Modus tollens
Encyclopedia
In classical logic
, modus tollens (or modus tollendo tollens) (Latin for "the way that denies by denying") has the following argument form:
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
.
where
represents the logical assertion
.
It can also be written as:
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, x is not in P.")
Also in first-order predicate logic
:
("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.
Consider an example:
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:
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).
and one use of transposition
to the premise which is a material implication. For example:
Likewise, every use of modus ponens can be converted to a use of modus tollens and transposition.
.
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.
Classical logic
Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. The class is sometimes called standard logic as well...
, modus tollens (or modus tollendo tollens) (Latin for "the way that denies by denying") has the following argument form:
- If P, then Q.
- Not Q.
- Therefore, not P.
It can also be referred to as denying the consequent, and is a valid
Validity
In logic, argument is valid if and only if its conclusion is entailed by its premises, a formula is valid if and only if it is true under every interpretation, and an argument form is valid if and only if every argument of that logical form is valid....
form of argument
Argument
In philosophy and logic, an argument is an attempt to persuade someone of something, or give evidence or reasons for accepting a particular conclusion.Argument may also refer to:-Mathematics and computer science:...
, unlike similarly named but invalid arguments such as affirming the consequent
Affirming the consequent
Affirming the consequent, sometimes called converse error, is a formal fallacy, committed by reasoning in the form:#If P, then Q.#Q.#Therefore, P....
or denying the antecedent
Denying the antecedent
Denying the antecedent, sometimes also called inverse error, is a formal fallacy, committed by reasoning in the form:The name denying the antecedent derives from the premise "not P", which denies the "if" clause of the conditional premise....
. Modus tollens is sometimes confused with proof by contradiction or proof by contrapositive
Proof by contrapositive
In logic, the contrapositive of a conditional statement of the form "if A then B" is formed by negating both terms and reversing the direction of inference...
. Evidence of absence
Evidence of absence
Evidence of absence is evidence of any kind that suggests the non-existence or non-presence of something. A simple example of evidence of absence: checking one's pocket for spare change and finding nothing but being confident that one would have found it if it were there...
applies modus tollens. A related valid form of argument is modus ponens
Modus ponens
In classical logic, modus ponendo ponens or implication elimination is a valid, simple argument form. It is related to another valid form of argument, modus tollens. Both Modus Ponens and Modus Tollens can be mistakenly used when proving arguments...
.
Formal notation
The modus tollens rule may be written in logical operator notation:where
represents the logical assertionLogical assertion
A logical assertion is a statement that asserts that a certain premise is true, and is useful for statements in proof. It is equivalent to a sequent with an empty antecedent....
.
It can also be written as:
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
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
:
("P is a subset of Q. x is not in Q. Therefore, x is not in P.")
Also in first-order predicate logic
Predicate logic
In mathematical logic, predicate logic is the generic term for symbolic formal systems like first-order logic, second-order logic, many-sorted logic or infinitary logic. This formal system is distinguished from other systems in that its formulae contain variables which can be quantified...
:
("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
Validity
In logic, argument is valid if and only if its conclusion is entailed by its premises, a formula is valid if and only if it is true under every interpretation, and an argument form is valid if and only if every argument of that logical form is 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
Karl Popper
Sir Karl Raimund Popper, CH FRS FBA was an Austro-British philosopher and a professor at the London School of Economics...
in his proposed response to the problem of induction
Problem of induction
The problem of induction is the philosophical question of whether inductive reasoning leads to knowledge. That is, what is the justification for either:...
, falsificationism
Falsifiability
Falsifiability or refutability of an assertion, hypothesis or theory is the logical possibility that it can be contradicted by an observation or the outcome of a physical experiment...
. 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 ponensModus ponens
In classical logic, modus ponendo ponens or implication elimination is a valid, simple argument form. It is related to another valid form of argument, modus tollens. Both Modus Ponens and Modus Tollens can be mistakenly used when proving arguments...
and one use of transposition
Transposition (logic)
In the methods of deductive reasoning in classical logic, transposition is the rule of inference that permits one to infer from the truth of "A implies B" the truth of "Not-B implies not-A", and conversely. Its symbolic expression is:...
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 tableTruth table
A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, boolean functions, and propositional calculus—to compute the functional values of logical expressions on each of their functional arguments, that is, on each combination of values taken by their...
.
| 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.
External links
- Modus Tollens at Wolfram MathWorld