Differential Propositional Calculus • 7

Special Classes of Propositions (concl.)

Last and literally least in extent, we examine the family of singular propositions in a 3-dimensional universe of discourse.

In our model of propositions as mappings of a universe of discourse to a set of two values, in other words, indicator functions of the form $f : X \to \mathbb{B},$ singular propositions are those singling out the minimal distinct regions of the universe, represented by single cells of the corresponding venn diagram.

Singular Propositions

In a universe of discourse based on three boolean variables, $p, q, r,$ there are $2^3 = 8$ singular propositions.  Their venn diagrams are shown in Figure 10.

$\text{Figure 10.} ~~ \text{Singular Propositions} : \mathbb{B}^3 \to \mathbb{B}$

At the top is the venn diagram for the singular proposition of rank 3, corresponding to the boolean product $pqr$ and identical with the positive proposition of rank 3.

Next are the venn diagrams for the three singular propositions of rank 2, which may be expressed by the following three forms, respectively:

$pr \texttt{(} q \texttt{)}, \qquad qr \texttt{(} p \texttt{)}, \qquad pq \texttt{(} r \texttt{)}.$

Next are the three singular propositions of rank 1, which may be expressed by the following three forms, respectively:

$q \texttt{(} p \texttt{)(} r \texttt{)}, \qquad p \texttt{(} q \texttt{)(} r \texttt{)}, \qquad r \texttt{(} p \texttt{)(} q \texttt{)}.$

At the bottom is the singular proposition of rank 0, which may be expressed by the following form:

$\texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)}.$

2 Responses to Differential Propositional Calculus • 7

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The passage describes the classification and representation of singular propositions in a 3-dimensional boolean universe, including their venn diagrams and boolean expressions. It does not contain any mathematical questions or answers.
no_question
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
Differential Propositional Calculus • 7

Special Classes of Propositions (concl.)

Last and literally least in extent, we examine the family of singular propositions in a 3-dimensional universe of discourse.

In our model of propositions as mappings of a universe of discourse to a set of two values, in other words, indicator functions of the form $f : X \to \mathbb{B},$ singular propositions are those singling out the minimal distinct regions of the universe, represented by single cells of the corresponding venn diagram.

Singular Propositions

In a universe of discourse based on three boolean variables, $p, q, r,$ there are $2^3 = 8$ singular propositions.  Their venn diagrams are shown in Figure 10.

$\text{Figure 10.} ~~ \text{Singular Propositions} : \mathbb{B}^3 \to \mathbb{B}$

At the top is the venn diagram for the singular proposition of rank 3, corresponding to the boolean product $pqr$ and identical with the positive proposition of rank 3.

Next are the venn diagrams for the three singular propositions of rank 2, which may be expressed by the following three forms, respectively:

$pr \texttt{(} q \texttt{)}, \qquad qr \texttt{(} p \texttt{)}, \qquad pq \texttt{(} r \texttt{)}.$

Next are the three singular propositions of rank 1, which may be expressed by the following three forms, respectively:

$q \texttt{(} p \texttt{)(} r \texttt{)}, \qquad p \texttt{(} q \texttt{)(} r \texttt{)}, \qquad r \texttt{(} p \texttt{)(} q \texttt{)}.$

At the bottom is the singular proposition of rank 0, which may be expressed by the following form:

$\texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)}.$

2 Responses to Differential Propositional Calculus • 7

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The passage describes the classification and representation of singular propositions in a 3-dimensional boolean universe, including their venn diagrams and boolean expressions. It does not contain any mathematical questions or answers.

TYPE: NO_QUESTION