Vacuous Truth

  Almost two weeks have passed since I stepped in the first lecture of CSC165. I must admit the course is not really what I expected…… and I AM LOVING IT (so far).

  I remember in our first tutorial, our TA jokingly mentioned that there are people who would spend their time debating over logic. Sitting there, I thought to myself, “Well that sounds like something I would do!” As of today, September 18^th, we have covered topics including quantifiers, if-then statements, and converses and contrapositives. Much of the topics discussed overlaps with the syllabus of MAT137, hence sometimes the lectures feel like a review for another class. However, there are several interesting problems I encountered, one of which being the case of vacuous truth.

  At first, I found the idea of vacuous truth very abstract and absurd. How can something that is false (or unrelated) in the first place make a logical if-then statement? However, after going through some examples in my head, the concept of a vacuous truth began to make sense. If the first statement of the implication is false, the if-then statement will always be true, whether the second statement is true of false. In my own interpretation, this is because the first condition of the implication (the if…. part) is false, the implication itself will have no value, and everything implied would be possible, hence true. 

  What really shocked me was the application of this concept in our tutorial questions. In one particular case where we had to draw a Venn diagram of the false case of the statement “some programs (out of 3) pass the test”. I forgot to include a “O” in the region exclusive of the 3 programs. I realized my mistake in the tutorial. The O must be included because in the case where there are no programs all statements are true, hence the application of the concept of vacuous truth.