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) = f_{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.

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&nbsp;52 shows <math>\operatorname{Syll}</math> as a relational dataset.

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

|

 

<pre>

{| 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%"

Table 25-a. Syllogism Relation

|+ style="height:30px" | <math>\text{Table 52.} ~~ \text{Syllogism Relation}</math>

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

|- style="height:40px"

|   p    |   q   |   r   |

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

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

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

|   0         0         0   |

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

|   0         0         1   |

|-

|   0         1         1   |

| <math>0\!</math> || <math>0\!</math> || <math>0\!</math>

|   1         1         1   |

|-

| <math>0\!</math> || <math>0\!</math> || <math>1\!</math>

</pre>

|-

| <math>0\!</math> || <math>1\!</math> || <math>1\!</math>

|-

| <math>1\!</math> || <math>1\!</math> || <math>1\!</math>

|}

|}

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.