DeMorgan’s Rule

So it turns out that digital logic and philosophical logic have slightly different implementations of DeMorgan’s rule, in terms of the strictness of shape conformity.

In digital logic, the rule is

!(ab) <-> !a + !b

This is a functional mapping, so it matches any pattern based on the main connective (AND or OR (+)), independent of the sign set by the unary NOT (!) operator. Thus MN <-> !(!M + !N), and !(M!N) <-> !M + N, and so forth.

However, in philosophical logic, the rules are

~(p & q) :: ~p v ~q
~(p v q) :: ~p & ~q

The key difference, ignoring the differences in notation (AND = &, OR = v, NOT = ~), is that this pair of rules only match patterns whose signs match. That is, the unary NOT operator is given as much credence as the binary AND and OR operators in determining the shape. This means that, for example, ~(M & ~N) :: ~M v N is not a valid application of DeMorgan’s rule in Philosophy, even though it worked above in Engineering.

The end result of all of this is that my last homework for Logic, instead of getting it completely correct because I already knew how to apply DeMorgan’s rule, I got it completely wrong. Yes.

This allows me to segue into a rabid rant about standards compliance. This is something that I have become a huge fan of when it comes to web development. At the same time, the absence of it has driven me crazy throughout my scientific and mathematical education. It would not be hard for some group to sit down and standardize notation across all fields. It would be something like the IPA, I suppose, except for mathematical and logical notation.

Just think of all of the problems it would solve: no more confusion about what the unit vectors in 3-space are (I prefer i-hat, j-hat, and k-hat), or how to represent NOT (prepended tilde? prepended bang? superscripted bar? the word “not”?), or using the same name for subtly different terms, or any number of other symbolic representation issues. I think that we certainly have enough symbols right now, and it would just be a matter of collapsing the set in a mutually agreeable way while also making the representational system consistent across different disciplines.

So, what do you think?

10 comments on “DeMorgan’s Rule
  1. carnap says:

    Isn’t that a valid application of DeMorgan, even in obnoxious intro logic do-every-damn-step mode? (When I took the logic final to pass out of the logic requirement for philosophy majors, I lost points for commuting and associating in the same step.) I guess you have to do:

    ~(M & ~N) :: ~M & ~~N :: ~M & N

    which is obnoxious to write out, because it’s really freaking obvious, but still valid.

  2. rose_garden says:

    Yes! Someone needs to figure out what the symbols are for spherical coordinates.

    Specifically, in all my math classes, phi is the polar angle and theta is the azimuthal angle, whereas in all my physics classes, phi is the azimuthal angle and theta is the polar angle. In addition, rho is the distance from the origin in math, but rho needs to be density in physics, so physics uses r for the spherical distance and also for the cylindrical coordinates distance, even though these are very different distances. (Except David Griffiths, a text author who uses s as the cylindrical distance coordinate.)

  3. rose_garden says:

    Does :: mean “implies”?

    I thought it was
    ~(M & ~N) :: ~M v ~~N.

  4. carnap says:

    I thought it was
    ~(M & ~N) :: ~M v ~~N.

    It is; I was just being sloppy. I think :: is “is logically equivalent to”, but either will work in this case.

  5. stormwynd says:

    I agree — I got nailed by ‘s spherical coordinates dilemma when I was in college, and my brain is still hurting from the experience.

    But can you imagine trying to get (for example) mathematicians, physicists, engineers, and philosophers to all agree on a single set of standards? *shudders* Or maybe I’m just being cynical.

  6. Hopefully we could also come up with a sensible representation in the process. I find ! to be much easier than ¬, simply because it’s both an everyday character and just screams “NO! NO! NO!” at you. I’d also gladly take just the word NOT. But ¬ just sort of looks like some kind of weird minus sign to me. Which I guess makes sense, because it’s sort of like a negation, but not quite. Still, the way it looks just confuses me.

    Don’t even get me started on superscript bars for NOT. Superscript bars are used for way too many things already.

    Using addition and multiplication to denote OR and AND also annoys me. I mean, you get things like (A AND B) OR C = (A)(B) + C. So if A, B, and C are all true, then (A)(B) = 1 and 1 + C = 2? There’s no 2. It’s 1. 1 + C = 1. But then what the hell kind of number space thingy (behold my mastery of complicated math terms) are you working in?

    It’s not Z mod 2. It’s like some kind of trippy place where you’ve redefined +. So why don’t we, instead of redefining +, just not call it + and save ourselves a lot of misery?

    & makes a lot of sense for AND, I think. I’m less happy with | for OR, although used to it. And that bar does sort of say “one or the other”. Some nice symbols for NOR (like !| perhaps), NAND (!&), and XOR (uh… I have no idea) would be nice.

    The best solution would probably be to standardize all of those symbols to something that’s visually sensible and easily typable but then emphasize the use of the words more.

  7. flurious says:

    I don’t understand why everyone can’t just agree to defer to the engineers. We do shit. Current is i and the imaginary axis is j. That’s all there is to it.

  8. Engineers have really shitty notation, in my experience.

    I’d much rather trust physicists to make notation than engineers. I mean, they’re the ones making new notation most of the time, anyway, since they’re forging ahead in terms of physical and mathematical concepts. Engineers only have to deal with the stuff later when it actually gains some sort of practical applications. Ditto for mathematicians, who certainly do spend a lot of time proving stuff that’s just sort of assumed and brushed under the rug by physicists who just really needed a certain mathematical concept to make something work.

    That said, math notation is pretty nice. It’s too austere, in my experience, however.

    Engineering notation seems too messy.

    I think logic should be some sort of happy medium between math and CS, and physical notation should probably be the province of physics. Although I’d much rather my angle from the z axis in a sphere be called phi.

  9. arctangent says:

    I would have written it thusly:

    If it’s not true that, taken together, one statement and another statement are both true, then that means that either the first statement, taken separately, isn’t true, or the second statement isn’t true, or maybe both aern’t true.

    “English, boy! English!”

Nurd Up!