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Modus tollens
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In propositional logic, modus tollens^{[1]}^{[2]}^{[3]}^{[4]} (or modus tollendo tollens and also denying the consequent)^{[5]} (Latin for "the way that denies by denying")^{[6]} is a valid argument form and a rule of inference. It is an application of the general truth that if a statement is true, then so is its contra-positive.
The first to explicitly state the argument form modus tollens were the Stoics.^{[7]}
The inference rule modus tollens, also known as the law of contrapositive, validates the inference from <math>P</math> implies <math>Q</math> and the contradictory of <math>Q</math>, to the contradictory of <math>P</math>.
The modus tollens rule can be stated formally as:
- <math>\frac{P \to Q, \neg Q}{\therefore \neg P}</math>
where <math>P \to Q</math> stands for the statement "P implies Q" (and <math>{\neg Q}{\to \neg P}</math> is called the "contrapositive"). <math>\neg Q</math> stands for "it is not the case that Q" (or in brief "not Q"). Then, whenever "<math>P \to Q</math>" and "<math>\neg Q</math>" each appear by themselves as a line of a proof, then "<math>\neg P</math>" can validly be placed on a subsequent line. The history of the inference rule modus tollens goes back to antiquity.^{[8]}
Modus tollens is closely related to modus ponens. There are two similar, but invalid, forms of argument: affirming the consequent and denying the antecedent. See also contraposition and proof by contraposition.
Contents
Formal notation
The modus tollens rule may be written in sequent notation:
- <math>P\to Q, \neg Q \vdash \neg P</math>
where <math>\vdash</math> is a metalogical symbol meaning that <math>\neg P</math> is a syntactic consequence of <math>P \to Q</math> and <math>\neg Q</math> in some logical system;
or as the statement of a functional tautology or theorem of propositional logic:
- <math>((P \to Q) \and \neg Q) \to \neg P</math>
where <math>P</math> and <math>Q</math> are propositions expressed in some formal system;
or including assumptions:
- <math>\frac{\Gamma \vdash P\to Q ~~~ \Gamma \vdash \neg Q}{\Gamma \vdash \neg P}</math>
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:
- <math>P\subseteq Q</math>
- <math>x\notin Q</math>
- <math>\therefore x\notin P</math>
("P is a subset of Q. x is not in Q. Therefore, x is not in P.")
Also in first-order predicate logic:
- <math>\forall x:~P(x) \to Q(x)</math>
- <math>\exists x:~\neg Q(x)</math>
- <math>\therefore \exists x:~\neg P(x)</math>
("For all x if x is P then x is Q. There exists some x that is not Q. Therefore, there exists some x that 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 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 | <math>P\rightarrow Q</math> | Given |
2 | <math>\neg Q</math> | Given |
3 | <math>\neg P\or Q</math> | Material implication (1) |
4 | <math>\neg P</math> | Disjunctive syllogism (2,3) |
Via reductio ad absurdum
Step | Proposition | Derivation |
---|---|---|
1 | <math>P\rightarrow Q</math> | Given |
2 | <math>\neg Q</math> | Given |
3 | <math>P</math> | Assumption |
4 | <math>Q</math> | Modus ponens (1,3) |
5 | <math>Q \and \neg Q</math> | Conjunction introduction (2,4) |
6 | <math>\neg P</math> | Reductio ad absurdum (3,5) |
See also
Notes
- ^ University of North Carolina, Philosophy Department, Logic Glossary. Accessdate on 31 October 2007.
- ^ Copi and Cohen
- ^ Hurley
- ^ Moore and Parker
- ^ 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."
- ^ Stone, Jon R. 1996. Latin for the Illiterati: Exorcizing the Ghosts of a Dead Language. London, UK: Routledge: 60.
- ^ "Stanford Encyclopedia of Philosophy: Ancient Logic: The Stoics"
- ^ Susanne Bobzien (2002). "The Development of Modus Ponens in Antiquity", Phronesis 47.
External link
- Modus Tollens at Wolfram MathWorld