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By way of reviewing notation and preparing to extend it to higher order universes of discourse, let us first consider the universe of discourse <math>X^\circ = [X] = [x_1, x_2] = [x, y],</math> based on two logical features or boolean variables <math>x\!</math> and <math>y.\!</math>
By way of reviewing notation and preparing to extend it to higher order universes of discourse, let us first consider the universe of discourse <math>X^\circ = [X] = [x_1, x_2] = [x, y],</math> based on two logical features or boolean variables <math>x\!</math> and <math>y.\!</math>
−{|
+{| cellpadding="4"
−| align=right width=36 | 1.
+| align="right" width="36" | 1.
| The points of <math>X^\circ</math> are collected in the space:
| The points of <math>X^\circ</math> are collected in the space:
|-
|-
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|
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|-
−| align=right width=36 | 2.
+| align="right" width="36" | 2.
| The propositions of <math>X^\circ</math> make up the space:
| The propositions of <math>X^\circ</math> make up the space:
|-
|-
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The intention of this operator is that we evaluate the proposition <math>q\!</math> on each model of the proposition <math>p\!</math> and combine the results according to the method indicated by the connective parameter <math>r.\!</math> In principle, the index <math>r\!</math> might specify any connective on as many as <math>2^k\!</math> arguments, but usually we have in mind a much simpler form of combination, most often either collective products or collective sums. By convention, each of the accessory indices <math>p, r\!</math> is assigned a default value that is understood to be in force when the corresponding argument place is left blank, specifically, the constant proposition <math>1 : \mathbb{B}^k \to \mathbb{B}</math> for the lower index <math>p,\!</math> and the continued conjunction or continued product operation <math>\textstyle\prod</math> for the upper index <math>r.\!</math> Taking the upper default value gives license to the following readings:
The intention of this operator is that we evaluate the proposition <math>q\!</math> on each model of the proposition <math>p\!</math> and combine the results according to the method indicated by the connective parameter <math>r.\!</math> In principle, the index <math>r\!</math> might specify any connective on as many as <math>2^k\!</math> arguments, but usually we have in mind a much simpler form of combination, most often either collective products or collective sums. By convention, each of the accessory indices <math>p, r\!</math> is assigned a default value that is understood to be in force when the corresponding argument place is left blank, specifically, the constant proposition <math>1 : \mathbb{B}^k \to \mathbb{B}</math> for the lower index <math>p,\!</math> and the continued conjunction or continued product operation <math>\textstyle\prod</math> for the upper index <math>r.\!</math> Taking the upper default value gives license to the following readings:
−{|
+{| cellpadding="4"
−| align=right width=36 | 1.
+| align="right" width="36" | 1.
−| Υ<sub>''p''</sub> ''q'' = Υ(''p'', ''q'') = Υ(''p'', ''q'', Π).
+| <math>\Upsilon_p q = \Upsilon (p, q) = \Upsilon (p, q, \textstyle\prod).</math>
|-
|-
−| align=right width=36 | 2.
+| align="right" width="36" | 2.
−| Υ<sub>''p''</sub> = Υ(''p'', __, Π) : ('''B'''<sup>''k''</sup> → '''B''') → '''B'''.
+| <math>\Upsilon_p = \Upsilon (p, \_\_, \textstyle\prod) : (\mathbb{B}^k \to \mathbb{B}) \to \mathbb{B}.</math>
|}<br>
|}<br>
−This means that Υ<sub>''p''</sub> ''q'' = 1 if and only if ''q'' holds for all models of ''p''. In propositional terms, this is tantamount to the assertion that ''p'' ⇒ ''q'', or that _(p (q))_ = 1.
+This means that <math>\Upsilon_p q = 1\!</math> if and only if <math>q\!</math> holds for all models of <math>p.\!</math> In propositional terms, this is tantamount to the assertion that <math>p \Rightarrow q,</math> or that <math>(\!| p (\!| q |\!) |\!) = 1.</math>
Throwing in the lower default value permits the following abbreviations:
Throwing in the lower default value permits the following abbreviations:
