In
classical logic, disjunctive syllogism (historically known as ''modus tollendo ponens'' (MTP),
Latin
Latin (, or , ) is a classical language belonging to the 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 the power of the ...
for "mode that affirms by denying") is a
valid argument form which is a
syllogism having a
disjunctive statement for one of its
premise
A premise or premiss is a true or false statement that helps form the body of an argument, which logically leads to a true or false conclusion. A premise makes a declarative statement about its subject matter which enables a reader to either agre ...
s.
An example in
English
English usually refers to:
* English language
* English people
English may also refer to:
Peoples, culture, and language
* ''English'', an adjective for something of, from, or related to England
** English national ide ...
:
# The breach is a safety violation, or it is not subject to fines.
# The breach is not a safety violation.
# Therefore, it is not subject to fines.
Propositional logic
In
propositional logic
Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations b ...
, disjunctive syllogism (also known as disjunction elimination and or elimination, or abbreviated ∨E), is a valid
rule of inference. If we are told that at least one of two statements is true; and also told that it is not the former that is true; we can
infer that it has to be the latter that is true. If ''P'' is true or ''Q'' is true and ''P'' is false, then ''Q'' is true. The reason this is called "disjunctive syllogism" is that, first, it is a syllogism, a three-step
argument, and second, it contains a logical disjunction, which simply means an "or" statement. "P or Q" is a disjunction; P and Q are called the statement's ''disjuncts''. The rule makes it possible to eliminate a
disjunction
In logic, disjunction is a logical connective typically notated as \lor and read aloud as "or". For instance, the English language sentence "it is raining or it is snowing" can be represented in logic using the disjunctive formula R \lor S ...
from a
logical proof. It is the rule that:
:
where the rule is that whenever instances of "
", and "
" appear on lines of a proof, "
" can be placed on a subsequent line.
Disjunctive syllogism is closely related and similar to
hypothetical syllogism
In classical logic, a hypothetical syllogism is a valid argument form, a syllogism with a conditional statement for one or both of its premises.
An example in English:
:If I do not wake up, then I cannot go to work.
:If I cannot go to work, then ...
, in that it is also a type of syllogism, and also the name of a rule of inference. It is also related to the
law of noncontradiction
In logic, the law of non-contradiction (LNC) (also known as the law of contradiction, principle of non-contradiction (PNC), or the principle of contradiction) states that contradictory propositions cannot both be true in the same sense at the sa ...
, one of the
three traditional laws of thought.
Formal notation
For a
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 form ...
that validates it, the ''disjunctive syllogism'' may be written in
sequent
In mathematical logic, a sequent is a very general kind of conditional assertion.
: A_1,\,\dots,A_m \,\vdash\, B_1,\,\dots,B_n.
A sequent may have any number ''m'' of condition formulas ''Ai'' (called " antecedents") and any number ''n'' of ass ...
notation:
:
where
is a
metalogic
Metalogic is the study of the metatheory of logic. Whereas ''logic'' studies how logical systems can be used to construct valid and sound arguments, metalogic studies the properties of logical systems.Harry GenslerIntroduction to Logic Routledge, ...
al symbol meaning that
is a
syntactic consequence
Logical consequence (also entailment) is a fundamental concept in logic, which describes the relationship between statement (logic), statements that hold true when one statement logically ''follows from'' one or more statements. A Validity (lo ...
of
, and
.
It may be expressed as a truth-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 ...
in the object language 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 form ...
.
Natural language examples
Here is an example:
# I will choose soup or I will choose salad.
# I will not choose soup.
# Therefore, I will choose salad.
Here is another example:
# It is red or it is blue.
# It is not blue.
# Therefore, it is red.
Inclusive and exclusive disjunction
Please observe that the disjunctive syllogism works whether 'or' is considered 'exclusive' or 'inclusive' disjunction. See below for the definitions of these terms.
There are two kinds of logical disjunction:
* ''
inclusive'' means "and/or"—at least one of them is true, or maybe both.
* ''
exclusive'' ("xor") means exactly one must be true, but they cannot both be.
The widely used English language concept of ''or'' is often ambiguous between these two meanings, but the difference is pivotal in evaluating disjunctive arguments.
This argument:
# P or Q.
# Not P.
# Therefore, Q.
is valid and indifferent between both meanings. However, only in the ''exclusive'' meaning is the following form valid:
# Either (only) P or (only) Q.
# P.
# Therefore, not Q.
With the ''inclusive'' meaning you could draw no conclusion from the first two premises of that argument. See
affirming a disjunct
The formal fallacy of affirming a disjunct also known as the fallacy of the alternative disjunct or a false exclusionary disjunct occurs when a deductive argument takes the following logical form:
:A or B
:A
:Therefore, not B
Or in logical op ...
.
Related argument forms
Unlike ''
modus ponens'' and ''
modus ponendo tollens
''Modus ponendo tollens'' (MPT; Latin: "mode that denies by affirming") is a valid rule of inference for propositional logic. It is closely related to ''modus ponens'' and ''modus tollendo ponens''.
Overview
MPT is usually described as having t ...
'', with which it should not be confused, disjunctive syllogism is often not made an explicit rule or axiom of
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 form ...
s, as the above arguments can be proven with a combination of
reductio ad absurdum
In logic, (Latin for "reduction to absurdity"), also known as (Latin for "argument to absurdity") or ''apagogical arguments'', is the form of argument that attempts to establish a claim by showing that the opposite scenario would lead to absu ...
and
disjunction elimination
In propositional logic, disjunction elimination (sometimes named proof by cases, case analysis, or or elimination), is the valid argument form and rule of inference that allows one to eliminate a disjunctive statement from a logical proof. It ...
.
Other forms of syllogism include:
*
hypothetical syllogism
In classical logic, a hypothetical syllogism is a valid argument form, a syllogism with a conditional statement for one or both of its premises.
An example in English:
:If I do not wake up, then I cannot go to work.
:If I cannot go to work, then ...
*
categorical 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 true ...
Disjunctive syllogism holds in classical propositional logic and
intuitionistic logic
Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems ...
, but not in some
paraconsistent logics.
[Chris Mortensen]
Inconsistent Mathematics
''Stanford encyclopedia of philosophy'', First published Tue Jul 2, 1996; substantive revision Thu Jul 31, 2008
See also
*
Stoic logic
Stoic logic is the system of propositional logic developed by the Stoic philosophers in ancient Greece.
It was one of the two great systems of logic in the classical world. It was largely built and shaped by Chrysippus, the third head of the Stoi ...
References
{{reflist
Rules of inference
Theorems in propositional logic
Classical logic
Paraconsistent logic