Disjunction introduction or addition (also called or introduction)
is a
rule of inference of
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 ...
and almost every other
deduction 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 ...
. The rule makes it possible to introduce
disjunctions to
logical proofs. It is the
inference that if ''P'' is true, then ''P or Q'' must be true.
An example in
English:
:Socrates is a man.
:Therefore, Socrates is a man or pigs are flying in formation over the English Channel.
The rule can be expressed as:
:
where the rule is that whenever instances of "
" appear on lines of a proof, "
" can be placed on a subsequent line.
More generally it's also a simple
valid argument form, this means that if the premise is true, then the conclusion is also true as any rule of inference should be, and an
immediate inference, as it has a single proposition in its premises.
Disjunction introduction is not a rule in some
paraconsistent logics because in combination with other rules of logic, it leads to
explosion (i.e. everything becomes provable) and paraconsistent logic tries to avoid explosion and to be able to reason with contradictions. One of the solutions is to introduce disjunction with over rules. See .
Formal notation
The ''disjunction introduction'' rule 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 of
in some
logical system;
and 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 ...
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 ...
.
References
{{DEFAULTSORT:Disjunction Introduction
Rules of inference
Paraconsistent logic
Theorems in propositional logic