that is today known as disjunctive syllogism:...
  • Modus ponendo tollensModus ponendo tollens

    Modus ponendo tollens is a valid rule of inference, sometimes abbreviated MPT....
  • Affirming the consequentAffirming the consequent

    Affirming the consequent is a logical fallacy in the form of a hypothetical proposition....
  • Denying the antecedentDenying the antecedent

    Denying the antecedent, also known as vacuous implication, is a type of logical fallacy....
  • Falsificationism
  • Non sequitur (logic)Non sequitur (logic)

    Non sequitur is Latin for "it does not follow." In formal logic, an argument is a non sequitur if the conclusion does n...


External links

  • at Wolfram MathWorld