In propositional logic, modus tollens (/ˈmoʊdəs ˈtɒlɛnz/) (MT), also known as modus tollendo tollens (Latin for "mode that by denying denies")[2] and denying the consequent,[3] is a deductive argument form and a rule of inference.
Modus tollens is a mixed hypothetical syllogism that takes the form of "If P, then Q.
Therefore, not P." It is an application of the general truth that if a statement is true, then so is its contrapositive.
The history of the inference rule modus tollens goes back to antiquity.
[4] The first to explicitly describe the argument form modus tollens was Theophrastus.
There are two similar, but invalid, forms of argument: affirming the consequent and denying the antecedent.
The form of a modus tollens argument is a mixed hypothetical syllogism, with two premises and a conclusion: The first premise is a conditional ("if-then") claim, such as P implies Q.
The second premise is an assertion that Q, the consequent of the conditional claim, is not the case.
From these two premises it can be logically concluded that P, the antecedent of the conditional claim, is also not the case.
For example: Supposing that the premises are both true (the dog will bark if he or she detects an intruder, and does indeed not bark), it logically 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 dog detects an intruder".
The modus tollens rule can be stated formally as: where
The modus tollens rule may be written in sequent notation: where
are propositions expressed in some formal system; or including assumptions: 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 is a subset of Q. x is not in Q.
Therefore, y is not P.") Strictly speaking these are not instances of modus tollens, but they may be derived from modus tollens using a few extra steps.
The validity of modus tollens can be clearly demonstrated through a truth table.
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.
Modus tollens represents an instance of the law of total probability combined with Bayes' theorem expressed as:
are obtained with (the extended form of) Bayes' theorem expressed as:
Hence, the law of total probability combined with Bayes' theorem represents a generalization of modus tollens.
[6] Modus tollens represents an instance of the abduction operator in subjective logic expressed as:
denotes a pair of binomial conditional opinions, as expressed by source
, i.e. in addition to assigning TRUE or FALSE the source
is an absolute TRUE opinion is equivalent to source
is an absolute FALSE opinion is equivalent to source
of subjective logic produces an absolute FALSE abduced opinion
Hence, subjective logic abduction represents a generalization of both modus tollens and of the Law of total probability combined with Bayes' theorem.