Changes

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''.

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''.

Table 33 is an attempt to outline the model-theoretic relationships that are involved in our study of transitive inference.  In it I use a number of abbreviated notations:

Table 33 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:

# I use the forms ''X''<b>:</b>''Y''<b>:</b>''Z'' and ''x''<b>:</b>''y''<b>:</b>''z'' as alternative notations for the cartesian product ''X'' &times; ''Y'' &times; ''Z'' and the tuple (''x'', ''y'', ''z''), respectively.

# In situations where we have products like ''X''''':'''''Y''''':'''''Z'' with ''X'' = ''Y'' = ''Z'' = '''B''', and relations like ''L'' &sube; ''X''<b>:</b>''Y'', ''M'' &sube; ''X''<b>:</b>''Z'', ''N'' &sube; ''Y''<b>:</b>''Z'', I will use forms like ''L'' &sube; '''B:B:~''', ''M'' &sube; '''B:~:B''', ''N'' &sube; '''~:B:B''' to remind us that we are considering particular ways of situating ''L'', ''M'', ''N'' within the product space ''X''<b>:</b>''Y''<b>:</b>''Z''.

| The forms <math>X:Y:Z\!</math> and <math>x:y:z\!</math> are used as alternative notations for the cartesian product <math>X \times Y \times Z</math> and the tuple <math>(x, y, z),\!</math> respectively.

| In situations where we have products like <math>X:Y:Z\!</math> with <math>X = Y = Z = \mathbb{B},</math> and relations like <math>L \subseteq X:Y,</math> &nbsp; <math>M \subseteq X:Z,</math> &nbsp; <math>N \subseteq Y:Z,</math> the forms <math>L \subseteq \mathbb{B}:\mathbb{B}:-,</math> &nbsp; <math>M \subseteq \mathbb{B}:-:\mathbb{B},</math> &nbsp; <math>N \subseteq -:\mathbb{B}:\mathbb{B}</math> are used to remind us that we are considering particular ways of situating <math>L, M, N\!</math> within the product space <math>X:Y:Z.\!</math>

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