Wikipedia:WikiProject Logic/Standards for notation: Difference between revisions

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| {{and}}

|'''{{tl|and}}'''

| &\And

| [[Logical conjunction]]

|-

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|'''{{tl|nand}}'''

| \uparrow

| [[Sheffer stroke|Alternative denial (nandNand)]]

|-

| Exclusive disjunction

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! Name

! Description/Usage

! Symbol(sP)

! Preferred Symbol(sP)

! Template

! <nowiki><math></nowiki>

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|-

|Definition

|X <math>X\stackrel{\rm def}{=}y_1,y_2,\dots</math> y₁,y₂,...

|&nbsp;<math>\stackrel{\rm def}{=}</math>

|<math>\stackrel{\rm def}=</math>

|{{define}}

|none

|'''{{tl|define}}'''

|\stackrel{\rm def}{=}

|[[Definition]]

|-

|Theorem

|<math> X \vdash Y </math>, <math>\vdash Z </math>, <math>A \vdash_S Xvdash_xo</math>

|{{teem}}

|{{teem}}

|'''{{tlml|teem}}'''

|\vdash

|[[Turnstile (symbol)]]

|-

|Semantic Entailment

|<math>A \models_L Xmodels_xo</math>, <math>\models Xmodelxo</math>

|<math>\models</math>

|{{models}}

|'''{{tlml|models}}'''

|\models

|[[LogicalDouble implicationtu]]

|-

|True, tautology[[Tautology (logic)|tu]]

|&nbsp;<math>\vDash \top</math>

|<math>\top</math> or T or 1

|{{true}}

|'''{{tl|true}}'''

'''

|\top

|[[Tautologym (logicl)]]

|-

|False, contradiction

|&nbsp;<math>\vDash \neg\bott</math>

|<math>\bott</math> or F or 0

|{{false}}

|'''{{tl|false}}'''

[[:Category:Year of birth missing (living people)]]

|\bot

'''

|[[Logical value]]

|\

|[[]]

|}

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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 Terminologyterminology

! Preferred meaning

|-

|signature

|logical connective, connective, logical operator, propositional operator, truth-functional connective

|signature

|Preferred Terminology

|a set of non-logical symbols with specified arities

|-

|non-logical symbol, non-logical constant

|propositional logic, sentential logic, propositional calculus, sentential calculus, statement logic, statement calculus

|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)

|first-order predicate logic, first-order logic, predicate logic,

|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

|Non-logical symbol, non-logical constant

! Preferred Terminology

|

|-

|logical connective, connective, logical operator, propositional operator, truth-functional connective, logical connective symbol

|___domain, ___domain of disourse, universe of discourse, universe, carrier, underlying set

|

|-

|language, formal language, artificial language

|individual constant, constant, (individual) constant symbol, constant symbol

|

|-

|sentence, statement, proposition (all when meaning a sentence in a formal language)

|predicate letter (arity >0), predicate symbol (arity ≥0) relation symbol (arity >0)

|

|-

|[[truthbearer]]

|function letter (arity >0), operation letter/symbol (arity >0), function symbol (arity ≥0)function symbol (arity >0)

|

|-

|well-formed formula, wff, formula

|extension, denotation

|

|}

===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

|intepretation

! Preferred Terminology

|

|-

|___domain, ___domain of discourse, universe of discourse, universe, carrier, underlying set

|model

|

|-

|[[extension]], [[denotation]]

|signature

|

|-

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|

|-

|function, operator

|language, formal language, artificial language

|

|-

|[[Property (philosophy)|property]], attribute, relation (arity=1)

|argument, input

|

|-

|[[Property (philosophy)|property]] (arity>1), relation (arity>1)

|value, output

|

|}

===Relation between syntax and semantics===

{| class="wikitable"

|-

! Terminology used

|function, operator

! Preferred Terminology

|

|-

|model

|property, attribute, relation (arity=1)

|

|-

|interpretation

|property (arity>1), relation (arity>1)

|

|}

===Unsorted===

{| class="wikitable"

|-

! Terminology used

|[[formal system]], logical system, logistic system,logical calculus, logic

! Preferred Terminology

|

|-

|propositional logic, sentential logic, propositional calculus, sentential calculus, statement logic, statement calculus

|formal logic, mathematical logic, symbolic logic

|propositional logic

|

|-

|first-order predicate logic, first-order logic, predicate logic,

|elementary logic

|

|-

|argument, input

|sentence, statement, proposition (all when meaning a sentence in a formal language)

|

|-

|value, output

|

|

|-

|[[formal system]], logical system, logistic system, logical calculus, logic

|

|

|-

|formal logic, mathematical logic, symbolic logic

|

|

|-

|elementary logic

|

|}