Contraction and Protraction in Philosophy: Part 02
Revisiting my earlier explanation about the relationship between contraction and protraction in philosophy, I thought to give a much more rigorous and less interpretive example. In logic, the phrase "x and y" is a conjunction. It is sometimes symbolized as:
x && y
Similarly, the phrase "not x" is a negation. It is sometimes expressed this way:
!x
In like fashion, the phrase "x or y, but not both" is referred to as an exclusive disjunction. It is also often symbolized as "XOR" in logical languages. So, if I were to write:
x XOR y
I would, in more basic terms (using only negation and conjunction) mean:
(!(!x & !y) && !(x && y))
This, in essence, reads "it is not the case that both not x and not y can be true, and it is not the case that both x and y can be true". This is quite a mouthful, however logically correct, so we often just say "x or y, but not both".
The first statement is a protraction of the second. The second statement is a contraction of the first. Stated another way,
x XOR y
is a contraction of:
(!(!x & !y) && !(x && y))
which is a protraction of:
x XOR y
So far, so good, right? You can see why we contract ideas--they're tough to communicate otherwise. But our minds can unpack a contracted idea and reconstitute its original meaning and implications. Our minds can also contract a protracted idea so that it can be more easily communicated.
But, does the contraction really save us that much effort? Maybe not with the simple example above, with the XOR statement. But there's a problem with the word "or". Exclusive OR is only one possible interpretation of the word "or". There's a similar concept "x or y, or both" that's called an "inclusive disjunction" or "inclusive or", which in logical languages is sometimes written like so:
x OR y
Now, this seems pretty straightforward, right? But how do we define this in terms of negations and conjunctions? Well, we really want to be specific, so we'd have to write:
((x && !y) XOR (!x && y)) XOR (x && y)
This reads "either x and not y, or y and not x, or both". It's pretty simple to understand now that we have a concept of exclusive disjunction established. But, recalling that XOR is itself a contraction of other conjunctions and negations, we'd have to unpack all of the XOR statements as well. If we were to do this, we would end up with the following:
!(
!(
!(
!(
x && !y
) && (
x && !y
)
) && !(
(
x && !y
) && !(
x && !y
)
) && !(
x && y
)
) && !(
(
!(
!(
x && !y
) && (
x && !y
)
) && !(
(
x && !y
) && !(
x && !y
)
) && (
x && y
)
)
.
And that is a protracted version of the expression "x OR y". In review, our mind can contract ideas to more communicable forms, such as language or symbols. Conversely, it can protract these contracted forms back to their original ideas.
Now, here's a challenge. If you have a language "x" with a finite number of words and grammatical constructions, how do you logically express an idea for which there has never been a word or expression? Mull over it.