Changes

===Note 5===

===Note 5===

We have come to the point of making a connection, at a very primitive level, between propositional logic and the classes of mathematical structures that are employed in mathematical systems theory to model dynamical systems of very general sorts.

We have come to the point of making a connection,

at a very primitive level, between propositional

logic and the classes of mathematical structures

that are employed in mathematical systems theory

to model dynamical systems of very general sorts.

Here is a flash montage of what has gone before,

Here is a flash montage of what has gone before, retrospectively touching on just the highpoints, and highlighting mostly just Figures and Tables, all directed toward the aim of ending up with a novel style of pictorial diagram, one that will serve us well in the future, as I have found it readily adaptable and steadily more trustworthy in my previous investigations, whenever we have to illustrate these very basic sorts of dynamic scenarios to ourselves, to others, to computers.

retrospectively touching on just the highpoints,

and highlighting mostly just Figures and Tables,

all directed toward the aim of ending up with a

novel style of pictorial diagram, one that will

serve us well in the future, as I have found it

readily adaptable and steadily more trustworthy

in my previous investigations, whenever we have

to illustrate these very basic sorts of dynamic

scenarios to ourselves, to others, to computers.

We typically start out with a proposition of interest,

We typically start out with a proposition of interest, for example, the proposition <math>q : X \to \mathbb{B}</math> depicted here:

for example, the proposition q : X -> B depicted here:

o-------------------------------------------------o

o-------------------------------------------------o

| q                                              |

| q                                              |

|            (( u v )( u w )( v w ))            |

|            (( u v )( u w )( v w ))            |

o-------------------------------------------------o

o-------------------------------------------------o

The proposition q is properly considered as an "abstract object",

The proposition <math>q\!</math> is properly considered as an ''[[abstract object]]'', in some acceptation of those very bedevilled and egging-on terms, but it enjoys an interpretation as a function of a suitable type, and all we have to do in order to enjoy the utility of this type of representation is to observe a decent respect for what befits.

in some acceptation of those very bedevilled and egging-on terms,

but it enjoys an interpretation as a function of a suitable type,

and all we have to do in order to enjoy the utility of this type

of representation is to observe a decent respect for what befits.

I will skip over the details of how to do this for right now.

I will skip over the details of how to do this for right now. I started to write them out in full, and it all became even more tedious than my usual standard, and besides, I think that everyone more or less knows how to do this already.

I started to write them out in full, and it all became even

more tedious than my usual standard, and besides, I think

that everyone more or less knows how to do this already.

Once we have survived the big leap of re-interpreting these

Once we have survived the big leap of re-interpreting these abstract names as the names of relatively concrete dimensions of variation, we can begin to lay out all of the familiar sorts of mathematical models and pictorial diagrams that go with these modest dimensions, the functions that can be formed on them, and the transformations that can be entertained among this whole crew.

abstract names as the names of relatively concrete dimensions

of variation, we can begin to lay out all of the familiar sorts

of mathematical models and pictorial diagrams that go with these

modest dimensions, the functions that can be formed on them, and

the transformations that can be entertained among this whole crew.

Here is the venn diagram for the proposition q.

Here is the venn diagram for the proposition <math>q\!</math>.

o-----------------------------------------------------------o

o-----------------------------------------------------------o

| X                                                        |

| X                                                        |

o-----------------------------------------------------------o

o-----------------------------------------------------------o

Figure 1.  Venn Diagram for the Proposition q

Figure 1.  Venn Diagram for the Proposition q

By way of excuse, if not yet a full justification, I probably ought to give

By way of excuse, if not yet a full justification, I probably ought to give an account of the reasons why I continue to hang onto these primitive styles of depiction, even though I can hardly recommend that anybody actually try to draw them, at least, not once the number of variables climbs much higher than three or four or five at the utmost.   One of the reasons would have to be this: that in the relationship between their continuous aspect and their discrete aspect, venn diagrams constitute a form of "iconic" reminder of a very important fact about all ''finite information depictions'' (FID's) of the larger world of reality, and that is the hard fact that we deceive ourselves to a degree if we imagine that the lines and the distinctions that we draw in our imagination are all there is to reality, and thus, that as we practice to categorize, we also manage to discretize, and thus, to distort, to reduce, and to truncate the richness of what there is to the poverty of what we can sieve and sift through our senses, or what we can draw in the tangled webs of our own very tenuous and tinctured distinctions.

an account of the reasons why I continue to hang onto these primitive styles

of depiction, even though I can hardly recommend that anybody actually try to

draw them, at least, not once the number of variables climbs much higher than

three or four or five at the utmost. One of the reasons would have to be this:

that in the relationship between their continuous aspect and their discrete aspect,

venn diagrams constitute a form of "iconic" reminder of a very important fact about

all "finite information depictions" (FID's) of the larger world of reality, and that

is the hard fact that we deceive ourselves to a degree if we imagine that the lines

and the distinctions that we draw in our imagination are all there is to reality,

and thus, that as we practice to categorize, we also manage to discretize, and

thus, to distort, to reduce, and to truncate the richness of what there is to

the poverty of what we can sieve and sift through our senses, or what we can

draw in the tangled webs of our own very tenuous and tinctured distinctions.

    

    

Another common scheme for description and evaluation of a proposition

Another common scheme for description and evaluation of a proposition is the so-called ''truth table'' or the ''semantic tableau'', for example:

is the so-called "truth table" or the "semantic tableau", for example:

Table 2.  Truth Table for the Proposition q

Table 2.  Truth Table for the Proposition q

o---------------o-----------o-----------o-----------o-------o

o---------------o-----------o-----------o-----------o-------o

|              |          |          |          |      |

|              |          |          |          |      |

o---------------o-----------o-----------o-----------o-------o

o---------------o-----------o-----------o-----------o-------o

Reading off the shaded cells of the venn diagram or the

Reading off the shaded cells of the venn diagram or the rows of the truth table that have a "1" in the q column, we see that the ''models'', or satisfying interpretations, of the proposition <math>q\!</math> are the four that can be expressed, in either the ''additive'' or the ''multiplicative'' manner, as follows:

rows of the truth table that have a "1" in the q column,

we see that the "models", or satisfying interpretations,

of the proposition q are the four that can be expressed,

in either the "additive" or the "multiplicative" manner,

as follows:

1.  The points of the space X that are assigned the coordinates:

# The points of the space <math>X\!</math> that are assigned the coordinates:<br><math>(u, v, w)\!</math> = <math>(0, 1, 1)\!</math> or <math>(1, 0, 1)\!</math> or <math>(1, 1, 0)\!</math> or <math>(1, 1, 1)\!</math>.

    <u, v, w> = <0, 1, 1> or <1, 0, 1> or <1, 1, 0> or <1, 1, 1>.

# The points of the space <math>X\!</math> that have the conjunctive descriptions:<br><code>(u) v w</code> or <code>u (v) w</code> or <code>u v (w)</code> or <code>u v w</code>, where "<code>(x)</code>" is "not&nbsp;<code>x</code>".

 

2.  The points of the space X that have the conjunctive descriptions:

    "(u) v w", "u (v) w", "u v (w)", "u v w", where "(x)" is "not x".

The next thing that one typically does is to consider the effects

The next thing that one typically does is to consider the effects

of various "operators" on the proposition of interest, which may

of various "operators" on the proposition of interest, which may