On the questions of what the difference is between mathematics and logic, and whether quantum mechanics defies logic.

As usual, words like 'logic' mean different things to different people. Very frequently, people use it for a series of statements that 'seem to make sense', but really this is what used to be called rhetoric, before that term was abused out of all meaning. When people doubt whether quantum mechanics is logical, they really mean they can't see how it makes sense. But this is nothing to with logic. In actuality, quantum mechanics is absolutely logical: from the starting observations like the two slit experiment and applying nothing but formal logic encapsulated in the mathematics, you end up at quantum mechanics. Nothing to do with 'it making sense', I'm afraid.

The question of the relationship between mathematics and logic is an interesting one. To begin with, there is 'predicate logic', which is basically the same as the syllogisms of Socrates in an algebraic form. So taking something like:

All men are mortal
Socrates is a man
Therefore Socrates is mortal

this can clearly be generalised to:

For all X and Y,
All X are Y
x is a X
Therefore x is Y

Add more symbols, like an upside-down A to mean 'for all', and a back-to-front E for 'there exists' you can turn this into a pure piece of algebra.

In this sense, then logic is just one more branch of mathematics to set alongside geometry, topology, group theory and all the rest. Logic in this form is nothing to do with truth, it is all about pattern replacement. Common to all of these branches is a single key concept:   Given a sequence of symbols containing a particular pattern, it can be replaced by a new sequence containing a transformed pattern.

For example given the sequence

A > B > C   (using the conventional meaning of > to be be 'greater than') this can be replaced by the pattern A > C

So one way of looking at mathematics is as a pattern replacement system. There are a large number of such replacement rules, and as a child we are taught that, for example, we can replace the pattern 2 + 2 by the simpler pattern 4, but there is no real concept of truth when you think of mathematics like this. Instead, there is a concept of consistency. The set of rules are consistent if, when properly applying them, they always end up with the same ultimate pattern. For example, in "ordinary arithmetic" if we applied the 2 + 2 pattern along with others to the sequence 2 + 2 + 2 we should end up with 6 whether we began be replacing the first 2 + 2 instance or the second.

So mathematics, and in particular predicate logic, can be thought of as pattern replacement rules. ("Real mathematics", i.e. what people employed as mathematicians do, is not this: it is the development of the patterns in the first place)

All of this is the branch of mathematics that I love and work with, known as 'pure mathematics'. Whether or not it has any relationship with 'the world' is irrelevant: it is what it is. "Applied mathematics" is essentially when you notice (or create) a mathematical model that is arelationship between mathematics and the observed world and make the assumption that given a good enough mathematical model, what the model predicts is what will be seen in the world. Fundamentally, that is flawed, even though all science depends upon it. You can never really be certain the model is good enough, which is why it is absolutely essential scientists are prepared to abandon or modify their theories (= their mathematical models) whenever a new observation is made that doesn't fit. The old model is still as mathematically valid as it ever was, it simply not longer adequately models the world.

Back to quantum mechanics. A mathematical model has been built that very accurately describes and can predict what happens in the real world. It has been built using the pattern-replacement rules of logic and mathematics. It is 'logical'. But it does not correspond to 'common sense'. Why should it: that was never the goal!