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{{DISPLAYTITLE:Differential Propositional Calculus}}

{{DISPLAYTITLE:Differential Propositional Calculus}}

A '''differential propositional calculus''' is a [[propositional calculus]] extended by a set of terms for describing aspects of change and difference, for example, processes that take place in a [[universe of discourse]] or transformations that map a source universe into a target universe.

A '''differential propositional calculus''' is a [[propositional calculus]] extended by a set of terms for describing aspects of change and difference, for example, processes that take place in a [[universe of discourse]] or transformations that map a source universe into a target universe.

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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>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&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.

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

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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>

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>

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.

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&nbsp;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&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:

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: