List of logic symbols Integration of Computers 464 pasar.pts-ptn.net

List of logic symbols

Basic logic symbols

Symbol	Name	Explanation	Examples	Unicode Value	HTML Entity	LaTeX symbol
	Should be read as
	Category
⇒ → ⊃	material implication	A ⇒ B is true just in the case that either A is false or B is true, or both. → may mean the same as ⇒ (the symbol may also indicate the domain and codomain of a function; see table of mathematical symbols). ⊃ may mean the same as ⇒ (the symbol may also mean superset).	x = 2 ⇒ x² = 4 is true, but x² = 4 ⇒ x = 2 is in general false (since x could be −2).	U+21D2 U+2192 U+2283	⇒ → ⊃	$\Rightarrow$ \Rightarrow $\to$ \to $\supset$ \supset
	implies; if .. then
	propositional logic, Heyting algebra
⇔ ≡ ↔	material equivalence	A ⇔ B is true just in case either both A and B are false, or both A and B are true.	x + 5 = y +2 ⇔ x + 3 = y	U+21D4 U+2261 U+2194	⇔ &equiv; ↔	$\Leftrightarrow$ \Leftrightarrow $\equiv$ \equiv $\leftrightarrow$ \leftrightarrow
	if and only if; iff; means the same as
	propositional logic
¬ ˜ !	negation	The statement ¬A is true if and only if A is false. A slash placed through another operator is the same as "¬" placed in front.	¬(¬A) ⇔ A x ≠ y ⇔ ¬(x = y)	U+00AC U+02DC	¬ &tilde; ~	$\neg$ \lnot or \neg $\sim$ \sim
	not
	propositional logic
∧ • &	logical conjunction	The statement A ∧ B is true if A and B are both true; else it is false.	n < 4 ∧ n >2 ⇔ n = 3 when n is a natural number.	U+2227 U+0026	&and; &	$\wedge$ \wedge or \land \&^[1]
	and
	propositional logic
∨ + ǀǀ	logical disjunction	The statement A ∨ B is true if A or B (or both) are true; if both are false, the statement is false.	n ≥ 4 ∨ n ≤ 2 ⇔ n ≠ 3 when n is a natural number.	U+2228	&or;	$\lor$ \lor or \vee
	or
	propositional logic
⊕ ⊻	exclusive disjunction	The statement A ⊕ B is true when either A or B, but not both, are true. A ⊻ B means the same.	(¬A) ⊕ A is always true, A ⊕ A is always false.	U+2295 U+22BB	&oplus;	$\oplus$ \oplus $\veebar$ \veebar
	xor
	propositional logic, Boolean algebra
⊤ T 1	Tautology	The statement ⊤ is unconditionally true.	A ⇒ ⊤ is always true.	U+22A4	T	$\top$ \top
	top, verum
	propositional logic, Boolean algebra
⊥ F 0	Contradiction	The statement ⊥ is unconditionally false.	⊥ ⇒ A is always true.	U+22A5	&perp; F	$\bot$ \bot
	bottom, falsum
	propositional logic, Boolean algebra
∀ ()	universal quantification	∀ x: P(x) or (x) P(x) means P(x) is true for all x.	∀ n ∈ N: n² ≥ n.	U+2200	∀	$\forall$ \forall
	for all; for any; for each
	first-order logic
∃	existential quantification	∃ x: P(x) means there is at least one x such that P(x) is true.	∃ n ∈ N: n is even.	U+2203	∃	$\exists$ \exists
	there exists
	first-order logic
∃!	uniqueness quantification	∃! x: P(x) means there is exactly one x such that P(x) is true.	∃! n ∈ N: n + 5 = 2n.	U+2203 U+0021	∃ !	$\exists !$ \exists !
	there exists exactly one
	first-order logic
:= ≡ :⇔	definition	x := y or x ≡ y means x is defined to be another name for y (but note that ≡ can also mean other things, such as congruence). P :⇔ Q means P is defined to be logically equivalent to Q.	cosh x := (1/2)(exp x + exp (−x)) A XOR B :⇔ (A ∨ B) ∧ ¬(A ∧ B)	U+2254 (U+003A U+003D) U+2261 U+003A U+229C	:= : &equiv; ⇔	$:=$ := $\equiv$ \equiv $\Leftrightarrow$ \Leftrightarrow
	is defined as
	everywhere
( )	precedence grouping	Perform the operations inside the parentheses first.	(8/4)/2 = 2/2 = 1, but 8/(4/2) = 8/2 = 4.	U+0028 U+0029	( )	$(~)$ ( )
	precedence grouping
	everywhere
⊢	Turnstile	x ⊢ y means y is provable from x (in some specified formal system).	A → B ⊢ ¬B → ¬A	U+22A2	⊢	$\vdash$ \vdash
	provable
	propositional logic, first-order logic
⊨	double turnstile	x ⊨ y means x semantically entails y	A → B ⊨ ¬B → ¬A	U+22A8	⊨	$\models$ \models
	entails
	propositional logic, first-order logic

Advanced and rarely used logical symbols

These symbols are sorted by their Unicode value:

U+00B7 · middle dot, an outdated way for denoting AND^{[citation needed]}, still in use in electronics; for example "A·B" is the same as "A&B"
·: Center dot with a line above it (using HTML style). Outdated way for denoting NAND, for example "A·B" is the same as "A NAND B" or "A|B" or "¬(A & B)". See also Unicode U+22C5 ⋅ dot operator.

U+0305 ̅ combining overline, used as abbreviation for standard numerals. For example, using HTML style "4̅" is a shorthand for the standard numeral "SSSS0".
Overline, is also a rarely used format for denoting Gödel numbers, for example "AVB" says the Gödel number of "(AVB)"
Overline is also an outdated way for denoting negation, still in use in electronics; for example "AVB" is the same as "¬(AVB)"

U+2191 ↑ upwards arrow or U+007C | vertical line: Sheffer stroke, the sign for the NAND operator.
U+2201 ∁ complement
U+2204 ∄ there does not exist: strike out existential quantifier same as "¬∃"
U+2234 ∴ therefore
U+2235 ∵ because
U+22A7 ⊧ models: is a model of
U+22A8 ⊨ true: is true of
U+22AC ⊬ does not prove: negated ⊢, the sign for "does not prove", for example T ⊬ P says "P is not a theorem of T"
U+22AD ⊭ not true: is not true of
U+22BC ⊼ nand: another NAND operator, can also be rendered as ∧
U+22BD ⊽ nor: another NOR operator, can also be rendered as V
U+22C4 ⋄ diamond operator: modal operator for "it is possible that", "it is not necessarily not" or rarely "it is not provable not" (in most modal logics it is defined as "¬�-�¬")
U+22C6 ⋆ star operator: usually used for ad-hoc operators
U+22A5 ⊥ up tack or U+2193 ↓ downwards arrow: Webb-operator or Peirce arrow, the sign for NOR. Confusingly, "⊥" is also the sign for contradiction or absurdity.

U+2310 ⌐ reversed not sign

U+231C ⌜ top left corner and U+231D ⌝ top right corner: corner quotes, also called "Quine quotes"; the standard symbol 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. And 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+25FB �-� white medium square or U+25A1 □ white square: modal operator for "it is necessary that" (in modal logic), or "it is provable that" (in provability logic), or "it is obligatory that" (in deontic logic), or "it is believed that" (in doxastic logic).

Note that the following operators are rarely supported by natively installed fonts. If you wish to use these in a web page, you should always embed the necessary fonts so the page viewer can see the web page without having the necessary fonts installed in their computer.

U+27E1 ⟡ white concave-sided diamond
U+27E2 ⟢ white concave-sided diamond with leftwards tick: modal operator for was never
U+27E3 ⟣ white concave-sided diamond with rightwards tick: modal operator for will never be
U+27E4 ⟤ white square with leftwards tick: modal operator for was always
U+27E5 ⟥ white square with rightwards tick: modal operator for will always be
U+297D ⥽ right fish tail: sometimes 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 hook is also used as strict implication by C.I.Lewis $p$ ⥽ $q \equiv \Box(p\rightarrow q)$ , the corresponding LaTeX macro is \strictif. See here for an image of glyph. Added to Unicode 3.2.0.

Notes

^ Although this character is available in LaTeX, the Mediawiki TeX system doesn't support this character.

External links

Named character entities in HTML 4.0.

Logic

Overview

Academic areas	Argumentation theory Axiology Critical thinking Computability theory Formal semantics History of logic Informal logic Logic in computer science Mathematical logic Mathematics Metalogic Metamathematics Model theory Philosophical logic Philosophy Philosophy of logic Philosophy of mathematics Proof theory Set theory

Foundations	Abduction Analytic truth Antinomy A priori Deduction Definition Description Induction Inference Logical form Logical consequence Logical truth Name Necessity Meaning Paradox Possible world Presupposition Probability Reason Reasoning Reference Semantics Statement Strict implication Substitution Syntax Truth Truth value Validity

Philosophical logic

Critical thinking Informal logic	Analysis Ambiguity Argument Belief Bias Credibility Evidence Explanation Explanatory power Fact Fallacy Inquiry Opinion Parsimony Premise Propaganda Prudence Reasoning Relevance Rhetoric Rigor Vagueness

Theories of deduction	Constructivism Dialetheism Fictionalism Finitism Formalism Intuitionism Logical atomism Logicism Nominalism Platonic realism Pragmatism Realism

Metalogic Metamathematics

Cantor's theorem Church's theorem Church's thesis Consistency Effective method Foundations of mathematics Gödel's completeness theorem Gödel's incompleteness theorems Soundness Completeness Decidability Interpretation Löwenheim–Skolem theorem Metatheorem Satisfiability Independence Type–token distinction Use–mention distinction

Mathematical logic

General	Formal language Formation rule Formal system Deductive system Formal proof Formal semantics Well-formed formula Set Element Class Classical logic Axiom Natural deduction Rule of inference Relation Theorem Logical consequence Axiomatic system Type theory Symbol Syntax Theory

Traditional logic	Proposition Inference Argument Validity Cogency Syllogism Square of opposition Venn diagram

Propositional calculus Boolean logic	Boolean functions Propositional calculus Propositional formula Logical connectives Truth tables

Predicate logic	First-order Quantifiers Predicate Second-order Monadic predicate calculus

Set theory	Set Empty set Enumeration Extensionality Finite set Function Subset Power set Countable set Recursive set Domain Range Ordered pair Uncountable set

Model theory	Model Interpretation Non-standard model Finite model theory Truth value Validity

Proof theory	Formal proof Deductive system Formal system Theorem Logical consequence Rule of inference Syntax

Computability theory	Recursion Recursive set Recursively enumerable set Decision problem Church–Turing thesis Computable function Primitive recursive function

Non-classical logic

Modal logic	Alethic Axiologic Deontic Doxastic Epistemic Temporal

Intuitionism	Intuitionistic logic Constructive analysis Heyting arithmetic Intuitionistic type theory Constructive set theory

Fuzzy logic	Degree of truth Fuzzy rule Fuzzy set Fuzzy finite element Fuzzy set operations

Substructural logic	Structural rule Relevance logic Linear logic

Paraconsistent logic	Dialetheism

Description logic	Ontology Ontology language

Logicians

Anderson Aristotle Averroes Avicenna Bain Barwise Bernays Boole Boolos Cantor Carnap Church Chrysippus Curry De Morgan Frege Geach Gentzen Gödel Hilbert Kleene Kripke Leibniz Löwenheim Peano Peirce Putnam Quine Russell Schröder Scotus Skolem Smullyan Tarski Turing Whitehead William of Ockham Wittgenstein Zermelo

Lists

Topics	Index of logic articles Mathematical logic Boolean algebra Set theory

Other	Logicians Rules of inference Paradoxes Fallacies Logic symbols

Common logical symbols

& ∨ ¬ ~ → ⊃ ≡ \| ∀ ∃ ⊤ ⊥ ⊢ ⊨ ∴ ∵

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List of logic symbols

Basic logic symbols

Advanced and rarely used logical symbols

See also

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