
Name: polly mackenzie
Question:
what is the difference between logic and math logic? thanks
Hi Polly,
Many fields have their accepted forms of reasoning and
reaching conclusions. In principle, philosophy studies
all these 'modes of reasoning'. Some are more reliable
than others. For example, we do 'reason' from many examples
to at least tenative conclusions (something philosophers
call induction). We keep looking at pieces of evidence
so see if it matches our conclusion but are never sure
what will happen with the next example. In that spirit,
the philosophers have a standard example. If the
statement in question is: all gulls are white, then a black
car is evidence for the statement! Why? Because if you
saw it in the distance as a black blob, you might wonder
whether it was a black seagull  violating your conclusion.
When you get close and see it is a car  it does not
contradict your statment that gulls are white (and things
which are not white are not gulls).
Mathematical logic is that part of logic which is
applied by mathematicians to mathematical objects, concepts,
examples, ideas. In mathematics, the philopsphers 'induction'
is NOT a proof  merely evidence for a conjecture.
Even millions of examples are not a proof.
Mathematicians have a different idea of induction:
proving something for all natural numbers (1,2,3,...,n,...)
by a direct argument including steps from samller numbers
(already checked) to larger numbers in a systematic way.
What a mathematician calls logical ( a proof) will probably be
accepted by any other logician as logical. But others will
need and use additional principles which a mathematician will
reject.
It is my sense that a computer scientist has the same standard
of 'logic' and 'proof'. (Many computer science departments
require their students to study mathematical logic.) However,
they will still have to impliment programs that may have gaps
or may fail in certain circumstances. They know the difference.
In other fields, even in statistics, there will be principles
of reasoning that do not stand up to the standards of mathematical
logic. Most human decisions are not made based solely on
mathematical logic and it is worthwhile to consider the reasoning
used and how to make it better. Work on artificial intelligence
which tries to duplicate (or exceed) human decision making will
have to include some additional principles and processes beyond
mathematical logic!
I will admit that there are specific issues on which even mathematicians
disagree. They include issues like: can a proof by contradiction
show that something exists  or must you actually have a process to
find or create the object to say it exists (constructivism).
These disagreements do not concern most people doing mathematics
but they do keep some logicians thinking hard about the borders
of our knowledge and experience.
Walter Whiteley
(Former logician, now a geometer)
Go to Math Central

