The boundaries of potential surprise in the economic decision provide a space between ℜ and μ such that intermediate point of ℜ and μ are Herbert Simon’s bounded (Katzner, 1986; Herbert, 1987; Kreeps, 2002) . We shall proceed by showing that the space of potential surprise bounding economic decision be a function f ( ℘ s ) and measurable in the interval [ ℜ , μ ] .

Lemma 1: If E = E 1 ∪ E 2 ∪ ⋯ where E 1 , E 2 , ⋯ are mutually disjointed, then,

∫ E f ( x ) d x = ∫ E 1 f ( x ) d x + ∫ E 2 f ( x ) d x + ⋯

Proof: If “S” is an upper sum related to any partition and “s” is also a lower sum related to same or different partition, then,

S ≥ I ≥ J ≥ s .

With I as the greatest lower bound of all upper sums, as a consequence, S ≥ I . Taking J as the least upper bound of all lower sums, as a consequence, J ≥ s . Corresponding, I ≥ J for the case of not mutually disjoint.

Complimentary,

∫ E f ( x ) d x = ∫ E 1 f ( x ) d x + ∫ E 2 f ( x ) d x .

Holds for

S ≥ I ≥ J ≥ s .

Lemma 2: If f ( x ) and g ( x ) are bounded and measurable on E, and

∫ E [ f ( x ) + g ( x ) ] d x = ∫ E f ( x ) d x + ∫ E g ( x ) d x .

Then f ( x ) is Lebesque integrable if and only if for any E > 0 there exist a partition with upper and lower sums S , s for which S − s < ε .

Proof: In a Lebesgue metric space, if for ε > 0 there correspond a partition such that S − s < E . Borrowing from Lemma 1, in which,

S ≥ I ≥ J ≥ s .

Then, 0 ≤ I − J ≤ S − s < ε .

Putting I = J with ε as any arbitrary small gaps, then f ( x ) is Lebesgue integrable. Conversely, for I = J and if ε > 0 there exist a partition such that

S < I + ε 2 .

Since I = g . l . b of all upper sums and s > J − ε 2 with J = l . u . b of all lower

sums.

As a consequence,

S − s < ( I + ε 2 ) − ( J − ε 2 ) = ε .

Taking any two real outcome, say γ and Γ such that γ < f ( ℘ s ) < Γ , we can have γ and Γ as subextremes of ℜ and μ and slicing it into n-decision groups by taking such values as, α 1 , α 2 , ⋯ , α n − 1 so that

γ = α 0 < α 1 < α 2 < ⋯ < α n − 1 = Γ .

Taking a particular set of sub decision group as a cluster of potential surprise and letting, ξ i , i = 1 , 2 , ⋯ , n the group of all ℘ s in [ ℜ , μ ] so that α i − 1 ≤ f ( ℘ s ) < α i , then,

ξ i = { ℘ s : α i − 1 ≤ f ( ℘ s ) < α i } with i = 1 , 2 , ⋯ , k (1)

Supposing all ξ i is quantifiable ( � e ) in “k”, for which,

� e ( k ) ≥ � e ( k ∩ ξ i ) + � e ( k ∩ ξ ¯ i )

and are verified to be disjoint. Taking the upper and lower bounds in ℜ and μ as outcomes as awful/implausible and astounding/implausible limits, then.

{ S a / 1 = ∑ i = 1 n α i � e ( ξ i ) : S a / 1 = Upperboundforawful / implausiblepossibility s a / 2 = ∑ i = 1 n α i − 1 � e ( ξ i ) : s a / 2 = Lowerboundforastounding / implausiblepossibility

By randomizing all sub-decision group in the cluster, limiting value, of I and J can be obtained for S a / 1 and s a / 2 taking I = greater lower bound of all values of S a / 1 for all sub-decision group J = least upper bound of all values of s a / 2 for all sub-decision group.

Taking the bounds of potential surprise in the bounds of the decision interval [ ℜ , μ ] with intermediate points as subjective decisions arrays limiting in I and J of [ ℜ and μ] we have that S a / 1 and s a / 2 being upper and lower sums corresponding to a sub-decision group λ, with ϕ and ψ as least upper and greater lower bonds of f ( ℘ s ) in [ ℜ , μ ] , there exist,

ψ ( μ − ℜ ) ≤ s a / 2 ≤ S a / 1 ≤ ϕ ( μ − ℜ ) (2)

Using the nation S a / 1 = ϕ 1 ( ℘ s 1 − ℘ s 0 ) + ⋯ + ϕ n ( ℘ s n − ℘ s n − 1 ) = ∑ ϕ i Δ ℘ s i

s a / 2 = ψ 1 ( ℘ s 1 − ℘ s 0 ) + ⋯ + ψ n ( ℘ s n − ℘ s n − 1 ) = ∑ ψ i Δ ℘ s i

S a / 1 = ∑ i = 1 n ϕ i Δ ℘ s i ,     s a / 2 = ∑ i = 1 n ψ i Δ ℘ s i .

Given that ψ ≤ ψ i ≤ ϕ i ≤ ϕ and on multiplicative operation on Δ ℘ s i and summing over i from 1 to n, we have

| ∑ i = 1 n ψ Δ ℘ s i ≤ ∑ i = 1 n ψ i Δ ℘ s i | ≤ | ∑ i = 1 n ϕ i Δ ℘ s i ≤ ∑ i = 1 n ϕ Δ ℘ s i |

ψ ∑ i = 1 n Δ ℘ s i ≤ s a / 2 ≤ S a / 1 ≤ ϕ ∑ i = 1 n Δ ℘ s i (3)

Showing equations (2) = (3) and following the existence of I and J for S a / 1 and S a / 2 of f ( ℘ s ) on the space of [ ℜ and μ] then

I [ S a / 1 ] = ∫ ℜ μ ¯ f ( ℘ s ) d ( ℘ s ) ,     J [ s a / 2 ] = ∫ ℜ ¯ μ f ( ℘ s ) d ( ℘ s ) .

For which I [ S a / 1 ] ≠ J [ s a / 2 ] = > f ( ℘ s ) is Lebesgue measurable and integrable in [ ℜ , μ ] with a common value of

∫ ℜ μ f ( ℘ s ) d ( ℘ s ) .

Since f ( ℘ s ) is bounded and quantifiable space in ℜ and μ, it follows specifically that ξ is a quantifiable sequence in [ ℜ , μ ] and as a property of lebesgue summation f n ( ℘ s ) on ξ is:

∫ ξ f ( ℘ s ) d ℘ s = ∫ ℜ μ g ( ℘ s ) d ℘ s .

With g ( ℘ s ) = { f ( ℘ s ) if   ℘ s ∈ ξ 0 if   ℘ s ∉ ξ

For which according to the sequence space sums of the upper and lower values of S a / 1 and s a / 2 given,

ξ i = { ℘ s : ℘ s ∈ ξ , α i − 1 ≤ f ( ℘ ) < α i }

This reduces the expression above to ξ = [ ℜ , μ ] . With 〈 f n ( ℘ ) 〉 as sequence of functions quantifiable ( � e ) on ξ, and if ϕ ( S a / 1 ) = Least upper bound f n ( ℘ s ) and ψ ( S a / 2 ) = greatest lower bound f n ( ℘ s ) . The ϕ ( S a / 1 ) and ψ ( S a / 2 )

are quantifiable in ξ. Since ϕ ( S a / 1 ) = l . u . b { f n ( ℘ s ) } = l . u . b { f i ( ℘ s ) , f 2 ( ℘ s ) , ⋯ } and taking ψ ( S a / 2 ) = g . l . b { f n ( ℘ s ) } = g . l . b { f i ( ℘ s ) , f 2 ( ℘ s ) , ⋯ } , then

� e [ ϕ ( S a / 1 ) ≥ k ] = ∪ i = 1 α � e [ f n ( ℘ s ) ≥ k ] as a consequence from Equation (1) since a

Countable union of quantifiable potential surprises group is also quantifiable.

We have proposed that economic decisions are bounded in the interval of [ ℜ , μ ] and therein with intermediate point provide potential for surprise decision over time as alternative forgone within the sequence (Shackle, 1949; Carter, 2015; Kreeps 2002) . The sequence of events within the space suggests that possibilities are components of the potential surprise.

Lemma 3: Every convergent sequence is Cauchy sequence.

Proof: Given the sequence of real numbers ( a n ) converges to L. Therefore for

any ε > 0 , we obtain no such that | a p − l | < ε 2 for all p > n 0 .

And, | a p − l | for all q > n 0 .

Similarly, for both set absorptivity of P > n 0 and q > n 0 , we have

| a p − a q | = | ( a p − l ) + ( l − a q ) | ≤ | a p − l | + | l − a q | < ε 2 + ε 2 = ε .

If f ( ℘ s ) → [ ℜ , μ ] , the sequence of potential outcome, as surprises in the bounded space [ ℜ , μ ] with,

α i − 1 ≤ f ( ℘ s ) < α i is the sequence of ξ, for which

ξ i = { ℘ s : α i − 1 ≤ f ( ℘ s ) < α i }

That this sequence is bounded and convergent is shown by

ξ i = | ξ i − ξ + ξ | ≤ | ξ i − ξ | + | ξ | .

And having a real positive number � such that | ξ i | < q for all i. We introduce a limiting point i o that makes | ξ i − ξ | < ε any positive number for all i > i o .

Hence, | ξ i | < ε + | ξ | for all i > i o .

Correspondingly relate that | ξ i | < q for all i if � is taken as one of the largest numbers ξ 1 , ξ 2 , ⋯ , ξ i o , ε + | ξ | . The potential surprise outcome is obtainable from the set of sub decision in the interval of [ ℜ i , μ i ] with i = 1 , 2 , 3 , ⋯ , Having ℜ 1 ≤ ℜ i ≤ μ i ≤ μ 1 bounded and monotonically increasing and decreasing sequences converges to ℜ and μ.

To show that an exert surprise exist is to put ℜ = μ. This is done by making.

μ − ℜ = ( μ − μ i ) + ( μ i − ℜ i ) + ( ℜ i − ℜ )

| μ − ℜ | ≤ | μ − μ i | + | μ i − ℜ i | + | ℜ i − ℜ | (4)

At the condition ε > 0, we deterime i 0 → i > i 0 .

| μ − μ i | < ε 3 ,     | μ i − ℜ i | < ε 3 ,     | ℜ i − ℜ | < ε 3

From (4) | μ − ℜ | < ε , with ε as any positive number, then μ − ℜ = 0, or ℜ = μ.

The potency and appeal of Shackle’s theory of potential surprise is laced with limitation critique economic decisions based on determinism and went ahead to propose innovation and surprise as a symptomatic response for indeterminacy (Basili & Zappia, 2009) . Such indeterminacy is bounded in the subjective space of surprise which time seems to validate for the decision taker (Shackle, 1968; Zongzhi, 2009) . Being not a laboratory science, economic science allude credence to subjective expectations with several national dimensions of past time (found memories), future time (expectations) and present time(decision). Resolving these dimensions with their associated indeterminacy is profoundly found in isolating decision variable and treating the competing past and future variables in simultaneity within potential surprise space (Carter, 2015) . The potential surprise space with avalanche of subjective and divergent thoughts is epistemically resolved in bounds of ℜ (awful and implausible outcome) and μ (astounding and implausible outcome). Within this bound which has semblance with Herbert Simon’s bound properties have been shown to have sequence of event outcomes between S a / 1 to s a / 2 to be potential for surprise along the sequence. With ℜ and μ showing characterization of conjugate properties, the sequence of subjective outcomes for potential surprise converging to a decision point ( Ω D ) can be deductively obtained by given any ε > 0 , (ε = any positive number) in a sequence of potential surprise outcomes ( ξ i ) converging to decision point ( Ω D ). We can get k o such that:

| ξ θ − Ω D | < ε 2       if   all   θ > k o .

Also true for | ξ λ − Ω D | < ε 2 if all λ = k o .

For both conditions of θ > k o and λ = k o and by means of factor difference

| ξ θ − ξ λ | = | ( ξ θ − Ω D ) + ( Ω D − ξ λ ) | ≤ | ξ θ − Ω D | + | Ω D − ξ λ | < ε 2 + ε 2 = ε

This shows that the sequence of ξ i = { ℘ s : α i − 1 ≤ f ( ℘ s ) < α i } bounded by [ ℜ , μ ] in the functions of f ( ℘ s ) are Cauchy sequence by the conjugate limits of ℜ and μ.

If we interchange the bounds of the pool of potential surprise by resizing them as conjugate multiple, in the form | ℜ | μ and | μ | ℜ , then operating their products gives

O ≤ ( | ℜ | μ − | μ | ℜ ) ⋅ ( | ℜ | μ − | μ | ℜ ) .

And as can be seen reduces to:

| ℜ , μ | ≤ | ℜ | μ | (5)

Provided ℜ and μ are bounds linearly related.

Shackle’s Potential Surprise theory in economic science has not received the desired application in economic problems owing to the subjective reasoning demands required by the theory. The efforts to making it a computable science has witnessed birth of probability and set theory by multi-valued mapping approaches as seen in the works of Dempster (1967, 1968) . Shafer progressively, in Shafer (1976, 1979) and (1982) empirically by Fioretti (2001) provided evidence theory for the construction of coherent picture of reality from evidence in an individual’s mind yet laced with subjectivity of measurements in probability space for new possibilities. Fuzzy logic computable method was also introduced by Fioretti (2001) with inductive use of interval logics to represent extremums. In all of these efforts, none has shown promise of computable process save for subjective beliefs from set theories and non-deterministic methods based on probabilities. Consequently this paper theorizes the use of measure theory to offer computation of the Shackelian potential surprise in an attempt to providing the theory with a quantitifiable schema.

Economic thoughts in favour of attempts at extinguishing Shackle’s potential surprise theory in economic decision are profoundly calling for exclusion of perfect possible for the realization of Shackle’s zero potential surprise outcomes (Starmer, 1993; Smith, 1961) . But Shackle’s strong point is that Statisticians results are based on probabilities of frequentative outcomes that are exogenous to other human contingent subjective needs and preference which are root sources of potential surprise in economic decisions (Carter, 2015) . Shackle’s theory draws from the strength that economic decisions are purely unique and it is a function of a state of mind. This alludes indeterminacy to economic decisions and rather favours predictability of economic decision variables within some boundaries (Jaffray & Philippe, 1997) . Indeterminacy of economic decision is further favoured with theoretics of economic unexpected economic collapse. In recent times most economic decisions taken on the basis of strong statistical results and observed frequency are in awe of Covid-19 interference that took economic decisions in “potential surprise”. This is also evident with decision made in Ukraine and Russia arising from the infliction of “potential surprise” from the war against previous economic decisions made.

In foregoing review of indeterminacy of economic decisions as a result of possibility of potential surprise, this paper shall proceed to show that within the bounds of subjectively awful and implausible outcome ( ℜ ) and subjectively astounding implausible outcomes (μ) a series of potential surprise outcomes exists with indeterminate properties (Shackle, 1980, 1983; Jaffray & Philippe, 1997).

Potential theory

It is impossible to determine economic decisions outside the bound of ℜ and μ which act as potential surprise space in economic decisions.

Proof

Suppose we have bounded space in length ℘ s [ ℜ , μ ] given the geometric representation

Let σ ℜ , σ μ be outside of bound of ℘ s [ ℜ , μ ] such that σ ℜ , σ μ are deviated variables from the sequence with chaotic property randomly dispersed outside of ℘ s space with dynamic time (Knudsen, 2000) index T as

H ( ℵ ) ≡ lim T → ∞ 1 T H T ( ℘ s 1 , ℘ s 2 , ⋯ , ℘ s T )

If the process normalizes in the bounds of ℜ , μ as a stationary sequence from threshold point acting.

With preceding conditional chaos ( H * ) to stationary property of sequences of ℘ s in ξ λ as;

H * ( ℵ ) ≡ lim T → ∞ 1 T H T ( ℘ s T | ℘ s T − 1 , ℘ s T − 2 , ⋯ , ℘ s 1 )

Then

H ( ℵ ) = H * ( ℵ ) = H T ( ℘ s 2 | ℘ s 1 ) ⋯ H ( ℘ s T | ℘ s T − 1 )

Taking average deviations from decision point ( Ω o ) by one to one mapping points in ℜ and μ as ℘ s , in which, D ℜ i = ℜ i − Ω D and D μ j = μ j − Ω D so

that ℜ i = 1 and μ j = 1 , gives 1 2 ( D ℜ i = 1 + D μ j = 1 ) = ℘ s 1 . With H ( ℵ ) and H * ( ℵ )

(Golan, 2006) as space state normalization metrics, for boundary condition stationarity occurring at ℜ , μ boundaries, than we have the standard deviations of ℜ , μ been in chaos from decision point ( Ω D ) as σ ℜ , σ μ conjugated variables. Therefore,

σ ℜ 2 = 〈 ( Ω D − 〈 Ω D 〉 2 ) 〉 = 〈 Ω D 2 〉 − 〈 Ω D 〉 2 (6)

With σ ℜ 2 as the average squared difference from the decision point ( Ω D )

within the sequence ( ξ i ) of expectations values ( ℜ ^ − 〈 ℜ ^ 〉 ) 2 overtime (T) in

chaotic state normalization metrics ( H T * ) having chaotic rate of dispersion from Ω D as:

H ( ℘ S ) = lim T → ∞ H ( ℘ s T ) T ,

℘ s been a consequence of time (T). Rewriting the Expression (6) above with expectation outcome plausibility, we have:

σ ℜ 2 = 〈 H T * | ( ℜ ^ − 〈 ℜ ^ 〉 ) 2 | H T * 〉

Having Ω D in bounds of ℜ and μ on Herbert Simon bound, we set: ω = ( ℜ ^ − 〈 ℜ ^ 〉 ) ξ i as σ ℜ 2 = 〈 ω | ω 〉 at ℜ -bound, and for μ-bound, we set,

Λ = ( μ ^ − 〈 μ ^ 〉 ) H *       as   σ μ 2 = 〈 Λ | Λ 〉

Recalling expression (5) and substituting for σ ℜ 2 and σ μ 2 above we have

σ ℜ 2 σ μ 2 = 〈 ω | ω 〉 〈 Λ | Λ 〉 ≤ | 〈 ω | ω 〉 | 2 (7)

Obtaining expression for (7) in their potential surprise values gives,

〈 ω | Λ 〉 = 〈 ( ℜ ^ − 〈 ℜ ^ 〉 ) H T * | ( μ ^ − 〈 μ ^ 〉 ) H T * 〉 = 〈 H T * ( ℜ ^ − 〈 ℜ ^ 〉 ) ( μ ^ − 〈 μ ^ 〉 ) H T * 〉 = 〈 H T * | ℜ ^ μ ^ H T * 〉 − 〈 μ 〉 〈 H T * | ℜ ^ H T * 〉 − 〈 ℜ 〉 〈 H T * | μ ^ H T * 〉 + 〈 ℜ 〉 〈 μ 〉 〈 H T * | H T * 〉 = 〈 ℜ ^ μ ^ 〉 − 〈 μ ^ 〉 〈 ℜ ^ 〉

And by deploying the conjugate of 〈 ω | Λ 〉 as test of symmetricity property we obtain

〈 Λ | ω 〉 = 〈 μ ^ ℜ ^ 〉 − 〈 ℜ ^ 〉 〈 μ ^ 〉

By standardizing the expression with complex number property of

Z * Z ≥ [ 1 2 i ( Z − Z * ) ] 2

Then 〈 ω | Λ 〉 is substituted by Z and Z * for 〈 Λ | ω 〉 to give,

σ ℜ 2 σ μ 2 ≥ [ 1 2 i ( 〈 ω | Λ 〉 − 〈 Λ | ω 〉 ) ] 2 ≥ [ 1 2 i ( 〈 ℜ ^ μ ^ 〉 − 〈 μ ^ 〉 〈 ℜ ^ 〉 − 〈 μ ^   ℜ ^ 〉 − 〈 ℜ ^ 〉 〈 μ ^ 〉 ) ] 2

≥ | 1 2 i 〈 | ℜ ^ , μ ^ | 〉 | 2

If ℜ ^ and μ ^ in symmetricity commute exclusively, then [ ℜ ^ μ ^ ] = i H T * and by obtaining the square root of both sides, we have the indeterminacy expression for economic decision within the bounds of ℜ | μ as

σ ℜ σ μ ≥ H T * 2     or     1 2 H T * , (8)

H T * been an Arbitrary chaotic economic state wherein T is the dynamic time of the process (Knudsen, 2000) for the expression in (8) suggests that economic decisions are bounded with surprises between an awful/implausible possibility ( ℜ ) and astounding/implausible possibility (μ). Between ℜ and μ presents us with ocean of potential surprises with indeterminate propensities. Implying that economic decisions cannot make outcome of both awful and astounding possibilities. If economic decisions is awful, then the astounding outcomes exists potentially and conjugatively if economic decisions is astounding, then an awful outcome exists potentially outcome exists potentially.

Drawing from the notion that information for economic decision are not finite (see Golan pg. 17/18) and shows spurious estimates over frequentative based probability outcomes, a tendency for entropy state (H) decisions arises with uncertainty outlook harping on the unpredictability expounded by the Shackelian

space, σ ℜ σ μ = 1 H T * .

In the light of such infinite propensities for economic outcomes from a probability dependent criterion, uncertainty therefore supports the state of indeterminacy from an entropic ( H T < ℵ ) information. This is especially the case when a decision maker knowledge or information catches is alien to the underlying characteristics behaviour of economic systems metric which often times induces entropy state from Zbili & Rama (2021) for a potential surprise as

H = − 〈 ∑ i     P ( x i ) T ∗ log 2 P ( x i ) T 〉 T (9)

In the real sense of information and entropy, no economic decision is absolute and free of economic skews or moments from a mortal economic force of imbalances and future occurences. This was the mind of G.L.S Shackle and it is overt to say that decision space in economics are not turbulence (entropy) free and are characterized by indeterminate state from incomplete reasoning that births potential surprise.

For example, decision for housing investment predicated on the developer’s budget based on certain economic decision metrics, as IRR, ARR, NPV, UNACOST, capitalized cost, etc. can as a matter of incomplete information be compelled to succumb to mortal economic force of indeterminacy. Suppose the developer above was foreclosed to other economic restraint in guiding his decisions, such outliers are deemed to be subjective ignorance to the decision maker, but yet governed by the information of the decision space with entropic probability attributes

H ( p ) = ∑ i = 1 m P i log 2 1 P i (10)

which does not rely on distinct realized randomized variables, x 1 , x 2 , x 3 , ⋯ , x m but on their probabilities, x 1 | p 1 , x 2 | p 2 , x 3 | p 3 , ⋯ , x m | p m . In our accompanying analysis below, we shall proceed to map decision metrics indices for a potential developer say IRR to a probability enabled entropy state computation as congruent parameters to obtaining the limit state of the developers economic decision. Based on computational advice from the developer’s budget, the developer obtains five (5) consecutive IRR for five (5) alternatives of housing A or B. on the basis of joint entropy dependence, we associate their probabilities as:

P ( A | B ) , letting P ( A = a i ) = P i , P ( B = b j ) = P j then,

P ( A = a i , B = b j ) = λ i j , P ( A | B ) = P ( A = a i | B = b j ) = P i | j

By commutating the probability values, we have P ( A | B ) , P ( B | A ) implying P ( B | A ) = P ( B = b j | A = a i ) = q j | i for which

P i = ∑ i = 1 i X i j ,     q i = ∑ j = 1 j λ j i

Then joint entropy of A and B given:

H ( A | B ) = ∑ i j     λ i j log 1 λ i j = − ∑ i j     λ i j log λ i j

On a more general note, entropy (Chernoff forms with α-) has been written for processes with incomplete random variable by Renyi (1970) with order α- as

H α R ( p ) = 1 1 − α log ∑ k     P k α (11)

Cross entropy between distributions as mentioned in Golan (2006) by Renyi (1970) for difference between two distributions, p and q of order α- was given as:

D α R ( x | y ) = D α R ( p | | q ) = 1 1 − α log ∑ k P k α q k α − 1 (12)

Construction economics investigation for a prospective developer with five housing alternative to decide from on the basis of budget returned advise of IRR having two scenarios of flat (or continuous IRR) or discontinued IRR on the five alternative investment are presented (see Table 1).

Lower entropy from the first scenario in Table 2 shows that investment decision in scenario is more realistic with less ignorance. However, according to G.L.S Shackles potential surprise theory, espitemic knowledge is bounded between lower and higher entropy beyond which the investment decision is indeterminate.

Ever since, Renyi (1970) work, there has been a systematic bibliography on generalized entropy reviewed in Golan (2006) for Cressie & Read (1984) and particularly by Tsallis (1988) as,

D α T ( x | y ) = D α T ( p | | q ) = 1 1 − α ( ∑ k P k α q k α − 1 − 1 ) (13)

Renyi (1970) and Tsallis (1988) entropies have been harmonized as stated in Golan (2006) in Tsallis (1988) and Holste et al. (1998) to be of the form;

H α R ( x ) = 1 1 − α log [ 1 + ( 1 − α ) log H α T ] (14)

By connecting Renyi (1970) , Tsallis (1988) and Holste et al. (1998) , Golan (2006) reported in Golan (2002) as stringing the three form of entropies, this form as:

D α + 1 R ( p | | q ) = − 1 α log [ 1 − α D α + 1 T ( p | | q ) ] (15)

For emphasis, over potential surprise space ( ℘ s ) with chaos (entropy) properly will continue to be denoted with H T for a function of indeterminacy from incomplete reasoning (information) owing to the stochastic property of the space.