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?