symmetry

Symmetric: The same, only different.

That'll do as a first approximation of a definition. Symmetrical objects aren't identical, but their shared structure (the "same" part) provides us with a short-cut to understanding them.

Logical operations have an abundance of symmetry. Lots of us are familiar with De Morgan's Law: not (P and Q) is equivalent to (not P) or (not Q): the not operation distributes over and, plus "toggles" it into an or. Symmetrically, you can replace every and by an or (and vice-versa) above, to get the other half of De Morgan's law.

There's probably some deep aesthetic pleasure and satisfaction we experience when we discover symmetry, and this probably helps us remember and use symmetrical concepts. However, occasionally the symmetry is so striking that we find it difficult to completely absorb and use. Perhaps this difficulty recedes when we become used to the symmetry, but it's certainly there at first.

I'm thinking of the distributive laws. One form states that P and (Q or R) is equivalent to (P and Q) or (P and R). We can certainly verify this using logical tools such as truth tables and Venn diagrams, and it is analogous to the distributive property in arithmetic, where multiplication is distributed over addition (just substitute multiplication for and and addition for or). However, the comfort and support of analogy evaporate when the second, symmetrical, form is considered: transform each and above into an or, and vice versa. In algebra, there is no distribution of addition over multiplication to help our intuition, but in logic and distributes over or, and or distributes over and.

I became forcefully aware of this working on an exercise I had posed to my students. In the middle of several transformations of some logical expressions, many of us ended up starting at something like:

(not P or R1) and (not P or R2)

Most of us saw one application of the distributive law --- distribute the middle and over the bracketed ors, corresponding to what we would call "expanding " in arithmetic:

((not P or R1) and not P) or ((not P or R1) and R2)

Unfortunately this approach, even when repeated on the bracketed expression on either side of the central or, didn't seem to lead very quick to the desired result. The interesting thing is that few of us saw that the original expression contained another application of the distributive law: the or following not P was distributed over the and, and this can be reversed, corresponding to what would call "factoring" in arithmetic. This approach yields:

not P or (R1 and R2)

... which happened to take us closer to the solution of the exercise.

What intrigues me is the tendency to "see" the first possible application of the distributive law, and not the second. An underlying difficulty is that there are twice as many distributive laws in logic (and over or plus or over and) as there are in arithmetic. This is probably aggravated by it being cognitively a bit harder to recognize the possibility of factoring compared to the possibility of expanding (gathering versus distributing, perhaps).

And we haven't (most of us, anyway) had several years of schoolwork preparing us to recognize these patterns. In some parallel universe, kids learn to manipulate logic symbols in grade one, and they just shake their heads ironically at our difficulty. But then, they are completely stumped by addition being commutative.