Hilbert notation For any formal language
L, extend
L by adding the epsilon operator to redefine quantification: • (\exists x) A(x)\ \equiv \ A(\epsilon x\ A) • (\forall x) A(x)\ \equiv \ \neg \exists x \neg A(x) \leftrightarrow \neg \big(\neg A(\epsilon x\ \neg A)\big) \leftrightarrow A(\epsilon x\ (\neg A)) The intended interpretation of ϵ
x A is
some x that satisfies
A, if it exists. In other words, ϵ
x A returns some
term t such that
A(
t) is true, otherwise it returns some default or arbitrary term. If more than one term can satisfy
A, then any one of these terms (which make
A true) can be
chosen, non-deterministically. Equality is required to be defined under
L, and the only rules required for
L extended by the epsilon operator are
modus ponens and the substitution of
A(
t) to replace
A(
x) for any term
t.
Bourbaki notation In tau-square notation from
N. Bourbaki's Theory of Sets, the quantifiers are defined as follows: • (\exists x) A(x)\ \equiv \ (\tau_x(A)|x)A • (\forall x) A(x)\ \equiv \ \neg (\tau_x(\neg A)|x)\neg A\ \equiv \ (\tau_x(\neg A)|x)A where
A is a relation in
L,
x is a variable, and \tau_x(A) juxtaposes a \tau at the front of
A, replaces all instances of
x with \square, and links them back to \tau. Then let
Y be an assembly,
(Y|x)A denotes the replacement of all variables
x in
A with
Y. This notation is equivalent to the Hilbert notation and is read the same. It is used by Bourbaki to define
cardinal assignment since they do not use the
axiom of replacement. Defining quantifiers in this way leads to great inefficiencies. For instance, the expansion of Bourbaki's original definition of the number one, using this notation, has length approximately 4.5 × 1012, and for a later edition of Bourbaki that combined this notation with the
Kuratowski definition of
ordered pairs, this number grows to approximately 2.4 × 1054. ==Modern approaches==