文档介绍：Foreword
Logic studies the way we draw conclusions and express ourselves, and deals
with how to formalise it. In Logic, propositions are elementary statement and
conclusion units; they are analysed both as to the form, namely their syntax, and
the interpretation, namely their semantics. The relation between the syntax and
the semantics is also examined.
The first steps in Logic are credited to the Ionian and Elean philosophers and
to the sophists. It is however Aristotle who is considered as the founder of Logic
as a science. Somehow, interest in Logic declined with the Roman prevalence
in the Mediterranean Sea. And in the Middle Ages, as most of the work of the
ancient philosophers except for Plato and Aristotle was already lost or had
disappeared, Aristotle's syllogistics became the priviledge of only a few monks.
Logic regained its interest as non-Euclidian geometries were discovered and as
the need for a theoretical foundation of analysis became evident. As soon as
1879, Frege presented the first formal language for Mathematics and Logic. But it
was the paradoxes of set theory, and the many conversations and disputes among
mathematicians of that period on that subject, that gave Logic its final impulsion.
In 1895 and 1897, Cantor published the first and the second part of his study on
cardinal and ordinal numbers. In this study, which constitutes the first foundation
of set theory, each collection of objects is regarded as a separate entity called a
set. Buralli-Forti discovered a paradox in Cantor's theory as soon as 1897. And
in 1899, Cantor himself published a paradox concerning his theory on cardinal
numbers. Russell published in 1902 a simpler form of that same paradox, known
in the history of mathematics as Russell's paradox:
In Cantor's theory, each set is defined by the characteristic property of
its elements. Let A be the set of all sets X defined by property X r X,
A = {XIXr
But then:
AEA