In
classical logic
Classical logic (or standard logic) or Frege–Russell logic is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy.
Characteristics
Each logical system in this c ...
, disjunctive syllogism (historically known as ''modus tollendo ponens'' (MTP),
Latin
Latin ( or ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken by the Latins (Italic tribe), Latins in Latium (now known as Lazio), the lower Tiber area aroun ...
for "mode that affirms by denying") is a
valid argument form which is a
syllogism
A syllogism (, ''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.
In its earliest form (defin ...
having a
disjunctive statement for one of its
premise
A premise or premiss is a proposition—a true or false declarative statement—used in an argument to prove the truth of another proposition called the conclusion. Arguments consist of a set of premises and a conclusion.
An argument is meaningf ...
s.
An example in
English:
# I will choose soup or I will choose salad.
# I will not choose soup.
# Therefore, I will choose salad.
Propositional logic
In
propositional logic
The propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. Sometimes, it is called ''first-order'' propositional logic to contra ...
, disjunctive syllogism (also known as disjunction elimination and or elimination, or abbreviated ∨E), is a valid
rule of inference
Rules of inference are ways of deriving conclusions from premises. They are integral parts of formal logic, serving as norms of the Logical form, logical structure of Validity (logic), valid arguments. If an argument with true premises follows a ...
. If it is known that at least one of two statements is true, and that it is not the former that is true; we can
infer that it has to be the latter that is true. Equivalently, if ''P'' is true or ''Q'' is true and ''P'' is false, then ''Q'' is true. The name "disjunctive syllogism" derives from its being a syllogism, a three-step
argument
An argument is a series of sentences, statements, or propositions some of which are called premises and one is the conclusion. The purpose of an argument is to give reasons for one's conclusion via justification, explanation, and/or persu ...
, and the use of a logical disjunction (any "or" statement.) For example, "P or Q" is a disjunction, where P and Q are called the statement's ''disjuncts''. The rule makes it possible to eliminate a
disjunction
In logic, disjunction (also known as logical disjunction, logical or, logical addition, or inclusive disjunction) is a logical connective typically notated as \lor and read aloud as "or". For instance, the English language sentence "it is ...
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, which is another rule of inference involving a syllogism. It is also related to the
law of noncontradiction, one of the
three traditional laws of thought.
Formal notation
For a
logical system
A formal system is an abstract structure and formalization of an axiomatic system used for deducing, using rules of inference, theorems from axioms.
In 1921, David Hilbert proposed to use formal systems as the foundation of knowledge in math ...
that validates it, the ''disjunctive syllogism'' may be written in
sequent notation as
:
where
is a
metalogic
Metalogic is 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. Logic concerns the truths that may be derived using a lo ...
al symbol meaning that
is a
syntactic consequence of
, and
.
It may be expressed as a truth-functional
tautology or
theorem
In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...
in the object language of propositional logic as
:
where
, and
are propositions expressed in some
formal system
A formal system is an abstract structure and formalization of an axiomatic system used for deducing, using rules of inference, theorems from axioms.
In 1921, David Hilbert proposed to use formal systems as the foundation of knowledge in ma ...
.
Natural language examples
Here is an example:
# It is red or it is blue.
# It is not blue.
# Therefore, it is red.
Here is another example:
# 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.
Strong form
''Modus tollendo ponens'' can be made stronger by using
exclusive disjunction
Exclusive or, exclusive disjunction, exclusive alternation, logical non-equivalence, or Logical_equality#Inequality, logical inequality is a Logical connective, logical operator whose negation is the logical biconditional. With two inputs, X ...
instead of inclusive disjunction as a premise:
:
Related argument forms
Unlike ''
modus ponens
In propositional logic, (; MP), also known as (), implication elimination, or affirming the antecedent, is a deductive argument form and rule of inference. It can be summarized as "''P'' implies ''Q.'' ''P'' is true. Therefore, ''Q'' must ...
'' and ''
modus ponendo tollens'', 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 and formalization of an axiomatic system used for deducing, using rules of inference, theorems from axioms.
In 1921, David Hilbert proposed to use formal systems as the foundation of knowledge in math ...
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 argument'', is the form of argument that attempts to establish a claim by showing that the opposite scenario would lead to absur ...
and
disjunction elimination.
Other forms of syllogism include:
*
hypothetical syllogism
*
categorical syllogism
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 logic
Paraconsistent logic is a type of non-classical logic that allows for the coexistence of contradictory statements without leading to a logical explosion where anything can be proven true. Specifically, paraconsistent logic is the subfield of log ...
s.
[Chris Mortensen]
Inconsistent Mathematics
''Stanford encyclopedia of philosophy'', First published Tue Jul 2, 1996; substantive revision Thu Jul 31, 2008
See also
*
Stoic logic
Stoicism is a school of Hellenistic philosophy that flourished in ancient Greece and Rome. The Stoics believed that the universe operated according to reason, ''i.e.'' by a God which is immersed in nature itself. Of all the schools of ancient p ...
*
Type of syllogism (disjunctive, hypothetical, legal, poly-, prosleptic, quasi-, statistical)
References
{{reflist
Rules of inference
Theorems in propositional logic
Classical logic
Paraconsistent logic