Name: polly mackenzie

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)

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