Changes

=====1.3.10.7.  Stretching Operations=====

=====1.3.10.7.  Stretching Operations=====

The preceding discussion of stretch operations is slightly more general than is called for in the present context, and so it is probably a good idea to draw out the particular implications that are needed right away.

The preceding discussion of stretch operations is slightly more general than is called for in the present context, and so it is probably a good idea to draw out the particular implications that are needed right away.

If F : Bk �> B is a boolean function on k variables, then it is possible to define a mapping F$ : (U �> B)k �> (U �> B), in effect, an operation that takes k propositions into a single proposition, where F$ satisfies the following conditions:

If F : Bk -> B is a boolean function on k variables, then it is possible to define a mapping F$ : (U -> B)k -> (U -> B), in effect, an operation that takes k propositions into a single proposition, where F$ satisfies the following conditions:

F$(f1, ..., fk) : U �> B

F$(f1, ..., fk) : U -> B

:

:

F$(f1, ..., fk)(u) = F(f(u))

F$(f1, ..., fk)(u) = F(f(u))

“0” Sign Higher Order Sign

“0” Sign Higher Order Sign

Set Proposition Sentence

Set Proposition Sentence

f�1(v) f "f"

f-1(v) f "f"

X 1 "1"

X 1 "1"

~X 0 "0"

~X 0 "0"

Strictly speaking, a proposition is too abstract to be a sign, and so the contents of Table 10 have to be taken with the indicated grains of salt.  Propositions, as indicator functions, are abstract mathematical objects, not any kinds of syntactic elements, and so propositions cannot literally constitute the orders of concrete signs that remain of ultimate interest in the pragmatic theory of signs, or in any theory of effective meaning.  Therefore, it needs to be understood that a proposition f can be said to "indicate" a set X only insofar as the values of 1 and 0 that it assigns to the elements of the universe U are positive and negative indications, respectively, of the elements in X, and thus indications of the set X and of its complement ~X = U � X, respectively.  It is actually these values, when rendered by a concrete implementation of the indicator function f, that are the actual signs of the objects that are inside the set X and the objects that are outside the set X, respectively.

Strictly speaking, a proposition is too abstract to be a sign, and so the contents of Table 10 have to be taken with the indicated grains of salt.  Propositions, as indicator functions, are abstract mathematical objects, not any kinds of syntactic elements, and so propositions cannot literally constitute the orders of concrete signs that remain of ultimate interest in the pragmatic theory of signs, or in any theory of effective meaning.  Therefore, it needs to be understood that a proposition f can be said to "indicate" a set X only insofar as the values of 1 and 0 that it assigns to the elements of the universe U are positive and negative indications, respectively, of the elements in X, and thus indications of the set X and of its complement ~X = U - X, respectively.  It is actually these values, when rendered by a concrete implementation of the indicator function f, that are the actual signs of the objects that are inside the set X and the objects that are outside the set X, respectively.

In order to deal with the HO sign relations that are involved in this situation, I introduce a couple of new notations:

In order to deal with the HO sign relations that are involved in this situation, I introduce a couple of new notations:

In particular, one can observe the following relations and formulas, all of a purely notational character:

In particular, one can observe the following relations and formulas, all of a purely notational character:

1. If the sentence S denotes the proposition P : U �> B, then [S] = P.

1. If the sentence S denotes the proposition P : U -> B, then [S] = P.

2. If the sentence S denotes the proposition P : U �> B

2. If the sentence S denotes the proposition P : U -> B

such that |P| = P�1(1) = X c U, then [S] = P = fX = {X}.

such that |P| = P-1(1) = X c U, then [S] = P = fX = {X}.

3. X = {u C U : u C X}

3. X = {u C U : u C X}

= |{X}| = {X}�1(1)

= |{X}| = {X}-1(1)

= |fX| = fX�1(1).

= |fX| = fX-1(1).

4. {X} = { {u C U : u C X} }

4. {X} = { {u C U : u C X} }

= fX.

= fX.

Now if a sentence S really denotes a proposition P, and if the notation "[S]" is merely meant to supply another name for the proposition that S already denotes, then why is there any need for the additional notation?  It is because the interpretive mind habitually races from the sentence S, through the proposition P that it denotes, and on to the set X = P�1(1) that the proposition P indicates, often jumping to the conclusion that the set X is the only thing that the sentence S is intended to denote.  This HO sign situation and the mind's inclination when placed within its setting calls for a linguistic mechanism or a notational device that is capable of analyzing the compound action and controlling its articulate performance, and this requires a way to interrupt the flow of assertion that typically takes place from S to P to X.

Now if a sentence S really denotes a proposition P, and if the notation "[S]" is merely meant to supply another name for the proposition that S already denotes, then why is there any need for the additional notation?  It is because the interpretive mind habitually races from the sentence S, through the proposition P that it denotes, and on to the set X = P-1(1) that the proposition P indicates, often jumping to the conclusion that the set X is the only thing that the sentence S is intended to denote.  This HO sign situation and the mind's inclination when placed within its setting calls for a linguistic mechanism or a notational device that is capable of analyzing the compound action and controlling its articulate performance, and this requires a way to interrupt the flow of assertion that typically takes place from S to P to X.

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