w-III

Here is a very nice and good post about week three in the course.

So the third week was about firstly, conjunction, disjunction, and negation, and then about truth tables and then about manipulation of these.

The first section is almost just introduction of conjunction and disjunction as extensions of the ideas of union and intersection. The interesting part comes with negation and the variety of equivalent statements that can be made by moving around the scope of the negation and making appropriate changes to quantifiers, etc. At this point, actually, it becomes simplest to work through these examples through natural language. We all have some sort of idea of how to explain the opposite of some situation, but now we have a better understanding and framework for moving easily between natural language and symbolic language so this becomes the simplest manner of manipulating these operators.

The bit about truth tables is always interesting because sometimes it is easy to want to look at an expression and have an answer but oftentimes a truth table lets us take a step back and slowly work through these expressions in a rigorous way to avoid making mistakes. I always liked the idea that if two truth tables are equivalent then the two expressions are equivalent in a complete and rigorous way. It is always entirely possible that two expressions are equivalent but have entirely different meanings. I would liken it to `the formula for prime numbers' which was mostly just a tricky way of enumerating values using an alternate formulation of Euler's totient function and while looked nice in some sort of closed form, provided no new or interesting information or insight. A truth table for two True/False arguments has only so many possibilities but we can formulate an infinity of absurd expressions that we could demonstrate to be equivalent to some obvious or simple expression. I'm not sure to what ends, but the rigour over a small number of elements is a very `easy' thing to investigate so equivalence proofs become all but trivial.

Finally, the idea that evaluating P=>Q for P False, always evaluating to True. We cannot evaluate this expression with respect to the implication of P when we have `not P', so we must call it true. It reminds me of concepts of open and closed in mathematical sets, maybe in topology. The idea is that sets can be open or closed, but they can also be both or neither. If a set's complement is open then that set is closed and if a set's complement is closed then the set is open. The implication of the converse is what describes the set and complement is a unitary/cyclic operator such that the complement of a set's complement is the set itself. It is just another interesting example about properties that we must be careful to define and manipulate in a very careful way, because while not easily relatable to physical or extant objects, the math itself is complete, consistent and rigorous. If we simply permit weirdness and shirk our human need to relate things to the physical world (an important part of learning about the quantum universe), we can use our math to learn more and extend our knowledge and understanding. We just need to have a little bit of faith. Which is actually pretty scary for a `scientist', I guess.

Okay, cool.

J