Changes

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.

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.

For ease of reference during 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) = q_{139}(p, q, r) = \texttt{(} p \texttt{(} q \texttt{))(} q \texttt{(} r \texttt{))}</math> as the ''syllogism mapping'', written as <math>\operatorname{syll} : \mathbb{B}^3 \to \mathbb{B},</math> and let us refer to the fiber <math>\operatorname{syll}^{-1}(1) \subseteq \mathbb{B}^3</math> as the ''syllogism relation'', written as <math>\operatorname{Syll} \subseteq \mathbb{B}^3.</math>  Table 25-a shows <math>\operatorname{Syll}</math> as a relational dataset.

For ease of reference during 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) = q_{139}(p, q, r) = \texttt{(} p \texttt{(} q \texttt{))(} q \texttt{(} r \texttt{))}</math> as the ''syllogism mapping'', written as <math>\operatorname{syll} : \mathbb{B}^3 \to \mathbb{B},</math> and let us refer to the fiber <math>\operatorname{syll}^{-1}(1) \subseteq \mathbb{B}^3</math> as the ''syllogism relation'', written as <math>\operatorname{Syll} \subseteq \mathbb{B}^3.</math>  Table&nbsp;25-a shows <math>\operatorname{Syll}</math> as a relational dataset.

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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.

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.

Table 25-b repeats the relation ''Syll'' in the first column, listing its 3-tuples in bit-string form, followed by the 2-adic or ''planar'' projections of ''Syll'' in the next three columns.  For instance, ''Syll''_''pq'' is the 2-adic projection of ''Syll'' on the ''pq'' plane that is arrived at by deleting the ''r'' column and counting each 2-tuple that results just one time.  Likewise, ''Syll''_''pr'' is obtained by deleting the ''q'' column and ''Syll''_''qr'' is derived by deleting the p column, ignoring whatever duplicate pairs may result.  The final row of the right three columns gives the propositions of the form ''f'' : '''B'''² &rarr; '''B''' that indicate the 2-adic relations that result from these projections.

Table&nbsp;25-b 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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