View source for Wikipedia:WikiProject Logic/Standards for notation

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In addition to the standards suggested for all Wikipedia articles, special attention to the following while expanding logic articles:

'''Guidelines for Philosophy articles'''
* [[Wikipedia:WikiProject Philosophy/readability|Guidelines for readability in philosophy articles]].
* [[Wikipedia:WikiProject Philosophy/style guide|Style guide for philosophy articles]].
* [[Wikipedia:WikiProject Philosophy/Proposal for criticisms|Requirements for criticisms in philosophy articles]]
* [[Wikipedia:Inline Citation|In-text citations]] should be made using the [[Wikipedia:Footnotes|Cite.php system]].

'''Guidelines for Mathematics articles'''

* [[Wikipedia:WikiProject Mathematics/Conventions]] (a working list of terminology conventions)
* [[Wikipedia:Naming conventions (theorems)]]
* [[Wikipedia:Manual of Style (mathematics)]]
* [[Wikipedia:Algorithms on Wikipedia]]
* [[Wikipedia:Scientific citation guidelines]]

These standards, as with all [[Wikipedia:Policies and guidelines|Wikipedia guidelines]], are not obligatory. However, it should be noted that any article that is seeking [[Wikipedia:What is a featured article?|featured article]] status should comply with these standards.

Note that new standards should be subjected to consensus building before being added here (a consensus should be reached on the discussion page).

== Standard ==
'''For consistency use the following preferred symbols and terminology in Logic articles'''

It is useful to have an agreed set of symbols and terminology. Not only do symbols vary from author to author, but any symbol may be written in a variety of fonts which may or may not appear on various browsers. The aim is consistency and legibility

== Symbols ==
For consistency use the following preferred symbols in Logic articles:

=== Truth Functional Connectives ===

{| class="wikitable"
|-
! Connective
! Name
! Symbol(s)
! Preferred Symbol(s)
! Template
! <nowiki><math></nowiki>
! See
|-
| Negation
| NOT
| &not; or {{not}} or ~
| {{not}}
| '''{{tl|not}}'''
| \neg
| [[Logical negation]]
|-
| Conjunction
| AND
| <math>\wedge</math> or &
| {{and}}
|'''{{tl|and}}'''
| \And
| [[Logical conjunction]]
|-
| Inclusive disjunction
| OR
| <math>\vee</math>
| {{or-}}
|'''{{tl|or-}}'''
| \vee
| [[Logical disjunction]]
|-
| Material implication
| IMPLIES
| <math>\rightarrow</math> or <math>\Rightarrow</math> or <math>\supset</math> or {{imp}}
| {{imp}}
|'''{{tl|imp}}'''
| \rightarrow
| [[Material conditional]]
|-
| Material equivalence (biconditional)
| EQV or XNOR
| {{eqv}} or <math>\Leftrightarrow</math> or = or {{ident}} (for definitions, := or :{{ident}} may be used)
| {{eqv}}
|'''{{tl|eqv}}'''
| \leftrightarrow
| [[Logical biconditional]], [[Logical equality]], [[Logical equivalence]]
|-
| Neither-nor (joint denial)
| NOR
| {{nor-}} or '''&nbsp;&darr;&nbsp;'''
| {{nor-}}
|'''{{tl|nor-}}'''
| \downarrow
| [[Logical NOR]]
|-
| Not both (alternative denial)
| NAND
| {{nand}}
| {{nand}}
|'''{{tl|nand}}'''
| \uparrow
| [[Sheffer stroke|Alternative denial (Nand)]]
|-
| Exclusive disjunction
| XOR
| {{xor}} or '''+''' or ''' <math>\oplus</math> ''' or '''&ne;'''
| {{xor}}
| '''{{tl|xor}}'''
| \nleftrightarrow
| [[XOR]]
|}

=== Quantifiers ===

{| class="wikitable"
|-
! Quantifier
! Description
! Symbols
! Preferred Symbol
! Template
! <nowiki><math></nowiki>
|-
| Universal
| For every x
| (x) or {{all}} ''x'' or <math> \forall x </math>
| <math> \forall x </math>
|'''{{tl|all}}'''
| <nowiki>\forall x</nowiki>
|-
| Existential
| There exists an x
| {{exist}}''x'' or <math>  \exists x </math>
| <math>  \exists x </math>
|'''{{tl|exist}}'''
| <nowiki>\exists x</nowiki>
|}

=== Metalogical Symbols ===

{| class="wikitable"
|-
! Name
! Description/Usage
! Symbol(P)
! Preferred Symbol(P)
! Template
! <nowiki><math></nowiki>
! See
|-
|Definition
|<math>X\stackrel{\rm def}=y_1,y_2,\dots</math>
|<math>\stackrel{\rm def}=</math>
|<math>\stackrel{\rm def}=</math>
|none
|\stackrel{\rm def}=
|[[Definition]]
|-
|Theorem
|<math> X \vdash Y </math>, <math>\vdash Z </math>, <math>A \vdash_xo</math>
|{{m}}
|{{m}}
|'''{{ml|m}}'''
|\vdash
|[[Turnstile (symbol)]]
|-
|Semantic Entailment
|<math>A \models_xo</math>,  <math>\modelxo</math>
|<math>\models</math>
|{{models}}
|'''{{ml|models}}'''
|\models
|[[Double tu]]
|-
|True, [[Tautology (logic)|tu]]
|&nbsp;<math>\vDash \top</math>
|<math>\top</math> or T or 
|{{}}
|'''
'''
|\top
|[[m (l)]]
|-
|False, 
|&nbsp;<math>\vDash \neg\t</math>
|<math>\t</math> or F or 0
|{{false}}
|'''
[[:Category:Year of birth missing (living people)]]
'''
|\
|[[]]
|}

==Terminology==
For consistency use the following terminology in Logic articles:
''Drafting in progress drafted'' cf [[Wikipedia talk:WikiProject Logic/Standards for notation#Terminology]]
<blockquote>
'''It's good to talk (and a common language can only help.)'''
</blockquote>

===Common basis for syntax and semantics===
One can talk about syntax while ignoring any possible semantics, or talk about semantics while ignoring that there might be a language describing them. The terms in the following table are common to both aspects.

{| class="wikitable"
|-
! Terminology used
! Preferred terminology
! Preferred meaning
|-
|signature
|signature
|a set of non-logical symbols with specified arities
|-
|non-logical symbol, non-logical constant
|non-logical symbol
|any of the symbols below
|-
|function letter (arity >0), operation letter/symbol (arity >0), function symbol (arity ≥0), function symbol (arity >0) 
|function symbol
|either arity >0, i.e. excl. constant symbols,<br>or arity ≥0, i.e. including constant symbols
|-
|individual constant, constant, (individual) constant symbol, constant symbol 
|constant symbol
|
|-
|predicate letter (arity >0), predicate symbol (arity ≥0), relation symbol (arity >0) 
|predicate symbol or relation symbol
|either arity >0, i.e. excl. symbols below<br>or arity ≥0, i.e. including symbols below
|-
|propositional variable, propositional letter, propositional symbol, sentential variable, sentential letter, sentential symbol 
|in propositional/sentential logic:<br>prop./sent. variable<br>in first-order logic:<br>nullary predicate/relation symbol
|
|}
'''Note:''' Nullary function symbols are constant symbols, and nullary predicate/relation symbols are propositional/sentential symbols. What differs about first-order logic between authors is 1) whether constant symbols are called (nullary) function symbols, and 2) whether proposition symbols are even allowed.

===Syntax===
The terms in the following table are used when working with syntax and are only marginally related to semantics.

{| class="wikitable"
|-
! Terminology used
! Preferred Terminology
|-
|logical connective, connective, logical operator, propositional operator, truth-functional connective, logical connective symbol
|
|-
|language, formal language, artificial language
|
|-
|sentence, statement, proposition (all when meaning a sentence in a formal language)
|
|-
|[[truthbearer]]
|
|-
|well-formed formula, wff, formula
|
|}

===Semantics===
The terms in the following table relate to semantics; they are not needed when discussing only syntax, although of course they motivate the syntax.

{| class="wikitable"
|-
! Terminology used
! Preferred Terminology
|-
|domain, domain of discourse, universe of discourse, universe, carrier, underlying set 
|
|-
|[[extension]], [[denotation]]
|
|-
|structure
|
|-
|function, operator
|
|-
|[[Property (philosophy)|property]], attribute, relation (arity=1)
|
|-
|[[Property (philosophy)|property]] (arity>1), relation (arity>1)
|
|}

===Relation between syntax and semantics===
{| class="wikitable"
|-
! Terminology used
! Preferred Terminology
|-
|model
|
|-
|interpretation
|
|}

===Unsorted===
{| class="wikitable"
|-
! Terminology used
! Preferred Terminology
|-
|propositional logic, sentential logic, propositional calculus, sentential calculus, statement logic, statement calculus
|propositional logic
|-
|first-order predicate logic, first-order logic, predicate logic, 
|
|-
|argument, input
|
|-
|value, output
|
|-
|[[formal system]], logical system, logistic system, logical calculus,  logic 
|
|-
|formal logic, mathematical logic, symbolic logic
|
|-
|elementary logic
|
|}