Changes

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>

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{| cellpadding="4"

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| 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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| &nbsp;

| &nbsp;

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| 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:

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| &Upsilon;_''p'' ''q'' = &Upsilon;(''p'', ''q'') = &Upsilon;(''p'', ''q'', &Pi;).

| <math>\Upsilon_p q = \Upsilon (p, q) = \Upsilon (p, q, \textstyle\prod).</math>

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| &Upsilon;_''p'' = &Upsilon;(''p'', __, &Pi;) : ('''B'''^''k'' &rarr; '''B''') &rarr; '''B'''.

| <math>\Upsilon_p = \Upsilon (p, \_\_, \textstyle\prod) : (\mathbb{B}^k \to \mathbb{B}) \to \mathbb{B}.</math>

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|}<br>

This means that &Upsilon;_''p'' ''q'' = 1 if and only if ''q'' holds for all models of ''p''.  In propositional terms, this is tantamount to the assertion that ''p'' &rArr; ''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: