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Showing posts from December, 2005

The Logic of Space part III

Ok, so last time we left off with a very informal discussion about Venn diagrams and how they relate to Boolean Logic.

First let us do a little set theory and then we'll start drawing connections with the previous post to make it a bit more rigorous. We will stick with completely finite sets, so keep that in mind.

A set is basically a collection of distinguishable objects. Sets have no notion of the number of times an object is in them. They simply contain an object or they do not. A set (if it is finite) can be writen in terms of its elements, for instance: S = {a,b,c} is the set S with elements a,b and c.

A map can be thought of as arrows that leave from one set (the domain) and arive at another (the codomain).

We will also introduce a special set 2^S which is an exponential or a "homset" called hom(S,2). S will be jus t a collection of elements as above, and 2 will be a collection of the elements {0,1}. We can think of a homset as a collection of all maps fro…

The Logic of Space Part II

I was checking my access logs and noticed that I was #1 on MSN for "logic of space" which is a bit embarassing since my last post had so little content, so I've decided to make it up to those who end up here.

We can start with Venn diagrams.

Venn diagrams are spatial representations of logic. They can be used to reason in boolean logic. A Boolean Algebra can be thought of as a tuple < S,∨,∧,¬,0,1 > where S is a set (a collection of elements), ∨,∧ are binary operators, ¬ is a unary operator, and 0,1 are nullary operators (if you don't know what I'm talking about yet don't worry). We can think of 0,1 as false and true respectively.

In the Diagram above S represents everything in the picture. A and B are subsets of S which we will write A<s meaning that A is a proper subset of S. White in our pictures will always represent things that are not in the set of interest.

When you look at these diagrams you can often make sense of them by replacing th…