§2. The logic of logicians
41
(a.) not P,
(b) P or Q,
(c) P => Q,
(d) (Vx(P)) where x appears in P as a free variable.
Some remarks need to be made on (d), the rules of formation (a), (b),
(c) presenting no problem other than that of determining the "primitive"
propositions starting from which all the others are to be formed with the aid
ofthe four preceding schemata. If P {x} is a proposition involving a variable
x representing an a priori indeterminate object, and maybe other variables,
the expressions
(VxP{x}) ,
(3xP{x})
are to be read, the first "for all x, P {x}" or "one has P {x} for all x" , the
second "there exists x such that P{x}" or "one has P{x} for some x", "an
x" signifying "at least one x". If P involves other variables y, z, . .. than x,
the assertion (VxP{x,y,z}) obtained by applying the quantifier Vx to P is
again an assertion involving the variables y, z, . .. but in which x has become
a. bound variable.
Since the free variables represent totally arbitrary objects, you can, in a
formula featuring the free variables y, z, . .. , replace them by other distinct
variables u, V, • .. , free or bound, apart from the y, z, . .. which appear in the
proposition: the assertions [(x> y) => (x+1 > y)] and [(x> z) => (x+l >
z)] are logically equivalent; on the contrary, the propositions [(x> y) =>
(x + 1 > y)] and [(y > y) => (y + 1 > y)] clearly are not; it is similarly
obvious that (3x(x E y)) is not equivalent to (3x(x EX».
We shall often say that a bound variable x is a phantom variable since one
can replace the letter x everywhere by a sign with no logical or mathematical
meaning, for example the signs $, %, 0, etc., with which at a stroke one can
replace x in the quantifier '
really contains "arbitrary variables" x and y, but that in the assertion
(3x[(x E JR) & (x 2 = 4) & (y + 1 > y)]),
the variable x does not play the same role as y; one could equally well write
~
,
(30[(0 E JR) & (0 2 = 4 & (y + 1 > y)]
as Bourbaki does. In the notation
X=UXi
iEI
41
(a.) not P,
(b) P or Q,
(c) P => Q,
(d) (Vx(P)) where x appears in P as a free variable.
Some remarks need to be made on (d), the rules of formation (a), (b),
(c) presenting no problem other than that of determining the "primitive"
propositions starting from which all the others are to be formed with the aid
ofthe four preceding schemata. If P {x} is a proposition involving a variable
x representing an a priori indeterminate object, and maybe other variables,
the expressions
(VxP{x}) ,
(3xP{x})
are to be read, the first "for all x, P {x}" or "one has P {x} for all x" , the
second "there exists x such that P{x}" or "one has P{x} for some x", "an
x" signifying "at least one x". If P involves other variables y, z, . .. than x,
the assertion (VxP{x,y,z}) obtained by applying the quantifier Vx to P is
again an assertion involving the variables y, z, . .. but in which x has become
a. bound variable.
Since the free variables represent totally arbitrary objects, you can, in a
formula featuring the free variables y, z, . .. , replace them by other distinct
variables u, V, • .. , free or bound, apart from the y, z, . .. which appear in the
proposition: the assertions [(x> y) => (x+1 > y)] and [(x> z) => (x+l >
z)] are logically equivalent; on the contrary, the propositions [(x> y) =>
(x + 1 > y)] and [(y > y) => (y + 1 > y)] clearly are not; it is similarly
obvious that (3x(x E y)) is not equivalent to (3x(x EX».
We shall often say that a bound variable x is a phantom variable since one
can replace the letter x everywhere by a sign with no logical or mathematical
meaning, for example the signs $, %, 0, etc., with which at a stroke one can
replace x in the quantifier '
really contains "arbitrary variables" x and y, but that in the assertion
(3x[(x E JR) & (x 2 = 4) & (y + 1 > y)]),
the variable x does not play the same role as y; one could equally well write
~
,
(30[(0 E JR) & (0 2 = 4 & (y + 1 > y)]
as Bourbaki does. In the notation
X=UXi
iEI