Wiki's list of logical symbols -- in Griceian key

We were wondering with J why Grice left ")(", i.e. iff, out of the picture. Here is wiki, "List of logical symbols". If any of you can copy and paste the thing in the nice format it displays in the page, compleat with background colour, do, or allow me to know how to copy and paste it. I will work with a boring plain text reversion:

Basic logic symbols

Symbol

Name

Explanation

Examples

Unicode

Value

HTML

Entity

LaTeX

symbol

Should be read as

Category

p ⇒ q

p → q

p ⊃ q

material implication

"A ⇒ B" means if A is true then B is also true;
if A is false then nothing is said about B.

p → q may mean the same as p ⇒ q (the symbol may also indicate the domain and codomain of a function; see table of mathematical symbols).

p ⊃ q may mean the same as p ⇒ q (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
\to
\supset

To be read as:

"implies"; "if .. then"

[Surely Grice would object. the horseshoe should NOT be read as "if p, THEN q". "Then" is an altogether different animal, and will devour the candid 'if' if you serve them in the same platter].

propositional logic, Heyting algebra

p ⇔ q

p ≡ q

p ↔ q

material equivalence

A ⇔ B means A is true if B is true and A is false if B is false.

x + 5 = y +2 ⇔ x + 3 = y

U+21D4
U+2261
U+2194 ⇔
≡
↔ \Leftrightarrow
\equiv
\leftrightarrow

To be read as:

"if and only if"; "iff" [do not spit when you pronounce 'ff'].

propositional logic

¬p

˜p

! negation

The statement ¬p is true if and only if p is false.

A slash placed through another operator is the same as "¬p" placed in front.

¬(¬p) ⇔ p

x ≠ y ⇔ ¬(x = y)

U+00AC
U+02DC ¬
˜
~ \lnot
˜\sim

To be read as:

"not"

[Strawson prefers: "it is not the case that..."]

propositional logic

p∧q
p•q
p&q

logical conjunction

The statement p∧q is true if p and q are both true; else it is false.

n < 4 ∧ n >2 ⇔ n = 3

when n is a natural number.
U+2227
U+0026 ∧
& \land
\&[1]

To be read as:

"and"

as in the recent film with Jennifer Lopes, "Plan B". "She married and she had a child".

propositional logic

p∨q
+

logical disjunction The statement p ∨ q is true if p or q (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 ∨ \lor

To be read as:

"or" or "eether p or q" ("You said 'eether' and I say "ewayther' -- let's call the whole thing off").

propositional logic

p ⊕ q
p⊻q

exclusive disjunction

The statement p ⊕ q is true when either p or q, but not both, are true. p ⊻ q means the same. (¬p) ⊕ p is always true,

p ⊕ p is always false. U+2295

U+22BB ⊕ \oplus
xor
propositional logic, Boolean algebra


⊤
T
1

Tautology The statement ⊤ is unconditionally true.

A ⇒ ⊤

is always true. U+22A4 T \top
top
propositional logic, Boolean algebra


⊥
F
0

Contradiction The statement ⊥ is unconditionally false.

⊥ ⇒ p

is always true. U+22A5 ⊥
F \bot
bottom
propositional logic, Boolean algebra


∀x

universal quantification

∀x:P(x) means

P(x) is true for all x.

∀ n ∈ N: n2 ≥ n.

U+2200 ∀ \forall
for all; for any; for each
predicate logic

-- {Oddly, Grice drops the 'for' which IS usually redundant: "What are you doing that FOR?". Better English: "Why are you doing it?". Grice suggests we read (x) as "all", not "for all".

∃
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
there exists

[Grice has this to be read as ""some" ("at least one")" -- I love his fastidiousness about 'at least one' -- Surely, "Some ghosts of Lady Godiva are seen in the streets of Coventry at midnight" is true even if it's just the old familiar one]

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 !
there exists exactly one

--- [Grice prefers: "read (ix) [iota operator], from 'i' of idiot, individual] as 'the'" -- as in "Thee uniquely existential three stooges"].

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+003A U+003D

U+2261
U+003A U+229C :=
: ≡
⇔ : = :=
\equiv
\Leftrightarrow

To be read as:

"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 ( ) ( )

everywhere

⊢

inference

p⊢q

means

q is derived from p.

A → B ⊢ ¬B → ¬A U+22A2 \vdash

"infers"

or

"is derived from"

[or as I prefer, using the Anglo-Saxonism, '... yields ...']. [Note the greengrocer plural too, to use 'infer' to mean 'imply', which is the horseshoe.


propositional logic, first-order logic

--- The wiki entry ends with a list of

"Advanced and rarely used logical symbols."

These symbols are sorted by their Unicode value:

U+00B7

·

: Center dot, an outdated way for denoting
AND, still in use in electronics; for example

p·q

is the same as "A&B"

·: Center dot with a line above it (using HTML style).
Outdated way for denoting NAND, for example

p·q

is the same as "A NAND B" or "A|B" or "¬(A & B)" See also Unicode "Dot operator" U+22C5

U+0305

̅

: overline, used as abbreviation for standard numerals. for example, using HTML style "4" is a shorthand for the standard numeral "SSSS0"

̅ : overline,

an outdated way for denoting negation, still in use in electronics; for example "AVB" is the same as "¬(AVB)"

̅ : overline,

a rarely used format for denoting Gödel numbers, for example "AVB" says the Gödel number of "(AVB)"

U+2191 ↑ or U+007C

p|q

Sheffer stroke, the sign for the NAND operator.
U+2201

∁

complement

U+2204

∄

strike out existential quantifier same as "¬∃"

U+2234

∴

to be read: "therefore"

U+2235

∵

to be read: "because"

U+22A7

p⊧q

q is a model of q

--- e.g. Kate Moss was a model of an American company, but no more.

U+22A8

⊨

is true of

U+22AC

⊬

strike out turnstile, the sign for "does not prove", for example

p⊬q says

"q is not a theorem of p"

U+22AD

p⊭q

is not true of
U+22BC

⊼

Another NAND operator, can also be rendered as ∧

U+22BD

p⊽q

Another NOR operator, can also be rendered as V

U+22C4

◊p

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 ⊥ or U+2193

p↓q

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

⌜p U+231D

⌝: 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

⌈p⌉

and (unicode 2308 and 2309) or by using a negation symbol and a reversed negation symbol

⌐p¬

in superscript mode. )

U+25FB

◻p or U+25A1 □:

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

⟡p

modal operator for never

U+27E2

⟢

modal operator for was never

⟣p

modal operator for "p will never be"

U+27E4

⟤p

modal operator for "was always"

U+27E5

⟥p

modal operator for "p will always be"

U+297D

⥽

right

"fishtail"

sign, sometimes used for "relation", also used for denoting various ad hoc relations (for example, for denoting "witnessing" in the context of Rosser's trick) See here for an image of glyph. Added to Unicode 3.2.0.

[edit] See also
Logic portal

Table of mathematical symbols

Polish notation

Logic Alphabet, a suggested set of logical symbols.

Unicode Mathematical Operators

Notes
1.^ 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.
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