Logic
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Here we go with this little *leit motif* of mine where I begin talking about something completely different and then it turns out that it is all connected. This time I will not merely transport you to a different branch of mathematics, but an entirely different discipline, namely *logic*. Or, to be more precise, intuitionistic logic. This discipline may seem to you as detached from what we have been talking about as it possibly can, but it is actually very close - I don't even have to start another chapter for it (althought it will make this one a bit lenghty).

Logic aims to study the *rules* by which knowing one thing leads you to conclude that some other thing is also true, and (most importantly) to do this in a formal way i.e. without regard of what these things are, or how you learned them.

Not only that, but logicians try to organize those rules in what are called *systems*, or *formal systems* - containing selections of rules that have the expressive ability to prove everything that you can see by intuition.

What does "prove" mean in this context? Simple, when we are able, using the rules of a given logical system, to transform one set of assertions **A** to another one **B** we say that we have proven that **A → B** in that logical system.

So let's get our balls again.


![Balls](balls.svg)

Follows
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Or and And 
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False and true
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Notice we didn't say anything about true and false. Logic is not about truth and falsity. 


Heyting algebra overview
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and say that it is an "atomic fact" as Wittgenstein calls it - it is not composed of any "more basic" facts, it is just the simplest thing that you can know.


Lattices
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Lattices and trees
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