When I say “mathematical logic” – I refer to Classical Aristotelian logic. Currently, there are different branches of mathematical logic. Fuzzy logic would be most notable example. I’m yet to see any reasonable application of all those complex disciplines to solve ANY real-life problem. If you need kill some time with “brain exercise” – just play chess or chinese Go. IMHO, Non-Aristotelian mathematical logic is just a big waste of time. There is no such problem with Aristotelian logic – it has been very useful tool for many centuries.
Classical Aristotelian logic operates on predicates. Every predicate can be either true or false. Everybody is familiar with Boolean algebraic expressions such as AB=C, A+B=C, etc. If you keep everything sane and don’t push things to “infinity” logic provides very useful tool.
Here is a simplistic Venn Diagram which is most commonly used to illustrate all those Boolean expressions.
It’s quite obvious that any Boolean expression can be represented using probabilities. For example, AB=>C can be express as P(AB)=P(C). Assuming that all probabilities are either 1(true) or 0(false). Well…that is the main problem with logic – it always assumes true or false. That is quite limiting and ignorant factor. Many people love “absolute” truths, but that’s not how real world operates. Logic pushes you into the “dogmatic” or “axiomatic” thinking.
Lets consider the following simplistic questions: “we know A is true – what can we say about B?”
Boolean logic can only answer that questions in 2 cases:
- A is subset of B ( A=>B )
- A and B have no overlap whatsoever (A => not B)
That is NOT a complex problem by any measure, but Boolean logic provides you with very retarded tool for real-life problems.
I like to think about any problem in terms of probabilities. I find this approach extremely useful when analyzing ALL real-life problem. I see probability as an extension of logic. It’s very common view among Bayesian statisticians.
And yes, it is hard to think in terms of probabilities because we all want to be 100% right. It take huge effort to keep yourself on that track.