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Revision as of 16:56, 24 January 2009
, 16:56, 24 January 2009→1.3.11. The Cactus Patch: + next segment
Let me now address one last question that may have occurred to some. What has happened, in this suggested scheme of functional reasoning, to the distinction that is quite pointedly made by careful logicians between (1) the connectives called ''conditionals'' and symbolized by the signs <math>(\rightarrow)</math> and <math>(\leftarrow),</math> and (2) the assertions called ''implications'' and symbolized by the signs <math>(\Rightarrow)</math> and <math>(\Leftarrow)</math>, and, in a related question: What has happened to the distinction that is equally insistently made between (3) the connective called the ''biconditional'' and signified by the sign <math>(\leftrightarrow)</math> and (4) the assertion that is called an ''equivalence'' and signified by the sign <math>(\Leftrightarrow)</math>? My answer is this: For my part, I am deliberately avoiding making these distinctions at the level of syntax, preferring to treat them instead as distinctions in the use of boolean functions, turning on whether the function is mentioned directly and used to compute values on arguments, or whether its inverse is being invoked to indicate the fibers of truth or untruth under the propositional function in question.
Let me now address one last question that may have occurred to some. What has happened, in this suggested scheme of functional reasoning, to the distinction that is quite pointedly made by careful logicians between (1) the connectives called ''conditionals'' and symbolized by the signs <math>(\rightarrow)</math> and <math>(\leftarrow),</math> and (2) the assertions called ''implications'' and symbolized by the signs <math>(\Rightarrow)</math> and <math>(\Leftarrow)</math>, and, in a related question: What has happened to the distinction that is equally insistently made between (3) the connective called the ''biconditional'' and signified by the sign <math>(\leftrightarrow)</math> and (4) the assertion that is called an ''equivalence'' and signified by the sign <math>(\Leftrightarrow)</math>? My answer is this: For my part, I am deliberately avoiding making these distinctions at the level of syntax, preferring to treat them instead as distinctions in the use of boolean functions, turning on whether the function is mentioned directly and used to compute values on arguments, or whether its inverse is being invoked to indicate the fibers of truth or untruth under the propositional function in question.
=====1.3.11.6. Stretching Exercises=====
The arrays of boolean connections described above, namely, the boolean functions <math>F^{(k)} : \underline\mathbb{B}^k \to \underline\mathbb{B},</math> for <math>k\!</math> in <math>\{ 0, 1, 2 \},\!</math> supply enough material to demonstrate the use of the stretch operation in a variety of concrete cases.
For example, suppose that <math>F\!</math> is a connection of the form <math>F : \underline\mathbb{B}^2 \to \underline\mathbb{B},</math> that is, any one of the sixteen possibilities in Table 18, while <math>p\!</math> and <math>q\!</math> are propositions of the form <math>p, q : X \to \underline\mathbb{B},</math> that is, propositions about things in the universe <math>X,\!</math> or else the indicators of sets contained in <math>X.\!</math>
Then one has the imagination <math>\underline{f} = (f_1, f_2) = (p, q) : (X \to \underline\mathbb{B})^2,</math> and the stretch of the connection <math>F\!</math> to <math>\underline{f}</math> on <math>X\!</math> amounts to a proposition <math>F^\$ (p, q) : X \to \underline\mathbb{B}</math> that may be read as the ''stretch of <math>F\!</math> to <math>p\!</math> and <math>q.\!</math>'' If one is concerned with many different propositions about things in <math>X,\!</math> or if one is abstractly indifferent to the particular choices for <math>p\!</math> and <math>q,\!</math> then one may detach the operator <math>F^\$ : (X \to \underline\mathbb{B}))^2 \to (X \to \underline\mathbb{B})),</math> called the ''stretch of <math>F\!</math> over <math>X,\!</math>'' and consider it in isolation from any concrete application.
When the cactus notation is used to represent boolean functions, a single <math>\$</math> sign at the end of the expression is enough to remind the reader that the connections are meant to be stretched to several propositions on a universe <math>X.\!</math>
For example, take the connection <math>F : \underline\mathbb{B}^2 \to \underline\mathbb{B}</math> such that:
: <math>F(x, y) ~=~ F_{6}^{(2)} (x, y) ~=~ \underline{(}~x~,~y~\underline{)}</math>
The connection in question is a boolean function on the variables <math>x, y\!</math> that returns a value of <math>\underline{1}</math> just when just one of the pair <math>x, y\!</math> is not equal to <math>\underline{1},</math> or what amounts to the same thing, just when just one of the pair <math>x, y\!</math> is equal to <math>\underline{1}.</math> There is clearly an isomorphism between this connection, viewed as an operation on the boolean domain <math>\underline\mathbb{B} = \{ \underline{0}, \underline{1} \},</math> and the dyadic operation on binary values <math>x, y \in \mathbb{B} = \operatorname{GF}(2)</math> that is otherwise known as <math>x + y\!.</math>
The same connection <math>F : \underline\mathbb{B}^2 \to \underline\mathbb{B}</math> can also be read as a proposition about things in the universe <math>X = \underline\mathbb{B}^2.</math> If <math>s\!</math> is a sentence that denotes the proposition <math>F,\!</math> then the corresponding assertion says exactly what one states in uttering the sentence <math>^{\backprime\backprime} \, x ~\operatorname{is~not~equal~to}~ y \, ^{\prime\prime}.</math> In such a case, one has <math>\downharpoonleft s \downharpoonright \, = F,</math> and all of the following expressions are ordinarily taken as equivalent descriptions of the same set:
{| align="center" cellpadding="4" style="text-align:left" width="90%"
|
|-
| <math>[| \downharpoonleft s \downharpoonright |]</math>
| <math>=\!</math>
| <math>[| F |]\!</math>
|-
|
| <math>=\!</math>
| <math>F^{-1} (\underline{1})</math>
|-
|
| <math>=\!</math>
| <math>\{~ (x, y) \in \underline\mathbb{B}^2 ~:~ s ~\}</math>
|-
|
| <math>=\!</math>
| <math>\{~ (x, y) \in \underline\mathbb{B}^2 ~:~ F(x, y) = \underline{1} ~\}</math>
|-
|
| <math>=\!</math>
| <math>\{~ (x, y) \in \underline\mathbb{B}^2 ~:~ F(x, y) ~\}</math>
|-
|
| <math>=\!</math>
| <math>\{~ (x, y) \in \underline\mathbb{B}^2 ~:~ \underline{(}~x~,~y~\underline{)} = \underline{1} ~\}</math>
|-
|
| <math>=\!</math>
| <math>\{~ (x, y) \in \underline\mathbb{B}^2 ~:~ \underline{(}~x~,~y~\underline{)} ~\}</math>
|-
|
| <math>=\!</math>
| <math>\{~ (x, y) \in \underline\mathbb{B}^2 ~:~ x ~\operatorname{exclusive~or}~ y ~\}</math>
|-
|
| <math>=\!</math>
| <math>\{~ (x, y) \in \underline\mathbb{B}^2 ~:~ \operatorname{just~one~true~of}~ x, y ~\}</math>
|-
|
| <math>=\!</math>
| <math>\{~ (x, y) \in \underline\mathbb{B}^2 ~:~ x ~\operatorname{not~equal~to}~ y ~\}</math>
|-
|
| <math>=\!</math>
| <math>\{~ (x, y) \in \underline\mathbb{B}^2 ~:~ x \nLeftrightarrow y ~\}</math>
|-
|
| <math>=\!</math>
| <math>\{~ (x, y) \in \underline\mathbb{B}^2 ~:~ x \neq y ~\}</math>
|-
|
| <math>=\!</math>
| <math>\{~ (x, y) \in \underline\mathbb{B}^2 ~:~ x + y ~\}.</math>
|-
|
|}
Notice the distinction, that I continue to maintain at this point, between the logical values <math>\{ \operatorname{falsehood}, \operatorname{truth} \}</math> and the algebraic values <math>\{ 0, 1 \}.\!</math> This makes it legitimate to write a sentence directly into the righthand side of a set-builder expression, for instance, weaving the sentence <math>s\!</math> or the sentence <math>^{\backprime\backprime} \, x ~\operatorname{is~not~equal~to}~ y \, ^{\prime\prime}</math> into the context <math>^{\backprime\backprime} \, \{ (x, y) \in \underline{B}^2 : \ldots \} \, ^{\prime\prime},</math> thereby obtaining the corresponding expressions listed above. It also allows us to assert the proposition <math>F(x, y)\!</math> in a more direct way, without detouring through the equation <math>F(x, y) = \underline{1},</math> since it already has a value in <math>\{ \operatorname{falsehood}, \operatorname{true} \},</math> and thus can be taken as tantamount to an actual sentence.
If the appropriate safeguards can be kept in mind, avoiding all danger of confusing propositions with sentences and sentences with assertions, then the marks of these distinctions need not be forced to clutter the account of the more substantive indications, that is, the ones that really matter. If this level of understanding can be achieved, then it may be possible to relax these restrictions, along with the absolute dichotomy between algebraic and logical values, which tends to inhibit the flexibility of interpretation.
This covers the properties of the connection <math>F(x, y) = \underline{(}~x~,~y~\underline{)},</math> treated as a proposition about things in the universe <math>X = \underline\mathbb{B}^2.</math> Staying with this same connection, it is time to demonstrate how it can be "stretched" to form an operator on arbitrary propositions.
To continue the exercise, let <math>p\!</math> and <math>q\!</math> be arbitrary propositions about things in the universe <math>X,\!</math> that is, maps of the form <math>p, q : X \to \underline\mathbb{B},</math> and suppose that <math>p, q\!</math> are indicator functions of the sets <math>P, Q \subseteq X,</math> respectively. In other words, we have the following data:
{| align="center" cellpadding="8" width="90%"
|
<math>\begin{matrix}
p
& = &
\upharpoonleft P \upharpoonright
& : &
X \to \underline\mathbb{B}
\\
\\
q
& = &
\upharpoonleft Q \upharpoonright
& : &
X \to \underline\mathbb{B}
\\
\\
(p, q)
& = &
(\upharpoonleft P \upharpoonright, \upharpoonleft Q \upharpoonright)
& : &
(X \to \underline\mathbb{B})^2
\\
\end{matrix}</math>
|}
Then one has an operator <math>F^\$,</math> the stretch of the connection <math>F\!</math> over <math>X,\!</math> and a proposition <math>F^\$ (p, q),</math> the stretch of <math>F\!</math> to <math>(p, q)\!</math> on <math>X,\!</math> with the following properties:
{| align="center" cellpadding="8" width="90%"
|
<math>\begin{array}{ccccl}
F^\$
& = &
\underline{(} \ldots, \ldots \underline{)}^\$
& : &
(X \to \underline\mathbb{B})^2 \to (X \to \underline\mathbb{B})
\\
\\
F^\$ (p, q)
& = &
\underline{(}~p~,~q~\underline{)}^\$
& : &
X \to \underline\mathbb{B}
\\
\end{array}</math>
|}
As a result, the application of the proposition <math>F^\$ (p, q)</math> to each <math>x \in X</math> returns a logical value in <math>\underline\mathbb{B},</math> all in accord with the following equations:
{| align="center" cellpadding="8" width="90%"
|
<math>\begin{matrix}
F^\$ (p, q)(x) & = & \underline{(}~p~,~q~\underline{)}^\$ (x) & \in & \underline\mathbb{B}
\\
\\
\Updownarrow & & \Updownarrow
\\
\\
F(p(x), q(x)) & = & \underline{(}~p(x)~,~q(x)~\underline{)} & \in & \underline\mathbb{B}
\\
\end{matrix}</math>
|}
For each choice of propositions <math>p\!</math> and <math>q\!</math> about things in <math>X,\!</math> the stretch of <math>F\!</math> to <math>p\!</math> and <math>q\!</math> on <math>X\!</math> is just another proposition about things in <math>X,\!</math> a simple proposition in its own right, no matter how complex its current expression or its present construction as <math>F^\$ (p, q) = \underline{(}~p~,~q~\underline{)}^\$</math> makes it appear in relation to <math>p\!</math> and <math>q.\!</math> Like any other proposition about things in <math>X,\!</math> it indicates a subset of <math>X,\!</math> namely, the fiber that is variously described in the following ways:
{| align="center" cellpadding="4" style="text-align:left" width="90%"
|
|-
| <math>[| F^\$ (p, q) |]</math>
| <math>=\!</math>
| <math>[| \underline{(}~p~,~q~\underline{)}^\$ |]</math>
|-
|
| <math>=\!</math>
| <math>(F^\$ (p, q))^{-1} (\underline{1})</math>
|-
|
| <math>=\!</math>
| <math>\{~ x \in X ~:~ F^\$ (p, q)(x) ~\}</math>
|-
|
| <math>=\!</math>
| <math>\{~ x \in X ~:~ \underline{(}~p~,~q~\underline{)}^\$ (x) ~\}</math>
|-
|
| <math>=\!</math>
| <math>\{~ x \in X ~:~ \underline{(}~p(x)~,~q(x)~\underline{)} ~\}</math>
|-
|
| <math>=\!</math>
| <math>\{~ x \in X ~:~ p(x) + q(x) ~\}</math>
|-
|
| <math>=\!</math>
| <math>\{~ x \in X ~:~ p(x) \neq q(x) ~\}</math>
|-
|
| <math>=\!</math>
| <math>\{~ x \in X ~:~ \upharpoonleft P \upharpoonright (x) ~\neq~ \upharpoonleft Q \upharpoonright (x) ~\}</math>
|-
|
| <math>=\!</math>
| <math>\{~ x \in X ~:~ x \in P ~\nLeftrightarrow~ x \in Q ~\}</math>
|-
|
| <math>=\!</math>
| <math>\{~ x \in X ~:~ x \in P\!-\!Q ~\operatorname{or}~ x \in Q\!-\!P ~\}</math>
|-
|
| <math>=\!</math>
| <math>\{~ x \in X ~:~ x \in P\!-\!Q ~\cup~ Q\!-\!P ~\}</math>
|-
|
| <math>=\!</math>
| <math>\{~ x \in X ~:~ x \in P + Q ~\}</math>
|-
|
| <math>=\!</math>
| <math>P + Q ~\subseteq~ X</math>
|-
|
| <math>=\!</math>
| <math>[|p|] + [|q|] ~\subseteq~ X</math>
|-
|
|}
==References==
==References==