| Line 362: |
Line 362: |
| | $\ldots$ | | $\ldots$ |
| | </pre> | | </pre> |
| − |
| |
| − | =Work In Progress=
| |
| − |
| |
| − | ==Casual introduction==
| |
| − |
| |
| − | Consider the situation represented by the venn diagram in Figure 1.
| |
| − |
| |
| − | <center><pre>
| |
| − | o-----------------------------------------------------------o
| |
| − | | X . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
| |
| − | | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
| |
| − | | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
| |
| − | | . . . . . . o-------------o . . . . . . . . . . . . . . . |
| |
| − | | . . h . . ./. . . . . . . .\. . . . . . . . . . . . . . . |
| |
| − | | . . @ . . / . . . . . . . . \ . . . . . . . . . . . . . . |
| |
| − | | . . . . ./. . i . . . . . . .\. . . . . . . . . . . . . . |
| |
| − | | . . . . / . . @ . . . . . . . \ . . . . . . . . . . . . . |
| |
| − | | . . . ./. . . . . . . . . . . .\. . . . . . . . . . . . . |
| |
| − | | . . . o . . . . . . . . . . j . o . . . . . . . . . . . . |
| |
| − | | . . . | . . . . . . . . . . @ . | . . . . . . . . . . . . |
| |
| − | | . . . | . . . . . . . . . . . . | . . . . . . . . . . . . |
| |
| − | | . . . | . . . . . .Q. . . . . . | . . . . . . . . . . . . |
| |
| − | | . . . | . . . . . . . . . . . . | . . . . . k . . . . . . |
| |
| − | | . . . | . . . . . . . . . . . . | . . . . . @ . . . . . . |
| |
| − | | . . . o . . . . . . . . . . . . o . . . . . . . . . . . . |
| |
| − | | . . . .\. . . . . . . . . . . ./. . . . . . . . . . . . . |
| |
| − | | . . . . \ . . . . . . . . . . / . . . . . . . . . . . . . |
| |
| − | | . . . . .\. . . . . . . . . ./. . . . . . . . . . . . . . |
| |
| − | | . . . . . \ . . . . . . . . / . . . . . . . . . . . . . . |
| |
| − | | . . . . . .\. . . . . . . ./. . . . . . . . . . . . . . . |
| |
| − | | . . . . . . o-------------o . . . . . . . . . . . . . . . |
| |
| − | | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
| |
| − | | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
| |
| − | | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
| |
| − | o-----------------------------------------------------------o
| |
| − | </pre>
| |
| − | '''Figure 1. Local Habitations, And Names'''
| |
| − | </center>
| |
| − |
| |
| − | The area of the rectangle represents a universe of discourse, <math>X.\!</math> 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 <math>q\!</math> or the locations that fall within the corresponding region <math>Q.\!</math> Four individuals, <math>h, i, j, k,\!</math> are singled out by name. It happens that <math>i\!</math> and <math>j\!</math> currently reside in region <math>Q\!</math> while <math>h\!</math> and <math>k\!</math> do not.
| |
| − |
| |
| − | Now consider the situation represented by the venn diagram in Figure 2.
| |
| − |
| |
| − | <center><pre>
| |
| − | o-----------------------------------------------------------o
| |
| − | | X . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
| |
| − | | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
| |
| − | | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
| |
| − | | . . . . . . o-------------o . . . . . . . . . . . . . . . |
| |
| − | | . . h . . ./. . . . . . . .\. . . . . . . . . . . . . . . |
| |
| − | | . . @ . . / . . . . . . . . \ . . . . . . . . . . . . . . |
| |
| − | | . . . . ./. . i . . . . . . .\. . . . . . . . . . . . . . |
| |
| − | | . . . . / . . @ . . . . . . . \ . . . . . . . . . . . . . |
| |
| − | | . . . ./. . . . . . . . . . . .\. . . . . . . . . . . . . |
| |
| − | | . . . o . . . . . . . . . . . . o . . . . . j . . . . . . |
| |
| − | | . . . | . . . . . . . . . . . . | . . . . . @ . . . . . . |
| |
| − | | . . . | . . . . . . . . . . . . | . . . . . . . . . . . . |
| |
| − | | . . . | . . . . . .Q. . . . . . | . . . . . . . . . . . . |
| |
| − | | . . . | . . . . . . . . . . k . | . . . . . . . . . . . . |
| |
| − | | . . . | . . . . . . . . . . @ . | . . . . . . . . . . . . |
| |
| − | | . . . o . . . . . . . . . . . . o . . . . . . . . . . . . |
| |
| − | | . . . .\. . . . . . . . . . . ./. . . . . . . . . . . . . |
| |
| − | | . . . . \ . . . . . . . . . . / . . . . . . . . . . . . . |
| |
| − | | . . . . .\. . . . . . . . . ./. . . . . . . . . . . . . . |
| |
| − | | . . . . . \ . . . . . . . . / . . . . . . . . . . . . . . |
| |
| − | | . . . . . .\. . . . . . . ./. . . . . . . . . . . . . . . |
| |
| − | | . . . . . . o-------------o . . . . . . . . . . . . . . . |
| |
| − | | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
| |
| − | | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
| |
| − | | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
| |
| − | o-----------------------------------------------------------o
| |
| − | </pre>
| |
| − | '''Figure 2. Same Names, Different Habitations'''
| |
| − | </center>
| |
| − |
| |
| − | Figure 2 differs from Figure 1 solely in the circumstance that the object <math>j\!</math> is outside the region <math>Q\!</math> while the object <math>k\!</math> is inside the region <math>Q.\!</math> 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 <math>h\!</math> and <math>i\!</math> have remained as they were with regard to quality <math>q\!</math> while <math>j\!</math> and <math>k\!</math> have changed their standings in that respect. In particular, <math>j\!</math> has moved from the region where <math>q\!</math> is <math>true\!</math> to the region where <math>q\!</math> is <math>false\!</math> while <math>k\!</math> has moved from the region where <math>q\!</math> is <math>false\!</math> to the region where <math>q\!</math> is <math>true.\!</math>
| |
| − |
| |
| − | Figure 1′ 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.
| |
| − |
| |
| − | <center><pre>
| |
| − | o-----------------------------------------------------------o
| |
| − | | X . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
| |
| − | | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
| |
| − | | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
| |
| − | | . . . . . . o-------------o . o-------------o . . . . . . |
| |
| − | | . . h . . ./. . . . . . . .\./. . . . . . . .\. . . . . . |
| |
| − | | . . @ . . / . . . . . . . . o . . . . . . . . \ . . . . . |
| |
| − | | . . . . ./. . i . . . . . ./.\. . . . . . . . .\. . . . . |
| |
| − | | . . . . / . . @ . . . . . / . \ . . . . . . . . \ . . . . |
| |
| − | | . . . ./. . . . . . . . ./. . .\. . . . . . . . .\. . . . |
| |
| − | | . . . o . . . . . . . . o . j . o . . . . . . . . o . . . |
| |
| − | | . . . | . . . . . . . . | . @ . | . . . . . . . . | . . . |
| |
| − | | . . . | . . . . . . . . | . . . | . . . . . . . . | . . . |
| |
| − | | . . . | . . . . . Q . . | . . . | . . dQ. . . . . | . . . |
| |
| − | | . . . | . . . . . . . . | . . . | . . . . . k . . | . . . |
| |
| − | | . . . | . . . . . . . . | . . . | . . . . . @ . . | . . . |
| |
| − | | . . . o . . . . . . . . o . . . o . . . . . . . . o . . . |
| |
| − | | . . . .\. . . . . . . . .\. . ./. . . . . . . . ./. . . . |
| |
| − | | . . . . \ . . . . . . . . \ . / . . . . . . . . / . . . . |
| |
| − | | . . . . .\. . . . . . . . .\./. . . . . . . . ./. . . . . |
| |
| − | | . . . . . \ . . . . . . . . o . . . . . . . . / . . . . . |
| |
| − | | . . . . . .\. . . . . . . ./.\. . . . . . . ./. . . . . . |
| |
| − | | . . . . . . o-------------o . o-------------o . . . . . . |
| |
| − | | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
| |
| − | | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
| |
| − | | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
| |
| − | o-----------------------------------------------------------o
| |
| − | </pre>
| |
| − | '''Figure 1′. Back, To The Future'''
| |
| − | </center>
| |
| − |
| |
| − | This new quality, <math>\operatorname{d}q,\!</math> 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 <math>X\!</math> outside and inside the region <math>\operatorname{d}Q.\!</math>
| |
| − |
| |
| − | Figure 1 represents a universe of discourse, <math>X,\!</math> together with a basis of discussion, <math>\{ q \},\!</math> for expressing propositions about the contents of that universe. Once the quality <math>q\!</math> is given a name, say, the symbol "<math>q\!</math>", we have the basis for a formal language that is specifically cut out for discussing <math>X\!</math> in terms of <math>q,\!</math> and this formal language is more formally known as the ''propositional calculus'' with alphabet <math>\{\!</math>"<math>q\!</math>"<math>\}.\!</math>
| |
| − |
| |
| − | In the context marked by <math>X\!</math> and <math>\{ q \}\!</math> there are but four different pieces of information that can be expressed in the corresponding propositional calculus, namely, the propositions: <math>false,\!</math> <math>\lnot q,\!</math> <math>q,\!</math> <math>true.\!</math> Referring to the sample of points in Figure 1, <math>false\!</math> holds of no points, <math>\lnot q\!</math> holds of <math>h\!</math> and <math>k,\!</math> <math>q\!</math> holds of <math>i\!</math> and <math>j,\!</math> and <math>true\!</math> 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, <math>\{ q, \operatorname{d}q \}.\!</math> In parallel fashion, the initial propositional calculus is extended by means of the enlarged alphabet, <math>\{\!</math>"<math>q\!</math>"<math>,\!</math> "<math>\operatorname{d}q\!</math>"<math>\}.\!</math> 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:
| |
| − |
| |
| − | :* <p><math>\overline{q}\ \overline{\operatorname{d}q}</math> describes <math>h\!</math></p>
| |
| − |
| |
| − | :* <p><math>\overline{q}\ \operatorname{d}q</math> describes <math>k\!</math></p>
| |
| − |
| |
| − | :* <p><math>q\ \overline{\operatorname{d}q}</math> describes <math>i\!</math></p>
| |
| − |
| |
| − | :* <p><math>q\ \operatorname{d}q</math> describes <math>j\!</math></p>
| |
| − |
| |
| − | Table 3 exhibits the rules of inference that give the differential quality <math>\operatorname{d}q\!</math> its meaning in practice.
| |
| − |
| |
| − | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:center; width:96%"
| |
| − | |+ '''Table 3. Differential Inference Rules'''
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:center; width:96%"
| |
| − | |
| |
| − | | From
| |
| − | | <math>\overline{q}\!</math>
| |
| − | | and
| |
| − | | <math>\overline{\operatorname{d}q}\!</math>
| |
| − | | infer
| |
| − | | <math>\overline{q}\!</math>
| |
| − | | next.
| |
| − | |
| |
| − | |-
| |
| − | |
| |
| − | | From
| |
| − | | <math>\overline{q}\!</math>
| |
| − | | and
| |
| − | | <math>\operatorname{d}q\!</math>
| |
| − | | infer
| |
| − | | <math>q\!</math>
| |
| − | | next.
| |
| − | |
| |
| − | |-
| |
| − | |
| |
| − | | From
| |
| − | | <math>q\!</math>
| |
| − | | and
| |
| − | | <math>\overline{\operatorname{d}q}\!</math>
| |
| − | | infer
| |
| − | | <math>q\!</math>
| |
| − | | next.
| |
| − | |
| |
| − | |-
| |
| − | |
| |
| − | | From
| |
| − | | <math>q\!</math>
| |
| − | | and
| |
| − | | <math>\operatorname{d}q\!</math>
| |
| − | | infer
| |
| − | | <math>\overline{q}\!</math>
| |
| − | | next.
| |
| − | |
| |
| − | |}
| |
| − | |}
| |
| − | <br>
| |
| − |
| |
| − | '''…'''
| |
| − |
| |
| − | Suppose we discover or begin to suspect that something we had been treating as a simple quality, <math>q,\!</math> is actually compounded of other qualities, <math>u\!</math> and <math>v\!</math>, according to a propositional formula <math>q = q(u, v).\!</math>
| |
| − |
| |
| − | '''…'''
| |
| − |
| |
| − | ==Transitional remarks==
| |
| − |
| |
| − | Up to this point we have been treating the universe of discourse <math>X,\!</math> the quality <math>q,\!</math> and the symbol "<math>q\!</math>" as all of one piece, almost as if the entire context marked by <math>X\!</math> and <math>q\!</math> and "<math>q\!</math>" amounted to the only way of viewing <math>X.\!</math> That is clearly not the case, but the fact is that people often use the term "universe of discourse" to cover a particular set of distinctions drawn in the space <math>X\!</math> and even sometimes a particular calculus or language for discussing the elements of <math>X.\!</math> If it were possible to coin a new phrase in this realm one might distinguish these latter components as the "discursive universe", but there is probably no escape from simply recognizing the equivocal senses of the terms already in use and trying to clarify the senses intended in context.
| |
| − |
| |
| − | '''…'''
| |
| − |
| |
| − | ==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.
| |
| − |
| |
| − | 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, <math>\mathfrak{A} = \lbrace\!</math> “<math>a_1\!</math>” <math>, \ldots,\!</math> “<math>a_n\!</math>” <math>\rbrace.\!</math> 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 <math>\mathfrak{A}</math> there is then a set of logical features, <math>\mathcal{A} = \{ a_1, \ldots, a_n \}.</math>
| |
| − |
| |
| − | A set of logical features, <math>\mathcal{A} = \{ a_1, \ldots, a_n \},</math> affords a basis for generating an <math>n\!</math>-dimensional universe of discourse, written <math>A^\circ = [ \mathcal{A} ] = [ a_1, \ldots, a_n ].</math> It is useful to consider a universe of discourse as a categorical object that incorporates both the set of points <math>A = \langle a_1, \ldots, a_n \rangle</math> and the set of propositions <math>A^\uparrow = \{ f : A \to \mathbb{B} \}</math> that are implicit with the ordinary picture of a venn diagram on <math>n\!</math> features. Accordingly, the universe of discourse <math>A^\circ</math> may be regarded as an ordered pair <math>(A, A^\uparrow)</math> having the type <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})),</math> and this last type designation may be abbreviated as <math>\mathbb{B}^n\ +\!\to \mathbb{B},</math> or even more succinctly as <math>[ \mathbb{B}^n ].</math> For convenience, the data type of a finite set on <math>n\!</math> elements may be indicated by either one of the equivalent notations, <math>[n]\!</math> or <math>\mathbf{n}.</math>
| |
| − |
| |
| − | Table 4 summarizes the notations that are needed to describe ordinary propositional calculi in a systematic fashion.
| |
| − |
| |
| − | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:96%"
| |
| − | |+ '''Table 4. Propositional Calculus : Basic Notation'''
| |
| − | |- style="background:ghostwhite"
| |
| − | ! Symbol
| |
| − | ! Notation
| |
| − | ! Description
| |
| − | ! Type
| |
| − | |-
| |
| − | | <math>\mathfrak{A}</math>
| |
| − | | <math>\lbrace\!</math> “<math>a_1\!</math>” <math>, \ldots,\!</math> “<math>a_n\!</math>” <math>\rbrace\!</math>
| |
| − | | Alphabet
| |
| − | | <math>[n] = \mathbf{n}</math>
| |
| − | |-
| |
| − | | <math>\mathcal{A}</math>
| |
| − | | <math>\{ a_1, \ldots, a_n \}</math>
| |
| − | | Basis
| |
| − | | <math>[n] = \mathbf{n}</math>
| |
| − | |-
| |
| − | | <math>A_i\!</math>
| |
| − | | <math>\{ \overline{a_i}, a_i \}\!</math>
| |
| − | | Dimension <math>i\!</math>
| |
| − | | <math>\mathbb{B}</math>
| |
| − | |-
| |
| − | | <math>A\!</math>
| |
| − | | <math>\langle \mathcal{A} \rangle</math><br>
| |
| − | <math>\langle a_1, \ldots, a_n \rangle</math><br>
| |
| − | <math>\{ (a_1, \ldots, a_n) \}\!</math>
| |
| − | <math>A_1 \times \ldots \times A_n</math><br>
| |
| − | <math>\textstyle \prod_i A_i\!</math>
| |
| − | | Set of cells,<br>
| |
| − | coordinate tuples,<br>
| |
| − | points, or vectors<br>
| |
| − | in the universe<br>
| |
| − | of discourse
| |
| − | | <math>\mathbb{B}^n</math>
| |
| − | |-
| |
| − | | <math>A^*\!</math>
| |
| − | | <math>(\operatorname{hom} : A \to \mathbb{B})</math>
| |
| − | | Linear functions
| |
| − | | <math>(\mathbb{B}^n)^* \cong \mathbb{B}^n</math>
| |
| − | |-
| |
| − | | <math>A^\uparrow</math>
| |
| − | | <math>(A \to \mathbb{B})</math>
| |
| − | | Boolean functions
| |
| − | | <math>\mathbb{B}^n \to \mathbb{B}</math>
| |
| − | |-
| |
| − | | <math>A^\circ</math>
| |
| − | | <math>[ \mathcal{A} ]</math><br>
| |
| − | <math>(A, A^\uparrow)</math><br>
| |
| − | <math>(A\ +\!\to \mathbb{B})</math><br>
| |
| − | <math>(A, (A \to \mathbb{B}))</math><br>
| |
| − | <math>[ a_1, \ldots, a_n ]</math>
| |
| − | | Universe of discourse<br>
| |
| − | based on the features<br>
| |
| − | <math>\{ a_1, \ldots, a_n \}</math>
| |
| − | | <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))</math><br>
| |
| − | <math>(\mathbb{B}^n\ +\!\to \mathbb{B})</math><br>
| |
| − | <math>[\mathbb{B}^n]</math>
| |
| − | |}
| |
| − | <br>
| |
| − |
| |
| − | An initial universe of discourse, <math>A^\circ,</math> supplies the groundwork for any number of further extensions, beginning with the ''first order differential extension'', <math>\operatorname{E}A^\circ.</math> This extends the initial alphabet, <math>\mathfrak{A} = \lbrace\!</math> “<math>a_1\!</math>” <math>, \ldots,\!</math> “<math>a_n\!</math>” <math>\rbrace,\!</math> by a ''first order differential alphabet'', <math>\operatorname{d}\mathfrak{A} = \lbrace\!</math> “<math>\operatorname{d}a_1\!</math>” <math>, \ldots,\!</math> “<math>\operatorname{d}a_n\!</math>” <math>\rbrace,\!</math> resulting in the ''first order extended alphabet'', <math>\operatorname{E}\mathfrak{A},</math> defined as follows:
| |
| − |
| |
| − | : <math>\operatorname{E}\mathfrak{A} = \mathfrak{A} \cup \operatorname{d}\mathfrak{A} = \lbrace\!</math> “<math>a_1\!</math>” <math>, \ldots,\!</math> “<math>a_n\!</math>”<math>,\!</math> “<math>\operatorname{d}a_1\!</math>” <math>, \ldots,\!</math> “<math>\operatorname{d}a_n\!</math>” <math>\rbrace.\!</math>
| |
| − |
| |
| − | '''…'''
| |
| − |
| |
| − | Table 5 summarizes the notations that are needed to describe the first order differential extensions of propositional calculi in a corresponding manner.
| |
| − |
| |
| − | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:96%"
| |
| − | |+ '''Table 5. Differential Extension : Basic Notation'''
| |
| − | |- style="background:ghostwhite"
| |
| − | ! Symbol
| |
| − | ! Notation
| |
| − | ! Description
| |
| − | ! Type
| |
| − | |-
| |
| − | | <math>\operatorname{d}\mathfrak{A}</math>
| |
| − | | <math>\lbrace\!</math> “<math>\operatorname{d}a_1</math>” <math>, \ldots,\!</math> “<math>\operatorname{d}a_n</math>” <math>\rbrace\!</math>
| |
| − | | Alphabet of<br>
| |
| − | differential<br>
| |
| − | symbols
| |
| − | | <math>[n] = \mathbf{n}</math>
| |
| − | |-
| |
| − | | <math>\operatorname{d}\mathcal{A}</math>
| |
| − | | <math>\{ \operatorname{d}a_1, \ldots, \operatorname{d}a_n \}</math>
| |
| − | | Basis of<br>
| |
| − | differential<br>
| |
| − | features
| |
| − | | <math>[n] = \mathbf{n}</math>
| |
| − | |-
| |
| − | | <math>\operatorname{d}A_i</math>
| |
| − | | <math>\{ \overline{\operatorname{d}a_i}, \operatorname{d}a_i \}</math>
| |
| − | | Differential<br>
| |
| − | dimension <math>i\!</math>
| |
| − | | <math>\mathbb{D}</math>
| |
| − | |-
| |
| − | | <math>\operatorname{d}A</math>
| |
| − | | <math>\langle \operatorname{d}\mathcal{A} \rangle</math><br>
| |
| − | <math>\langle \operatorname{d}a_1, \ldots, \operatorname{d}a_n \rangle</math><br>
| |
| − | <math>\{ (\operatorname{d}a_1, \ldots, \operatorname{d}a_n) \}</math><br>
| |
| − | <math>\operatorname{d}A_1 \times \ldots \times \operatorname{d}A_n</math><br>
| |
| − | <math>\textstyle \prod_i \operatorname{d}A_i</math>
| |
| − | | Tangent space<br>
| |
| − | at a point:<br>
| |
| − | Set of changes,<br>
| |
| − | motions, steps,<br>
| |
| − | tangent vectors<br>
| |
| − | at a point
| |
| − | | <math>\mathbb{D}^n</math>
| |
| − | |-
| |
| − | | <math>\operatorname{d}A^*</math>
| |
| − | | <math>(\operatorname{hom} : \operatorname{d}A \to \mathbb{B})</math>
| |
| − | | Linear functions<br>
| |
| − | on <math>\operatorname{d}A</math>
| |
| − | | <math>(\mathbb{D}^n)^* \cong \mathbb{D}^n</math>
| |
| − | |-
| |
| − | | <math>\operatorname{d}A^\uparrow</math>
| |
| − | | <math>(\operatorname{d}A \to \mathbb{B})</math>
| |
| − | | Boolean functions<br>
| |
| − | on <math>\operatorname{d}A</math>
| |
| − | | <math>\mathbb{D}^n \to \mathbb{B}</math>
| |
| − | |-
| |
| − | | <math>\operatorname{d}A^\circ</math>
| |
| − | | <math>[\operatorname{d}\mathcal{A}]</math><br>
| |
| − | <math>(\operatorname{d}A, \operatorname{d}A^\uparrow)</math><br>
| |
| − | <math>(\operatorname{d}A\ +\!\to \mathbb{B})</math><br>
| |
| − | <math>(\operatorname{d}A, (\operatorname{d}A \to \mathbb{B}))</math><br>
| |
| − | <math>[\operatorname{d}a_1, \ldots, \operatorname{d}a_n]</math>
| |
| − | | Tangent universe<br>
| |
| − | at a point of <math>A^\circ,</math><br>
| |
| − | based on the<br>
| |
| − | tangent features<br>
| |
| − | <math>\{ \operatorname{d}a_1, \ldots, \operatorname{d}a_n \}</math>
| |
| − | | <math>(\mathbb{D}^n, (\mathbb{D}^n \to \mathbb{B}))</math><br>
| |
| − | <math>(\mathbb{D}^n\ +\!\to \mathbb{B})</math><br>
| |
| − | <math>[\mathbb{D}^n]</math>
| |
| − | |}
| |
| − | <br>
| |
| − |
| |
| − | '''…'''
| |
| − |
| |
| − | ==Expository examples==
| |
| − |
| |
| − | '''…'''
| |
| − |
| |
| − | Consider the logical proposition represented by the following venn diagram:
| |
| − |
| |
| − | <center><pre>
| |
| − | o-----------------------------------------------------------o
| |
| − | | X . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
| |
| − | | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
| |
| − | | . . . . . . . . . . .o-------------o. . . . . . . . . . . |
| |
| − | | . . . . . . . . . . / . . . . . . . \ . . . . . . . . . . |
| |
| − | | . . . . . . . . . ./. . . . . . . . .\. . . . . . . . . . |
| |
| − | | . . . . . . . . . / . . . . . . . . . \ . . . . . . . . . |
| |
| − | | . . . . . . . . ./. . . . . . . . . . .\. . . . . . . . . |
| |
| − | | . . . . . . . . / . . . . . . . . . . . \ . . . . . . . . |
| |
| − | | . . . . . . . .o. . . . . . . . . . . . .o. . . . . . . . |
| |
| − | | . . . . . . . .|. . . . . . U . . . . . .|. . . . . . . . |
| |
| − | | . . . . . . . .|. . . . . . . . . . . . .|. . . . . . . . |
| |
| − | | . . . . . . . .|. . . . . . . . . . . . .|. . . . . . . . |
| |
| − | | . . . . . . . .|. . . . . . . . . . . . .|. . . . . . . . |
| |
| − | | . . . . . . . .|. . . . . . . . . . . . .|. . . . . . . . |
| |
| − | | . . . . . . o--o----------o . o----------o--o . . . . . . |
| |
| − | | . . . . . ./. . \%%%%%%%%%%\./%%%%%%%%%%/ . .\. . . . . . |
| |
| − | | . . . . . / . . .\%%%%%%%%%%o%%%%%%%%%%/. . . \ . . . . . |
| |
| − | | . . . . ./. . . . \%%%%%%%%/%\%%%%%%%%/ . . . .\. . . . . |
| |
| − | | . . . . / . . . . .\%%%%%%/%%%\%%%%%%/. . . . . \ . . . . |
| |
| − | | . . . ./. . . . . . \%%%%/%%%%%\%%%%/ . . . . . .\. . . . |
| |
| − | | . . . o . . . . . . .o--o-------o--o. . . . . . . o . . . |
| |
| − | | . . . | . . . . . . . . |%%%%%%%| . . . . . . . . | . . . |
| |
| − | | . . . | . . . . . . . . |%%%%%%%| . . . . . . . . | . . . |
| |
| − | | . . . | . . . . . . . . |%%%%%%%| . . . . . . . . | . . . |
| |
| − | | . . . | . . . .V. . . . |%%%%%%%| . . . .W. . . . | . . . |
| |
| − | | . . . | . . . . . . . . |%%%%%%%| . . . . . . . . | . . . |
| |
| − | | . . . o . . . . . . . . o%%%%%%%o . . . . . . . . o . . . |
| |
| − | | . . . .\. . . . . . . . .\%%%%%/. . . . . . . . ./. . . . |
| |
| − | | . . . . \ . . . . . . . . \%%%/ . . . . . . . . / . . . . |
| |
| − | | . . . . .\. . . . . . . . .\%/. . . . . . . . ./. . . . . |
| |
| − | | . . . . . \ . . . . . . . . o . . . . . . . . / . . . . . |
| |
| − | | . . . . . .\. . . . . . . ./.\. . . . . . . ./. . . . . . |
| |
| − | | . . . . . . o-------------o . o-------------o . . . . . . |
| |
| − | | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
| |
| − | | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
| |
| − | o-----------------------------------------------------------o
| |
| − | </pre>
| |
| − | '''Figure 1. Proposition''' <math>q : X \to \mathbb{B}</math>
| |
| − | </center>
| |
| − |
| |
| − | The following language is useful in describing the facts represented by the venn diagram.
| |
| − |
| |
| − | * The universe of discourse is a set, <math>X,\!</math> represented by the area inside the large rectangle.
| |
| − |
| |
| − | * The boolean domain is a set of two elements, <math>\mathbb{B} = \{ 0, 1 \},</math> represented by the two distinct shadings of the regions inside the rectangle.
| |
| − |
| |
| − | * According to the conventions observed in this context, the algebraic value 0 is interpreted as the logical value <math>\operatorname{false}</math> and represented by the lighter shading, while the algebraic value 1 is interpreted as the logical value <math>\operatorname{true}</math> and represented by the darker shading.
| |
| − |
| |
| − | * The universe of discourse <math>X\!</math> is the domain of three functions <math>u, v, w : X \to \mathbb{B}</math> called ''basic'', ''coordinate'', or ''simple'' propositions.
| |
| − |
| |
| − | * As with any proposition, <math>p : X \to \mathbb{B},</math> a simple proposition partitions <math>X\!</math> into two fibers, the fiber of 0 under <math>p,\!</math> defined as <math>p^{-1}(0) \subseteq X,</math> and the fiber of 1 under <math>p,\!</math> defined as <math>p^{-1}(1) \subseteq X.</math>
| |
| − |
| |
| − | * Each coordinate proposition is represented by a "circle", or a simple closed curve, that divides the rectangular region into the region exterior to the circle, representing the fiber of 0 under <math>p,\!</math> and the region interior to the circle, representing the fiber of 1 under <math>p.\!</math>
| |
| − |
| |
| − | * The fibers of 1 under the propositions <math>u, v, w\!</math> are the respective subsets <math>U, V, W \subseteq X.</math>
| |
| − |
| |
| − | '''…'''
| |
| | | | |
| | =Work Area= | | =Work Area= |
