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Directory:Jon Awbrey/Papers/Propositional Equation Reasoning Systems (view source)

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→‎Variations on a theme of transitivity: renumber tables

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|+ <math>\text{Table 46.}~~\text{Composite and Compiled Order Relations}</math>

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|+ <math>\text{Table 1.}~~\text{Composite and Compiled Order Relations}</math>

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|+ style="height:30px" | <math>\text{Table 51.} ~~ [| f_{207} |] ~=~ [| p \le q |]</math>

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|+ style="height:30px" | <math>\text{Table 2.} ~~ [| f_{207} |] ~=~ [| p \le q |]</math>

|- style="height:40px; background:#f0f0ff"

| style="border-bottom:1px solid black" | <math>p\!</math>

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|+ style="height:30px" | <math>\text{Table 52.} ~~ [| f_{187} |] ~=~ [| q \le r |]</math>

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|+ style="height:30px" | <math>\text{Table 3.} ~~ [| f_{187} |] ~=~ [| q \le r |]</math>

|- style="height:40px; background:#f0f0ff"

| style="border-bottom:1px solid black" | <math>p\!</math>

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|+ style="height:30px" | <math>\text{Table 53.} ~~ [| f_{175} |] ~=~ [| p \le r |]</math>

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|+ style="height:30px" | <math>\text{Table 4.} ~~ [| f_{175} |] ~=~ [| p \le r |]</math>

|- style="height:40px; background:#f0f0ff"

| style="border-bottom:1px solid black" | <math>p\!</math>

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|+ style="height:30px" | <math>\text{Table 54.} ~~ [| f_{139} |] ~=~ [| p \le q \le r |]</math>

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|+ style="height:30px" | <math>\text{Table 5.} ~~ [| f_{139} |] ~=~ [| p \le q \le r |]</math>

|- style="height:40px; background:#f0f0ff"

| style="border-bottom:1px solid black" | <math>p\!</math>

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These are just some of the initial observations that can be made about the dimensions of information and uncertainty in the conduct of logical inference, and there are many issues to be taken up as we get to the thick of it. In particular, we are taking propositions far too literally at the outset, reading their spots at face value, as it were, without yet considering their species character as fallible signs.

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For ease of reference in the rest of this discussion, let us refer to the propositional form <math>f : \mathbb{B}^3 \to \mathbb{B}</math> such that <math>f(p, q, r) = f_{139}(p, q, r) = \texttt{(} p \texttt{(} q \texttt{))(} q \texttt{(} r \texttt{))}</math> as the ''syllogism map'', written as <math>\operatorname{syll} : \mathbb{B}^3 \to \mathbb{B},</math> and let us refer to its fiber of truth <math>[| \operatorname{syll} |] = \operatorname{syll}^{-1}(1)</math> as the ''syllogism relation'', written as <math>\operatorname{Syll} \subseteq \mathbb{B}^3.</math> Table 51 shows <math>\operatorname{Syll}</math> as a relational dataset.

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For ease of reference in the rest of this discussion, let us refer to the propositional form <math>f : \mathbb{B}^3 \to \mathbb{B}</math> such that <math>f(p, q, r) = f_{139}(p, q, r) = \texttt{(} p \texttt{(} q \texttt{))(} q \texttt{(} r \texttt{))}</math> as the ''syllogism map'', written as <math>\operatorname{syll} : \mathbb{B}^3 \to \mathbb{B},</math> and let us refer to its fiber of truth <math>[| \operatorname{syll} |] = \operatorname{syll}^{-1}(1)</math> as the ''syllogism relation'', written as <math>\operatorname{Syll} \subseteq \mathbb{B}^3.</math> Table 6 shows <math>\operatorname{Syll}</math> as a relational dataset.

<br>

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|+ style="height:30px" | <math>\text{Table 55.} ~~ \text{Syllogism Relation}</math>

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|+ style="height:30px" | <math>\text{Table 6.} ~~ \text{Syllogism Relation}</math>

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| style="border-bottom:1px solid black" | <math>p\!</math>

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One of the first questions that we might ask about a 3-adic relation, in this case <math>\operatorname{Syll},</math> is whether it is ''determined by'' its 2-adic projections. I will illustrate what this means in the present case.

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Table 56 repeats the relation <math>\operatorname{Syll}</math> in the first column, listing its 3-tuples in bit-string form, followed by the 2-adic or ''planar'' projections of <math>\operatorname{Syll}</math> in the next three columns. For instance, <math>\operatorname{Syll}_{pq}</math> is the 2-adic projection of <math>\operatorname{Syll}</math> on the <math>pq\!</math> plane that is arrived at by deleting the <math>r\!</math> column and counting each 2-tuple that results just one time. Likewise, <math>\operatorname{Syll}_{pr}</math> is obtained by deleting the <math>q\!</math> column and <math>\operatorname{Syll}_{qr}</math> is derived by deleting the <math>p\!</math> column, ignoring whatever duplicate pairs may result. The final row of the right three columns gives the propositions of the form <math>f : \mathbb{B}^2 \to \mathbb{B}</math> that indicate the 2-adic relations that result from these projections.

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Table 7 repeats the relation <math>\operatorname{Syll}</math> in the first column, listing its 3-tuples in bit-string form, followed by the 2-adic or ''planar'' projections of <math>\operatorname{Syll}</math> in the next three columns. For instance, <math>\operatorname{Syll}_{pq}</math> is the 2-adic projection of <math>\operatorname{Syll}</math> on the <math>pq\!</math> plane that is arrived at by deleting the <math>r\!</math> column and counting each 2-tuple that results just one time. Likewise, <math>\operatorname{Syll}_{pr}</math> is obtained by deleting the <math>q\!</math> column and <math>\operatorname{Syll}_{qr}</math> is derived by deleting the <math>p\!</math> column, ignoring whatever duplicate pairs may result. The final row of the right three columns gives the propositions of the form <math>f : \mathbb{B}^2 \to \mathbb{B}</math> that indicate the 2-adic relations that result from these projections.

<br>

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|+ style="height:30px" | <math>\text{Table 56.} ~~ \text{Dyadic Projections of the Syllogism Relation}</math>

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|+ style="height:30px" | <math>\text{Table 7.} ~~ \text{Dyadic Projections of the Syllogism Relation}</math>

|- style="height:40px; background:#f0f0ff"

| <math>\operatorname{Syll}</math>

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Table 58 shows the 3-adic relation <math>\operatorname{Syll} \subseteq \mathbb{B}^3</math> again, and Figure 59 shows it plotted on a 3-cube template.

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Table 8 shows the 3-adic relation <math>\operatorname{Syll} \subseteq \mathbb{B}^3</math> again, and Figure 59 shows it plotted on a 3-cube template.

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<pre>

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Table 58. Syll c B^3

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Table 8. Syll c B^3

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

| p q r |

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<pre>

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Table 60. Syll c B^3

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Table 9. Syll c B^3

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

| p q r |

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<pre>

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Table 61. Dyadic Projections of Syll

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Table 10. Dyadic Projections of Syll

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

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<pre>

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Table 63. Syll c B^3

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Table 11. Syll c B^3

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

| p q r |

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<pre>

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Table 64. Dyadic Projections of Syll

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Table 12. Dyadic Projections of Syll

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

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<pre>

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Table 65. Tacit Extensions of Projections of Syll

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Table 13. Tacit Extensions of Projections of Syll

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

Jon Awbrey

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