Introduced by Gerhard Gentzen who based it on the Latin letter A, by analogy with ∃.

universal quantifier
universal quantifier

(mathematics, logic) The symbol used in predicate calculus, etc, to represent the universal quantifier, meaning “for all”.


Introduced by Giuseppe Peano in Formulaire de mathematiqués (1897), from the uppercase Latin letter E. The modern variable-binding syntax first appeared in Principia Mathematica (1910).

existential quantifier
existential quantifier

(mathematics, logic) The existential quantifier, meaning "there exists (at least one)".


Coordinate term: ∵


(logic) Intensional disjunction, as in some relevant logics.


~p


Introduced by Arend Heyting in “Die formalen Regeln der intuitionistischen Logik” (1930)

(x ≠ y) ↔ ¬(x = y)


Con

ZFC ⊨ Con(PA)

(mathematics, logic) consistency predicate


Q

English: max Q

Ancient Greek language
Egyptian language
Etruscan language
Phoenician language
Q

From the Etruscan letter 𐌒 (q, “cu”), from the Ancient Greek letter Ϙ (Q, “qoppa”), derived from the Phoenician letter 𐤒 (q, “qop”), from the Egyptian hieroglyph 𓃻.

(logic, mathematics) Robinson arithmetic.


Π

arithmetical hierarchy
arithmetical hierarchy

polynomial hierarchy
polynomial hierarchy

(mathematics, logic, computer science) A class (in some hierarchy of formulae, like the arithmetical hierarchy or polynomial hierarchy) of those formulae whose outermost quantifiers are universal quantifiers.


Coordinate term: ∴


elementary equivalence
elementary equivalence


(logic) intensional conjunction, as in some relevant logics.


head

∧ or #92;land (English symbol name wedge)

land (canonical)
wedge (English name symbol)

(mathematics, logic) The conjunction operator, forming a Boolean-valued function, typically with two arguments, returning true only if all of its arguments are true.


In its original sense "disjunction", probably from the initial letter of Latin vel (“or”).


+ (English symbol name plus sign or plus)

plus sign (English name symbol)
plus (English name symbol)

antonym(s) of “addition or positiveness”

From a ligature of Latin et (“and”). Compare & and ⁊ of same meaning.

inclusive disjunction
inclusive disjunction

(mathematics, logic) An inclusive disjunction.


(logic) A contradiction; a truth value corresponding to false.


Suppose that G ≅ H. Then there exists an isomorphism f, say, from G to H.

2011 May 5, Clive Reis, Abstract Algebra: An Introduction To Groups, Rings And Fields (in English), World Scientific Publishing Company, →ISBN, page 137:

(algebra, category theory, logic) Isomorphic to.


(logic, algebra) the n-ary disjunction or join operator


mathbf V_ω⊧ mathsf ZFCᶠⁱⁿ

T⊧φ if and only if T⊢φ

semantic consequence
semantic consequence

(mathematics, logic) semantic consequence


elementary substructure
elementary substructure

(logic) is an elementary substructure of


⊢ developed from ⊦, used to signify assertions in Principia Mathematica; in turn derived from idiosyncratic two-dimensional notation for logic employed in Begriffsschrift.

mathsf PA⊢∀a∀b:a+b=b+a


α ⧟ β means α ⊸ β and β ⊸ α


material implication
material implication#English

(logic, text-style only) material implication


`vert`

| (English symbol name vertical bar or vertical line or pipe or pipe key)

vertical bar (English name symbol)
vertical line (English name symbol)
pipe (English name symbol)
pipe key (English name symbol)

(logic) The Sheffer stroke: joins two statements of which not both are true (nand).


(logic) In linear logic, used to denote an intensional disjunction.


(logic) Denotes a definite descriptor in formal logic.


praeclarum theorema

So named by G.W. Leibniz in his unpublished papers of 1690 (later published as Leibniz: Logical Papers in 1966), meaning "splendid theorem" in Latin.

The praeclarum theorema can be seen to correspond with the rule R⊗ of linear logic; given two sequents A⊢B and C⊢D one may infer (through the said rule) that A,C⊢B⊗D. Then one may further infer, through the rule L⊗, that A⊗C⊢B⊗D.

propositional calculus
propositional calculus

(logic) The following theorem of propositional calculus: (A → B) ∧ (C → D) → (A ∧ C → B ∧ D).


The use of this notation in modal logic was developed by C.I. Lewis.

∀𝑥□𝐹𝑥 → □∀𝑥𝐹𝑥 ⇔ ◊∃𝑥𝐹𝑥 → ∃𝑥◊𝐹𝑥

(logic, unfilled) possibility, possible, possibly


Coordinate term: ◇

[…] obtained by adding the standard modal operators □ (necessity) and ◇ (possibility) to paralogics.


(algebra, logic) Boolean value corresponding to truth


MSO

(logic, computer science) Monadic second-order logic.


material equivalence
material equivalence

(logic) material equivalence; if and only if


ex falso quod libet

ex falso quodlibet (alternative)
ex falso sequitur quodlibet (alternative)
ex falso sequitur quod libet (alternative)

(logic) The principle or axiom of logic stating that if a contradiction or a false proposition is proven to be true, then it proves that everything is true. In symbols: ⊥→P


Coordinate term: □