Page 16 - Judaic Logic
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10 JUDAIC LOGIC
Some people are rather weak in practice, though well-informed theoretically. In any case,
study of the subject is bound to improve one’s skills.
Logic is traditionally divided into two - induction and deduction. Induction is
taken to refer to inference from particular data to general principles (often through the
medium of prior generalities); whereas deduction is taken to refer to inference from
general principles to special applications (or to other generalities). The processes ‘from the
particular to general’ and ‘from the general to the particular’ are rarely if ever purely one
way or the other. Knowledge does not grow linearly, up from raw data, down from
generalities, but in a complex interplay of the two; the result at any given time being a
thick web of mutual dependencies between the various items of one’s knowledge.
Logic theory has succeeded in capturing and expressing in formal terms many of
the specific logical processes we use in practice. Once properly validated, these processes,
whether inductive or deductive in description, become formally certain. But it must always
be kept in mind that, however impeccably these formalities have been adhered to - the
result obtained is only as reliable as the data on which it is ultimately based. In a sense,
the role of logic is to ponder information and assign it some probability rating between
zero and one hundred.
Advanced logic theory has shown that what ultimately distinguishes induction from
deduction is simply the number of alternative results offered as possible by given information: if
there is a choice, the result is inductive; if there is no choice, the result is deductive. Deductive
logic may seem to give more certain results, but only because it conceals its assumptions more; in
truth, it is merely passing on probability, its outputs being no more probable than the least probable
of its inputs. When inductive logic suggests some idea as the most likely to be true, compared to
any other idea, it is not really leaving us with much choice; it is telling us that in the present
context of knowledge, we decisively have to follow its suggestion. These are the reasons why the
word “proof” is often ambiguous; do we mean deductive proof or inductive proof, and does it
matter which we mean?
a. Some propositional forms and their interrelations.
The first task of logicians is to observe actual thought and speech, and take note of
recurring linguistic formulas. At first, the variety may seem bewildering; but, starting with
the most common and simple items, and gradually considering more detailed issues and
more complex cases, Logic has grown and matured. A great breakthrough, which we owe
to Aristotle (4th century BCE, Greece), was the discovery of an ingenious artifice, which
clarified all subsequent discussion. In everyday discourse, we make statements with
specific contents, like “swans are white”; Aristotle developed logical science by focusing
on forms, substituting variables like “X” and “Y” for specific values like “swans” and
“white”. Such a formal approach signifies that certain aspects of reasoning can be justified
without reference to content; they are abstract truths for all propositions of a certain kind.
We shall here first consider some of the simplest of the forms called categorical
propositions. (It is worth memorizing the symbols, traditionally used since the Middle
Ages to abbreviate theoretical discussions. A and I come from the word affirmo; E and O,
from nego - these are Latin words, whose meanings are obvious. Note that IO refers to the
sum of I and O.)