Modus tollens

Rules of inference
Propositional calculus
Modus ponens (A→B, A ⊢ B) Modus tollens (A→B, ¬B ⊢ ¬A) Modus ponendo tollens (¬(A∧B), A ⊢ ¬B) Conjunction introduction (A, B ⊢ A∧B) Simplification (A∧B ⊢ A) Disjunction introduction (A ⊢ A∨B) Disjunction elimination (A∨B, A→C, B→C ⊢ C) Disjunctive syllogism (A∨B, ¬A ⊢ B) Hypothetical syllogism (A→B, B→C ⊢ A→C) Constructive dilemma (A→P, B→Q, A∨B ⊢ P∨Q) Destructive dilemma (A→P, B→Q, ¬P∨¬Q ⊢ ¬A∨¬B) Biconditional introduction (A→B, B→A ⊢ A↔B) Biconditional elimination (A↔B ⊢ A→B)
Predicate calculus
Universal generalization Universal instantiation Existential generalization Existential instantiation
v · d · e

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.

Not Q.

Therefore, not 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. Modus tollens is sometimes confused with proof by contradiction or proof by contrapositive. Evidence of absence applies modus tollens. A related valid form of argument is modus ponens.

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:

$P\subseteq Q$

$x\notin Q$

$\therefore x\notin P$

("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 any x if x is P then x is Q.Some object x is such that x is not Q. Therefore, some object x 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 be 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.

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).

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.

See also

• Non sequitur (logic)
• Modus Ponens

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 31 October 2007.

External links

• Modus Tollens at Wolfram MathWorld