Continuous logic

Boolean operations where 0 is false and 1 is true can be extended to real numbers between 0 and 1.
Lets take two independent variables x and y.

and their ven diagram of all possibilities

The space of all possibilities adds up to one (because 1 is true),
We can avoid mentioning a, b, c, and d and just work with arithmetic operations on x and y.

not x = (1 - x)
x and y = (x*y)
x or y = (x + y - x*y)
x xor y = (x + y - 2xy)

DeMorgan's laws

x and y = not (not x or not y)
x or y = not (not y and not x)

There is one bug, "x and not x" should be 0 because not x is true only to the degree that x is false, however our formula gives us "x and not x = x - x*x".
Similarly "x or not x" should be always true because "x" and "not x" are exactly complementary. But when we plug it in we get "1 - x + x*x".

The solution is is to disallow higher powers, whenever we see something like x^2 or "x*x" we replace it with just plain x. This works because information from the source x should only be counted once, even if we receive it through two different paths.

examples

x xor y = (x or y) and not (x and y) = x + y - 2x*y
if we allowed higher powers we would have the incorrect result of x xor y = x + y - x*y - x*x*y - x*y*y + x*x*y*y

it works just as well when the two variables are not independent

w = 1/3x + 2/3y
z = 3/4x + 1/4y
w or z = 10/12x + 9/12y - 7/12y*x

words vs understanding

The other day me and my girlfriend were in a store buying a lightbulb for an old fluorescent lamp we'd found.  We both knew nothing about fluorescent lamps so we handed it to the clerk and asked him to find the right kind of bulb. When he found the right bulb (a standard 5 watt one)  he remarked that because the lamp was old it was very noisy.  I asked him if it was a ballast problem (when a fluorescent light had flickered and buzzed my dad had said it had bad ballast) he immediately brightened up and explained that it wasn't a problem with the ballast but simply the fact that the old style of ballast was noisier than the new style. After we left my girlfriend was indignant that I knew no more about fluorescent lights than she did but by saying the right words I could make everyone think I did.

Hillary Putnam pointed out that people can talk about things without knowing the details of what those words denote. His example was beach tree. He doesn't know what they look like, and he couldn't pick one out of a police lineup, but he has the verbal distinction and when someone tells him that beech nuts are edible he can pass that knowledge on, even though it has no grounding in his own experience.

From this perspective money serves a similar function in the economy, without money debt is grounded as a specific relationship between people, with money debts can be bought and sold and travel far beyond their original context.

Similarly there are two ways of doing algebra, one is to think about the objects and what operations would make sense on them, another is to move symbols around according to symbol manipulation rules.

Many facts about the world can appear in a languages as rules about how to combine words so that someone can then speak coherently about things they don't understand. There is a fuzzy boundary between pragmatics and grammar.

Citations

Putnam, H. (1975/1985) The meaning of 'meaning' . In Philosophical Papers, Vol. 2: Mind, Language and Reality. Cambridge University Press.