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.