SLOG week 3

   I have greatly enjoyed the course up to this point. This unit is on implication and boolean algebra. I enjoy using it to make concrete sense out of concepts that previously seemed to be very human. Of course, implication is something that is ubiquitous in human interactions and thought; almost any statement contains something resembling a "P implies Q" implication.
   I am surprised at how much I am enjoying the first assignment. The exhilaration of problem solving that Danny mentioned in the first class is very real; slowly pulling apart each statement given in assignment 1, discerning what is and what is not able to be said with logical certainty, is great.
   One of the things I noticed when it comes to this course is that Danny has taught us quite effectively about being conservative with our statements. When moving through each question, I am very confident in identifying what is and is not true. In previous years, I might have felt satisfied with merely suspecting the validity of my answers. In this course, I am able to analyse my thoughts on the questions, stopping after every antecedent, consequent, etc. to reflect on whether or not I could say each thing, not only with confidence, but  with certainty.
   

SLOG week 1

   The first week of CSC165 was not very challenging because it was a very, very simple introduction to the first unit. We learned about being precise with our words and mathematical symbols, and the proper symbols to use to be most precise. We went over the ways that the English language can fail us when we mean to be absolutely understood, with not even a hint of ambiguity. Some of it was amusing, like when we went over some newspaper headlines that attempted to capture some story or concept with a few words and utterly failed to do so.
   The biggest thing I can say at this point is that I am interesting to see where we go from here. The first week was, ironically, quite ambiguous. I do not quite understand what it has to do with whatever mathematical concepts are coming, but I look forward to refining my understanding of this week through the lens of whatever we learn in the course.

SLOG week 2

   It is the end of the second week of the fall semester, and I have now attended six lectures and one tutorial of Mathematical Expression and Reasoning for Computer Science (CSC165H1). The material so far has been simple, and so too, at this point, are my thoughts and reflections. Still, six lectures adds up to five hours total, and five hours adds up to many words spoken, topics discussed, and questions and answers considered, so there is at least a bit to say at this point.

   The concept of implication has so far been the most interesting and challenging part of the course. It is simultaneously sensible and at odds with my--and most people's, I'm sure--intuition. The fact that a statement can be made such as "All the orange snakes in the lecture hall have buckteeth", and this statement, from a logician's point of view, is correct, is bizarre. Of course, the quality of being true, from a logician's point of view, is different from that of your average Joe, who plays fast and loose with logic. To a logician, any claim can be made about the empty set because there are no counterexamples within it.

   A thought on this: it feels, in a way, like a "cop-out". The concept of vacuous truth is upheld because it fits with the logician and mathematician's established methods of proof and disproof. A universal claim is disproved only by a counterexample, but, in a sense, that is only because of the limited faculties of people. People cannot assess infinite or extremely large amounts of data to completion, checking if every example is one that fits the claim - if they could, we would do this to prove claims about sets. But because people simply cannot, the method adopted by logicians and mathematicians--who, just like the rest of us, have limited processing ability--is to look for a counter example and, if one cannot be found, declare a claim to be true. It is because logicians need to say their method of proof is universally applicable and infallible that they claim it is also applicable to the empty set--and this is why, with their method, the empty set can "truthfully" be assigned any quality. I say all this because, no matter how much it can be explored and verified with logical and mathematical language, it isn't really true, in the human sense (which should not be dismissed, though it might not have much place in this course), that all the orange snakes in the lecture hall have buckteeth.

   So, in a sense, I think that vacuous truth is sort of a bi-product not of genuine logic as much it is a bi-product of the human brain using its limited faculties to make an attempt at genuine logic. If we could not prove things, we would not be able to get very far in the fields of math or logic. To prove things, though, we can only use our measly little brains, and vacuous truth is implicit in knowing that our own methods work.