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List of Logic Symbols

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List of Logic Symbols 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, pronunciation, and the related field of mathematics. Additionally, the third column contains an informal definition, the fourth column gives a short example, the fifth and sixth give the unicode location and name for use in HTML documents.[1] The last column provides the LaTeX symbol. 1 Basic logic symbols Name Unicode HTML H Symbol Read as Explanation Examples Value Value E Category (hexdecimal) (decimal) (n A ⇒ B is true only in the case that material either A is implication false or B is true. → may mean ⇒ the same as ⇒ (the symbol implies; if .. U+21D2 &#8658; &rA may also x = 2 ⇒ x2 = 4 is true, then indicate the but x2 = 4 ⇒ x = 2 is U+2192 &#8594; &r → domain and in general false (since codomain of a x could be −2). U+2283 &#8835; &s function; see table of ⊃ mathematical symbols). propositional logic, Heyting ⊃ may mean algebra the same as ⇒ (the symbol may also mean superset). ⇔ material equivalence A ⇔ B is true U+21D4 &#8660; &hA if and only if; only if both A x + 5 = y + 2 ⇔ x + 3 = iff; means the and B are false, U+2261 &#8801; &e ≡ y same as or both A and B are true. U+2194 &#8596; &h propositional ↔ logic negation The statement ¬(¬A) ⇔ A U+00AC &#172; &n ¬ ¬A is true if x ≠ y ⇔ ¬(x = y) and only if A is U+02DC &#732; &t false. not U+0021 &#33; &ex ˜ A slash placed through another propositional operator is the logic ! same as "¬" placed in front. Name Unicode HTML H Symbol Read as Explanation Examples Value Value E Category (hexdecimal) (decimal) (n ∧ logical conjunction The statement U+2227 &#8743; &a and A ∧ B is true if n < 4 ∧ n >2 ⇔ n = 3 · A and B are when n is a natural U+00B7 &#183; &m propositional both true; else number. logic, it is false. U+0026 &#38; &am Boolean & algebra logical ∨ (inclusive) disjunction The statement or A ∨ B is true if U+2228 &#8744; &o A or B (or both) n ≥ 4 ∨ n ≤ 2 ⇔ n ≠ 3 + are true; if both when n is a natural U+002B &#43; propositional are false, the number. logic, statement is U+2225 &#8741; Boolean false. algebra ∥ exclusive The statement (¬A) ⊕ A is always U+2295 &#8853; &o disjunction A ⊕ B is true true, and A ⊕ A always xor when either A false, if vacuous truth U+22BB &#8891; ⊕ or B, but not is excluded. propositional both, are true. logic, A ⊻ B means Boolean the same. ⊻ algebra Tautology ⊤ top, verum The statement U+22A4 &#8868; ⊤ is A ⇒ ⊤ is always true. T propositional unconditionally logic, true. Boolean algebra 1 Name Unicode HTML H Symbol Read as Explanation Examples Value Value E Category (hexdecimal) (decimal) (n Contradiction ⊥ bottom, falsum, The statement falsity ⊥ is U+22A5 &#8869; &p unconditionally F false. (The ⊥ ⇒ A is always true. propositional symbol ⊥ may logic, also refer to Boolean perpendicular algebra lines.) 0 universal ∀ x: P(x) or ∀ n ∈ ℕ: n2 ≥ n. U+2200 &#8704; &f ∀ quantification (x) P(x) means for all; for any; for each P(x) is true for all x. first-order () logic existential ∃ x: P(x) quantification means there is ∃ there exists at least one x ∃ n ∈ ℕ: n is even. U+2203 &#8707; &ex first-order such that P(x) logic is true. uniqueness quantification ∃! x: P(x) means there is there exists &#8707; exactly one x ∃! n ∈ ℕ: n + 5 = 2n. U+2203 U+0021 ∃! exactly one &#33; such that P(x) first-order is true. logic Name Unicode HTML H Symbol Read as Explanation Examples Value Value E Category (hexdecimal) (decimal) (n definition x ≔ y or x ≡ y is defined as means x is defined to be ≔ another name for y (but note &#8788; U+2254 that ≡ can also cosh x ≔ (1/2)(exp x + (&#58; (U+003A U+003D) mean other exp (−x)) &#61;) things, such as ≡ U+2261 &e congruence). everywhere A XOR B :⇔ &#8801; (A ∨ B) ∧ ¬(A ∧ B) U+003A U+229C &hA P :⇔ Q means &#8860; :⇔ P is defined to be logically equivalent to Q. precedence Perform the (8 ÷ 4) ÷ 2 = 2 ÷ 2 = 1, U+0028 U+0029 &#40; ( ) grouping operations but 8 ÷ (4 ÷ 2) = 8 ÷ &#41; parentheses, inside the 2 = 4. brackets parentheses everywhere first. Turnstile x ⊢ y means y is provable provable from x (in A → B ⊢ ¬B → ¬A U+22A2 &#8866; ⊢ propositional some specified logic, first- formal order logic system). double turnstile x ⊨ y means x entails ⊨ semantically A → B ⊨ ¬B → ¬A U+22A8 &#8872; propositional entails y logic, first- order logic Name Unicode HTML H Symbol Read as Explanation Examples Value Value E Category (hexdecimal) (decimal) (n 2 Advanced and rarely used logical symbols These symbols are sorted by their Unicode value: U+00B7 · MIDDLE DOT, an outdated way for denoting AND,[3] still in use in electronics; for example "A · B" is the same as "A & B" · : Center dot with a line above it; 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 ⋅ . U+0305 ̅ COMBINING OVERLINE, used as abbreviation for standard numerals (Typographical Number Theory). 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, "A V B" says the Gödel number of "(A V B)". Overline is also an outdated way for denoting negation, still in use in electronics: for example, "A V B" is the same as "¬(A V B)". U+2191 ↑ UPWARDS ARROW or U+007C | VERTICAL LINE: Sheffer stroke, the sign for the NAND operator. U+2193 ↓ DOWNWARDS ARROW Peirce Arrow, the sign for the NOR operator. U+2201 ∁ C U+2204 ∄ THERE DOES NOT EXIST: strike out existential quantifier same as "¬∃" U+2234 ∴ T: Therefore U+2235 ∵ B: because U+22A7 ⊧ M: is a model of U+22A8 ⊨ T: 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 ⊭ N : is not true of U+22BC ⊼ NAND: NAND operator. In HTML, it can also be produced by <span style="text-decoration: overline">&and;</span> : ∧ U+22BD ⊽ N: NOR operator. In HTML, it can also be produced by <span style="text-decoration: overline">&or;</span> : ∨ U+25C7 ◇ WHITE DIAMOND: 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"; for quasi-quotation, i.e. quoting specific context of unspecified ("variable") expressions;[4] also used for denoting Gödel number;[5] 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); also as empty clause (alternatives: and ⊥). 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 , the corresponding LaTeX macro is \strictif. See here for an image of glyph. Added to Unicode 3.2.0. U+2A07 ⨇ TWO LOGICAL AND OPERATOR 2.1 Poland and Germany As of 2014 in Poland, the universal quantifier is sometimes written and the existential quantifier as . The same applies for Germany.[6] 3 See also Józef Maria Bocheński List of notation used in Principia Mathematica List of mathematical symbols Logic alphabet, a suggested set of logical symbols Logical connective Mathematical operators and symbols in Unicode Polish notation Truth function Truth table 4 References 1. "Named character references" . HTML 5.1 Nightly. W3C. Retrieved 9 September 2015. 2. Although this character is available in LaTeX, the MediaWiki TeX system doesn't support this character. 3. Brody, Baruch A.
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