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==Where I Left Off In June 2004==
==Where I Left Off In June 2004==
=====1.3.10.14. Syntactic Transformations (cont.)=====
<pre>
<pre>
As another example of a ROST, consider the
As another example of a ROST, consider the
following logical equivalence, that holds
following logical equivalence, that holds
=====1.3.10.14. Syntactic Transformations (cont.)=====
=====1.3.10.14. Syntactic Transformations (cont.)=====
<pre>
Besides linking rules together into extended sequences of equivalents,
Besides linking rules together into extended sequences of equivalents,
there is one other way that is commonly used to get new rules from old.
there is one other way that is commonly used to get new rules from old.
L2b15. [True] = (()) = 1 : U->B.
L2b15. [True] = (()) = 1 : U->B.
Geometric Translation Rule 2
Geometric Translation Rule 2
If X, Y c U
If X, Y c U
and P, Q U -> B, such that:
and P, Q U -> B, such that:
G2a. {X} = P and {Y} = Q,
G2a. {X} = P and {Y} = Q,
then the following equations hold:
then the following equations hold:
G2b00. {{}} = () = 0 : U->B.
G2b00. {{}} = () = 0 : U->B.
G2b01. {~X n ~Y} = ({X})({Y}) = (P)(Q).
G2b01. {~X n ~Y} = ({X})({Y}) = (P)(Q).
G2b02. {~X n Y} = ({X}){Y} = (P) Q.
G2b02. {~X n Y} = ({X}){Y} = (P) Q.
G2b03. {~X} = ({X}) = (P).
G2b03. {~X} = ({X}) = (P).
G2b04. {X n ~Y} = {X}({Y}) = P (Q).
G2b04. {X n ~Y} = {X}({Y}) = P (Q).
G2b05. {~Y} = ({Y}) = (Q).
G2b05. {~Y} = ({Y}) = (Q).
G2b06. {X + Y} = ({X}, {Y}) = (P, Q).
G2b06. {X + Y} = ({X}, {Y}) = (P, Q).
G2b07. {~(X n Y)} = ({X}.{Y}) = (P Q).
G2b07. {~(X n Y)} = ({X}.{Y}) = (P Q).
G2b08. {X n Y} = {X}.{Y} = P.Q.
G2b08. {X n Y} = {X}.{Y} = P.Q.
G2b09. {~(X + Y)} = (({X}, {Y})) = ((P, Q)).
G2b09. {~(X + Y)} = (({X}, {Y})) = ((P, Q)).
G2b10. {Y} = {Y} = Q.
G2b10. {Y} = {Y} = Q.
G2b11. {~(X n ~Y)} = ({X}({Y})) = (P (Q)).
G2b11. {~(X n ~Y)} = ({X}({Y})) = (P (Q)).
G2b12. {X} = {X} = P.
G2b12. {X} = {X} = P.
G2b13. {~(~X n Y)} = (({X}) {Y}) = ((P) Q).
G2b13. {~(~X n Y)} = (({X}) {Y}) = ((P) Q).
G2b14. {X u Y} = (({X})({Y})) = ((P)(Q)).
G2b14. {X u Y} = (({X})({Y})) = ((P)(Q)).
G2b15. {U} = (()) = 1 : U->B.
G2b15. {U} = (()) = 1 : U->B.
If v, w C B
If v, w C B
then "v = w" is a sentence about <v, w> C B2,
then "v = w" is a sentence about <v, w> C B2,
[v = w] is a proposition : B2 -> B,
[v = w] is a proposition : B2 -> B,
and the following are identical values in B:
and the following are identical values in B:
V1a. [ v = w ](v, w)
V1a. [ v = w ](v, w)
V1b. [ v <=> w ](v, w)
V1b. [ v <=> w ](v, w)
V1c. ((v , w))
V1c. ((v , w))
If v, w C B,
If v, w C B,
then the following are equivalent:
then the following are equivalent:
V1a. v = w.
V1a. v = w.
V1b. v <=> w.
V1b. v <=> w.
V1c. (( v , w )).
V1c. (( v , w )).
If v, w C B,
If v, w C B,
then the following are identical values in B:
then the following are identical values in B:
V1a. [ v = w ]
V1a. [ v = w ]
V1b. [ v <=> w ]
V1b. [ v <=> w ]
V1c. (( v , w ))
V1c. (( v , w ))
Value Rule 1
Value Rule 1
If f, g : U -> B,
If f, g : U -> B,
and u C U
and u C U
then the following are identical values in B:
then the following are identical values in B:
V1a. [ f(u) = g(u) ]
V1a. [ f(u) = g(u) ]
V1b. [ f(u) <=> g(u) ]
V1b. [ f(u) <=> g(u) ]
V1c. (( f(u) , g(u) ))
V1c. (( f(u) , g(u) ))
Value Rule 1
Value Rule 1
If f, g : U -> B,
If f, g : U -> B,
then the following are identical propositions on U:
then the following are identical propositions on U:
V1a. [ f = g ]
V1a. [ f = g ]
V1b. [ f <=> g ]
V1b. [ f <=> g ]
V1c. (( f , g ))$
V1c. (( f , g ))$
Evaluation Rule 1
Evaluation Rule 1
If f, g : U -> B
If f, g : U -> B
and u C U,
and u C U,
then the following are equivalent:
then the following are equivalent:
E1a. f(u) = g(u). :V1a
E1a. f(u) = g(u). :V1a
E1d. (( f , g ))$(u). :$1b
E1d. (( f , g ))$(u). :$1b
Evaluation Rule 1
Evaluation Rule 1
about things in the universe U,
about things in the universe U,
f, g are propositions: U -> B,
f, g are propositions: U -> B,
and u C U,
and u C U,
then the following are equivalent:
then the following are equivalent:
E1a. f(u) = g(u). :V1a
E1a. f(u) = g(u). :V1a
E1d. (( f , g ))$(u). :$1b
E1d. (( f , g ))$(u). :$1b
If X, Y c U,
If X, Y c U,
then the following are equivalent:
then the following are equivalent:
D2a. X = Y.
D2a. X = Y.
D2b. u C X <=> u C Y, for all u C U.
D2b. u C X <=> u C Y, for all u C U.
Definition 3
Definition 3
If f, g : U -> V,
If f, g : U -> V,
then the following are equivalent:
then the following are equivalent:
D3a. f = g.
D3a. f = g.
D3b. f(u) = g(u), for all u C U.
D3b. f(u) = g(u), for all u C U.
Definition 4
Definition 4
If X c U,
If X c U,
then the following are identical subsets of UxB:
D4a. {X}
D4a. {X}
D4b. {< u, v> C UxB : v = [u C X]}
D4b. {< u, v> C UxB : v = [u C X]}
Definition 5
Definition 5
If X c U,
If X c U,
then the following are identical propositions:
then the following are identical propositions:
D5a. {X}.
D5a. {X}.
D5b. f : U -> B
D5b. f : U -> B
: f(u) = [u C X], for all u C U.
: f(u) = [u C X], for all u C U.
for all j C J,
for all j C J,
then the following are equivalent:
then the following are equivalent:
D6a. Sj, for all j C J.
D6a. Sj, for all j C J.
D6b. For all j C J, Sj.
D6b. For all j C J, Sj.
D6c. Conj(j C J) Sj.
D6c. Conj(j C J) Sj.
D6d. ConjJ,j Sj.
D6d. ConjJ,j Sj.
D6e. ConjJj Sj.
D6e. ConjJj Sj.
Definition 7
Definition 7
about things in the universe U,
about things in the universe U,
then the following are equivalent:
then the following are equivalent:
D7a. S <=> T.
D7a. S <=> T.
D7b. [S] = [T].
D7b. [S] = [T].
Rule 5
Rule 5
If X, Y c U,
If X, Y c U,
then the following are equivalent:
then the following are equivalent:
R5a. X = Y. :D2a
R5a. X = Y. :D2a
R5e. {X} = {Y}. :D5a
R5e. {X} = {Y}. :D5a
Rule 6
Rule 6
If f, g : U -> V,
If f, g : U -> V,
then the following are equivalent:
then the following are equivalent:
R6a. f = g. :D3a
R6a. f = g. :D3a
R6c. ConjUu (f(u) = g(u)). :D6e
R6c. ConjUu (f(u) = g(u)). :D6e
Rule 7
Rule 7
If P, Q : U -> B,
If P, Q : U -> B,
then the following are equivalent:
then the following are equivalent:
R7a. P = Q. :R6a
R7a. P = Q. :R6a
R7f. ConjUu (( P , Q ))$(u). :$1b
R7f. ConjUu (( P , Q ))$(u). :$1b
Rule 8
Rule 8
about things in the universe U,
about things in the universe U,
then the following are equivalent:
then the following are equivalent:
R8a. S <=> T. :D7a
R8a. S <=> T. :D7a
If X, Y c U,
If X, Y c U,
then the following are equivalent:
then the following are equivalent:
R9a. X = Y. :R5a
R9a. X = Y. :R5a
R9g. ConjUu (( {X} , {Y} ))$(u). :R7f
R9g. ConjUu (( {X} , {Y} ))$(u). :R7f
Rule 10
Rule 10
If X, Y c U,
If X, Y c U,
then the following are equivalent:
then the following are equivalent:
R10a. X = Y. :D2a
R10a. X = Y. :D2a
R10h. ConjUu (( [u C X] , [u C Y] ))$(u). :R8g
R10h. ConjUu (( [u C X] , [u C Y] ))$(u). :R8g
Rule 11
Rule 11
If X c U
If X c U
then the following are equivalent:
then the following are equivalent:
R11a. X = {u C U : S}. :R5a
R11a. X = {u C U : S}. :R5a
R11e. {X} = [S]. :R
R11e. {X} = [S]. :R
An application of Rule 11 involves the recognition of an antecedent condition as a case under the Rule, that is, as a condition that matches one of the sentences in the Rule's chain of equivalents, and it requires the relegation of the other expressions to the production of a result. Thus, there is the choice of an initial expression that has to be checked on input for whether it fits the antecedent condition, and there is the choice of three types of output that are generated as a consequence, only one of which is generally needed at any given time. More often than not, though, a rule is applied in only a few of its possible ways. The usual antecedent and the usual consequents for Rule 11 can be distinguished in form and specialized in practice as follows:
An application of Rule 11 involves the recognition of an antecedent condition as a case under the Rule, that is, as a condition that matches one of the sentences in the Rule's chain of equivalents, and it requires the relegation of the other expressions to the production of a result. Thus, there is the choice of an initial expression that has to be checked on input for whether it fits the antecedent condition, and there is the choice of three types of output that are generated as a consequence, only one of which is generally needed at any given time. More often than not, though, a rule is applied in only a few of its possible ways. The usual antecedent and the usual consequents for Rule 11 can be distinguished in form and specialized in practice as follows:
a. R11a marks the usual starting place for an application of the Rule, that is, the standard form of antecedent condition that is likely to lead to an invocation of the Rule.
a. R11a marks the usual starting place for an application of the Rule, that is, the standard form of antecedent condition that is likely to lead to an invocation of the Rule.
b. R11b records the trivial consequence of applying the spiny braces to both sides of the initial equation.
b. R11b records the trivial consequence of applying the spiny braces to both sides of the initial equation.
c. R11c gives a version of the indicator function with {X} c UxB, called its "extensional form".
c. R11c gives a version of the indicator function with {X} c UxB, called its "extensional form".
d. R11d gives a version of the indicator function with {X} : U->B, called its "functional form".
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.
Applying Rule 9, Rule 8, and the Logical Rules to the special case where S <=> (X = Y), one obtains the following general fact.
If X,Y c U,
If X,Y c U,
then the following are equivalent:
then the following are equivalent:
F1a. S <=> X = Y. :R9a
F1a. S <=> X = Y. :R9a
::
::
F1h. [S] = ConjUu (( {X} , {Y} ))$(u). :$1b
F1h. [S] = ConjUu (( {X} , {Y} ))$(u). :$1b
///
///
///
///
</pre>
=====1.3.10.15 Derived Equivalence Relations=====
=====1.3.10.15 Derived Equivalence Relations=====
<pre>
One seeks a method of general application for approaching the individual sign relation, a way to select an aspect of its form, to analyze it with regard to its intrinsic structure, and to classify it in comparison with other sign relations. With respect to a particular sign relation, one approach that presents itself is to examine the relation between signs and interpretants that is given directly by its connotative component and to compare it with the various forms of derived, indirect, mediate, or peripheral relationships that can be found to exist among signs and interpretants by way of secondary considerations or subsequent studies. Of especial interest are the relationships among signs and interpretants that can be obtained by working through the collections of objects that they commonly or severally denote.
One seeks a method of general application for approaching the individual sign relation, a way to select an aspect of its form, to analyze it with regard to its intrinsic structure, and to classify it in comparison with other sign relations. With respect to a particular sign relation, one approach that presents itself is to examine the relation between signs and interpretants that is given directly by its connotative component and to compare it with the various forms of derived, indirect, mediate, or peripheral relationships that can be found to exist among signs and interpretants by way of secondary considerations or subsequent studies. Of especial interest are the relationships among signs and interpretants that can be obtained by working through the collections of objects that they commonly or severally denote.
If R c OxSxI,
If R c OxSxI,
then the following are identical subsets of SxI:
then the following are identical subsets of SxI:
D8a. RSI
D8a. RSI
D8b. ConR
D8b. ConR
D8c. Con(R)
D8c. Con(R)
D8d. PrSI(R)
D8d. PrSI(R)
D8e. {< s, i> C SxI : <o, s, i> C R for some o C O}
D8e. {< s, i> C SxI : <o, s, i> C R for some o C O}
If R c OxSxI,
If R c OxSxI,
then the following are identical subsets of IxS:
then the following are identical subsets of IxS:
D9a. RIS
D9a. RIS
D9b. RSI^
D9b. RSI^
D9c. ConR^
D9c. ConR^
D9d. Con(R)^
D9e. PrIS(R)
D9e. PrIS(R)
D9f. Conv(Con(R))
D9f. Conv(Con(R))
D9g. {< i, s> C IxS : <o, s, i> C R for some o C O}
D9g. {< i, s> C IxS : <o, s, i> C R for some o C O}
If R c OxSxI,
If R c OxSxI,
then the following are identical subsets of OxS:
then the following are identical subsets of OxS:
D10a. ROS
D10a. ROS
D10b. DenR
D10b. DenR
D10c. Den(R)
D10c. Den(R)
D10d. PrOS(R)
D10d. PrOS(R)
D10e. {<o, s> C OxS : <o, s, i> C R for some i C I}
D10e. {<o, s> C OxS : <o, s, i> C R for some i C I}
If R c OxSxI,
If R c OxSxI,
then the following are identical subsets of SxO:
then the following are identical subsets of SxO:
D11a. RSO
D11a. RSO
D11b. ROS^
D11b. ROS^
D11c. DenR^
D11c. DenR^
D11d. Den(R)^
D11d. Den(R)^
D11e. PrSO(R)
D11e. PrSO(R)
D11f. Conv(Den(R))
D11f. Conv(Den(R))
D11g. {< s, o> C SxO : <o, s, i> C R for some i C I}
D11g. {< s, o> C SxO : <o, s, i> C R for some i C I}
If R c OxSxI,
If R c OxSxI,
and x C S,
and x C S,
then the following are identical subsets of O:
then the following are identical subsets of O:
D12a. ROS.x
D12a. ROS.x
D12b. DenR.x
D12b. DenR.x
D12c. DenR|x
D12c. DenR|x
D12d. DenR(, x)
D12d. DenR(, x)
D12e. Den(R, x)
D12e. Den(R, x)
D12f. Den(R).x
D12f. Den(R).x
D12g. {o C O : <o, x> C Den(R)}
D12h. {o C O : <o, x, i> C R for some i C I}
D12h. {o C O : <o, x, i> C R for some i C I}
Signs are "equiferent" if they refer to all and only the same objects, that is, if they have exactly the same denotations. In other language for the same relation, signs are said to be "denotatively equivalent" or "referentially equivalent", but it is probably best to check whether the extension of this concept over the syntactic domain is really a genuine equivalence relation before jumpimg to the conclusions that are implied by these latter terms.
Signs are "equiferent" if they refer to all and only the same objects, that is, if they have exactly the same denotations. In other language for the same relation, signs are said to be "denotatively equivalent" or "referentially equivalent", but it is probably best to check whether the extension of this concept over the syntactic domain is really a genuine equivalence relation before jumpimg to the conclusions that are implied by these latter terms.
These definitions and notations are recorded in the following display.
These definitions and notations are recorded in the following display.
Definition 13
Definition 13
If R c OxSxI,
If R c OxSxI,
then the following are identical subsets of SxI:
then the following are identical subsets of SxI:
D13a. DerR
D13a. DerR
D13b. Der(R)
D13b. Der(R)
D13c. {<x,y> C SxI : DenR|x = DenR|y}
D13c. {<x,y> C SxI : DenR|x = DenR|y}
D13d. {<x,y> C SxI : Den(R, x) = Den(R, y)}
D13d. {<x,y> C SxI : Den(R, x) = Den(R, y)}
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.
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.
1. 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.
1. 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.
2. 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.
2. 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.
3. 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.
3. 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.
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.
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:
1. If E is an arbitrary equivalence relation,
1. If E is an arbitrary equivalence relation,
then the equation "x =E y" means that <x, y> C E.
then the equation "x =E y" means that <x, y> C E.
2. If R is a sign relation such that RSI is a SER on S = I,
2. If R is a sign relation such that RSI is a SER on S = I,
then the semiotic equation "x =R y" means that <x, y> C RSI.
then the semiotic equation "x =R y" means that <x, y> C RSI.
3. If R is a sign relation such that F is its DER on S = I,
3. If R is a sign relation such that F is its DER on S = I,
then the denotative equation "x =R y" means that <x, y> C F,
then the denotative equation "x =R y" means that <x, y> C F,
The uses of square brackets for denoting equivalence classes are recalled and extended in the following ways:
The uses of square brackets for denoting equivalence classes are recalled and extended in the following ways:
1. If E is an arbitrary equivalence relation,
1. If E is an arbitrary equivalence relation,
then "[x]E" denotes the equivalence class of x under E.
then "[x]E" denotes the equivalence class of x under E.
2. If R is a sign relation such that Con(R) is a SER on S = I,
2. If R is a sign relation such that Con(R) is a SER on S = I,
then "[x]R" denotes the SEC of x under Con(R).
then "[x]R" denotes the SEC of x under Con(R).
3. If R is a sign relation such that Der(R) is a DER on S = I,
3. If R is a sign relation such that Der(R) is a DER on S = I,
then "[x]R" denotes the DEC of x under Der(R).
then "[x]R" denotes the DEC of x under Der(R).
By applying the form of Fact 1 to the special case where X = Den(R, x) and Y = Den(R, y), one obtains the following facts.
By applying the form of Fact 1 to the special case where X = Den(R, x) and Y = Den(R, y), one obtains the following facts.
Fact 2.1
Fact 2.1
If R c OxSxI,
If R c OxSxI,
then the following are identical subsets of SxI:
then the following are identical subsets of SxI:
F2.1a. DerR :D13a
F2.1a. DerR :D13a
If R c OxSxI,
If R c OxSxI,
then the following are equivalent:
then the following are equivalent:
F2.2a. DerR = {<x, y> C SxI :
F2.2a. DerR = {<x, y> C SxI :
} :R11a
} :R11a
::
F2.2b. {DerR} = { {<x, y> C SxI :
F2.2b. {DerR} = { {<x, y> C SxI :
} :$
} :$
If R c OxSxI,
If R c OxSxI,
then the following are equivalent:
then the following are equivalent:
F2.3a. DerR = {<x, y> C SxI :
F2.3a. DerR = {<x, y> C SxI :
)) :D10a
)) :D10a
</pre>
=====1.3.10.16 Digression on Derived Relations=====
=====1.3.10.16 Digression on Derived Relations=====
<pre>
A better understanding of derived equivalence relations (DER's) can be achieved by placing their constructions within a more general context, and thus comparing the associated type of derivation operation, namely, the one that takes a triadic relation R into a dyadic relation Der(R), with other types of operations on triadic relations. The proper setting would permit a comparative study of all their constructions from a basic set of projections and a full array of compositions on dyadic relations.
A better understanding of derived equivalence relations (DER's) can be achieved by placing their constructions within a more general context, and thus comparing the associated type of derivation operation, namely, the one that takes a triadic relation R into a dyadic relation Der(R), with other types of operations on triadic relations. The proper setting would permit a comparative study of all their constructions from a basic set of projections and a full array of compositions on dyadic relations.
{P.Q}(x, y) = Disj(m C M) ( {P}(x, m) . {Q}(m, y) ).
{P.Q}(x, y) = Disj(m C M) ( {P}(x, m) . {Q}(m, y) ).
</pre>
1.4 Outlook of the Project: All Ways Lead to Inquiry
===1.4. Outlook of the Project : All Ways Lead to Inquiry===
<pre>
I am using the word "inquiry" in a way that is roughly synonymous with the
I am using the word "inquiry" in a way that is roughly synonymous with the
term "scientific method". Use of "inquiry" is more convenient, aside from
term "scientific method". Use of "inquiry" is more convenient, aside from
model, however conjectural, naive, uncritical, and unreflective it
model, however conjectural, naive, uncritical, and unreflective it
may seem.
may seem.
</pre>
====1.4.1 The Matrix of Inquiry====
====1.4.1. The Matrix of Inquiry====
<pre>
<pre>
|
|
| Plato, 'Laws', VII, 790D
| Plato, 'Laws', VII, 790D
Try as I may, I've never seen a way to develop a theory of inquiry from nothing:
Try as I may, I've never seen a way to develop a theory of inquiry from nothing:
The reader may take my apology for this style of presentation to be implicit in its dogmatic character. It is done this way in a first approach for the sake of avoiding an immense number of distractions, each of which is not being slighted but demands to be addressed in its own good time. I want to convey the general drift of my current model, however conjectural, naive, uncritical, and unreflective it may seem.
The reader may take my apology for this style of presentation to be implicit in its dogmatic character. It is done this way in a first approach for the sake of avoiding an immense number of distractions, each of which is not being slighted but demands to be addressed in its own good time. I want to convey the general drift of my current model, however conjectural, naive, uncritical, and unreflective it may seem.
</pre>
====1.4.1 The Matrix of Inquiry (2)====
====1.4.1 The Matrix of Inquiry (2)====
</blockquote>
</blockquote>
<pre>
Try as I might, I do not see a way to develop a theory of inquiry from nothing: To take for granted nothing more than is already given, to set out from nothing but absolutely certain beginnings, or to move forward with nothing but absolutely certain means of proceeding. In particular, the present inquiry into inquiry, y0 = y.y, ought not to be misconstrued as a device for magically generating a theory of inquiry from nothing. Like any other inquiry, it requires an agent to invest in a conjecture, to make a guess about the relevant features of the subject of interest, and to choose the actions, the aspects, and the attitudes with regard to the subject that are critical to achieving the objectives of the study.
Try as I might, I do not see a way to develop a theory of inquiry from nothing: To take for granted nothing more than is already given, to set out from nothing but absolutely certain beginnings, or to move forward with nothing but absolutely certain means of proceeding. In particular, the present inquiry into inquiry, y0 = y.y, ought not to be misconstrued as a device for magically generating a theory of inquiry from nothing. Like any other inquiry, it requires an agent to invest in a conjecture, to make a guess about the relevant features of the subject of interest, and to choose the actions, the aspects, and the attitudes with regard to the subject that are critical to achieving the objectives of the study.
There is already a model of inquiry that is implicit, at least partially, in the text of the above description. Let me see if I can tease out a few of its tacit assumptions.
There is already a model of inquiry that is implicit, at least partially, in the text of the above description. Let me see if I can tease out a few of its tacit assumptions.
</pre>
=====1.4.1.1 Inquiry as Conduct=====
=====1.4.1.1 Inquiry as Conduct=====
<pre>
First of all, inquiry is conceived to be a form of conduct.
First of all, inquiry is conceived to be a form of conduct.
This invokes the technical term "conduct", referring to the
This invokes the technical term "conduct", referring to the
If it is to have the properties that it is commonly thought to have, then reflection must be capable of running in parallel, and not interfering too severely, with the conduct on which it reflects. If this turns out to be an illusion of reflection that is not really possible in actuality, then reflection must be capable, at the very least, of reviewing the memory record of the conduct in question, in ways that appear concurrent with a replay of its action. But these are the abilities that reflection is "pre-reflectively" thought to have, that is, before the reflection on reflection can get under way. If reflection is truly a form of conduct, then it becomes conceivable as a project to reflect on reflection itself, and this reflection can even lead to the conclusion that reflection does not have all of the powers that it is commonly portrayed to have.
If it is to have the properties that it is commonly thought to have, then reflection must be capable of running in parallel, and not interfering too severely, with the conduct on which it reflects. If this turns out to be an illusion of reflection that is not really possible in actuality, then reflection must be capable, at the very least, of reviewing the memory record of the conduct in question, in ways that appear concurrent with a replay of its action. But these are the abilities that reflection is "pre-reflectively" thought to have, that is, before the reflection on reflection can get under way. If reflection is truly a form of conduct, then it becomes conceivable as a project to reflect on reflection itself, and this reflection can even lead to the conclusion that reflection does not have all of the powers that it is commonly portrayed to have.
</pre>
=====1.4.1.2 Types of Conduct=====
=====1.4.1.2. Types of Conduct=====
<pre>
The chief distinction that applies to different forms of conduct is whether the object is the same sort of thing as the states or whether it is something entirely different, a thing apart, of a wholly other order. Although I am using different words for objects and states, it is always possible that these words are indicative of different roles in a formal relation and not indicative of substantially different types of things. If objects and states are but formal points and naturally belong to the same domain, then it is conceivable that a temporal sequence of states can include the object in its succession, in other words, that a path through a state space can reach or pass through an object of conduct. But if a form of conduct has an object that is completely different from any one of its temporal states, then the role of the object in regard to the action cannot be like the end or goal of a temporal development.
The chief distinction that applies to different forms of conduct is whether the object is the same sort of thing as the states or whether it is something entirely different, a thing apart, of a wholly other order. Although I am using different words for objects and states, it is always possible that these words are indicative of different roles in a formal relation and not indicative of substantially different types of things. If objects and states are but formal points and naturally belong to the same domain, then it is conceivable that a temporal sequence of states can include the object in its succession, in other words, that a path through a state space can reach or pass through an object of conduct. But if a form of conduct has an object that is completely different from any one of its temporal states, then the role of the object in regard to the action cannot be like the end or goal of a temporal development.
What names can be given to these two orders of conduct?
What names can be given to these two orders of conduct?
</pre>
=====1.4.1.3 Perils of Inquiry=====
=====1.4.1.3 Perils of Inquiry=====
<pre>
Now suppose that making a hypothesis is a kind of action, no matter how covert, or that testing a hypothesis takes an action that is more overt. If entertaining a hypothesis in any serious way requires action, and if action is capable of altering the situation in which it acts, then what prevents this action from interfering with the subject of inquiry in a way that undermines, with positive or negative intentions, the very aim of inquiry, namely, to understand the situation as it is in itself?
Now suppose that making a hypothesis is a kind of action, no matter how covert, or that testing a hypothesis takes an action that is more overt. If entertaining a hypothesis in any serious way requires action, and if action is capable of altering the situation in which it acts, then what prevents this action from interfering with the subject of inquiry in a way that undermines, with positive or negative intentions, the very aim of inquiry, namely, to understand the situation as it is in itself?
Of course, a finite person can only take up so many causes in a single lifetime, and so there is always the excuse of time for not chasing down every conceivable hypothesis that comes to mind.
Of course, a finite person can only take up so many causes in a single lifetime, and so there is always the excuse of time for not chasing down every conceivable hypothesis that comes to mind.
</pre>
=====1.4.1.4 Forms of Relations=====
=====1.4.1.4 Forms of Relations=====
The next distingishing trait that I can draw out of this incipient treatise is its emphasis on the forms of relations. From a sufficiently "formal and relational" (FAR) point of view, many of the complexities that arise from throwing intentions, objectives, and purposes into the mix of discussion are conceivably due to the greater arity of triadic relations over dyadic relations, and do not necessarily implicate any differences of essence inhering in the entities and the states invoked. As far as this question goes, whether a dynamic object is essentially different from a deliberate object, I intend to remain as neutral as possible, at least, until forced by some good reason to do otherwise. In the meantime, the factors that are traceable to formal differences among relations are ready to be investigated and useful to examine. With this in mind, it it useful to make the following definition:
<pre>
The next distinguishing trait that I can draw out of this incipient treatise is its emphasis on the forms of relations. From a sufficiently "formal and relational" (FAR) point of view, many of the complexities that arise from throwing intentions, objectives, and purposes into the mix of discussion are conceivably due to the greater arity of triadic relations over dyadic relations, and do not necessarily implicate any differences of essence inhering in the entities and the states invoked. As far as this question goes, whether a dynamic object is essentially different from a deliberate object, I intend to remain as neutral as possible, at least, until forced by some good reason to do otherwise. In the meantime, the factors that are traceable to formal differences among relations are ready to be investigated and useful to examine. With this in mind, it it useful to make the following definition:
A "conduct relation" is a triadic relation involving a domain of objects and two domains of states. When a shorter term is desired, I refer to a conduct relation as a "conduit". A conduit is given in terms of its extension as a subset C c XxYxZ, where X is the "object domain" and where Y and Z are the "state domains". Typically, Y = Z.
A "conduct relation" is a triadic relation involving a domain of objects and two domains of states. When a shorter term is desired, I refer to a conduct relation as a "conduit". A conduit is given in terms of its extension as a subset C c XxYxZ, where X is the "object domain" and where Y and Z are the "state domains". Typically, Y = Z.
I doubt if there is any hard and fast answer to this question, but think that it depends on particular interpreters and particular observers, to what extent each one interprets a state as a sign, and to what degree each one recognizes a sign as a component of a state.
I doubt if there is any hard and fast answer to this question, but think that it depends on particular interpreters and particular observers, to what extent each one interprets a state as a sign, and to what degree each one recognizes a sign as a component of a state.
</pre>
=====1.4.1.5 Models of Inquiry=====
=====1.4.1.5 Models of Inquiry=====
My observations of inquiry in general, together with a few suggestions that seem apt to me, have led me to believe that inquiry begins with a "surprise" or a "problem". The way I understand these words, they refer to departures, differences, or discrepancies among various modalities of experience, in particular, among "observations", "expectations", and "intentions".
My observations of inquiry in general, together with a few suggestions that seem apt to me, have led me to believe that inquiry begins with a "surprise" or a "problem". The way I understand these words, they refer to departures, differences, or discrepancies among various modalities of experience, in particular, among "observations", "expectations", and "intentions".
1. A "surprise" is a departure of an observation from an expectation, and thus it invokes a comparison between present experience and past experience, since expectations are based on the remembered disposition of past experience.
1. A "surprise" is a departure of an observation from an expectation, and thus it invokes a comparison between present experience and past experience, since expectations are based on the remembered disposition of past experience.
2. A "problem" is a departure of an observation from an intention, and thus it invokes a comparison between present experience and future experience, since intentions choose from the envisioned disposition of future experience.
2. A "problem" is a departure of an observation from an intention, and thus it invokes a comparison between present experience and future experience, since intentions choose from the envisioned disposition of future experience.
With respect to these
With respect to these
