40
I - Sets and Functions
(3xP)
signifies
not [Vx(not P)];
(iii) variables x, y, ... , X, Y, ... , a, b, ... etc., in unlimited quantity, symbolising totally indeterminate objects;
(iv) logical punctuation signs such as (, ), [, ], etc., whose purpose is
to group one or other of the sub-propositions of which a given proposition
is composed. The professional logicians use these much less than we have
done 34 , but we are not addressing them.
In particular, the parentheses allow us to indicate the domains of application of quantifiers clearly. This last point leads to the fundamental distinction
between free variables and bound variables: a variable x is said to be bound
if it appears in the domain of application of a quantifier Vx or 3x, whose
domain of application is, in principle, delimited by the parentheses ( and );
a variable is said to be free if it is not bound. Although the professionallogicians do not hesitate to write statements in which the same letter x is bound
in certain parts of a proposition and free in others, for example
(Vx(x 2 > 0)) & ((x> 1) ==> (x 2 > x)) ,
one can only discourage the reader unfamiliar with logic from this usage; it
is much more prudent to write the preceding proposition in the form
[\'x(x 2 > 0)] & [(y > 1) ==> (y2 > y)]
to make clear the fact that it is composed of two unrelated assertions.
The use of these symbols 35 allows one to construct "propositions" or
assertions; these are finite sequences of letters and of signs taken from the
list above and a priori chosen arbitrarily, for example
(Vz((za)or((3y)b) ==> not(x(3y)x)
It goes without saying that to obtain meaningful sequences (not the case in
the example above), one must observe a certain number of generally obvious
rules of syntax. To be precise, the only syntactically correct propositions 36
are those one can obtain by repeated application of the following rules or
propositions, in which P and Q stand for propositions or assemblages of
signs which are already known to be syntactically correct:
34 The parentheses ( and) are in fact enough, as anyone will know who, for example,
has consulted a computerised catalogue in the library using key words.
35 To respect strict logical formalism is not in my programme - anyway I would
be incapable of it - apart from implication arrows and the signs "&" and "or"
which will serve sometimes to eliminate every ambiguity in a statement. Nor
is it to encourage the reader to avail himself of simple stenographic signs which
permit him, as one has seen so often, to write gobbledegook instead of expressing
himself plainly.
36 This does not mean "true". The relation 1 = 2 is syntactically correct. Likewise,
the syllogism "every man is immortal, Socrates is a man, therefore Socrates is
immortal" is perfectly correct.
I - Sets and Functions
(3xP)
signifies
not [Vx(not P)];
(iii) variables x, y, ... , X, Y, ... , a, b, ... etc., in unlimited quantity, symbolising totally indeterminate objects;
(iv) logical punctuation signs such as (, ), [, ], etc., whose purpose is
to group one or other of the sub-propositions of which a given proposition
is composed. The professional logicians use these much less than we have
done 34 , but we are not addressing them.
In particular, the parentheses allow us to indicate the domains of application of quantifiers clearly. This last point leads to the fundamental distinction
between free variables and bound variables: a variable x is said to be bound
if it appears in the domain of application of a quantifier Vx or 3x, whose
domain of application is, in principle, delimited by the parentheses ( and );
a variable is said to be free if it is not bound. Although the professionallogicians do not hesitate to write statements in which the same letter x is bound
in certain parts of a proposition and free in others, for example
(Vx(x 2 > 0)) & ((x> 1) ==> (x 2 > x)) ,
one can only discourage the reader unfamiliar with logic from this usage; it
is much more prudent to write the preceding proposition in the form
[\'x(x 2 > 0)] & [(y > 1) ==> (y2 > y)]
to make clear the fact that it is composed of two unrelated assertions.
The use of these symbols 35 allows one to construct "propositions" or
assertions; these are finite sequences of letters and of signs taken from the
list above and a priori chosen arbitrarily, for example
(Vz((za)or((3y)b) ==> not(x(3y)x)
It goes without saying that to obtain meaningful sequences (not the case in
the example above), one must observe a certain number of generally obvious
rules of syntax. To be precise, the only syntactically correct propositions 36
are those one can obtain by repeated application of the following rules or
propositions, in which P and Q stand for propositions or assemblages of
signs which are already known to be syntactically correct:
34 The parentheses ( and) are in fact enough, as anyone will know who, for example,
has consulted a computerised catalogue in the library using key words.
35 To respect strict logical formalism is not in my programme - anyway I would
be incapable of it - apart from implication arrows and the signs "&" and "or"
which will serve sometimes to eliminate every ambiguity in a statement. Nor
is it to encourage the reader to avail himself of simple stenographic signs which
permit him, as one has seen so often, to write gobbledegook instead of expressing
himself plainly.
36 This does not mean "true". The relation 1 = 2 is syntactically correct. Likewise,
the syllogism "every man is immortal, Socrates is a man, therefore Socrates is
immortal" is perfectly correct.