Changes

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, 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.

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.

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

Another common scheme for description and evaluation of a proposition 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

|- style="background:paleturquoise"

|   u   v   w   |   u & v   |   u & w   |   v & w   |   q   |

! style="width:20%" | ''u v w''

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

! style="width:20%" | ''u'' ∧ ''v''

! style="width:20%" | ''u'' ∧ ''w''

|   0   0   0   |     0    |     0     |     0    |   0   |

! style="width:20%" | ''v'' ∧ ''w''

|               |           |           |           |      |

! style="width:20%" | ''q''

|   0   0   1   |     0    |     0     |     0    |   0   |

|-

|               |           |           |          |      |

| 0 0 0 || 0 || 0 || 0 || 0

|   0   1   0   |     0    |     0     |     0    |   0   |

|-

|               |           |           |          |      |

| 0 0 1 || 0 || 0 || 0 || 0

|   0   1   1   |     0     |     0     |     1     |   1   |

|-

|               |          |          |          |      |

| 0 1 0 || 0 || 0 || 0 || 0

|   1   0   0   |     0    |     0     |     0    |   0   |

|-

|               |           |           |          |      |

| 0 1 1 || 0 || 0 || 1 || 1

|   1   0   1   |     0     |     1     |     0     |   1   |

|-

|               |          |          |          |      |

| 1 0 0 || 0 || 0 || 0 || 0

|   1   1   0   |     1     |     0     |     0     |   1   |

|-

|               |          |          |          |      |

| 1 0 1 || 0 || 1 || 0 || 1

|   1   1   1   |     1    |     1     |     1    |   1   |

|-

|               |           |           |          |      |

| 1 1 0 || 1 || 0 || 0 || 1

|-

| 1 1 1 || 1 || 1 || 1 || 1

|}

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:

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: