Directory talk:Jon Awbrey/Papers/Syntactic Transformations - Revision history

Jon Awbrey: /* 1.3.12.1. Syntactic Transformation Rules */ del merged text

2010-09-12T14:58:20Z

1.3.12.1. Syntactic Transformation Rules: del merged text

Jon Awbrey: /* 1.3.12.1. Syntactic Transformation Rules */ del merged text

2010-09-11T15:25:41Z

1.3.12.1. Syntactic Transformation Rules: del merged text

Jon Awbrey: /* 1.3.12.1. Syntactic Transformation Rules */ del merged text

2010-09-11T02:42:11Z

1.3.12.1. Syntactic Transformation Rules: del merged text

Jon Awbrey: /* Other Rules */ del merged text

2010-09-11T02:32:04Z

Other Rules: del merged text

Jon Awbrey: /* Definitions */

2010-09-11T02:21:58Z

Definitions

Jon Awbrey: /* Definitions */ del merged text

2010-09-11T02:21:33Z

Definitions: del merged text

Jon Awbrey: /* 1.3.12.1. Syntactic Transformation Rules */ sort out different types of rules

2010-09-10T21:58:07Z

1.3.12.1. Syntactic Transformation Rules: sort out different types of rules

Jon Awbrey: /* 1.3.12.1. Syntactic Transformation Rules */ del merged text

2010-09-10T20:40:13Z

1.3.12.1. Syntactic Transformation Rules: del merged text

Jon Awbrey: /* 1.3.12.2. Derived Equivalence Relations */ ✔ completed section

2010-09-10T19:02:19Z

1.3.12.2. Derived Equivalence Relations: ✔ completed section

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Jon Awbrey: /* 1.3.12.2. Derived Equivalence Relations */ del merged text

2010-09-10T18:14:14Z

1.3.12.2. Derived Equivalence Relations: del merged text

@@ Line 6: / Line 6: @@
 =====1.3.12.1.  Syntactic Transformation Rules=====
 <pre>
@@ Line 145: / Line 143: @@
 E1d.	(( f , g ))$(u).	:$1b
 </pre>

← Older revision		Revision as of 15:25, 11 September 2010
Line 145:		Line 145:

	E1d. (( f , g ))$(u). :$1b		E1d. (( f , g ))$(u). :$1b
−	~~</pre>~~
−
−	~~======Definitions======~~
−
−	~~<pre>~~
−	~~Definition 5~~
−
−	~~If X c U,~~
−
−	~~then the following are identical propositions:~~
−
−	~~D5a. {X}.~~
−
−	~~D5b. f : U -> B~~
−
−	~~: f(u) = [u C X], for all u C U.~~
	</pre>		</pre>

@@ Line 162: / Line 162: @@
 :	f(u)	=	[u C X], for all u C U.
 </pre>
 ======Facts======

← Older revision		Revision as of 02:32, 11 September 2010
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	======Other Rules======		======Other Rules======
−
−	~~<pre>~~
−	~~Rule 5~~
−
−	~~If X, Y c U,~~
−
−	~~then the following are equivalent:~~
−
−	~~R5a. X = Y. :D2a~~
−
−	::
−
−	~~R5b. u C X <=> u C Y, for all u C U. :D2b~~
−
−	~~:D7a~~
−
−	::
−
−	~~R5c. [u C X] = [u C Y], for all u C U. :D7b~~
−
−	~~:???~~
−
−	::
−
−	~~R5d. {< u, v> C UxB : v = [u C X]}~~
−
−	=
−
−	~~{< u, v> C UxB : v = [u C Y]}. :???~~
−
−	~~:D5b~~
−
−	::
−
−	~~R5e. {X} = {Y}. :D5a~~
−	~~</pre>~~
−
−	~~<pre>~~
−	~~Rule 6~~
−
−	~~If f, g : U -> V,~~
−
−	~~then the following are equivalent:~~
−
−	~~R6a. f = g. :D3a~~
−
−	::
−
−	~~R6b. f(u) = g(u), for all u C U. :D3b~~
−
−	~~:D6a~~
−
−	::
−
−	~~R6c. ConjUu (f(u) = g(u)). :D6e~~
−	~~</pre>~~
−
−	~~<pre>~~
−	~~Rule 7~~
−
−	~~If P, Q : U -> B,~~
−
−	~~then the following are equivalent:~~
−
−	~~R7a. P = Q. :R6a~~
−
−	::
−
−	~~R7b. P(u) = Q(u), for all u C U. :R6b~~
−
−	::
−
−	~~R7c. ConjUu (P(u) = Q(u)). :R6c~~
−
−	~~:P1a~~
−
−	::
−
−	~~R7d. ConjUu (P(u) <=> Q(u)). :P1b~~
−
−	::
−
−	~~R7e. ConjUu (( P(u) , Q(u) )). :P1c~~
−
−	~~:$1a~~
−
−	::
−
−	~~R7f. ConjUu (( P , Q ))$(u). :$1b~~
−	~~</pre>~~
−
−	~~<pre>~~
−	~~Rule 8~~
−
−	~~If S, T are sentences~~
−
−	~~about things in the universe U,~~
−
−	~~then the following are equivalent:~~
−
−	~~R8a. S <=> T. :D7a~~
−
−	::
−
−	~~R8b. [S] = [T]. :D7b~~
−
−	~~:R7a~~
−
−	::
−
−	~~R8c. [S](u) = [T](u), for all u C U. :R7b~~
−
−	::
−
−	~~R8d. ConjUu ( [S](u) = [T](u) ). :R7c~~
−
−	::
−
−	~~R8e. ConjUu ( [S](u) <=> [T](u) ). :R7d~~
−
−	::
−
−	~~R8f. ConjUu (( [S](u) , [T](u) )). :R7e~~
−
−	::
−
−	~~R8g. ConjUu (( [S] , [T] ))$(u). :R7f~~
−	~~</pre>~~
−
−	For instance, the observation that expresses the equality of sets in terms of their indicator functions can be formalized according to the pattern in Rule 9, namely, at lines (a, b, c), and these components of Rule 9 can be cited in future uses as "R9a", "R9b", "R9c", respectively. Using Rule 7, annotated as "R7", to adduce a few properties of indicator functions to the account, it is possible to extend Rule 9 by another few steps, referenced as "R9d", "R9e", "R9f", "R9g".
−
−	~~<pre>~~
−	~~Rule 9~~
−
−	~~If X, Y c U,~~
−
−	~~then the following are equivalent:~~
−
−	~~R9a. X = Y. :R5a~~
−
−	::
−
−	~~R9b. {X} = {Y}. :R5e~~
−
−	~~:R7a~~
−
−	::
−
−	~~R9c. {X}(u) = {Y}(u), for all u C U. :R7b~~
−
−	::
−
−	~~R9d. ConjUu ( {X}(u) = {Y}(u) ). :R7c~~
−
−	::
−
−	~~R9e. ConjUu ( {X}(u) <=> {Y}(u) ). :R7d~~
−
−	::
−
−	~~R9f. ConjUu (( {X}(u) , {Y}(u) )). :R7e~~
−
−	::
−
−	~~R9g. ConjUu (( {X} , {Y} ))$(u). :R7f~~
−	~~</pre>~~

	<pre>		<pre>

@@ Line 162: / Line 162: @@
 :	f(u)	=	[u C X], for all u C U.
 </pre>
 ======Other Rules======

@@ Line 148: / Line 148: @@
 ======Definitions======
 <pre>
@@ Line 199: / Line 163: @@
 </pre>
 <pre>
 ======Other Rules======

@@ Line 6: / Line 6: @@
 =====1.3.12.1.  Syntactic Transformation Rules=====
 <pre>
@@ Line 144: / Line 146: @@
 E1d.	(( f , g ))$(u).	:$1b
 </pre>
 <pre>
@@ Line 232: / Line 236: @@
 D7b.	[S] = [T].
 </pre>
 <pre>
@@ Line 480: / Line 486: @@
 d.  R11d gives a version of the indicator function with {X} : U->B, called its "functional form".
 Applying Rule 9, Rule 8, and the Logical Rules to the special case where S <=> (X = Y), one obtains the following general fact.

← Older revision		Revision as of 20:40, 10 September 2010
Line 6:		Line 6:

	=====1.3.12.1. Syntactic Transformation Rules=====		=====1.3.12.1. Syntactic Transformation Rules=====
−
−	~~<pre>~~
−	~~Logical Translation Rule 1~~
−
−	~~If S is a sentence~~
−
−	~~about things in the universe U~~
−
−	~~and P is a proposition : U -> B, such that:~~
−
−	~~L1a. [S] = P,~~
−
−	~~then the following equations hold:~~
−
−	~~L1b00. [False] = () = 0 : U->B.~~
−
−	~~L1b01. [Not S] = ([S]) = (P) : U->B.~~
−
−	~~L1b10. [S] = [S] = P : U->B.~~
−
−	~~L1b11. [True] = (()) = 1 : U->B.~~
−	~~</pre>~~
−
−	~~<pre>~~
−	~~Geometric Translation Rule 1~~
−
−	~~If X c U~~
−
−	~~and P : U -> B, such that:~~
−
−	~~G1a. {X} = P,~~
−
−	~~then the following equations hold:~~
−
−	~~G1b00. {{}} = () = 0 : U->B.~~
−
−	~~G1b10. {~X} = ({X}) = (P) : U->B.~~
−
−	~~G1b01. {X} = {X} = P : U->B.~~
−
−	~~G1b11. {U} = (()) = 1 : U->B.~~
−	~~</pre>~~
−
−	~~<pre>~~
−	~~Logical Translation Rule 2~~
−
−	~~If S, T are sentences~~
−
−	~~about things in the universe U~~
−
−	~~and P, Q are propositions: U -> B, such that:~~
−
−	~~L2a. [S] = P and [T] = Q,~~
−
−	~~then the following equations hold:~~
−
−	~~L2b00. [False] = () = 0 : U->B.~~
−
−	~~L2b01. [Neither S nor T] = ([S])([T]) = (P)(Q).~~
−
−	~~L2b02. [Not S, but T] = ([S])[T] = (P) Q.~~
−
−	~~L2b03. [Not S] = ([S]) = (P).~~
−
−	~~L2b04. [S and not T] = [S]([T]) = P (Q).~~
−
−	~~L2b05. [Not T] = ([T]) = (Q).~~
−
−	~~L2b06. [S or T, not both] = ([S], [T]) = (P, Q).~~
−
−	~~L2b07. [Not both S and T] = ([S].[T]) = (P Q).~~
−
−	~~L2b08. [S and T] = [S].[T] = P.Q.~~
−
−	~~L2b09. [S <=> T] = (([S], [T])) = ((P, Q)).~~
−
−	~~L2b10. [T] = [T] = Q.~~
−
−	~~L2b11. [S => T] = ([S]([T])) = (P (Q)).~~
−
−	~~L2b12. [S] = [S] = P.~~
−
−	~~L2b13. [S <= T] = (([S]) [T]) = ((P) Q).~~
−
−	~~L2b14. [S or T] = (([S])([T])) = ((P)(Q)).~~
−
−	~~L2b15. [True] = (()) = 1 : U->B.~~
−	~~</pre>~~
−
−	~~<pre>~~
−	~~Geometric Translation Rule 2~~
−
−	~~If X, Y c U~~
−
−	~~and P, Q U -> B, such that:~~
−
−	~~G2a. {X} = P and {Y} = Q,~~
−
−	~~then the following equations hold:~~
−
−	~~G2b00. {{}} = () = 0 : U->B.~~
−
−	~~G2b01. {~X n ~Y} = ({X})({Y}) = (P)(Q).~~
−
−	~~G2b02. {~X n Y} = ({X}){Y} = (P) Q.~~
−
−	~~G2b03. {~X} = ({X}) = (P).~~
−
−	~~G2b04. {X n ~Y} = {X}({Y}) = P (Q).~~
−
−	~~G2b05. {~Y} = ({Y}) = (Q).~~
−
−	~~G2b06. {X + Y} = ({X}, {Y}) = (P, Q).~~
−
−	~~G2b07. {~(X n Y)} = ({X}.{Y}) = (P Q).~~
−
−	~~G2b08. {X n Y} = {X}.{Y} = P.Q.~~
−
−	~~G2b09. {~(X + Y)} = (({X}, {Y})) = ((P, Q)).~~
−
−	~~G2b10. {Y} = {Y} = Q.~~
−
−	~~G2b11. {~(X n ~Y)} = ({X}({Y})) = (P (Q)).~~
−
−	~~G2b12. {X} = {X} = P.~~
−
−	~~G2b13. {~(~X n Y)} = (({X}) {Y}) = ((P) Q).~~
−
−	~~G2b14. {X u Y} = (({X})({Y})) = ((P)(Q)).~~
−
−	~~G2b15. {U} = (()) = 1 : U->B.~~
−	~~</pre>~~

	<pre>		<pre>

← Older revision		Revision as of 18:14, 10 September 2010
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	=====1.3.12.2. Derived Equivalence Relations=====		=====1.3.12.2. Derived Equivalence Relations=====
−
−	The relation Der(R) is defined and the notation "x =R y" is meaningful in every situation where Den(-,-) makes sense, but it remains to check whether this relation enjoys the properties of an equivalence relation.
−
−	# Reflexive property. Is it true that x =R x for every x C S = I? By definition, x =R x if and only if Den(R, x) = Den(R, x). Thus, the reflexive property holds in any setting where the denotations Den(R, x) are defined for all signs x in the syntactic domain of R.
−	# Symmetric property. Does x =R y => y =R x for all x, y C S? In effect, does Den(R, x) = Den(R, y) imply Den(R, y) = Den(R, x) for all signs x and y in the syntactic domain S? Yes, so long as the sets Den(R, x) and Den(R, y) are well-defined, a fact which is already being assumed.
−	# Transitive property. Does x =R y & y =R z => x =R z for all x, y, z C S? To belabor the point, does Den(R, x) = Den(R, y) and Den(R, y) = Den(R, z) imply Den(R, x) = Den(R, z) for all x, y, z in S? Yes, again, under the stated conditions.
−
−	It should be clear at this point that any question about the equiference of signs reduces to a question about the equality of sets, specifically, the sets that are indexed by these signs. As a result, so long as these sets are well-defined, the issue of whether equiference relations induce equivalence relations on their syntactic domains is almost as trivial as it initially appears.
−
−	Taken in its set-theoretic extension, a relation of equiference induces a "denotative equivalence relation" (DER) on its syntactic domain S = I. This leads to the formation of "denotative equivalence classes" (DEC's), "denotative partitions" (DEP's), and "denotative equations" (DEQ's) on the syntactic domain. But what does it mean for signs to be equiferent?
−
−	Notice that this is not the same thing as being "semiotically equivalent", in the sense of belonging to a single "semiotic equivalence class" (SEC), falling into the same part of a "semiotic partition" (SEP), or having a "semiotic equation" (SEQ) between them. It is only when very felicitous conditions obtain, establishing a concord between the denotative and the connotative components of a sign relation, that these two ideas coalesce.
−
−	In general, there is no necessity that the equiference of signs, that is, their denotational equivalence or their referential equivalence, induces the same equivalence relation on the syntactic domain as that defined by their semiotic equivalence, even though this state of accord seems like an especially desirable situation. This makes it necessary to find a distinctive nomenclature for these structures, for which I adopt the term "denotative equivalence relations" (DER's). In their train they bring the allied structures of "denotative equivalence classes" (DEC's) and "denotative partitions" (DEP's), while the corresponding statements of "denotative equations" (DEQ's) are expressible in the form "x =R y".

	The uses of the equal sign for denoting equations or equivalences are recalled and extended in the following ways:		The uses of the equal sign for denoting equations or equivalences are recalled and extended in the following ways: