Editing Logical disjunction (section)

==Classical disjunction==

=== Semantics ===

In the [[semantics of logic]], classical disjunction is a [[truth function]]al [[logical operation|operation]] which returns the [[truth value]] ''true'' unless both of its arguments are ''false''. Its semantic entry is standardly given as follows:{{efn|For the sake of generality across classical systems, this entry suppresses the parameters of evaluation. The [[double turnstile]] [[List of logic symbols|symbol]] <math> \models </math> here is intended to mean "semantically entails".<!--this is hardly readable without context, it means  "(X ⊨  phi ∨ psi) if [(X  ⊨  phi) or (X  ⊨  psi)], for any X", in other words, " '∨' means 'or' ".-->}}

:: <math> \models \phi \lor \psi</math> &nbsp; &nbsp; if &nbsp; &nbsp; <math> \models \phi</math> &nbsp; &nbsp; or &nbsp; &nbsp; <math>\models \psi</math> &nbsp; &nbsp; or &nbsp; &nbsp; both

This semantics corresponds to the following [[truth table]]:<ref name=":1" />

{{2-ary truth table|0|1|1|1|<math>A \lor B</math>}}

===Defined by other operators===
In [[classical logic]] systems where logical disjunction is not a primitive, it can be defined in terms of the primitive ''[[logical conjunction|and]]'' (<math>\land</math>) and ''[[logical negation|not]]'' (<math>\lnot</math>) as:

:<math>A \lor B =  \neg ((\neg A) \land (\neg B))</math>.

Alternatively, it may be defined in terms of ''[[material conditional|implies]]'' (<math>\to</math>) and ''not'' as:<ref>{{cite book
 |last=Walicki
 |first=Michał
 |author-link=
 |date=2016
 |title=Introduction to Mathematical Logic
 |url=https://www.worldscientific.com/doi/abs/10.1142/9783
 |publisher=WORLD SCIENTIFIC
 |page=150
 |doi=10.1142/9783
 |isbn=978-9814343879
}}</ref>
:<math>A \lor B = (\lnot A) \to B</math>.
The latter can be checked by the following truth table:

{{2-ary truth table
|1|1|0|0|<math>\neg A</math>
|thick
|0|1|1|1|<math>\neg A \rightarrow B</math>
|
|0|1|1|1|<math>A \or B</math>
}}

It may also be defined solely in terms of <math>\to</math>:
:<math>A \lor B = (A \to B) \to B</math>.
It can be checked by the following truth table:

{{2-ary truth table
|1|1|0|1|<math>A \rightarrow B</math>
|thick
|0|1|1|1|<math>(A \rightarrow B) \rightarrow B</math>
|
|0|1|1|1|<math>A \or B</math>
}}

<!-- === Proof theory === -->

===Properties===

The following properties apply to disjunction:

*[[Associativity]]: <math>a \lor (b \lor c) \equiv (a \lor b) \lor c </math>'''<ref name=":13">{{Cite book |last=Howson |first=Colin |title=Logic with trees: an introduction to symbolic logic |date=1997 |publisher=Routledge |isbn=978-0-415-13342-5 |location=London; New York |pages=38}}</ref>'''
*[[Commutativity]]: <math>a \lor  b \equiv b \lor a </math>
*[[Distributivity]]: <math>(a \land (b \lor c)) \equiv ((a \land b) \lor (a \land c))</math>
:::<math>(a \lor (b \land c)) \equiv ((a \lor b) \land (a \lor c))</math>
:::<math>(a \lor (b \lor c)) \equiv ((a \lor b) \lor (a \lor c))</math>
:::<math>(a \lor (b \equiv c)) \equiv ((a \lor b) \equiv (a \lor c))</math>

*[[Idempotency]]: <math>a \lor a \equiv a </math>
*[[Monotonicity]]: <math>(a \rightarrow b) \rightarrow ((c \lor a) \rightarrow (c \lor b))</math>
:::<math>(a \rightarrow b) \rightarrow ((a \lor c) \rightarrow (b \lor c))</math>

*''Truth-preserving'': The interpretation under which all variables are assigned a [[truth value]] of 'true', produces a truth value of 'true' as a result of disjunction.
*''Falsehood-preserving'': The interpretation under which all variables are assigned a [[truth value]] of 'false', produces a truth value of 'false' as a result of disjunction.