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

Guidelines for Mathematics articles

These standards, as with all Wikipedia guidelines, are not obligatory. However, it should be noted that any article that is seeking 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

Connective Name Symbol(s) Preferred Symbol(s) Template [itex] See
Negation NOT ¬ or ¬ or ~ ¬ {{not}} \neg Logical negation
Conjunction AND $\wedge$ or & & {{and}} \And Logical conjunction
Inclusive disjunction OR $\vee$ $\lor$ {{or-}} \vee Logical disjunction
Material implication IMPLIES $\rightarrow$ or $\Rightarrow$ or $\supset$ or → {{imp}} \rightarrow Material conditional
Material equivalence (biconditional) EQV or XNOR $\leftrightarrow$ or $\Leftrightarrow$ or = or $\equiv$ (for definitions, := or :$\equiv$ may be used) $\leftrightarrow$ {{eqv}} \leftrightarrow Logical biconditional, Logical equality, Logical equivalence
Neither-nor (joint denial) NOR or  ↓  {{nor-}} \downarrow Logical NOR
Not both (alternative denial) NAND {{nand}} \uparrow Alternative denial (nand)
Exclusive disjunction XOR $\nleftrightarrow$ or + or $\oplus$ or $\nleftrightarrow$ {{xor}} \nleftrightarrow XOR

Quantifiers

Quantifier Description Symbols Preferred Symbol Template [itex]
Universal For every x (x) or $\forall$ x or $\forall x$ $\forall x$ {{all}} \forall x
Existential There exists an x $\exists$x or $\exists x$ $\exists x$ {{exist}} \exists x

Metalogical Symbols

Name Description/Usage Symbol(s) Preferred Symbol(s) Template [itex] See
Definition $X\stackrel{\rm def}=y_1,y_2,\dots$ $\stackrel{\rm def}=$ $\stackrel{\rm def}=$ none \stackrel{\rm def}= Definition
Theorem $X \vdash Y$, $\vdash Z$, $A \vdash_S X$ $\vdash$ $\vdash$ {{tee}} \vdash Turnstile (symbol)
Semantic Entailment $A \models_L X$, $\models X$ $\models$ $\models$ {{models}} \models Double turnstile
True, tautology  $\vDash \top$ $\top$ or T or 1 $\top$ {{true}} \top Tee (symbol)
False, contradiction  $\vDash \neg\bot$ $\bot$ or F or 0 $\bot$ {{false}} \bot Falsum

Terminology

For consistency use the following terminology in Logic articles: Drafting in progress drafted cf Wikipedia talk:WikiProject Logic/Standards for notation#Terminology

It's good to talk (and a common language can only help.)

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.

Terminology used Preferred terminology Preferred meaning
signature signature a set of non-logical symbols
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,
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
or arity ≥0, i.e. including symbols below
propositional variable, propositional letter, propositional symbol, sentential variable, sentential letter, sentential symbol in propositional/sentential logic:
prop./sent. variable
in first-order logic:
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.

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.

Terminology used Preferred Terminology
domain, domain of discourse, universe of discourse, universe, carrier, underlying set
extension, denotation
structure
function, operator
property, attribute, relation (arity=1)
property (arity>1), relation (arity>1)

Relation between syntax and semantics

Terminology used Preferred Terminology
model
interpretation

Unsorted

Terminology used Preferred Terminology
propositional logic, sentential logic, propositional calculus, sentential calculus, statement logic, statement calculus
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