Predicate calculus is heavily involved in the cliological frameworks, but it seems, in its conventional form, to be a bit too clunky to use. An alternative symbolic system has been developed that uses diacratic marks to denote various properties of sets and predicates, and is used to simplify logical statements and make blob and arrow graphs easier to manipulate and read.
The essence is the symbolism of set theory; one based on Boolean truth evalusations. The extensions presented here are used to indicate the size of the set and how many elements of that set are true. Rather than size being a numerical value, some basic inequalities are used: equal to zero, greater than zero, equal to one, greater than one, and a few others. Likewise a few basic quantifiers are employed: none, one, some, all, all but one, etc. In addition, some other symbols are introduced or borrowed from other calculi.
In the spirit of cliology, this symbolic system is intended for ease and practicality where suitable, rather than for mathematical purity. It isnt known, nor necessarity to know, whether certain expressions cannot be formed of if the system is coherent. Those issues are left to mathematicians rather than engineers.
A predicate calculus statement that all members of a set S are true according to a predicate P may be written
∀ s∈S⋅ P(s)
We could apply a predicate to the whole set and evaluate which set members A are true (A is chosen here because it has diacratics in the symbols character set).
∀ s∈S⋅ P(s) → s∈A
Or, alternativly we could just say A by making the predicate implicit or defining in rather ill-formed symbolism:
P(S) = A
So A ⊆ S. In the proposed system, diacratic marks would then be applied to A to indicate properties of A. Â would denote all were true, Ã would denote none are true. Mathematicians would rightly argue that this is not proper set theory. Binary notation might help.
Suppose A was a string of 8 bits, a byte in computing then we could for example write something like:
A={01101001}
If A was {11111111} then we could say Â. If A was {00000000} then we could say Ã. Other diacratic marks would be used for expressions like only one member is true, or all but one member is true, and so on. By introducing the Not operator (¬) or modals (like ◊) could then explore the truth relationships between the diacratically symbolised sets – such as by the square of opposition.
Cliology uses Boolean values extensivly to determine the presence or asbsense of a trait in some cultural item. Binary strings are a convenient way of denoting those traits. As they can be converted to bits and bytes, then this system facilitates thier encoding and manipulation in a software environment. Numerical taxonomy, the form of cluster analysis employed in cliology, when performed on a sizable set of cultural items, can really only be done by hard number crunching, to which bit string representations are well suited.
Unfortunatly, without a good maths forula plug-in, certain diacratics are unavailable clumsey in raw text. This can be circumvented by using pre and post super and sub script formatting. Other expressions are just unwheildly and ugly (eg. “some but not all”). Without going too deep here, some of (and there are other more exhotic and compound ones) the proposed markings are:
∧ – Â – All eg. {11111111}
~ – Ã – None eg. {00000000} Alternatively ∅
⋅ – Ȧ – Only one eg. {00010000}
၀ – Å – All but one eg. {11110111}
∨ – Ǎ – Some (one or more are true) eg. {00010110}
The superscript prefix variations can also be used where necessary (nb. the above can also be written this way):
∧A – All eg. {11111111}
>A – Most eg. {00011111}
<A – Less eg. {00000111}
Diacratics below the set symbol can be used to denote the size of the set according to inequalities. Again many expressions are more suited to a whiteboad than text. The inequality can be written, but some simpler symbol convention is recommended; the following are tentative possibilitities.
A≥2
empty {~, A~, ø, =0}
any size {A, ≥0} no diacratic marking
only one {⋅, Ạ, =1}
one or more {≥1; >0}
many {…, m, >1 , ≥2}
lots etc. {», ∞, ◊, b, l} (b= byte, l = 64bit)
Diacratics above and below can be combiined to create fuller expressions
∨A… – some of a set of more than one item are true
Again, this is not complete nor mathematically pure, but rather a shorthand intended for whiteboard doodling.