Wikipedia:WikiProject Logic/Standards for notation

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	<math>	See
Negation	NOT	¬ or $\neg$ or ~	$\neg$	{{not}}	\neg	Logical negation
Conjunction	AND	$\wedge$ or &	$\wedge$	{{and}}	\And	Logical conjunction
Inclusive disjunction	OR	$\vee$	$\lor$	{{or-}}	\vee	Logical disjunction
Material implication	IMPLIES	$\rightarrow$ or $\Rightarrow$ or $\supset$ or $\to$	$\to$	{{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	$\downarrow$ or ↓	$\downarrow$	{{nor-}}	\downarrow	Logical NOR
Not both (alternative denial)	NAND	$\uparrow$	$\uparrow$	{{nand}}	\uparrow	Alternative denial (Nand)
Exclusive disjunction	XOR	$\nleftrightarrow$ or + or $\oplus$ or ≠	$\nleftrightarrow$	{{xor}}	\nleftrightarrow	XOR

Quantifiers

Quantifier	Description	Symbols	Preferred Symbol	Template	<math>
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	<math>	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 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, 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	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