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

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→‎Variations on a theme of transitivity: re-number figures & tables

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{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"

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

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|+ <math>\text{Table 51.}~~\text{Composite and Compiled Order Relations}</math>

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{| align="center" cellpadding="6" style="text-align:center"

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| [[Image:Venn Diagram (P (Q)).jpg|500px]] || (51)

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| [[Image:Venn Diagram (P (Q)).jpg|500px]] || (52)

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| <math>f_{207}(p, q, r) ~=~ \texttt{(} p \texttt{~(} q \texttt{))}</math>

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| [[Image:Venn Diagram (Q (R)).jpg|500px]] || (52)

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| [[Image:Venn Diagram (Q (R)).jpg|500px]] || (53)

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| <math>f_{187}(p, q, r) ~=~ \texttt{(} q \texttt{~(} r \texttt{))}</math>

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| [[Image:Venn Diagram (P (R)).jpg|500px]] || (53)

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| [[Image:Venn Diagram (P (R)).jpg|500px]] || (54)

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| <math>f_{175}(p, q, r) ~=~ \texttt{(} p \texttt{~(} r \texttt{))}</math>

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| [[Image:Venn Diagram (P (Q)) (Q (R)).jpg|500px]] || (54)

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| [[Image:Venn Diagram (P (Q)) (Q (R)).jpg|500px]] || (55)

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| <math>f_{139}(p, q, r) ~=~ \texttt{(} p \texttt{~(} q \texttt{))~(} q \texttt{~(} r \texttt{))}</math>

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|+ style="height:30px" | <math>\text{Table 6.} ~~ [| f_{207} |] ~=~ [| p \le q |]</math>

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|+ style="height:30px" | <math>\text{Table 56.} ~~ [| 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 7.} ~~ [| f_{187} |] ~=~ [| q \le r |]</math>

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|+ style="height:30px" | <math>\text{Table 57.} ~~ [| 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 8.} ~~ [| f_{175} |] ~=~ [| p \le r |]</math>

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|+ style="height:30px" | <math>\text{Table 58.} ~~ [| 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 9.} ~~ [| f_{139} |] ~=~ [| p \le q \le r |]</math>

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|+ style="height:30px" | <math>\text{Table 59.} ~~ [| 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 10 shows <math>\operatorname{Syll}</math> as a relational dataset.

+

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 60 shows <math>\operatorname{Syll}</math> as a relational dataset.

<br>

{| align="center" cellpadding="8" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:40%"

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|+ style="height:30px" | <math>\text{Table 10.} ~~ \text{Syllogism Relation}</math>

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|+ style="height:30px" | <math>\text{Table 60.} ~~ \text{Syllogism Relation}</math>

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

| 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 11 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 61 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>

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"

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|+ style="height:30px" | <math>\text{Table 11.} ~~ \text{Dyadic Projections of the Syllogism Relation}</math>

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|+ style="height:30px" | <math>\text{Table 61.} ~~ \text{Dyadic Projections of the Syllogism Relation}</math>

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

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

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But the more I survey the problem setting the more it looks like we need better ways to bring our visual intuitions to play on the scene, and so let us next lay out some visual schemata that are designed to facilitate that.

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Figure 12 shows the familiar picture of a boolean 3-cube, where the points of <math>\mathbb{B}^3</math> are coordinated as bit strings of length three. Looking at the functions <math>f : \mathbb{B}^3 \to \mathbb{B}</math> and the relations <math>L \subseteq \mathbb{B}^3</math> on this pattern, one views the construction of either type of object as a matter of coloring the nodes of the 3-cube with choices from a pair of colors that stipulate which points are in the relation <math>L = [| f |]\!</math> and which points are out of it. Bowing to common convention, we may use the color <math>1\!</math> for points that are ''in'' a given relation and the color <math>0\!</math> for points that are ''out'' of the same relation. However, it will be more convenient here to indicate the former case by writing the coordinates in the place of the node and to indicate the latter case by plotting the point as an unlabeled node "o".

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Figure 62 shows the familiar picture of a boolean 3-cube, where the points of <math>\mathbb{B}^3</math> are coordinated as bit strings of length three. Looking at the functions <math>f : \mathbb{B}^3 \to \mathbb{B}</math> and the relations <math>L \subseteq \mathbb{B}^3</math> on this pattern, one views the construction of either type of object as a matter of coloring the nodes of the 3-cube with choices from a pair of colors that stipulate which points are in the relation <math>L = [| f |]\!</math> and which points are out of it. Bowing to common convention, we may use the color <math>1\!</math> for points that are ''in'' a given relation and the color <math>0\!</math> for points that are ''out'' of the same relation. However, it will be more convenient here to indicate the former case by writing the coordinates in the place of the node and to indicate the latter case by plotting the point as an unlabeled node "o".

{| align="center" cellpadding="10" style="text-align:center; width:90%"

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

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Figure 12. Boolean 3-Cube B^3

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Figure 62. Boolean 3-Cube B^3

</pre>

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| (55)

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| (62)

|}

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Table 13 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 63 shows the 3-adic relation <math>\operatorname{Syll} \subseteq \mathbb{B}^3</math> again, and Figure 64 shows it plotted on a 3-cube template.

{| align="center" cellpadding="10" style="text-align:center; width:90%"

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<pre>

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Table 13. Syll c B^3

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Table 63. Syll c B^3

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

| p q r |

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

</pre>

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| (56)

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| (63)

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

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Figure 14. Triadic Relation Syll c B^3

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Figure 64. Triadic Relation Syll c B^3

</pre>

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| (57)

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| (64)

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<pre>

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Table 15. Syll c B^3

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Table 65. Syll c B^3

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

| p q r |

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

</pre>

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| (58)

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| (65)

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<pre>

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Table 16. Dyadic Projections of Syll

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Table 66. Dyadic Projections of Syll

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

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

</pre>

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| (59)

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| (66)

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In showing the 2-adic projections of a 3-adic relation <math>L \subseteq \mathbb{B}^3,</math> I will translate the coordinates of the points in each relation to the plane of the projection, there dotting out with a dot "." the bit of the bit string that is out of place on that plane.

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Figure 17 shows <math>\operatorname{Syll}</math> and its three 2-adic projections:

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Figure 67 shows <math>\operatorname{Syll}</math> and its three 2-adic projections:

{| align="center" cellpadding="10" style="text-align:center; width:90%"

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

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Figure 17. Syll c B^3 and its Dyadic Projections

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Figure 67. Syll c B^3 and its Dyadic Projections

</pre>

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| (60)

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| (67)

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<pre>

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Table 18. Syll c B^3

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Table 68. Syll c B^3

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

| p q r |

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

</pre>

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| (61)

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| (68)

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<pre>

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Table 19. Dyadic Projections of Syll

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Table 69. Dyadic Projections of Syll

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

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

</pre>

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| (62)

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| (69)

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<pre>

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Table 20. Tacit Extensions of Projections of Syll

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Table 70. Tacit Extensions of Projections of Syll

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

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

</pre>

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| (70)

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

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Figure 21. Tacit Extension te_12_3 (Syll_12)

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Figure 71. Tacit Extension te_12_3 (Syll_12)

</pre>

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| (71)

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

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Figure 22. Tacit Extension te_13_2 (Syll_13)

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Figure 72. Tacit Extension te_13_2 (Syll_13)

</pre>

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

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Figure 23. Tacit Extension te_23_1 (Syll_23)

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Figure 73. Tacit Extension te_23_1 (Syll_23)

</pre>

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| (66)

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| (73)

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The reader may wish to contemplate Figure 24 and use it to verify the following two facts:

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The reader may wish to contemplate Figure 74 and use it to verify the following two facts:

{| align="center" cellpadding="8" width="90%"

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

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Figure 24. Syll = te(Syll_12) |^| te(Syll_23)

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Figure 74. Syll = te(Syll_12) |^| te(Syll_23)

</pre>

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| (67)

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| (74)

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In accord with my experimental way, I will stick with the case of transitive inference until I have pinned it down thoroughly, but of course the real interest is much more general than that.

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At first sight, the relationships seem easy enough to write out. Figure 25 shows how the various logical expressions are related to each other: The expressions <math>{}^{\backprime\backprime} \texttt{(} p \texttt{~(} q \texttt{))} {}^{\prime\prime}</math> and <math>{}^{\backprime\backprime} \texttt{(} q \texttt{~(} r \texttt{))} {}^{\prime\prime}</math> are conjoined in a purely syntactic fashion — much in the way that one might compile a theory from axioms without knowing what either the theory or the axioms were about — and the best way to sum up the state of information implicit in taking them together is just the expression <math>{}^{\backprime\backprime} \texttt{(} p \texttt{~(} q \texttt{))~(} q \texttt{~(} r \texttt{))}{}^{\prime\prime}</math> that would the canonical result of an equational or reversible rule of inference. From that equational inference, one might arrive at the implicational inference <math>{}^{\backprime\backprime} \texttt{(} p \texttt{~(} r \texttt{))} {}^{\prime\prime}</math> by the most conventional implication.

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At first sight, the relationships seem easy enough to write out. Figure 75 shows how the various logical expressions are related to each other: The expressions <math>{}^{\backprime\backprime} \texttt{(} p \texttt{~(} q \texttt{))} {}^{\prime\prime}</math> and <math>{}^{\backprime\backprime} \texttt{(} q \texttt{~(} r \texttt{))} {}^{\prime\prime}</math> are conjoined in a purely syntactic fashion — much in the way that one might compile a theory from axioms without knowing what either the theory or the axioms were about — and the best way to sum up the state of information implicit in taking them together is just the expression <math>{}^{\backprime\backprime} \texttt{(} p \texttt{~(} q \texttt{))~(} q \texttt{~(} r \texttt{))}{}^{\prime\prime}</math> that would the canonical result of an equational or reversible rule of inference. From that equational inference, one might arrive at the implicational inference <math>{}^{\backprime\backprime} \texttt{(} p \texttt{~(} r \texttt{))} {}^{\prime\prime}</math> by the most conventional implication.

<pre>

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

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Figure 25. Expressive Aspects of Transitive Inference

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Figure 75. Expressive Aspects of Transitive Inference

</pre>

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Despite how simple the formulaic aspects of transitive inference might appear on the surface, there are problems that wait for us just beneath the syntactic surface, as we quickly discover if we turn to considering the kinds of objects, abstract and concrete, that these formulas are meant to denote, and all the more so if we try to do this in a context of computational implementations, where the "interpreters" to be addressed take nothing on faith. Thus we engage the ''denotative semantics'' or the ''model theory'' of these extremely simple programs that we call ''propositions''.

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Figure 26 is an attempt to outline the model-theoretic relationships that are involved in our study of transitive inference. A couple of alternative notations are introduced in this Table:

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Figure 76 is an attempt to outline the model-theoretic relationships that are involved in our study of transitive inference. A couple of alternative notations are introduced in this Table:

{| align="center" cellpadding="8" width="90%"

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

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Figure 26. Denotative Aspects of Transitive Inference

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Figure 76. Denotative Aspects of Transitive Inference

</pre>

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An abstract reference to a point of <math>X\!</math> is a triple in <math>\mathbb{B}^3.</math> A concrete reference to a point of <math>X\!</math> is a conjunction of signs from the dimensions <math>P^\ddagger, Q^\ddagger, R^\ddagger,</math> picking exactly one sign from each dimension.

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To illustrate the use of concrete coordinates for points and concrete types for spaces and propositions, Figure 27 translates the contents of Figure 26 into the new language.

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To illustrate the use of concrete coordinates for points and concrete types for spaces and propositions, Figure 77 translates the contents of Figure 76 into the new language.

<pre>

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

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Figure 27. Denotative Aspects of Transitive Inference

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Figure 77. Denotative Aspects of Transitive Inference

</pre>

Jon Awbrey

12,080

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