Changes

Given an ordered pair of propositions <math>e, f : \langle u, v \rangle \to \mathbb{B}</math> as arguments, the relative operator reports the value 1 if the first implies the second, otherwise 0.

Given an ordered pair of propositions <math>e, f : \langle u, v \rangle \to \mathbb{B}</math> as arguments, the relative operator reports the value 1 if the first implies the second, otherwise 0.

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{| align="center" cellpadding="8" style="text-align:center"

| <math>\Upsilon (e, f) = 1\!</math>

| <math>\Upsilon (e, f) = 1\!</math>

| <math>\operatorname{if~and~only~if}</math>

| <math>\operatorname{if~and~only~if}</math>

Expressing it another way, we may also write:

Expressing it another way, we may also write:

{| align="center" cellpadding="8"

{| align="center" cellpadding="8" style="text-align:center"

| <math>\Upsilon (e, f) = 1\!</math>

| <math>\Upsilon (e, f) = 1\!</math>

| <math>\Leftrightarrow</math>

| <math>\Leftrightarrow</math>

In writing this, however, it is important to notice that the 1's appearing on the left and right have different meanings.  Filling in the details, we have:

In writing this, however, it is important to notice that the 1's appearing on the left and right have different meanings.  Filling in the details, we have:

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{| align="center" cellpadding="8" style="text-align:center"

| <math>\Upsilon (e, f) = 1 \in \mathbb{B}</math>

| <math>\Upsilon (e, f) = 1 \in \mathbb{B}</math>

| <math>\Leftrightarrow</math>

| <math>\Leftrightarrow</math>

Writing types as subscripts and using the fact that <math>X = \langle u, v \rangle,</math> it is possible to express this a little more succinctly as follows:

Writing types as subscripts and using the fact that <math>X = \langle u, v \rangle,</math> it is possible to express this a little more succinctly as follows:

{| align="center" cellpadding="8"

{| align="center" cellpadding="8" style="text-align:center"

| <math>\Upsilon (e, f) = 1_\mathbb{B}</math>

| <math>\Upsilon (e, f) = 1_\mathbb{B}</math>

| <math>\Leftrightarrow</math>

| <math>\Leftrightarrow</math>

As a special application of this operator, we next define the absolute umpire operator, also called the ''umpire measure''.  This is a higher order proposition <math>\Upsilon_1 : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}</math> which is given by the relation <math>\Upsilon_1 (f) = \Upsilon (1, f).\!</math>  Here, the subscript 1 on the left and the argument 1 on the right both refer to the constant proposition <math>1 : \mathbb{B}^2 \to \mathbb{B}.</math>  In most contexts where <math>\Upsilon_1\!</math> is actually applied the subscript 1 is safely omitted, since the number of arguments indicates which type of operator is intended.  Thus, we have the following identities and equivalents:

As a special application of this operator, we next define the absolute umpire operator, also called the ''umpire measure''.  This is a higher order proposition <math>\Upsilon_1 : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}</math> which is given by the relation <math>\Upsilon_1 (f) = \Upsilon (1, f).\!</math>  Here, the subscript 1 on the left and the argument 1 on the right both refer to the constant proposition <math>1 : \mathbb{B}^2 \to \mathbb{B}.</math>  In most contexts where <math>\Upsilon_1\!</math> is actually applied the subscript 1 is safely omitted, since the number of arguments indicates which type of operator is intended.  Thus, we have the following identities and equivalents:

{| align="center" cellpadding="8"

{| align="center" cellpadding="10" style="text-align:center"

| <math>\Upsilon f = \Upsilon_1 (f) = 1 \in \mathbb{B} \quad \Leftrightarrow \quad (1 (f)) = 1 \quad \Leftrightarrow \quad f = 1 : \mathbb{B}^2 \to \mathbb{B}.</math>

| <math>\Upsilon f = \Upsilon_1 (f) = 1 \in \mathbb{B}</math>

| <math>\Leftrightarrow</math>

| <math>\underline{(1~(f))} = \underline{1}</math>

| <math>\Leftrightarrow</math>

| <math>f = 1 : \mathbb{B}^2 \to \mathbb{B}.</math>

|}

|}

In order to get a handle on the space of higher order propositions and eventually to carry out a functional approach to quantification theory, it serves to construct some specialized tools.  Specifically, I define a higher order operator <math>\Upsilon,\!</math> called the ''umpire operator'', which takes up to three propositions as arguments and returns a single truth value as the result.  Formally, this so-called ''multi-grade'' property of <math>\Upsilon\!</math> can be expressed as a union of function types, in the following manner:

In order to get a handle on the space of higher order propositions and eventually to carry out a functional approach to quantification theory, it serves to construct some specialized tools.  Specifically, I define a higher order operator <math>\Upsilon,\!</math> called the ''umpire operator'', which takes up to three propositions as arguments and returns a single truth value as the result.  Formally, this so-called ''multi-grade'' property of <math>\Upsilon\!</math> can be expressed as a union of function types, in the following manner:

: <math>\Upsilon : \bigcup_{\ell = 1, 2, 3} ((\mathbb{B}^k \to \mathbb{B})^\ell \to \mathbb{B}).</math>

{| align="center" cellpadding="8" style="text-align:center"

| <math>\Upsilon : \bigcup_{\ell = 1, 2, 3} ((\mathbb{B}^k \to \mathbb{B})^\ell \to \mathbb{B}).</math>

In contexts of application the intended sense can be discerned by the number of arguments that actually appear in the argument list.  Often, the first and last arguments appear as indices, the one in the middle being treated as the main argument while the other two arguments serve to modify the sense of the operation in question.  Thus, we have the following forms:

In contexts of application the intended sense can be discerned by the number of arguments that actually appear in the argument list.  Often, the first and last arguments appear as indices, the one in the middle being treated as the main argument while the other two arguments serve to modify the sense of the operation in question.  Thus, we have the following forms:

: <math>\Upsilon_p^r q  =  \Upsilon (p, q, r)\!</math>

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| <math>\Upsilon_p^r q  =  \Upsilon (p, q, r)\!</math>

: <math>\Upsilon_p^r : (\mathbb{B}^k \to \mathbb{B}) \to \mathbb{B}</math>

| <math>\Upsilon_p^r : (\mathbb{B}^k \to \mathbb{B}) \to \mathbb{B}</math>

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="center" cellpadding="8" style="text-align:center"

| align="right" width="36" | 1.

| <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>

|}<br>

|}

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:

{| cellpadding="4"

{| align="center" cellpadding="8" style="text-align:center"

| align="right" width="36" | 3.

| <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>

|}<br>

|}

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.