Differential Propositional Calculus
Author: Jon Awbrey
A differential propositional calculus is a propositional calculus extended by a set of terms for describing aspects of change and difference, for example, processes that take place in a universe of discourse or transformations that map a source universe into a target universe.
1. Casual Introduction
Consider the situation represented by the venn diagram in Figure 1.
The area of the rectangle represents a universe of discourse, This might be a population of individuals having various additional properties or it might be a collection of locations that various individuals occupy. The area of the “circle” represents the individuals that have the property or the locations that fall within the corresponding region Four individuals, are singled out by name. It happens that and currently reside in region while and do not.
Now consider the situation represented by the venn diagram in Figure 2.
Figure 2 differs from Figure 1 solely in the circumstance that the object is outside the region while the object is inside the region So far, there is nothing that says that our encountering these Figures in this order is other than purely accidental, but if we interpret the present sequence of frames as a “moving picture” representation of their natural order in a temporal process, then it would be natural to say that and have remained as they were with regard to quality while and have changed their standings in that respect. In particular, has moved from the region where is to the region where is while has moved from the region where is to the region where is
Figure 3 reprises the situation shown in Figure 1, but this time interpolates a new quality that is specifically tailored to account for the relation between Figure 1 and Figure 2.
This new quality, is an example of a differential quality, since its absence or presence qualifies the absence or presence of change occurring in another quality. As with any other quality, it is represented in the venn diagram by means of a “circle” that distinguishes two halves of the universe of discourse, in this case, the portions of outside and inside the region
Figure 1 represents a universe of discourse, together with a basis of discussion, for expressing propositions about the contents of that universe. Once the quality is given a name, say, the symbol we have the basis for a formal language that is specifically cut out for discussing in terms of and this formal language is more formally known as the propositional calculus with alphabet
In the context marked by and there are but four different pieces of information that can be expressed in the corresponding propositional calculus, namely, the propositions: Referring to the sample of points in Figure 1, the constant proposition holds of no points, the proposition holds of and the proposition holds of and and the constant proposition holds of all points in the sample.
Figure 3 preserves the same universe of discourse and extends the basis of discussion to a set of two qualities, In parallel fashion, the initial propositional calculus is extended by means of the enlarged alphabet, Any propositional calculus over two basic propositions allows for the expression of 16 propositions all together. Just by way of salient examples in the present setting, we can pick out the most informative propositions that apply to each of our sample points. Using overlines to express logical negation, these are given as follows:

describes


describes


describes


describes

Table 4 exhibits the rules of inference that give the differential quality its meaning in practice.

2. Cactus Calculus
Table 5 outlines a syntax for propositional calculus based on two types of logical connectives, both of variable ary scope.
 A bracketed list of propositional expressions in the form indicates that exactly one of the propositions is false.
 A concatenation of propositional expressions in the form indicates that all of the propositions are true, in other words, that their logical conjunction is true.
 



 

 

 

 





All other propositional connectives can be obtained through combinations of these two forms. Strictly speaking, the concatenation form is dispensable in light of the bracket form, but it is convenient to maintain it as an abbreviation for more complicated bracket expressions. While working with expressions solely in propositional calculus, it is easiest to use plain parentheses for logical connectives. In contexts where parentheses are needed for other purposes “teletype” parentheses or barred parentheses may be used for logical operators.
The briefest expression for logical truth is the empty word, abstractly denoted or in formal languages, where it forms the identity element for concatenation. It may be given visible expression in this context by means of the logically equivalent form or, especially if operating in an algebraic context, by a simple Also when working in an algebraic mode, the plus sign may be used for exclusive disjunction. For example, we have the following paraphrases of algebraic expressions:

It is important to note that the last expressions are not equivalent to the triple bracket
For more information about this syntax for propositional calculus, see the entries on minimal negation operators, zeroth order logic, and Table A1 in Appendix 1.
3. Formal Development
The preceding discussion outlined the ideas leading to the differential extension of propositional logic. The next task is to lay out the concepts and terminology that are needed to describe various orders of differential propositional calculi.
3.1. Elementary Notions
Logical description of a universe of discourse begins with a set of logical signs. For the sake of simplicity in a first approach, assume that these logical signs are collected in the form of a finite alphabet, Each of these signs is interpreted as denoting a logical feature, for instance, a property that objects in the universe of discourse may have or a proposition about objects in the universe of discourse. Corresponding to the alphabet there is then a set of logical features,
A set of logical features, affords a basis for generating an dimensional universe of discourse, written It is useful to consider a universe of discourse as a categorical object that incorporates both the set of points and the set of propositions that are implicit with the ordinary picture of a venn diagram on features. Accordingly, the universe of discourse may be regarded as an ordered pair having the type and this last type designation may be abbreviated as or even more succinctly as For convenience, the data type of a finite set on elements may be indicated by either one of the equivalent notations, or
Table 6 summarizes the notations that are needed to describe ordinary propositional calculi in a systematic fashion.






3.2. Special Classes of Propositions
A basic proposition, coordinate proposition, or simple proposition in the universe of discourse is one of the propositions in the set
Among the propositions in are several families of propositions each that take on special forms with respect to the basis Three of these families are especially prominent in the present context, the linear, the positive, and the singular propositions. Each family is naturally parameterized by the coordinate tuples in and falls into ranks, with a binomial coefficient giving the number of propositions that have rank or weight

The linear propositions, may be written as sums:

The positive propositions, may be written as products:

The singular propositions, may be written as products:
In each case the rank ranges from to and counts the number of positive appearances of the coordinate propositions in the resulting expression. For example, for the linear proposition of rank is the positive proposition of rank is and the singular proposition of rank is
The basic propositions are both linear and positive. So these two kinds of propositions, the linear and the positive, may be viewed as two different ways of generalizing the class of basic propositions.
Finally, it is important to note that all of the above distinctions are relative to the choice of a particular logical basis For example, a singular proposition with respect to the basis will not remain singular if is extended by a number of new and independent features. Even if one keeps to the original set of pairwise options to pick out a new basis, the sets of linear propositions and positive propositions are both determined by the choice of basic propositions, and this whole determination is tantamount to the purely conventional choice of a cell as origin.
3.3. Differential Extensions
An initial universe of discourse, supplies the groundwork for any number of further extensions, beginning with the first order differential extension, The construction of can be described in the following stages:

The initial alphabet, is extended by a first order differential alphabet, resulting in a first order extended alphabet, defined as follows:

The initial basis, is extended by a first order differential basis, resulting in a first order extended basis, defined as follows:

The initial space, is extended by a first order differential space or tangent space, at each point of resulting in a first order extended space or tangent bundle space, defined as follows:

Finally, the initial universe, is extended by a first order differential universe or tangent universe, at each point of resulting in a first order extended universe or tangent bundle universe, defined as follows:
This gives the type:
A proposition in a differential extension of a universe of discourse is called a differential proposition and forms the analogue of a system of differential equations in ordinary calculus. With these constructions, the first order extended universe and the first order differential proposition we have arrived, in concept at least, at the foothills of differential logic.
Table 7 summarizes the notations that are needed to describe the first order differential extensions of propositional calculi in a systematic manner.










…
Appendices
Appendix 1. Propositional Forms and Differential Expansions
Table A1. Propositional Forms on Two Variables












Table A2. Propositional Forms on Two Variables






























Table A3. Ef Expanded Over Differential Features



 






























Table A4. Df Expanded Over Differential Features



 






























Table A5. Ef Expanded Over Ordinary Features



 






























Table A6. Df Expanded Over Ordinary Features



 




 




 




 




 




 
Appendix 2. Differential Forms
The actions of the difference operator and the tangent operator on the 16 bivariate propositions are shown in Tables A7 and A8.
Table A7 expands the differential forms that result over a logical basis:

This set consists of the singular propositions in the first order differential variables, indicating mutually exclusive and exhaustive cells of the tangent universe of discourse. Accordingly, this set of differential propositions may also be referred to as the cellbasis, pointbasis, or singular differential basis. In this setting it is frequently convenient to use the following abbreviations:
and 
Table A8 expands the differential forms that result over an algebraic basis:
This set consists of the positive propositions in the first order differential variables, indicating overlapping positive regions of the tangent universe of discourse. Accordingly, this set of differential propositions may also be referred to as the positive differential basis.
Table A7. Differential Forms Expanded on a Logical Basis


 


 


 


 


 
Table A8. Differential Forms Expanded on an Algebraic Basis


 





 


 


 
Table A9. Tangent Proposition as Pointwise Linear Approximation



















Table A10. Taylor Series Expansion Df = df + d^{2}f






 




 




 




 




 
Table A11. Partial Differentials and Relative Differentials


















Table A12. Detail of Calculation for the Difference Map



 




 




 




 




 




 




 




 




 




 




 




 




 




 




 
Appendix 3. Computational Details
Operator Maps for the Logical Conjunction f_{8}(u, v)
Computation of εf_{8}

Computation of Ef_{8}


Computation of Df_{8}



Computation of df_{8}

Computation of rf_{8}

Computation Summary for Conjunction

Operator Maps for the Logical Equality f_{9}(u, v)
Computation of εf_{9}

Computation of Ef_{9}

Computation of Df_{9}


Computation of df_{9}

Computation of rf_{9}

Computation Summary for Equality

Operator Maps for the Logical Implication f_{11}(u, v)
Computation of εf_{11}

Computation of Ef_{11}

Computation of Df_{11}


Computation of df_{11}

Computation of rf_{11}

Computation Summary for Implication

Operator Maps for the Logical Disjunction f_{14}(u, v)
Computation of εf_{14}

Computation of Ef_{14}

Computation of Df_{14}


Computation of df_{14}

Computation of rf_{14}

Computation Summary for Disjunction

Appendix 4. Source Materials
Appendix 5. Various Definitions of the Tangent Vector
References
 Ashby, William Ross (1956/1964), An Introduction to Cybernetics, Chapman and Hall, London, UK, 1956. Reprinted, Methuen and Company, London, UK, 1964.
 Awbrey, J., and Awbrey, S. (1989), "Theme One : A Program of Inquiry", Unpublished Manuscript, 09 Aug 1989. Microsoft Word Document.
 Edelman, Gerald M. (1988), Topobiology : An Introduction to Molecular Embryology, Basic Books, New York, NY.
 Leibniz, Gottfried Wilhelm, Freiherr von, Theodicy : Essays on the Goodness of God, The Freedom of Man, and The Origin of Evil, Austin Farrer (ed.), E.M. Huggard (trans.), based on C.J. Gerhardt (ed.), Collected Philosophical Works, 1875–1890, Routledge and Kegan Paul, London, UK, 1951. Reprinted, Open Court, La Salle, IL, 1985.
 McClelland, James L., and Rumelhart, David E. (1988), Explorations in Parallel Distributed Processing : A Handbook of Models, Programs, and Exercises, MIT Press, Cambridge, MA.
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