(repost by request)

Given a six sided die, and the single operation “roll the die”, we can produce a noisy distribution of :

1(x1), 2(x1), 3(x1), 4(x1), 5(x1), 6(x1).

Given two six sided dice, and the single operation “roll the dice and sum the results”, we can produce a noisy distribution of:

2(x1), 3(x2), 4(x3), 5(x4), 6(x5), 7(x6), 8(x5), 9(x4),
10(x3), 11(x2), 12(x1).

The difference between the one-die and two-die distributions is that while the results of rolling one die are equidistributed between 1 and 6, with two dice the results of rolling can produce more combinations that sum to 7 than there are that sum to 2 and 12, and therefor the results are normally distributed: in a bell curve.

We can produce the same results with logic instead of numbers: For example, we can take the two words “Even” and “Odd”, and define two operations: “addition” and “multiplication”. Then apply the operations to all pairs:

Even + Even = Even,
Even + Odd = Odd + Even = Odd,
Odd + Odd = Even,
Even x Even = Even x Odd = Odd x Even = Even,
Odd x Odd = Odd.

And we can produce the same set of results with *any grammatically correct operations on a set, given the operations possible on the set*; including the set of Ordinary Language using Ordinary Language grammar. Although, unlike our simple examples using dice, the set of combinations of ordinary language is not closed, and so the number of combinations is infinite.

So any grammar allows us to produce a distribution of results, and a density (frequency) of result.

In mathematics this result set is called a ‘field’. A field consists of all the possible results of a set of operations on a set’s members, that are selected from the range of possible operations on those set members.

So in any set of results there will be a range of very dense, less dense, sparse, and empty spaces in the set’s distribution.