set:  A, ~A
elaboration: A, ~A implies opposite A,~A 
	|—> A, ~A (X, ~X)  

		‘opposite’ extends to all terms so also 
		|—> A, ~A (X, ~X) [Z, ~Z] A, ~A (X, ~X) ~{A,~A (X,~X)}  

and we can see how quickly this gets complicated.  

The set and elaboration count as synthetic.  i.e. not always does it make sense to say that ~A of A implies opposites, but we stipulate in this case that it does.  i.e. the opposition toward/away.  

	An elaboration of the example as in the elaboration of the set states that ‘toward’ has (A,~A) in it already by dint of the existential opposite of it ‘away’ but we want to carry this further for some reason, perhaps because we note another implication. 

Toward/Away implies Tangential by extension of the implication of opposites.  

	Tangential works as the opposite of toward and the opposite of away.  
	if someone said, did he walk toward (as with away) from you, you could say, no - he walked along a tangential path which definitionally stands in an opposite relation to 		points C and C2.   

	This means that we have TangentialX and TangentialX2: tangential, the opposite of toward and tangential the opposite of away; (X,~X) 

		In the terminology we extended opposition from (X, ~X) into [Z, ~Z].  This makes sense to do actually.  
		
		A detour in our explanation will help to make sense of this. The extension of A,~A figures tautologically as A, ~A ~(A , ~A) and finally A, ~A ~(A, ~A) ~[~(A, ~A)] which amounts to A, ~A ~{A, ~A) | ~[A, ~A ~(A, ~A]}.The notion that toward (really the prime tautological ‘A’ in this situation) implies away by opposition generates through extension the notion that the pair toward/away imply by opposition another pair of which we have only mentioned ‘tangential’.  We know the pair of tangential as the denomination parallel. 
			toward/away|tangential/parallel 

To make sense of the extension of ‘toward/away|tangential/parallel’ otherwise known as A, ~A (X, ~X) we have two as of yet unexplained formulas.   

		A, ~A (X, ~X) [Z, ~Z] A, ~A (X, ~X) ~{A, ~A (X, ~X)} 
and		A, ~A ~{A, ~A) | ~[A, ~A ~(A, ~A]}

	What this says at the root has to do with that when toward went to toward/away it doubled just as when toward/away went to toward/away|tangential/parallel it doubled. The bottom formula has eight terms for the reason that we shall expect another doubling if we can preserve everything. We expect a our set to looks something like this:			\___\___/___/___|___\___\___/___/.  We also know, however, that the parallel in tangential/parallel applies as an opposite not only to tangential, but also as an opposite of both toward and away for the same definitional reasons.  
	Essentially the extension of formula one to 
		A, ~A (X, ~X) [Z, ~Z] A, ~A (X, ~X [Z, ~Z]) ~{A,~A (X, ~X [Z, ~Z])}  
	establishes a frame where some unknown denomination answers for its opposite in every X term.  It serves us to remember at this point that we used toward/away as a convenient example of a thing that could extend the rule of opposites to as many terms 	as it did, but in fact we seek only to describe a system that extends opposition along indefinitely to an indefinite number of terms.  We can do this in hypothetical form as we 		have done in formula two, but what we learn has to do with the fact that without knowledge yielded by exemplification we fall into abstraction.  I will illustrate the problem.

		In formula 2, Z seeks to describe the opposition nature it entails to all X variables. This means three sets of two Z variables.  We know however that we in fact need four sets to render the eight positive variables we should end up with.  So we devolve at this juncture into pure abstraction from whence we cannot return.  

			||  A, ~A (X, ~X) [Z, ~Z] A, ~A (X, ~X [Z, ~Z]) ~{A,~A (X, ~X [Z, ~Z])}  ||??||

			such that 
			
			|| Z, ~Z (Z, ~Z) [Z, ~Z] {Z, ~Z} ||??|| 

			whatever that implies such that 

			|| θ, ~θ (θ, ~θ) [θ, ~θ] {θ, ~θ} ((θ, ~θ)) [[θ, ~θ]] {{θ, ~θ}} <θ, ~θ>  ||??||
In other words a language of pure opposition.  
	We can invent terms that satisfy our example, but since the example uses loose terminology even if we opposed, say, ‘the point’ along with ‘the fourth dimension’ to satisfy part of the eight piece puzzle, the final two elements find their situation out of our reality of experience.  
		
		i.e. toward/away/tangential/parallel/‘point’/4thdimension/_____/______

		This kind of loose analysis happens due to the fact that toward and away figure as opposites on a linear continuum.  Tangential/parallel figure loosely as opposites on a three dimensional continuum relative linearity.  Noticing this pattern makes us hypothesize that the pattern continues.

\4thdimensionquality\tangential/toward/point | anti-point\away\parallel/anti-4thdimensionquality/

			It arises, knowing what a point means that we seek what the opposite of a point could amongst all of the empirical evidence of our sensorium amount to.  It bothers us that points do not exist (take up space) according to the dictum of mathematics but in another sense do take up space if considered as a physical object  If a point can not exist empirically or spatially but have a veritable use describing reality we can 	extend this notion to other non existent mathematical ideas; they would, perhaps, also make describing reality easier.   I consider that a non-trivial point.  Either this amounts to truth and has value, or amounts to a misuse of the capacity to differentiate.