Axiomatic Transactional Analysis

Axiomatic Transactional Analysis
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TA Relation
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A and B are objects in A nnnn B whereas nnnn is a TA relation so that, counting from 0:
0th n means "I'm Not OK, You're Not OK",
1st n means "I'm Not OK, You're OK",
2nd n means "I'm OK, You're Not OK",
3rd n means "I'm OK, You're OK".

A false value of n is substituted by 0 and a true value of n is substituted by 1. Undefined values are substituted by x. x is not a variable. That is, the multiple instances of x in A xxxx B do not necessarily refer to the same value. In order to refer to the same value, p is used.



Moral hierarcy
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Generally, omitting variables and using = for quantitative (not qualitative) moral value, the favourability of relations adheres to the following rules.
0000 = 1111
0110 = 1000
0111 = 1001
1000 < 0111, 1000 < 1001
0111 < 0001, 1001 < 0001

This shouldn't affect neutrality of evaluation. Furthermore, for A:
A 1xxx B  ≤  A x1xx B
A x1xx B  ≤  A xx1x B
A xx1x B  ≤  A xxx1 B
A 0xxx B  ≥  A x0xx B
A x0xx B  ≥  A xx0x B
A xx0x B  ≥  A xxx0 B



Hierarchical equality
---------------------

Let P refer to a defined truth value other than p. For A and B:
A ppPP B  =  B PPpp A



Symmetric relations
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A 0000 B
A 0001 B
A 0110 B
A 0111 B
A 1000 B
A 1001 B
A 1110 B
A 1111 B

For each symmetric TA relation A nnnn B it holds that B nnnn A so that nnnn is the same relation. This also applies in case of undefined values so that:
A xxxp B  ⇒  B xxxp A
A xppx B  ⇒  B xppx A
A pxxx B  ⇒  B pxxx A



Directed relations
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A 0010 B
A 0011 B
A 0100 B
A 0101 B
A 1010 B
A 1011 B
A 1100 B
A 1101 B

For directed TA relations it holds that:
A xxpx b  ⇒  B ppxx A
A xpxx B  ⇒  B xxpp A



Passive denial
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For any TA relation denoted by nnnn it's possible to refer to another relation in which each defined value of n is changed. That relation is denoted by NNNN.

Furthermore, the morality, validity, soundness, relevance, feasibility or other such vital criterion of any TA relation A nnnn B can be called into question. Such an event is called denial and denoted as a function so that !(A nnnn B) = o in which o is the output.

Passive denial is passive in the linguistic sense, that is, it isn't specified who imposes the denial:
!(A nnnn B)  =  B NNNN A

Examples:
!(A xxxp B)  =  B xxxP A
!(A xppx B)  =  B xPPx A
!(A pxxx B)  =  B Pxxx A
!(A xxpx B)  =  B PPxx A
!(A xpxx B)  =  B xxPP A



Active denial
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Let A(B nnnn C) mean "A causes the relation B nnnn C". In this expression A is referred to as a function, but A, B and/or C can all refer to the same object. For symmetric relations:
A(B nnnn C) 			   =  B nnnn A ∧ C nnnn A
A(B nnnn C) ∧ !(B nnnn C)  =  B NNNN A ∧ C NNNN A

For directed relations:
A(B xxpx C)  =  B xxpp A ∧ C xxPP A
A(B xpxx C)  =  B ppxx A ∧ C PPxx A



TA matrix
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Let TAM(A,B) output four TA relations from A to B in the form of a matrix so that:
TAM(A,B) = [[nnnn],
			[nnnn],
			[nnnn],
			[nnnn]]

The rows of letters n stand for different TA relations so that:
1st relation is in a goal-oriented context,
2nd relation is in a cultural context,
3rd relation is in a situational context,
4th relation is in a personal context.

The index number of the relation correlates directly with short-term priority and inversely with long-term priority.



Evaluation of TA matrix
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Let TAE(i,nnnn) accept a TA relation of TAM(A,B) as input, accompanied by that relation's index number i so that i < 3. The output of TAE(i,nnnn) is a TA relation of the same matrix whose index number is i + 1.

TAE(i,xpxp)  =  (i + 1, ppPP)
In this case A is or isn't being given a chance by B, depending on the value of p. But to need to take a chance means A isn't OK.

TAE(i,pxpx)  =  (i + 1, PppP)
In this case A has or doesn't have a chance to exploit B. In order to figure out whether the chance is worth taking, A should evaluate the competitive aspect of the situation.

TAE(i,xxxp)  =  (i + 1, pPPp)
TAE(i,pxxx)  =  (i + 1, pPPp)
In this case A and B have or don't have an opportunity for cooperation. Different situations can involve different preferences for competition and cooperation. In case of a tie, it's safer for own immediate and narrow-minded self-interest to choose competition over cooperation. However, needing to cooperate doesn't mean one isn't OK.

This method can be used for defining all values of TAM(A,B) from some smaller subsets of defined values.