Disjunction introduction or addition (also called or introduction)
is 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 in ...
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
Inferences are steps in reasoning, moving from premises to logical consequences; etymologically, the word '' infer'' means to "carry forward". Inference is theoretically traditionally divided into deduction and induction, a distinction that in ...
that if ''P'' is true, then ''P or Q'' must be true.
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 ...
:
: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
In logic, logical form of a statement is a precisely-specified semantic version of that statement in a formal system. Informally, the logical form attempts to formalize a possibly ambiguous statement into a statement with a precise, unambiguo ...
, 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 An immediate inference is an inference which can be made from only one statement or proposition. For instance, from the statement "All toads are green", the immediate inference can be made that "no toads are not green" or "no toads are non-green" ( ...
, as it has a single proposition in its premises.
Disjunction introduction is not a rule in some
paraconsistent logic
A paraconsistent logic is an attempt at a logical system to deal with contradictions in a discriminating way. Alternatively, paraconsistent logic is the subfield of logic that is concerned with studying and developing "inconsistency-tolerant" syste ...
s because in combination with other rules of logic, it leads to
explosion
An explosion is a rapid expansion in volume associated with an extreme outward release of energy, usually with the generation of high temperatures and release of high-pressure gases. Supersonic explosions created by high explosives are known ...
(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 asse ...
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
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 form ...
;
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 th ...
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