Saturday, 3 December 2005

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 the letters A B and S with familiar heirarchical objects. For example take S as Animals, A as mammals and B as animals with legs. The statement A<S can be expressed as "All mammals are also animals". Since it is < (proper subset) and not ≤ the statement also implies that there are animals which are not mammals.

The symbols '∨' and '∧' can be used to produce new sets from subsets of S. '∨' is called 'join', but in our case it can be read as 'or'. '∧' is called 'meet' and can be read as 'and'.

In the above image we see A∨B. This can be read A or B, as in "Animals with legs or Mammals".

The above diagram in light of or Animals example would represent mamals with legs (which of course is just about all mammals excluding pinepeds).

Next we have the unary operator ¬ also called complementation or negation. In our case it gives us set complementation by which we mean ¬A represents all elements that are in the set S but not in the set A. With the animals example ¬A is any animal that is not a mammal which includes birds and reptiles.

With our Animals example we pictorially described the situation 'Not a mammal or not an animal with legs'. This set (¬A)∨(¬B) would have such elements as whales and lizards but would not include dogs.

The DeMorgan's law: '(¬A)∨(¬B)=¬(A∧B)' is true for any A or B in S in a Boolean algebra (but also in other algebras including orthomodular algebras). This law is obeyed by our intuitive notion of inclusion with finite sets. Since ¬ is an involution in Boolean algebra (¬¬x=x; read 'not not x' is equivelent to 'x') we can also derive the dual formula: '(¬A)∧(¬B)=¬(A∨B)'.

Another important concept that will be more central in our next discussion, because complementation/negation is not a fundmental operation in Heyting algebras, is implication. In a Boolean Algebra we can define the symbol → as 'A → B = (¬A) ∨ B'. At first this formula might be a bit difficult to interpret. It does however follow our intuitive notions of implication. If ¬A is true then A is false so B must be true.

All of this discussion so far has been very informal and has been meant as an introduction to the ideas I plan to present next. In the next post I'll talk about the topology of boolean algebras and heyting algebras and how we can think of them pictorially.

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