**Title: Potency**

__Subtitle__: Informational Efficacy of Symbolisation to Exponentiated Bases

**Introduction: Positional Notation**

In positional numerical notation, we are used to as many different and distinguishable symbols in a set or inventory of characters as the cardinal number of the base to the first power being available for each place or position representing any power of that base. In this case, the number of items, say \(N\), that may be referred to as numbers, names, or words by a set of symbols, say \(S\), is the number \(\# S\) of symbols or characters in the set \(S\) equal to the base \(B\) raised to the power of the number \(n\) of times that different symbols \({}^{S}X\) appear in the name or word \({}^{N}X_{i}\) that represents or is used to call something. Thus, \(N = \# S^{n} = B^{n}\). In this case, the base is \(B = \# S\) the number of symbols available, an identity which does not apply to other forms of notation for symbolisation than positional notation whereby the indication of place value of a symbol by its position in a permutation allows the recycling of the characters \(^{S}X\) from one place to another in the \(i^{\text{th}}\) word or number \(^{N}X_{i}\).

**Extension to Permutations**

A permutation is a sequence, not necessarily linearly arranged visually but which may nevertheless be mapped to a linear sequence, in which the position of a symbol or character in the sequence affects the value of the appellational number or word produced by the permutational sequence. Unlike positional notation, the number of symbols available to each position need not be the same, although positional notation is a subcategory of permutational sequences. In the case of general permutations, the number \(N\) of possible numbers standing for words is equal to the product \(\prod_{i}^{n} \text{#}S_i\) of the cardinal numbers of the sets \(S_i\) of symbols available at the places \(i\). The cardinal numbers #\(S_i\) of the sets \(S_i\) of symbols \(^{S}X\) for positions \(i\) may be called subbases \(b_i\) of the number \(N\) of possible word numbers as though it were an overall base. Thus, \[N = \prod_{i}^{n}b_{i}\] However, the total number of symbols available from the sum \(\sum_{i}^{n}b_i\) of all the cardinal numbers \(\# S_i\) provided there are no symbols \(^{S}X_{i}\) common to any of the sets \(S_i\) may be interpreted as a base \(B\) in the sense that it is analogous to the number of symbols available in positional notation and it is less than the number \(N\) of words that may be formed out of it. If there are symbols \(^{S}X_{i}\) common to any of the sets \(S_{i}\), the base \(B\) would be the cardinal number of symbols of the union of the sets \(S_i\) without counting any of the symbols \(^{S}X_{i}\) more than once. I may represent this as $$B = \# \cup S_i$$

**Limitations in Zeroless Notational Systems Under Constraints**

Consider the Braille system of writing. For each permutation composing a glyph \(^{N}X_i\), there are six positions or places, and in each position there are two possibilities: either a dot or no dot. These are the only two symbols \(^{S}X_i\) and they are common to all positions. Thus, the base of the positions is binary and the number \(N\) of possible glyphs \(^{N}X_i\) from the equation \(N = B^{n}\) given above is \(2^{6} = 8^{2}\).

However, while one of the symbols of the binary base is a dot, the other is no mark. This has the implication that this system does not represent a value of zero in the positions within the glyphs (as opposed to by the glyphs) by any symbol. We may interpret this particular version of a binary symbolic set \(S\) as containing the dot as one element and a null element (normally called the null set) as its second member. That is, \(S = \{ \{\}, \bullet \} \).

The number of useful glyphs in the Braille system of writing under conditions in which a small number of glyphs may appear in a statement is reduced by some of them being equivalent under translational operations or shifting of the glyphs. Hence, the actual number \(N\) of glyphs is less than \(B^{n}\). The amount of information that can be usefully encoded under such a constraint is less than would be expected from the number of elementary symbols and the number of positions within the glyph.

To quantify the efficacy of a base \(B\) and a system of formation of words from its symbols \(^{S}X\) in representing information \(^{N}X\), I introduce a test which I call potency, \(P\). The rationale of the potency \(P\) test or quantification is to find the exponent equivalent to \( \log_{B}{N}\) to which the base \(B\) must be raised to give the number \(N\) of different word numbers \(^{N}X_{i}\). This exponent represents the number of positions or number of times symbols ought to appear in the words \(^{N}X_{i}\). If this hypothetical number of symbols per word \(^{N}X_{i}\) is less than the number \(n\) of symbols \(^{S}X_i\) that actually occur in making up the words \(^{N}X\), then their ratio or fraction \(P\) is taken: \[P = \frac{\log_{B}{N}}{n}\] For example, if there are only 44 =

*(To be continued …)*

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