differential propositional calculus

A differential propositional calculus is a propositional calculusMathworldPlanetmath (http://planetmath.org/PropositionalCalculus) extended by a set of terms for describing aspects of change and differencePlanetmathPlanetmath, for example, processes that take place in a universe of discourse or transformationsMathworldPlanetmath that map a source universe into a target universe.

1 Casual introduction

Consider the situation represented by the venn diagramMathworldPlanetmath in Figure 1.

Figure 1. Local Habitations, And Names

The area of the rectangleMathworldPlanetmathPlanetmath represents a universe of discourse, X. This might be a population of individuals having various additional properties or it might be a collectionMathworldPlanetmath of locations that various individuals occupy. The area of the “circle” represents the individuals that have the property q or the locations that fall within the corresponding region Q. Four individuals, a,b,c,d, are singled out by name. It happens that b and c currently reside in region Q while a and d do not.

Now consider the situation represented by the venn diagram in Figure 2.

Figure 2. Same Names, Different Habitations

Figure 2 differs from Figure 1 solely in the circumstance that the object c is outside the region Q while the object d is inside the region Q. 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 a and b have remained as they were with regard to quality q while c and d have changed their standings in that respect. In particular, c has moved from the region where q is true to the region where q is false while d has moved from the region where q is false to the region where q is 𝑡𝑟𝑢𝑒.

Figure 1 reprises the situation shown in Figure 1, but this time interpolates a new quality that is specifically tailored to account for the relationMathworldPlanetmathPlanetmath between Figure 1 and Figure 2.

Figure 1. Back, To The Future

This new quality, dq, 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 X outside and inside the region dQ.

Figure 1 represents a universe of discourse, X, together with a basis of discussion, {q}, for expressing propositionsPlanetmathPlanetmathPlanetmath about the contents of that universe. Once the quality q is given a name, say, the symbol ``q", we have a basis for a formal languageMathworldPlanetmath that is specifically cut out for discussing X in terms of q, and this formal language is more formally known as the propositional calculus with alphabetMathworldPlanetmath {``q"}.

In the context marked by X and {q} there are but four different pieces of information that can be expressed in the corresponding propositional calculus, namely, the propositions: 𝑓𝑎𝑙𝑠𝑒,¬q,q,𝑡𝑟𝑢𝑒. Referring to the sample of points in Figure 1, false holds of no points, ¬q holds of a and d, q holds of b and c, and true holds of all points in the sample.

Figure 1 preserves the same universe of discourse and extends the basis of discussion to a set of two qualities, {q,dq}. In parallelMathworldPlanetmathPlanetmath fashion, the initial propositional calculus is extended by means of the enlarged alphabet, {``q",``dq"}. 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 negationMathworldPlanetmath, these are given as follows:

  • q¯dq¯ describes a

  • q¯dq describes d

  • qdq¯ describes b

  • qdq describes c

Table 3 exhibits the rules of inferenceMathworldPlanetmath that give the differential quality dq its meaning in practice.

Table 3. Differential Inference Rules
From q¯ and dq¯ infer q¯ next.
From q¯ and dq infer q next.
From q and dq¯ infer q next.
From q and dq infer q¯ next.

2 Cactus calculus

Table 4 outlines a syntax for propositional calculus based on two types of logical connectives, both of variable k-ary scope.

  • A bracketed list of propositional expressions in the form (e1,e2,,ek-1,ek) indicates that exactly one of the propositions e1,e2,,ek-1,ek is false.

  • A concatenationMathworldPlanetmath of propositional expressions in the form e1e2ek-1ek indicates that all of the propositions e1,e2,,ek-1,ek are true, in other words, that their logical conjunction is true.

Table 4. Syntax and Semantics of a Propositional Calculus
Expression InterpretationMathworldPlanetmathPlanetmath Other Notations
True 1
() False 0
x x x
(x) Notx xx~¬x
xyz xandyandz xyz
((x)(y)(z)) xoryorz xyz
(x(y)) ximpliesyIfxtheny xy
(x,y) xnotequaltoyxexclusiveory xyx+y
((x,y)) xisequaltoyxifandonlyify x=yxy
(x,y,z) Justoneofx,y,zisfalse. xyzxyzxyz
((x),(y),(z)) Justoneofx,y,zistrue.Partitionallintox,y,z. xyzxyzxyz
((x,y),z)(x,(y,z)) Oddlymanyofx,y,zaretrue. x+y+z=xyzxyzxyzxyz
(w,(x),(y),(z)) Partitionwintox,y,z.Genuswcomprisesspeciesx,y,z. wxyzwxyzwxyzwxyz

All other propositional connectives can be obtained through combinationsMathworldPlanetmathPlanetmath 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 of more complicated bracket expressions. The briefest expression for logical truth is the empty wordPlanetmathPlanetmathPlanetmath, abstractly denoted ε or λ in formal languages, where it forms the identity elementMathworldPlanetmath for concatenation. It can be given visible expression in this context by means of the logically equivalent expression ``(())", or, especially if operating in an algebraic context, by a simple ``1". 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 by bracket expressions:


It is important to note that the last expressions are not equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to the triple bracket (x,y,z).

For more information about this syntax for propositional calculus, see the entries on minimal negation operators (http://planetmath.org/MinimalNegationOperator), zeroth order logic (http://planetmath.org/ZerothOrderLogic), and Table A1 in Appendix 1 (http://planetmath.org/DifferentialPropositionalCalculusAppendices).

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, 𝔄={``a1",,``an"}. 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 𝔄 there is then a set of logical features, 𝒜={a1,,an}.

A set of logical features, 𝒜={a1,,an}, affords a basis for generating an n-dimensional universe of discourse, written A=[𝒜]=[a1,,an]. It is useful to consider a universe of discourse as a categorical (http://planetmath.org/Category) object that incorporates both the set of points A=a1,,an and the set of propositions A={f:A𝔹} that are implicit with the ordinary picture of a venn diagram on n features. Accordingly, the universe of discourse A may be regarded as an ordered pairMathworldPlanetmath (A,A) having the type (𝔹n,(𝔹n𝔹)), and this last type designation may be abbreviated as 𝔹n+𝔹, or even more succinctly as [𝔹n]. For convenience, the data type of a finite setMathworldPlanetmath on n elementsMathworldMathworld may be indicated by either one of the equivalent notations, [n] or 𝐧.

Table 5 summarizes the notations that are needed to describe ordinary propositional calculi in a systematic fashion.

Table 5. Propositional Calculus : Basic Notation

Notation Description Type

{``a1",,``an"} Alphabet [n]=𝐧

{a1,,an} Basis [n]=𝐧

{ai¯,ai} Dimension i 𝔹

𝒜 Set of cells, 𝔹n
a1,,an coordinate tuples,
{(a1,,an)} points, or vectors
A1××An in the universe
i=1nAi of discourse

(hom:A𝔹) Linear functions (𝔹n)*𝔹n

(A𝔹) Boolean functions 𝔹n𝔹

[𝒜] Universe of discourse (𝔹n,(𝔹n𝔹))
(A,A) based on the features (𝔹n+𝔹)
(A+𝔹) {a1,,an} [𝔹n]

3.2 Special classes of propositions

A basic proposition, coordinate proposition, or simple proposition in the universe of discourse [a1,,an] is one of the propositions in the set {a1,,an}.

Among the 22n propositions in [a1,,an] are several families of 2n propositions each that take on special forms with respect to the basis {a1,,an}. 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 n-tuples in 𝔹n and falls into n+1 ranks, with a binomial coefficient (nk) giving the number of propositions that have rank or weight k.

  • The linear propositions, {:𝔹n𝔹}=(𝔹n𝔹), may be expressed as sums:


  • The positive propositions, {p:𝔹n𝔹}=(𝔹n𝑝𝔹), may be expressed as productsPlanetmathPlanetmath:


  • The singular propositions, {𝐱:𝔹n𝔹}=(𝔹n𝑠𝔹), may be expressed as products:


In each case the rank k ranges from 0 to n and counts the number of positive appearances of the coordinate propositions a1,,an in the resulting expression. For example, for n=3, the linear proposition of rank 0 is 0, the positive proposition of rank 0 is 1, and the singular proposition of rank 0 is (a1)(a2)(a3).

The basic propositions ai:𝔹n𝔹 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 𝒜={a1,,an}. For example, a singular proposition with respect to the basis 𝒜 will not remain singular if 𝒜 is extended by a number of new and independent features. Even if one keeps to the original set of pairwise options {ai}{(ai)} 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, A, supplies the groundwork for any number of further extensionsPlanetmathPlanetmathPlanetmath, beginning with the first order differential extension, EA. The construction of EA can be described in the following stages:

  • The initial alphabet, 𝔄={``a1",,``an"}, is extended by a first order differential alphabet, d𝔄={``da1",,``dan"}, resulting in a first order extended alphabet, E𝔄, defined as follows:


  • The initial basis, 𝒜={a1,,an}, is extended by a first order differential basis, d𝒜={da1,,dan}, resulting in a first order extended basis, E𝒜, defined as follows:


  • The initial space, A=a1,,an, is extended by a first order differential space or tangent space, dA=da1,,dan, at each point of A, resulting in a first order extended space or tangent bundle space, EA, defined as follows:


  • Finally, the initial universe, A=[a1,,an], is extended by a first order differential universe or tangent universe, dA=[da1,,dan], at each point of A, resulting in a first order extended universe or tangent bundle universe, EA, defined as follows:


    This gives EA 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 (http://planetmath.org/Calculus). With these constructions, the first order extended universe EA and the first order differential proposition f:EA𝔹, we have arrived, in concept at least, at the foothills of differential logic.

Table 6 summarizes the notations that are needed to describe the first order differential extensions of propositional calculi in a systematic manner.

Table 6. Differential Extension : Basic Notation

Notation Description Type

{``da1",,``dan"} Alphabet of differential symbols [n]=𝐧

{da1,,dan} Basis of differential features [n]=𝐧

{dai¯,dai} Differential dimension i 𝔻

d𝒜 Tangent space at a point: 𝔻n
da1,,dan Set of changes,
{(da1,,dan)} motions, steps,
dA1××dAn tangent vectors
i=1ndAi at a point

(hom:dA𝔹) Linear functions on dA (𝔻n)*𝔻n

(dA𝔹) Boolean functions on dA 𝔻n𝔹

[d𝒜] Tangent universe (𝔻n,(𝔻n𝔹))
(dA,dA) at a point of A, (𝔻n+𝔹)
(dA+𝔹) based on the [𝔻n]
(dA,(dA𝔹)) tangent features
[da1,,dan] {da1,,dan}
Title differential propositional calculus
Canonical name DifferentialPropositionalCalculus
Date of creation 2013-08-22 21:54:38
Last modified on 2013-08-22 21:54:38
Owner Jon Awbrey (15246)
Last modified by Jon Awbrey (15246)
Numerical id 69
Author Jon Awbrey (15246)
Entry type Definition
Classification msc 53A40
Classification msc 39A12
Classification msc 34G99
Classification msc 03B44
Classification msc 03B42
Classification msc 03B05
Synonym differential extension of propositional calculus
Related topic Derivation
Related topic DifferentialField
Related topic DifferentialGeometry
Related topic DifferentialLogic
Related topic FiniteDifference
Related topic FiniteField
Related topic MinimalNegationOperator
Related topic PropositionalCalculus
Related topic ZerothOrderLogic
Defines differential basis
Defines differential extension
Defines differential feature
Defines differential inference
Defines differential proposition
Defines differential quality
Defines differential variable
Defines logical transformation
Defines source universe
Defines target universe
Defines tangent universe
Defines basic proposition