Editing Propositional calculus (section)

{{Short description|Branch of logic}}
{{Distinguish|Propositional analysis|predicate calculus}}
{{Use dmy dates|date=February 2021}}
The '''propositional calculus'''{{refn|group=lower-alpha|Many sources write this with a definite article, as '''the''' propositional calculus, while others just call it propositional calculus with no article.}} is a branch of [[logic]].<ref name=":1" /> It is also called '''propositional logic''',<ref name=":2"/> '''statement logic''',<ref name=":1" /> '''sentential calculus''',<ref name=":18"/> '''sentential logic''',<ref name="Hilbert_Ackermann"/><ref name=":1" /> or sometimes '''zeroth-order logic'''.{{efn|Zeroth-order logic is sometimes used to denote a [[quantifier (logic)|quantifier]]-free predicate logic. That is, propositional logic extended with functions, relations, and constants.<ref name="tao"/>}}<ref name="ms52"/><ref name=":10"/><ref name=":11"/> Sometimes, it is called '''''first-order'' propositional logic'''<ref name="ms53"/> to contrast it with [[System F]], but it should not be confused with [[first-order logic]]. It deals with [[propositions]]<ref name=":1" /> (which can be [[Truth value|true or false]])<ref name=":22"/> and relations between propositions,<ref name="ms54"/> including the construction of arguments based on them.<ref name="ms55"/> Compound propositions are formed by connecting propositions by [[logical connective]]s representing the [[truth function]]s of [[Logical conjunction|conjunction]], [[Logical disjunction|disjunction]], [[Material conditional|implication]], [[Logical biconditional|biconditional]], and [[negation]].<ref name=":5"/><ref name=":0"/><ref name=":3"/><ref name=":12"/> Some sources include other connectives, as in the table below.

Unlike [[first-order logic]], propositional logic does not deal with non-logical objects, predicates about them, or [[Quantifier (logic)|quantifiers]]. However, all the machinery of propositional logic is included in first-order logic and higher-order logics. In this sense, propositional logic is the foundation of first-order logic and higher-order logic.

Propositional logic is typically studied with a [[formal language]],{{efn|For propositional logic, the formal language used is a ''propositional language''.}} in which propositions are represented by letters, which are called ''[[propositional variable]]s''. These are then used, together with symbols for connectives, to make ''[[propositional formula]]''. Because of this, the propositional variables are called ''[[atomic formula]]s'' of a formal propositional language.<ref name=":0" /><ref name=":2" /> While the atomic propositions are typically represented by letters of the [[alphabet]],{{efn|Not to be confused with the formal language's [[Formal_language#Words_over_an_alphabet|alphabet]].}}<ref name=":0" /> there is a variety of notations to represent the logical connectives. The following table shows the main notational variants for each of the connectives in propositional logic.

{| class="wikitable"
|+ Notational variants of the connectives{{efn|See all possible [[Truth_table#Sentential_operator_truth_tables|connectives on truth-functional propositional logic]] with some of their properties.}}<ref name="ms56"/><ref name=":23"/>
|-
! Connective
! Symbol
|-
| [[Logical conjunction|AND]]
| <math>A \land B</math>, <math>A \cdot B</math>, <math>AB</math>, <math>A \& B</math>, <math>A \&\& B</math>
|-
| [[Logical biconditional|equivalent]]
| <math>A \equiv B</math>, <math>A \Leftrightarrow B</math>, <math>A \leftrightharpoons B</math>
|-
| [[Material conditional|implies]]
| <math>A \Rightarrow B</math>, <math>A \supset B</math>, <math>A \rightarrow B</math>
|-
| [[Sheffer stroke|NAND]]
| <math>A \overline{\land} B</math>, <math>A \mid B</math>, <math>\overline{A \cdot B}</math>
|-
| nonequivalent
| <math>A \not\equiv B</math>, <math>A \not\Leftrightarrow B</math>, <math>A \nleftrightarrow B</math>
|-
| [[Logical NOR|NOR]]
| <math>A \overline{\lor} B</math>, <math>A \downarrow B</math>, <math>\overline{A+B}</math>
|-
| [[Negation|NOT]]
| <math>\neg A</math>, <math>-A</math>, <math>\overline{A}</math>, <math>\sim A</math>
|-
| [[Logical disjunction|OR]]
| <math>A \lor B</math>, <math>A + B</math>, <math>A \mid B</math>, <math>A \parallel B</math>
|-
| [[XNOR gate|XNOR]]
| <math>A \odot B</math>
|-
| [[Exclusive or|XOR]]
| <math>A \underline{\lor} B</math>, <math>A \oplus B</math>
|}

The most thoroughly researched branch of propositional logic is '''classical truth-functional propositional logic''',<ref name=":1" /> in which formulas are interpreted as having precisely one of two possible [[truth value]]s, the truth value of ''true'' or the truth value of ''false''.<ref name="ms57"/> The [[principle of bivalence]] and the [[law of excluded middle]] are upheld. By comparison with [[first-order logic]], truth-functional propositional logic is considered to be ''zeroth-order logic''.<ref name=":10" /><ref name=":11" />