Changes

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>\operatorname{true}\!</math> to the region where <math>q\!</math> is <math>\operatorname{false}\!</math> while <math>k\!</math> has moved from the region where <math>q\!</math> is <math>\operatorname{false}\!</math> to the region where <math>q\!</math> is <math>\operatorname{true}.\!</math>

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>\operatorname{true}\!</math> to the region where <math>q\!</math> is <math>\operatorname{false}\!</math> while <math>k\!</math> has moved from the region where <math>q\!</math> is <math>\operatorname{false}\!</math> to the region where <math>q\!</math> is <math>\operatorname{true}.\!</math>

Figure&nbsp;1&prime; reprises the situation shown in Figure&nbsp;1, but adduces a new quality for the purpose of explaining what we now know we'll see in Figure&nbsp;2.

Figure&nbsp;1&prime; reprises the situation shown in Figure&nbsp;1, but configures a new quality designed to explain the sequel to come in Figure&nbsp;2.

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Within the pale of <math>X\!</math> and <math>\{ q \}\!</math> there are but four different pieces of information that can be given expression in the corresponding propositional calculus, namely, the propositions:  <math>\operatorname{false},\!</math> <math>\lnot q,\!</math> <math>q,\!</math> <math>\operatorname{true}.\!</math>  Referring to the sample of points in Figure&nbsp;1, <math>\operatorname{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>\operatorname{true}\!</math> holds of all points in the sample.

Within the pale of <math>X\!</math> and <math>\{ q \}\!</math> there are but four different pieces of information that can be given expression in the corresponding propositional calculus, namely, the propositions:  <math>\operatorname{false},\!</math> <math>\lnot q,\!</math> <math>q,\!</math> <math>\operatorname{true}.\!</math>  Referring to the sample of points in Figure&nbsp;1, <math>\operatorname{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>\operatorname{true}\!</math> holds of all points in the sample.

Figure 1&prime; maintains the same universe of discourse and extends the basis of discussion up to a set of two qualities, <math>\{ q, \operatorname{d}q \}.\!</math>  In parallel fashion the initial propositional calculus is extended in the medium of a larger alphabet, <math>\{\!</math>"<math>q\!</math>"<math>,\!</math>&nbsp;"<math>\operatorname{d}q\!</math>"<math>\}.\!</math>

Figure&nbsp;1&prime; maintains the same universe of discourse and extends the basis of discussion up to a set of two qualities, <math>\{ q, \operatorname{d}q \}.\!</math>  In parallel fashion, the initial propositional calculus is extended in the medium of a larger alphabet, <math>\{\!</math>"<math>q\!</math>"<math>,\!</math> "<math>\operatorname{d}q\!</math>"<math>\}.\!</math>

 

Figure&nbsp;1&prime; 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>q\ \operatorname{d}q</math> describes <math>j\!</math></p>

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