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# Modus tollens

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 Title: Modus tollens Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Modus tollens

remises. The first premise is a conditional or "if-then" statement, for example that if P then Q. The second premise is that it is not the case that Q . From these two premises, it can be logically concluded that it is not the case that P.

Consider an example:

If the watch-dog detects an intruder, the watch-dog will bark.
The watch-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 that no intruder has been detected. This is a valid argument since it is not possible for the conclusion to be false if the premises are true. (It is conceivable that there may have been an intruder that the dog did not detect, but that does not invalidate the argument; the first premise is "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 can use an axe.
I cannot use an axe.
Therefore, I am not the axe murderer.

## 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 not Q , then not P. (derived by transposition)
Not Q . (premise)
Therefore, not P. (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.

## Formal proof

### Via disjunctive syllogism

Step Proposition Derivation
1 P\rightarrow Q Given
2 \neg Q Given
3 \neg P\or Q Material implication (1)
4 \neg P Disjunctive syllogism (2,3)

Step Proposition Derivation
1 P\rightarrow Q Given
2 \neg Q Given
3 P Assumption
4 Q Modus ponens (1,3)
5 Q \and \neg Q Conjunction introduction (2,4)
6 \neg P Reductio ad absurdum (3,5)

## Notes

1. ^ University of North Carolina, Philosophy Department, Logic Glossary. Accessdate on 31 October 2007.
2. ^ Copi and Cohen
3. ^ Hurley
4. ^ Moore and Parker
5. ^ 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."
6. ^ Stone, Jon R. 1996. Latin for the Illiterati: Exorcizing the Ghosts of a Dead Language. London, UK: Routledge: 60.
7. ^ "Ancient Logic: The Stoics"Stanford Encyclopedia of Philosophy:
8. ^ Susanne Bobzien (2002). "The Development of Modus Ponens in Antiquity", Phronesis 47.