Week 2 : Quantifiers, Sentences, Symbols, Implications

        This past week began with to topic of quantifiers. Quantifiers are something that I'm familiar with from before taking this course, so they weren't particularly challenging to grasp and get a hang of the symbols involved in them. What was helpful, as I mentioned in the previous post, was the use of venn diagrams to determine how to make it false and how to make it true. I think it's a very useful approach to see how something can be verified and how something can be falsified, and not just one or the other, and the venn diagrams make that much more visual than written English or symbols. I've learned set notation in the past and only some of the symbols were new to me, but I still feel the need to always translate the symbols to english or a diagram to make sense of it. I'm not sure if that's a good thing or very inefficient, but I feel like I should work on being more fluent with these symbols, especially when it comes time to actually translate complex statements to them (simpler statements are fine).

     Moving on from quantifiers, there was the topic of statements vs sentences, predicates, and then implications. The difference between a sentence and statement was pretty straight forward at least by definition, although I had always thought they were synonymous (but that's in english, I just didn't know what they meant in logic). This lead into predicate and how they can be turned into a statement by quantifying them (L(x) -> L(something)). As for implications, most of it was straightforward, but but whatever reason the notion of a contra-positive went over my head during lecture so I had to look it up. Turned out that I was confusing it with an inversion, which does not depend on the truth value, but contrapositives do. The difference symbolically is like the combination of an inversion and converse. I just went through a few statements (in english) on my own to write out the converse, inverse, negation, and contrapositive just to make sure I have a solid idea of how they are different (and how they are related) . So something like this:

statement: All penguins are birds
converse: All birds are penguins (this is false and the statement is still true)
inversion: If it is not a penguin, then it is not a bird (also false, and the statement is still true)
negation: There exists a penguin that is not a bird (if this is true, then the statement is false and vise-versa)
contrapositive: If it isn't a bird, then it isn't a penguin (relies on the statement also being true)

I at least hope the above logic is correct.

Moving on, vacuous truth was discussed, and the idea of equivalence where a statement that is an empty set can be equivalent to another statement that is an empty set . . . . I think.
It wasn't really tough to grasp the concept, even it doesn't seem very intuitive (my initial thought on a vacuous truth would be to say it is false), but I can see how there is a distinction. The one thing I'm curious about is where this idea becomes useful/necessary because right now I can only think of weird examples of irrelevant stuff like an empty set of minotaurs who also tap dance.