List of logic symbols

In logic, a set of symbols is commonly used to express logical representation. The following table lists many common symbols, together with their name, how they should be read out loud, and the related field of mathematics. In addition, the subsequent columns contain an informal explanation, a short example, the Unicode location, the name for use in HTML documents, and the LaTeX symbol. __TOC__

Propositional logic

+Logical connectives Logic Name Symbol Unicodevalue(hexadecimal) HTMLcodes LaTeXsymbol Read as Explanation Diagram Negation NOT or Conjunction or AND Although this character is available in LaTeX, the MediaWiki TeX system does not support it. Inclusive disjunction OR Exclusive disjunction XOR, either ... or ... Non-conjunction (Sheffer stroke) NAND, neither ... nor ... , Non-disjunction \downarrowNOR \overline{\vee}Conditional ⇒ Implies,thenIn other mathematical contexts may indicate the domain and codomain of a function \to or \rightarrowIn other mathematical contexts may mean superset. \supsetBiconditional If and only if, iff, XNOR U+2299 \odot

+ Other symbols Logic Name Symbol Unicodevalue(hexadecimal) HTMLcodes LaTeXsymbol Read as Category Explanation Examples true (tautology) top, truth, tautology, verum, full clause propositional logic, Boolean algebra, first-order logic denotes a proposition that is always true. The proposition is always true since at least one of the two is unconditionally true. false (contradiction) bottom, falsity, contradiction, falsum, empty clause propositional logic, Boolean algebra, first-order logic denotes a proposition that is always false. The symbol ⊥ may also refer to perpendicular lines. The proposition is always false since at least one of the two is unconditionally false. precedence grouping ( ) parentheses; brackets almost all logic syntaxes, as well as metalanguage Perform the operations inside the parentheses first. , but .

Advanced or rarely used + SymbolUnicodevalue(hexadecimal)Logic Name ⌐U+2310reversed not sign ⨇U+2A07two logical AND operator

First-order logic + Symbol Unicodevalue(hexadecimal) HTMLcodes LaTeXsymbol Logic Name Read as Category Explanation Examples ∀ universal quantification given any, for all, for every, for each, for any first-order logic or says "given any , has property ." existential quantification there exists, for some first-order logic says "there exists an (at least one) such that has property ." n is even. ∃! uniqueness quantification there exists exactly one first-order logic (abbreviation) says "there exists exactly one such that has property ." Only and are part of formal logic. is an abbreviation for

Advanced or rarely used first-order symbols + SymbolUnicodevalue(hexadecimal)HTMLvalue(decimal)HTMLentity(named)LaTeXsymbolLogic NameRead asCategoryExplanation ∄U+2204\nexiststhere does not existStrike out existential quantifier. "¬∃" is sometimes used instead.

Metalanguage and metalogic + Symbol Unicodevalue(hexadecimal) HTMLcodes LaTeXsymbol Logic Name Read as Category Explanation Examples &Dopf; domain of discourse domain of discourse metalanguage (first-order logic semantics) &vdash; syntactic consequence proves, syntactically entails (single) turnstile metalanguage (metalogic) says " is a theorem of ". In other words, proves via a deductive system. (eg. by using natural deduction) semantic consequence or satisfaction (semantically) entails or satisfies, models double turnstile metalanguage (metalogic) says "in every model, it is not the case that is true and is false".

says a formula

is true in a model with

variable assignment . (eg. by using truth tables) logical equivalence is logically equivalent to metalanguage (metalogic) It's when and . Whether a symbol means a material biconditional or a logical equivalence, depends on the author's style. ⊬ does not syntactically entail (does not prove) metalanguage (metalogic) says " is not a theorem of ". In other words, is not derivable from via a deductive system. does not semantically entail metalanguage (metalogic) says " does not guarantee the truth of ". In other words, does not make true. therefore therefore metalanguage abbreviation for "therefore" ∵ because because metalanguage abbreviation for "because". ≔ definition is defined as metalanguage means "from now on, is defined to be another name for ." This is a statement in the metalanguage, not the object language. The notation may occasionally be seen in physics, meaning the same as . \stackrel{ \scriptscriptstyle \mathrm{def}}{=}

Advanced or rarely used metalogical symbols + SymbolUnicodevalue(hexadecimal)LaTeXsymbolLogic NameExplanation ̅U+0305\overline{x}combining overlineUsed format for denoting Gödel numbers. Using HTML style "4̅" is an abbreviation for the standard numeral "SSSS0". It may also denote a negation (used primarily in electronics). ⌜⌝U+231CU+231D\ulcorner \urcornertop left cornertop right cornerCorner quotes, also called "Quine quotes"; for quasi-quotation, i.e. quoting specific context of unspecified ("variable") expressions;Quine, W.V. (1981): Mathematical Logic, §6 also used for denoting Gödel number;. for example "⌜G⌝" denotes the Gödel number of G. (Typographical note: although the quotes appears as a "pair" in unicode (231C and 231D), they are not symmetrical in some fonts. In some fonts (for example Arial) they are only symmetrical in certain sizes. Alternatively the quotes can be rendered as ⌈ and ⌉ (U+2308 and U+2309) or by using a negation symbol and a reversed negation symbol ⌐ ¬ in superscript mode.) ⟛U+27DBleft and right tack"Proves and is proved by". ⊩U+22A9\VdashforcesOne of this symbol's uses is to mean "truthmakes" in the truthmaker theory of truth. It is also used to mean "forces" in the set theory method of forcing. ⋆U+22C6\starstar operatorMay sometimes be used for ad-hoc operators.

Modal logic + Symbol Unicodevalue(hexadecimal) HTMLcodes LaTeXsymbol Logic Name Read as Category Explanation Examples necessity (in a model) box; it is necessary that modal logic modal operator for "it is necessary that"in alethic logic, "it is provable that"in provability logic, "it is obligatory that"in deontic logic, "it is believed that"in doxastic logic, "it is known that"in autoepistemic logic. says "it is necessary that everything has property " possibility (in a model) diamond;it is possible that modal logic modal operator for "it is possible that", (in most modal logics it is defined as "¬□¬", "it is not necessarily not"). says "it is possible that something has property "

Advanced or rarely used modal symbols + SymbolUnicodevalue(hexadecimal)HTMLvalue(decimal)HTMLentity(named)LaTeXsymbolLogic NameRead asCategoryExplanation ⥽U+297D\strictifright fish tailSometimes used for "relation", also used for denoting various ad hoc relations (for example, for denoting "witnessing" in the context of Rosser's trick). The fish tail is also used as strict implication by C.I.Lewis ⥽ . ⟡U+27E1white concave-sided diamondnevermodal operator ⟢U+27E2white concave-sided diamond with leftwards tickwas nevermodal operator ⟣U+27E3white concave-sided diamond with rightwards tickwill never bemodal operator ⟤U+25A4white square with leftwards tickwas alwaysmodal operator ⟥U+25A5white square with rightwards tickwill always bemodal operator

List of logic symbols

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