w-II

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THIS IS A POST ABOUT WEEK TWO IN THE COURSE

I'M NOT YELLING, THIS IS JUST HOW COVER PAGES
FOR TOP SECRET DOCUMENTS LOOK

I JUST IMAGINE GEORGE W. BUSH IN HIS TEXAS DRAWL:
"HOW COULD ANYONE IGNORE ALL THESE CAPITAL LETTERS? IT'S IMPOSSIBLE!"

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Week two in the course was touching on material that was new to me which is always nice because I like that sort of thing. I am perfectly familiar and comfortable with set notation and the like but predicates are definitely a new thing.

The rest of the lecture was a lot of parsing statements to identify P and Q if they were to be made to match the mathematical statement P=>Q. This started off slow and easy but later progressed to some rather challenging examples that required careful consideration to understand what was being said in plain english and understand the implications in a mathematical (P=>Q) sense. Overall pretty interesting but will likely require me to go review some of the trickier examples. Another horrible failing of the english language. All these equivalent ways of describing a P=>Q situation with some much less clear than the others.

Also a fun example of vacuous truth. Such an interesting mathematical/philosophical phenomenon, and there's another good one about it in the course notes. I guess the idea is that we might run into such a situation when we are programming. Nonetheless they are important to know about and understand.

My final note about the lecture is that there were two characters seated immediately behind me who seemed to think that their unsolicited sarcastic comments were in some way valuable or insightful. Sadly they were mistaken. It was pretty annoying. Almost as annoying as me griping about it on my SLOG, but just not quite.

The final piece of things we did this week that I feel in some way about it the venn diagram with 'x' and 'o' marking. The idea is to draw a venn diagram of two subsets of a third larger set and place 'x' in any area where we know that no element can exist, and 'o' in any area where we know some element must exist. In our tutorial, we were always interested in a single one of the two subsets, and placing indicators 'x' or 'o' in either its intersection with the other subset or its are disjoint with the other subset. This one set in which we were interested, was known to have three elements. Because it is known to be non-empty, when either of the disjoint or intersection (with the other subset) areas has 'x', it is known that the other must have 'o'. Because the union of the disjoint and the intersection section is the whole (sub)set, if one section is empty, the other must necessarily contain any any elements that are a part of that (sub)set, which is at least one (indicated by 'o') by the (sub)set's known non-emptiness. A set's non-emptiness is a required condition to produce a situation to which this 'x' - 'o' complement applies. For our example problems, we can and we must use this to place 'o' (in its correct spot) in any situation where we have placed 'x'. Reviewing the other examples, we typically see that sets are rarely given as non-empty. It is necessary then to take caution when attempting to apply this 'rule'.

I guess my final point is the role that our use of natural language takes in understanding and thus solving these problems. Suppose we pose the question:

If "some men are doctors", and "some doctors are tall", does it follow that "some men are tall"?

It is maybe not entirely clear, but the answer to the question (in this case) "Are some men tall?" is "no". There is an entire psychological aspect to this problem where if we simply ask "In real life, are some men tall?" everyone would be answering "yes". It is true. We can construct a reasonable definition for what constitutes "tall" based on human height measurements and distributions, and then look at the set of professional basketball players and find at least one single example to demonstrate that there exists a man (an element in the set of men) who is tall. No problem, all done. But at the same time, if we simply replace "men", "doctors" and "tall" and we instead ask:

If "some cats are pets", and "some pets are dogs", does it follow that "some cats are dogs"?  

It might be easier to evoke a correct response of "Of course not! What a ridiculous question! Why would you waste my time even asking that?". But this is because the question that we ask with the foreword "does it follow" is false even with the foreword "in real life". What about further abstraction? What happens when we ask:

If "some A are B", and "some B are C", does it follow that "some A are C"?  

Maybe we can prompt more advanced or specific thought about the question rather than about men and/or medical degrees and/or height and/or cats and/or dogs and/or pets. Just some thoughts. I guess the point is that abstraction to more concrete statements can help us solve questions in our natural language that can often be unclear or confusing for reasons of familiarity and association of these words with extant things or concepts. What about asking:

If "A ∩ B ≠ ∅", and "B ∩ C ≠ ∅", does it follow that "A ∩ C ≠ ∅"?

I'm not sure, but I think it's interesting for sure.

Please someone let me know if I'm on the right track with this whole SLOG thing.

That's all for now.

Thanks for slig-sluggin' it with me,

J