§2. The logic of logicians
39
§2. The logic of logicians 33
In the version of logic more or less universally adopted the basic material
comprises the following elements:
(i) expressions which the Greek philosophers and geometers already used
in plain language and which one represents by the sign ===? ("implies", ''thus'',
"it follows that", "it results that", "if ... then", etc.), the sign V which
resembles the sign U and which one writes "or" (the nonexclusive logical
disjunction of two assertions), and finally the sign ..." "not", the negation of
an assertion ("it is false that ... "); to these three fundamental signs one adds
two other signs which are also useful but are only convenient abbreviations:
signifies: not[(not P) or (not Q)),
signifies: (P ===? Q) /\ (Q ===? P);
we always write P or Q instead of P V Q, P&Q instead of P 1\ Q and "not"
instead of the sign ...,; there are already enough cabbalistic signs in Mathematics not to want to add to them;
(ii) the quantifiers V ("for all" or "whatever") and :3 ("there exists",
"there is at least one"), such as appear in the following statement: for any
positive number x, there exists an number y such that x = y2 (generally
there exist even two ... ), which one writes
(Vx[(x E JR) & (x > O)J ===? {:3y[(y E JR) & (y2 = x)]});
by convention,
33 My feeble attempts to find an exposition of logic, either in French or in English, at
once accessible and readable, have not been fruitful, with one sole exception: Rene
Cori and Daniel Lascar, Logique mathlimatique, transl. as Mathematical Logic: A
Course with Exercises (OUP, 2001), which also presents the axiomatic theory of
sets. Patrick Suppes, Axiomatic Set Theory (Van Nostrand, 1960, reprint Dover,
1972), without comparing it to Cori and Lascar, has been of great use, but already
assumes the reader to be familiar with the basic principles of logic and, indeed,
with the naive theory and its usages. Paul R. Halmos, Naive Set Theory (Van
Nostrand, 1960 or Springer, 1974) is very readable but ignores Logic completely,
which is not necessarily an inconvenience to mathematicians. There are also the
volumes on the Theory of Sets by N. Bourbaki; Cori and Lascar write that logic
''is not the forte" of Bourbaki, which can be explained by the fact that the
treatise was written by mathematicians for mathematicians. Since, before the
war, only two people were interested in logic in France - Jacques Herbrand and
the philosopher Jean Cavailles - who both departed prematurely (the first in
a mountaineering accident, resistance to German occupation took the second),
one should at least ascribe the credit to Bourbaki for having spread the subject
widely in France, even if the French Cardinals of logic have a much more elaborate
conception of it, as is normal. As far as I am concerned, reading Bourbaki's
Fascicule de resultats on Set Theory as soon as it was published at the beginning
of thf' 1940s allowed me to learn in a few weeks everything I ever needed in this
respect.
39
§2. The logic of logicians 33
In the version of logic more or less universally adopted the basic material
comprises the following elements:
(i) expressions which the Greek philosophers and geometers already used
in plain language and which one represents by the sign ===? ("implies", ''thus'',
"it follows that", "it results that", "if ... then", etc.), the sign V which
resembles the sign U and which one writes "or" (the nonexclusive logical
disjunction of two assertions), and finally the sign ..." "not", the negation of
an assertion ("it is false that ... "); to these three fundamental signs one adds
two other signs which are also useful but are only convenient abbreviations:
signifies: not[(not P) or (not Q)),
signifies: (P ===? Q) /\ (Q ===? P);
we always write P or Q instead of P V Q, P&Q instead of P 1\ Q and "not"
instead of the sign ...,; there are already enough cabbalistic signs in Mathematics not to want to add to them;
(ii) the quantifiers V ("for all" or "whatever") and :3 ("there exists",
"there is at least one"), such as appear in the following statement: for any
positive number x, there exists an number y such that x = y2 (generally
there exist even two ... ), which one writes
(Vx[(x E JR) & (x > O)J ===? {:3y[(y E JR) & (y2 = x)]});
by convention,
33 My feeble attempts to find an exposition of logic, either in French or in English, at
once accessible and readable, have not been fruitful, with one sole exception: Rene
Cori and Daniel Lascar, Logique mathlimatique, transl. as Mathematical Logic: A
Course with Exercises (OUP, 2001), which also presents the axiomatic theory of
sets. Patrick Suppes, Axiomatic Set Theory (Van Nostrand, 1960, reprint Dover,
1972), without comparing it to Cori and Lascar, has been of great use, but already
assumes the reader to be familiar with the basic principles of logic and, indeed,
with the naive theory and its usages. Paul R. Halmos, Naive Set Theory (Van
Nostrand, 1960 or Springer, 1974) is very readable but ignores Logic completely,
which is not necessarily an inconvenience to mathematicians. There are also the
volumes on the Theory of Sets by N. Bourbaki; Cori and Lascar write that logic
''is not the forte" of Bourbaki, which can be explained by the fact that the
treatise was written by mathematicians for mathematicians. Since, before the
war, only two people were interested in logic in France - Jacques Herbrand and
the philosopher Jean Cavailles - who both departed prematurely (the first in
a mountaineering accident, resistance to German occupation took the second),
one should at least ascribe the credit to Bourbaki for having spread the subject
widely in France, even if the French Cardinals of logic have a much more elaborate
conception of it, as is normal. As far as I am concerned, reading Bourbaki's
Fascicule de resultats on Set Theory as soon as it was published at the beginning
of thf' 1940s allowed me to learn in a few weeks everything I ever needed in this
respect.