Tuesday, December 6, 2016

Propositional Logic: ifs, ands, ors, buts(new!)

1. I especially appreciate the symbol for subjunction, thanks for a comic in notation.

I feel like there are probably seriously considered modal logics that look a lot like some of these; doxastic (belief logic) comes to mind in that it can model "a conceited reasoner" who "believes his or her beliefs are never inaccurate."

A few nits (unless I'm missing a joke): in subjunction and universal quantifier, < and > are glossed as "greater" and "less", respectively; liberal bias quantifier is not drawn rotated in the example eq.; I think Leibniz's Law goes both ways (and if it doesn't your use of it is the reverse of your definition).

1. Gosh, thanks for the close read. I've been really interested in maths and their related disciplines for the last few years, but I'm still not very good at them. I'm not attentive enough to detail, as these fudges demonstrated. (Fixed the first three.)

As for fixing the proof...hmm. Took another go at it just now. Does it work any better? If nothing else, THAT one should at least sort of be legit. I'm open to advice.

I've been reading Tarski's Introduction to Logic and to the Methodology of Deductive Sciences lately. I've gleaned a bit, but unfortunately it's not quite the same as learning from someone who can correct your mistakes and answer questions. (Hmm. Didn't Socrates have something to say about this in the Phaedrus?)

2. Ah, I see, I think I missed the point that x is b, y is a, and Pz is z = c. The mistake was my own.

I meant to point out that 1) equality and 2) all-propositions-are-equivalent-ness are taken to be equivalent, not just 1 implies 2. But it seems like 2 implies 1 is easily provable with just the reflexivity of =:

if, for all P, Px <=> Py, then with Pz defined as z = y, Py is true (y = y by reflexivity of =), so Px is true (x = y).

I was supposed to have learned the basics a few times in school but never became comfortable with it, so I'm unreasonably defensive sometimes. Sorry for picking at your post, and wrongly at that. I appreciate it in the spirit it is presented, thanks again, and best of luck with logic!

3. DON'T APOLOGIZE. There is no need. I always appreciate feedback/corrections from someone who's more familiar with a subject than me. And I understand that the impulse to nitpick often comes from a genuine desire to help (at least that's why I do it), so no offense was taken.

2. In graduate school I took a course in Foundations of Mathematics, where I proved, among other fascinating results, that for every open paren "(" there is a corresponding close paren. After spending a few weeks on such things, you build up some defenses. You say "well, it isn't exactly surprising, but we'd be in a lot of trouble if it wasn't true." So it took me a little while to start laughing at this. I think you should take that as a compliment. The best jokes are the ones that hit a little too close to home.

If you're interested in learning more about mathematics, here's a tip: mathematicians often build their edifices from the top floors downward towards the foundation. That's to say, they start with some intuitions and their consequences, and only build up a rigorous theory to justify them later. The intuitions are where the action is; the theory work is only undertaken grudgingly, as a kind of mopping up. But, for reasons that boil down to ideology, the theory is always presented first, even though it outright obscures the intuitions that are the living heart of the subject.

So in your reading, especially at the beginning, when you're confronting a lot of definitions and formalism and you're trying to get your bearings, remember that what you're reading is the last and least important part, and that your eventual goal is to see beyond it. So you may not have to reproduce every definition and theorem in detail. A passing familiarity can be enough.

I know I would have been a better mathematician if someone had told me this on day one. I hope it's helpful, or at least interesting.

1. It is! I also remember reading similar comments about philosophers: often all the reasoning and dialectics are just so much justification of a gut feeling.

As an undergrad I weaseled out of a math gen ed requirement by taking a similar course called...I think it was "Development of Mathematics." Basically it was an overview of the history of mathematics, geometry, et cetra. (It was a VERY long time ago, so I don't recall many specifics.)

Kind of a brilliant idea on that professor's part: snag all the undergraduates who don't want to bother with a math course, and spend a semester nurturing their interest in the concepts, if not the calculations.