Modus tollens

From Wikipedia, the free encyclopedia

"Indirect proof" redirects here. For proof by contradiction, see Reductio ad absurdum.

In logic, modus tollendo tollens[1] (Latin for "the way that denies by denying")[2] is the formal name for indirect proof or proof by contraposition (contrapositive inference), often abbreviated to MT or modus tollens.[3][4] 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). It is closely related to another valid form of argument, modus ponens or "affirming the antecedent".

Modus tollens has the following argument form:

If P, then Q.
¬Q
Therefore, ¬P.[5]

Contents

[edit] 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.

Or in set-theoretic form:

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.")

It can also be written as:

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

[edit] 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.

[edit] 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.

[edit] 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.

[edit] See also

[edit] 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. ^ Politzer, Guy & Carles, Laure. 2001. 'Belief Revision and Uncertain Reasoning'. Thinking and Reasoning, 7:217-234.
  4. ^ Suppes, Patrick & Hill, Shirley A. 1964. First Course in Mathematical Logic. Dover Publications:54-55.
  5. ^ University of North Carolina, Philosophy Department, Logic Glossary. Accessdate on 31 October 2007.

[edit] External links