Changes

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

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

|-

|-

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

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

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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>

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>\underline{(p (q))} = \underline{1}.</math>

Throwing in the lower default value permits the following abbreviations:

Throwing in the lower default value permits the following abbreviations:

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| <math>\Upsilon  q  = \Upsilon (q) = \Upsilon_1 (q) = \Upsilon (1, q, \textstyle\prod).</math>

| <math>\Upsilon  q  = \Upsilon (q) = \Upsilon_1 q = \Upsilon (1, q, \textstyle\prod).</math>

|-

|-

| <math>\Upsilon = \Upsilon (1, \underline{~~}, \textstyle\prod)) : (\mathbb{B}^k\ \to \mathbb{B}) \to \mathbb{B}.</math>

| <math>\Upsilon = \Upsilon (1, \underline{~~}, \textstyle\prod)) : (\mathbb{B}^k\ \to \mathbb{B}) \to \mathbb{B}.</math>

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This means that <math>\Upsilon q = 1\!</math> if and only if <math>q\!</math> holds for the whole universe of discourse in question, that is, if and only <math>q\!</math> is the constantly true proposition <math>1 : \mathbb{B}^k \to \mathbb{B}.</math>  The ambiguities of this usage are not a problem so long as we distinguish the context of definition from the context of application and restrict all shorthand notations to the latter.

This means that <math>\Upsilon q = 1\!</math> if and only if <math>q\!</math> holds for the whole universe of discourse in question, that is, if and only <math>q\!</math> is the constantly true proposition <math>1 : \mathbb{B}^k \to \mathbb{B}.</math>  The ambiguities of this usage are not a problem so long as we distinguish the context of definition from the context of application and restrict all shorthand notations to the latter.