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 readability in philosophy articles. Style guide for philosophy articles. Requirements for criticisms in philosophy articles In-text citations should be made using the 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 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 <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