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

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→‎Variations on a theme of transitivity: use arabic numbering of figures & tables, but sequential within sections

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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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| [[Image:Venn Diagram (P (Q)).jpg|500px]] || (47)

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| [[Image:Venn Diagram (P (Q)).jpg|500px]] || (2)

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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]] || (48)

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| [[Image:Venn Diagram (Q (R)).jpg|500px]] || (3)

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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]] || (49)

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| [[Image:Venn Diagram (P (R)).jpg|500px]] || (4)

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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]] || (50)

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| [[Image:Venn Diagram (P (Q)) (Q (R)).jpg|500px]] || (5)

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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 51.} ~~ [| f_{207} |] ~=~ [| p \le q |]</math>

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|+ style="height:30px" | <math>\text{Table 6.} ~~ [| 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 7.} ~~ [| 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 8.} ~~ [| 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 9.} ~~ [| 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.

+

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.

<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 10.} ~~ \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 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.

<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 56.} ~~ \text{Dyadic Projections of the Syllogism Relation}</math>

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|+ style="height:30px" | <math>\text{Table 11.} ~~ \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 57 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 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".

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

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

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Figure 56. Boolean 3-Cube B^3

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Figure 12. Boolean 3-Cube B^3

</pre>

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| (57)

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| (12)

|}

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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 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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<pre>

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Table 58. Syll c B^3

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Table 13. 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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| (13)

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

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Figure 59. Triadic Relation Syll c B^3

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Figure 14. Triadic Relation Syll c B^3

</pre>

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| (59)

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| (14)

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<pre>

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Table 60. Syll c B^3

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Table 15. Syll c B^3

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

| p q r |

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

</pre>

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| (60)

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| (15)

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<pre>

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Table 61. Dyadic Projections of Syll

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Table 16. 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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| (61)

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| (16)

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

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

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

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Figure 62. Syll c B^3 and its Dyadic Projections

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

</pre>

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| (62)

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| (17)

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<pre>

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Table 63. Syll c B^3

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Table 18. Syll c B^3

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

| p q r |

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

</pre>

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| (63)

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| (18)

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<pre>

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Table 64. Dyadic Projections of Syll

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Table 19. 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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| (64)

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| (19)

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<pre>

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Table 65. Tacit Extensions of Projections of Syll

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Table 20. 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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| (65)

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| (20)

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

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Figure 66. Tacit Extension te_12_3 (Syll_12)

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Figure 21. Tacit Extension te_12_3 (Syll_12)

</pre>

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| (66)

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| (21)

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

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Figure 67. Tacit Extension te_13_2 (Syll_13)

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Figure 22. Tacit Extension te_13_2 (Syll_13)

</pre>

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| (22)

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

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Figure 68. Tacit Extension te_23_1 (Syll_23)

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Figure 23. Tacit Extension te_23_1 (Syll_23)

</pre>

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| (68)

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| (23)

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The reader may wish to contemplate Figure 69 and use it to verify the following two facts:

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

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Figure 69. Syll = te(Syll_12) |^| te(Syll_23)

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

</pre>

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| (69)

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| (24)

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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 70 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 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.

<pre>

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

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Figure 70. Expressive Aspects of Transitive Inference

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Figure 25. 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 71 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 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:

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

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

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Figure 71. Denotative Aspects of Transitive Inference

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Figure 26. 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 72 translates the contents of Figure 71 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 27 translates the contents of Figure 26 into the new language.

<pre>

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

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Figure 72. Denotative Aspects of Transitive Inference

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Figure 27. Denotative Aspects of Transitive Inference

</pre>

Jon Awbrey

12,080

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