In
propositional logic, ''modus tollens'' () (MT), also known as ''modus tollendo
tollens'' (
Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through ...
for "method of removing by taking away") and denying the consequent, is a
deductive argument form
In logic, logical form of a Statement (logic), statement is a precisely-specified Semantics, semantic version of that statement in a formal system. Informally, the logical form attempts to formalize a possibly Syntactic ambiguity, ambiguous sta ...
and a
rule of inference
In the philosophy of logic, a rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their syntax, and returns a conclusion (or conclusions). For example, the rule of ...
. ''Modus tollens'' takes the form of "If P, then Q. Not Q. Therefore, not P." It is an application of the general truth that if a statement is true, then so is its
contrapositive
In logic and mathematics, contraposition refers to the inference of going from a conditional statement into its logically equivalent contrapositive, and an associated proof method known as proof by contraposition. The contrapositive of a stat ...
. The form shows that
inference
Inferences are steps in reasoning, moving from premises to logical consequences; etymologically, the word ''wikt:infer, infer'' means to "carry forward". Inference is theoretically traditionally divided into deductive reasoning, deduction and in ...
from ''P implies Q'' to ''the negation of Q implies the negation of P'' is a
valid argument.
The history of the inference rule ''modus tollens'' goes back to
antiquity
Antiquity or Antiquities may refer to:
Historical objects or periods Artifacts
*Antiquities, objects or artifacts surviving from ancient cultures
Eras
Any period before the European Middle Ages (5th to 15th centuries) but still within the histo ...
. The first to explicitly describe the argument form ''modus tollens'' was
Theophrastus
Theophrastus (; grc-gre, Θεόφραστος ; c. 371c. 287 BC), a Greek philosopher and the successor to Aristotle in the Peripatetic school. He was a native of Eresos in Lesbos.Gavin Hardy and Laurence Totelin, ''Ancient Botany'', Routle ...
.
''Modus tollens'' is closely related to ''
modus ponens
In propositional logic, ''modus ponens'' (; MP), also known as ''modus ponendo ponens'' (Latin for "method of putting by placing") or implication elimination or affirming the antecedent, is a deductive argument form and rule of inference ...
''. There are two similar, but
invalid, forms of argument:
affirming the consequent and
denying the antecedent. See also
contraposition and
proof by contrapositive.
Explanation
The form of a ''modus tollens'' argument resembles a
syllogism
A syllogism ( grc-gre, συλλογισμός, ''syllogismos'', 'conclusion, inference') is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two propositions that are asserted or assumed to be tru ...
, with two premises and a conclusion:
:If ''P'', then ''Q''.
:Not ''Q''.
:Therefore, not ''P''.
The first premise is a
conditional
Conditional (if then) may refer to:
*Causal conditional, if X then Y, where X is a cause of Y
*Conditional probability, the probability of an event A given that another event B has occurred
*Conditional proof, in logic: a proof that asserts a co ...
("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:
:If the dog detects an intruder, the dog will bark.
:The dog did not bark.
:Therefore, no intruder was detected by the 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 dog ''detects'' an intruder". The thing of importance is that the dog detects or does not detect an intruder, not whether 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.
Another example:
:If Rex is a chicken, then he is a bird.
:Rex is not a bird.
:Therefore, Rex is not a chicken.
Relation to ''modus ponens''
Every use of ''modus tollens'' can be converted to a use of ''
modus ponens
In propositional logic, ''modus ponens'' (; MP), also known as ''modus ponendo ponens'' (Latin for "method of putting by placing") or implication elimination or affirming the antecedent, is a deductive argument form and rule of inference ...
'' 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.
Formal notation
The ''modus tollens'' rule can be stated formally as:
:
where
stands for the statement "P implies Q".
stands for "it is not the case that Q" (or in brief "not Q"). Then, whenever "
" and "
" each appear by themselves as a line of a
proof
Proof most often refers to:
* Proof (truth), argument or sufficient evidence for the truth of a proposition
* Alcohol proof, a measure of an alcoholic drink's strength
Proof may also refer to:
Mathematics and formal logic
* Formal proof, a c ...
, then "
" can validly be placed on a subsequent line.
The ''modus tollens'' rule may be written in
sequent notation:
:
where
is a
metalogical symbol meaning that
is a
syntactic consequence of
and
in some
logical system
A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system.
A for ...
;
or as the statement of a functional
tautology or
theorem
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of t ...
of propositional logic:
:
where
and
are propositions expressed in some
formal system
A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system.
A fo ...
;
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 is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...
:
:
:
:
("P is a subset of Q. x is not in Q. Therefore, x is not in P.")
Also in first-order
predicate logic:
:
:
:
("For all x if x is P then x is Q. y is not 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.
Justification via truth table
The validity of ''modus tollens'' can be clearly demonstrated through a
truth table
A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, boolean functions, and propositional calculus—which sets out the functional values of logical expressions on each of their functional arg ...
.
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
Via ''reductio ad absurdum''
Via contraposition
Correspondence to other mathematical frameworks
Probability calculus
''Modus tollens'' represents an instance of the
law of total probability combined with
Bayes' theorem
In probability theory and statistics, Bayes' theorem (alternatively Bayes' law or Bayes' rule), named after Thomas Bayes, describes the probability of an event, based on prior knowledge of conditions that might be related to the event. For examp ...
expressed as:
,
where the conditionals
and
are obtained with (the extended form of)
Bayes' theorem
In probability theory and statistics, Bayes' theorem (alternatively Bayes' law or Bayes' rule), named after Thomas Bayes, describes the probability of an event, based on prior knowledge of conditions that might be related to the event. For examp ...
expressed as:
and
.
In the equations above
denotes the probability of
, and
denotes the
base rate (aka.
prior probability) of
. The
conditional probability generalizes the logical statement
, i.e. in addition to assigning TRUE or FALSE we can also assign any probability to the statement. Assume that
is equivalent to
being TRUE, and that
is equivalent to
being FALSE. It is then easy to see that
when
and
. This is because
so that
in the last equation. Therefore, the product terms in the first equation always have a zero factor so that
which is equivalent to
being FALSE. Hence, the
law of total probability combined with
Bayes' theorem
In probability theory and statistics, Bayes' theorem (alternatively Bayes' law or Bayes' rule), named after Thomas Bayes, describes the probability of an event, based on prior knowledge of conditions that might be related to the event. For examp ...
represents a generalization of ''modus tollens''.
Subjective logic
''Modus tollens'' represents an instance of the abduction operator in
subjective logic expressed as:
,
where
denotes the subjective opinion about
, and
denotes a pair of binomial conditional opinions, as expressed by source
. The parameter
denotes the
base rate (aka. the
prior probability) of
. The abduced marginal opinion on
is denoted
. The conditional opinion
generalizes the logical statement
, i.e. in addition to assigning TRUE or FALSE the source
can assign any subjective opinion to the statement. The case where
is an absolute TRUE opinion is equivalent to source
saying that
is TRUE, and the case where
is an absolute FALSE opinion is equivalent to source
saying that
is FALSE. The abduction operator
of
subjective logic produces an absolute FALSE abduced opinion
when the conditional opinion
is absolute TRUE and the consequent opinion
is absolute FALSE. Hence, subjective logic abduction represents a generalization of both ''modus tollens'' and of the
Law of total probability combined with
Bayes' theorem
In probability theory and statistics, Bayes' theorem (alternatively Bayes' law or Bayes' rule), named after Thomas Bayes, describes the probability of an event, based on prior knowledge of conditions that might be related to the event. For examp ...
.
[Audun Jøsang 2016:p.92]
See also
*
*
*
*
*
*
*
*
*
Notes
Sources
* Audun Jøsang, 2016,
Subjective Logic; A formalism for Reasoning Under Uncertainty' Springer, Cham,
External links
*
' at Wolfram MathWorld
{{DEFAULTSORT:Modus Tollens
Classical logic
Rules of inference
Latin logical phrases
Theorems in propositional logic