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<div class="nonumtoc">__TOC__</div>
==1. Introduction==
==1. Introduction==
</pre>
</pre>
====1.3.1. Initial Analysis of Inquiry -- Allegro Aperto====
====1.3.1. Initial Analysis of Inquiry — Allegro Aperto====
<pre>
<pre>
</pre>
</pre>
====1.3.8. Rondeau — Tempo di Menuetto====
====1.3.8. Rondeau — Tempo di Menuetto====
<pre>
<pre>
</pre>
</pre>
====1.3.11. The Cactus Patch====
<pre>
<pre>
</pre>
</pre>
=====1.3.10.9. The Cactus Language : Syntax=====
=====1.3.11.1. The Cactus Language : Syntax=====
<pre>
<pre>
</pre>
</pre>
=====1.3.10.10. The Cactus Language : Stylistics=====
=====1.3.11.2. Generalities About Formal Grammars=====
=====1.3.11.3. The Cactus Language : Stylistics=====
<pre>
<pre>
</pre>
</pre>
=====1.3.10.11. The Cactus Language : Mechanics=====
=====1.3.11.4. The Cactus Language : Mechanics=====
<pre>
<pre>
</pre>
</pre>
=====1.3.10.12. The Cactus Language : Semantics=====
=====1.3.11.5. The Cactus Language : Semantics=====
<pre>
<pre>
</pre>
</pre>
=====1.3.10.13. Stretching Exercises=====
=====1.3.11.6. Stretching Exercises=====
<pre>
<pre>
</pre>
</pre>
====1.3.12. Syntactic Transformations====
We have been examining several distinct but closely related notions of ''indication''. To discuss the import of these ideas in greater depth, it serves to establish a number of logical relations and set-theoretic identities that can be found to hold among their roughly parallel arrays of conceptions and constructions. Facilitating this task requires in turn a number of auxiliary concepts and notations.
We have been examining several distinct but closely related
notions of indication. To discuss the import of these ideas
The diverse notions of indication presently under discussion are expressed in a variety of different notations, enumerated as follows:
in greater depth, it serves to establish a number of logical
relations and set-theoretic identities that can be found to
# The functional language of propositions
hold among their roughly parallel arrays of conceptions and
# The logical language of sentences
constructions. Facilitating this task, in turn, requires
# The geometric language of sets
a number of auxiliary concepts and notations.
Correspondingly, one way to explain the relationships that exist among the various notions of indication is to describe the translations that they induce among the associated families of notation.
=====1.3.12.1. Syntactic Transformation Rules=====
A good way to summarize the necessary translations between different styles of indication, and along the way to organize their use in practice, is by means of the ''rules of syntactic transformation'' (ROSTs) that partially formalize the translations in question.
summarize the necessary translations between different styles of
indication, and along the way to organize their use in practice,
is by means of the "rules of syntactic transformation" (ROST's)
that partially formalize the translations in question.
Rudimentary examples of ROST's are readily mined from the
Rudimentary examples of ROSTs are readily mined from the raw materials that are already available in this area of discussion. To begin as near the beginning as possible, let the definition of an indicator function be recorded in the following form:
raw materials that are already available in this area of
discussion. To begin as near the beginning as possible,
let the definition of an indicator function be recorded
in the following form:
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| Definition 1. Indicator Function |
| Definition 1. Indicator Function |
| |
| |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
In practice, a definition like this is commonly used to substitute
In practice, a definition like this is commonly used to substitute one of two logically equivalent expressions or sentences for the other in a context where the conditions of using the definition in this way are satisfied and where the change is perceived as potentially advancing a proof. The employment of a definition in this way can be expressed in the form of a ROST that allows one to exchange two expressions of logically equivalent forms for one another in every context where their logical values are the only consideration. To be specific, the ''logical value'' of an expression is the value in the boolean domain %B% = {%0%, %1%} that the expression represents to its context or that it stands for in its context.
one of two logically equivalent expressions or sentences for the
other in a context where the conditions of using the definition
in this way are satisfied and where the change is perceived as
potentially advancing a proof. The employment of a definition
in this way can be expressed in the form of a ROST that allows
one to exchange two expressions of logically equivalent forms
for one another in every context where their logical values are
the only consideration. To be specific, the "logical value" of
an expression is the value in the boolean domain %B% = {%0%, %1%}
that the expression represents to its context or that it stands for
in its context.
In the case of Definition 1, the corresponding ROST permits one
In the case of Definition 1, the corresponding ROST permits one to exchange a sentence of the form "x in Q" with an expression of the form "-{Q}-(x)" in any context that satisfies the conditions of its use, namely, the conditions of the definition that lead up to the stated equivalence. The relevant ROST is recorded in Rule 1. By way of convention, I list the items that fall under a rule in rough order of their ascending conceptual subtlety or their increasing syntactic complexity, without regard for the normal or the typical orders of their exchange, since this can vary from widely from case to case.
to exchange a sentence of the form "x in Q" with an expression of
the form "-{Q}-(x)" in any context that satisfies the conditions of
its use, namely, the conditions of the definition that lead up to the
stated equivalence. The relevant ROST is recorded in Rule 1. By way
of convention, I list the items that fall under a rule in rough order
of their ascending conceptual subtlety or their increasing syntactic
complexity, without regard for the normal or the typical orders of
their exchange, since this can vary from widely from case to case.
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| Rule 1 |
| Rule 1 |
| |
| |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
</pre>
Conversely, any rule of this sort, properly qualified by the conditions under which it applies, can be turned back into a summary statement of the logical equivalence that is involved in its application. This mode of conversion between a static principle and a transformational rule, in other words, between a statement of equivalence and an equivalence of statements, is so automatic that it is usually not necessary to make a separate note of the "horizontal" versus the "vertical" versions of what amounts to the same abstract principle.
<pre>
As another example of a ROST, consider the following logical equivalence, that holds for any <math>X \subseteq U\!</math> and for all <math>u \in U.</math>
: -{X}-(u) <=> -{X}-(u) = 1.
In practice, this logical equivalence is used to exchange an expression of the form "-{X}-(u)" with a sentence of the form "-{X}-(u) = 1" in any context where one has a relatively fixed X c U in mind and where one is conceiving u in U to vary over its whole domain, namely, the universe U. This leads to the ROST that is given in Rule 2.
In practice, this logical equivalence is used to exchange
an expression of the form "-{X}-(u)" with a sentence of the
form "-{X}-(u) = 1" in any context where one has a relatively
fixed X c U in mind and where one is conceiving u in U to vary
over its whole domain, namely, the universe U. This leads to
the ROST that is given in Rule 2.
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| Rule 2 |
| Rule 2 |
| |
| |
o-------------------------------------------------o
o-------------------------------------------------o
</pre>
Rules like these can be chained together to establish extended
Rules like these can be chained together to establish extended rules, just so long as their antecedent conditions are compatible. For example, Rules 1 and 2 combine to give the equivalents that are listed in Rule 3. This follows from a recognition that the function -{X}- : U -> %B% that is introduced in Rule 1 is an instance of the function f : U -> %B% that is mentioned in Rule 2. By the time one arrives in the "consequence box" of either Rule, then, one has in mind a comparatively fixed X c U, a proposition f or -{X}- about things in U, and a variable argument u in U.
rules, just so long as their antecedent conditions are compatible.
For example, Rules 1 and 2 combine to give the equivalents that are
listed in Rule 3. This follows from a recognition that the function
-{X}- : U -> %B% that is introduced in Rule 1 is an instance of the
function f : U -> %B% that is mentioned in Rule 2. By the time one
arrives in the "consequence box" of either Rule, then, one has in
mind a comparatively fixed X c U, a proposition f or -{X}- about
things in U, and a variable argument u in U.
<pre>
o-------------------------------------------------o---------o
o-------------------------------------------------o---------o
| Rule 3 | |
| Rule 3 | |
| | |
| | |
o-------------------------------------------------o---------o
o-------------------------------------------------o---------o
</pre>
A large stock of rules can be derived in this way, by chaining together
A large stock of rules can be derived in this way, by chaining together segments that are selected from a stock of previous rules, with perhaps the whole process of derivation leading back to an axial body or a core stock of rules, with all recurring to and relying on an axiomatic basis. In order to keep track of their derivations, as their pedigrees help to remember the reasons for trusting their use in the first place, derived rules can be annotated by citing the rules from which they are derived.
segments that are selected from a stock of previous rules, with perhaps
the whole process of derivation leading back to an axial body or a core
stock of rules, with all recurring to and relying on an axiomatic basis.
In order to keep track of their derivations, as their pedigrees help to
remember the reasons for trusting their use in the first place, derived
rules can be annotated by citing the rules from which they are derived.
In the present discussion, I am using a particular style of annotation
In the present discussion, I am using a particular style of annotation for rule derivations, one that is called "proof by grammatical paradigm" or "proof by syntactic analogy". The annotations in the right margin of the Rule box can be read as the "denominators" of the paradigm that is being employed, in other words, as the alternating terms of comparison in a sequence of analogies. This can be illustrated by considering the derivation Rule 3 in detail. Taking the steps marked in the box one at a time, one can interweave the applications in the central body of the box with the annotations in the right margin of the box, reading "is to" for the ":" sign and "as" for the "::" sign, in the following fashion:
for rule derivations, one that is called "proof by grammatical paradigm"
or "proof by syntactic analogy". The annotations in the right margin of
the Rule box can be read as the "denominators" of the paradigm that is
being employed, in other words, as the alternating terms of comparison
in a sequence of analogies. This can be illustrated by considering the
derivation Rule 3 in detail. Taking the steps marked in the box one at
a time, one can interweave the applications in the central body of the
box with the annotations in the right margin of the box, reading "is to"
for the ":" sign and "as" for the "::" sign, in the following fashion:
<pre>
R3a. "u in X" is to R1a, namely, "u in X",
R3a. "u in X" is to R1a, namely, "u in X",
R3c. "{X}(u) = 1" is to R2b, namely, "f(u) = 1".
R3c. "{X}(u) = 1" is to R2b, namely, "f(u) = 1".
</pre>
Notice how the sequence of analogies pivots on the item R3b,
Notice how the sequence of analogies pivots on the item R3b, viewing it first under the aegis of R1b, as the second term of the first analogy, and then turning to view it again under the guise of R2a, as the first term of the second analogy.
viewing it first under the aegis of R1b, as the second term of
the first analogy, and then turning to view it again under the
guise of R2a, as the first term of the second analogy.
By way of convention, rules that are tailored to a particular application, case, or subject, and rules that are adapted to a particular goal, object, or purpose, I frequently refer to as "Facts".
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. Novel starting points for rules can be obtained by extracting pairs of equivalent expressions from a sequence that falls under an established rule, and then by stating their equality in the proper form of equation. For example, by extracting the equivalent expressions that are annotated as "R3a" and "R3c" in Rule 3 and by explictly stating their equivalence, one obtains the specialized result that is recorded in Corollary 1.
there is one other way that is commonly used to get new rules from old.
Novel starting points for rules can be obtained by extracting pairs of
equivalent expressions from a sequence that falls under an established
rule, and then by stating their equality in the proper form of equation.
For example, by extracting the equivalent expressions that are annotated
as "R3a" and "R3c" in Rule 3 and by explictly stating their equivalence,
one obtains the specialized result that is recorded in Corollary 1.
<pre>
Corollary 1
Corollary 1
C1a. u C X <=> {X}(u) = 1. R3a=R3c
C1a. u C X <=> {X}(u) = 1. R3a=R3c
</pre>
There are a number of issues, that arise especially in establishing the proper use of STR's, that are appropriate to discuss at this juncture. The notation "[S]" is intended to represent "the proposition denoted by the sentence S". There is only one problem with the use of this form. There is, in general, no such thing as "the" proposition denoted by S. Generally speaking, if a sentence is taken out of context and considered across a variety of different contexts, there is no unique proposition that it can be said to denote. But one is seldom ever speaking at the maximum level of generality, or even found to be thinking of it, and so this notation is usually meaningful and readily understandable whenever it is read in the proper frame of mind. Still, once the issue is raised, the question of how these meanings and understandings are possible has to be addressed, especially if one desires to express the regulations of their syntax in a partially computational form. This requires a closer examination of the very notion of "context", and it involves engaging in enough reflection on the "contextual evaluation" of sentences that the relevant principles of its successful operation can be discerned and rationalized in explicit terms.
There are a number of issues, that arise especially in establishing the proper use of ROSTs, that are appropriate to discuss at this juncture. The notation "[S]" is intended to represent "the proposition denoted by the sentence S". There is only one problem with the use of this form. There is, in general, no such thing as "the" proposition denoted by S. Generally speaking, if a sentence is taken out of context and considered across a variety of different contexts, there is no unique proposition that it can be said to denote. But one is seldom ever speaking at the maximum level of generality, or even found to be thinking of it, and so this notation is usually meaningful and readily understandable whenever it is read in the proper frame of mind. Still, once the issue is raised, the question of how these meanings and understandings are possible has to be addressed, especially if one desires to express the regulations of their syntax in a partially computational form. This requires a closer examination of the very notion of "context", and it involves engaging in enough reflection on the "contextual evaluation" of sentences that the relevant principles of its successful operation can be discerned and rationalized in explicit terms.
A sentence that is written in a context where it represents a value of 1 or 0 as a function of things in the universe U, where it stands for a value of "true" or "false", depending on how the signs that constitute its proper syntactic arguments are interpreted as denoting objects in U, in other words, where it is bound to lead its interpreter to view its own truth or falsity as determined by a choice of objects in U, is a sentence that might as well be written in the context "[ ... ]", whether or not this frame is explicitly marked around it.
A sentence that is written in a context where it represents a value of 1 or 0 as a function of things in the universe U, where it stands for a value of "true" or "false", depending on how the signs that constitute its proper syntactic arguments are interpreted as denoting objects in U, in other words, where it is bound to lead its interpreter to view its own truth or falsity as determined by a choice of objects in U, is a sentence that might as well be written in the context "[ ... ]", whether or not this frame is explicitly marked around it.
More often than not, the context of interpretation fixes the denotations of most of the signs that make up a sentence, and so it is safe to adopt the convention that only those signs whose objects are not already fixed are free to vary in their denotations. Thus, only the signs that remain in default of prior specification are subject to treatment as variables, with a decree of functional abstraction hanging over all of their heads.
More often than not, the context of interpretation fixes the denotations of most of the signs that make up a sentence, and so it is safe to adopt the convention that only those signs whose objects are not already fixed are free to vary in their denotations. Thus, only the signs that remain in default of prior specification are subject to treatment as variables, with a decree of functional abstraction hanging over all of their heads.
[u C X] = Lambda (u, C, X).(u C X).
: [u C X] = Lambda (u, C, X).(u C X).
As it is presently stated, Rule 1 lists a couple of manifest sentences, and it authorizes one to make exchanges in either direction between the syntactic items that have these two forms. But a sentence is any sign that denotes a proposition, and thus there are a number of less obvious sentences that can be added to this list, extending the number of items that are licensed to be exchanged. Consider the sense of equivalence among sentences that is recorded in Rule 4.
As it is presently stated, Rule 1 lists a couple of manifest sentences, and it authorizes one to make exchanges in either direction between the syntactic items that have these two forms. But a sentence is any sign that denotes a proposition, and thus there are a number of less obvious sentences that can be added to this list, extending the number of items that are licensed to be exchanged. Consider the sense of equivalence among sentences that is recorded in Rule 4.
<pre>
Rule 4
Rule 4
R4e. {X}(u) = 1.
R4e. {X}(u) = 1.
</pre>
The first and last items on this list, namely, the sentences "u C X" and "{X}(u) = 1" that are annotated as "R4a" and "R4e", respectively, are just the pair of sentences from Rule 3 whose equivalence for all u C U is usually taken to define the idea of an indicator function {X} : U -> B. At first sight, the inclusion of the other items appears to involve a category confusion, in other words, to mix the modes of interpretation and to create an array of mismatches between their own ostensible types and the ruling type of a sentence. On reflection, and taken in context, these problems are not as serious as they initially seem. For instance, the expression "[u C X]" ostensibly denotes a proposition, but if it does, then it evidently can be recognized, by virtue of this very fact, to be a genuine sentence. As a general rule, if one can see it on the page, then it cannot be a proposition, but can be, at best, a sign of one.
The first and last items on this list, namely, the sentences "u C X" and "{X}(u) = 1" that are annotated as "R4a" and "R4e", respectively, are just the pair of sentences from Rule 3 whose equivalence for all u C U is usually taken to define the idea of an indicator function {X} : U -> B. At first sight, the inclusion of the other items appears to involve a category confusion, in other words, to mix the modes of interpretation and to create an array of mismatches between their own ostensible types and the ruling type of a sentence. On reflection, and taken in context, these problems are not as serious as they initially seem. For instance, the expression "[u C X]" ostensibly denotes a proposition, but if it does, then it evidently can be recognized, by virtue of this very fact, to be a genuine sentence. As a general rule, if one can see it on the page, then it cannot be a proposition, but can be, at best, a sign of one.
The use of the basic connectives can be expressed in the form of a STR as follows:
The use of the basic connectives can be expressed in the form of a ROST as follows:
<pre>
Logical Translation Rule 0
Logical Translation Rule 0
L0c. [SurcJj Sj] = SurjJj [Sj] = SurjJj Pj.
L0c. [SurcJj Sj] = SurjJj [Sj] = SurjJj Pj.
</pre>
As a general rule, the application of a STR involves the recognition of an antecedent condition and the facilitation of a consequent condition. The antecedent condition is a state whose initial expression presents a match, in a formal sense, to one of the sentences that are listed in the STR, and the consequent condition is achieved by taking its suggestions seriously, in other words, by following its sequence of equivalents and implicants to some other link in its chain.
As a general rule, the application of a ROST involves the recognition of an antecedent condition and the facilitation of a consequent condition. The antecedent condition is a state whose initial expression presents a match, in a formal sense, to one of the sentences that are listed in the STR, and the consequent condition is achieved by taking its suggestions seriously, in other words, by following its sequence of equivalents and implicants to some other link in its chain.
Generally speaking, the application of a rule involves the recognition of an antecedent condition as a case that falls under a clause of the rule. This means that the antecedent condition is able to be captured in the form, conceived in the guise, expressed in the manner, grasped in the pattern, or recognized in the shape of one of the sentences in a list of equivalents or a chain of implicants.
Generally speaking, the application of a rule involves the recognition of an antecedent condition as a case that falls under a clause of the rule. This means that the antecedent condition is able to be captured in the form, conceived in the guise, expressed in the manner, grasped in the pattern, or recognized in the shape of one of the sentences in a list of equivalents or a chain of implicants.
A condition is "amenable" to a rule if any of its conceivable expressions formally match any of the expressions that are enumerated by the rule. Further, it requires the relegation of the other expressions to the production of a result. Thus, there is the choice of an initial expression that needs to be checked on input for whether it fits the antecedent condition and there are several types of output that are generated as a consequence, only a few of which are usually needed at any given time.
A condition is "amenable" to a rule if any of its conceivable expressions formally match any of the expressions that are enumerated by the rule. Further, it requires the relegation of the other expressions to the production of a result. Thus, there is the choice of an initial expression that needs to be checked on input for whether it fits the antecedent condition and there are several types of output that are generated as a consequence, only a few of which are usually needed at any given time.
<pre>
Logical Translation Rule 1
Logical Translation Rule 1
L1b11. [True] = (()) = 1 : U->B.
L1b11. [True] = (()) = 1 : U->B.
</pre>
<pre>
Geometric Translation Rule 1
Geometric Translation Rule 1
G1b11. {U} = (()) = 1 : U->B.
G1b11. {U} = (()) = 1 : U->B.
</pre>
<pre>
Logical Translation Rule 2
Logical Translation Rule 2
L2b15. [True] = (()) = 1 : U->B.
L2b15. [True] = (()) = 1 : U->B.
</pre>
<pre>
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)).
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.
</pre>
<pre>
Value Rule 1
Value Rule 1
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))
</pre>
<pre>
Value Rule 1
Value Rule 1
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 )).
</pre>
A rule that allows one to turn equivalent sentences into identical propositions:
A rule that allows one to turn equivalent sentences into identical propositions:
(S <=> T) <=> ([S] = [T])
: (S <=> T) <=> ([S] = [T])
Consider [ v = w ](v, w) and [ v(u) = w(u) ](u)
Consider [ v = w ](v, w) and [ v(u) = w(u) ](u)
<pre>
Value Rule 1
Value Rule 1
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 ))
</pre>
<pre>
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) ))
</pre>
<pre>
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 ))$
</pre>
<pre>
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). :$1b
</pre>
<pre>
Evaluation Rule 1
Evaluation Rule 1
about things in the universe U,
about things in the universe U,
f, g are propositions: U -> B,
f, g are propositions: U -> B,
and u C U,
and u C U,
then the following are equivalent:
then the following are equivalent:
E1a. f(u) = g(u). :V1a
E1a. f(u) = g(u). :V1a
E1d. (( f , g ))$(u). :$1b
E1d. (( f , g ))$(u). :$1b
</pre>
<pre>
Definition 2
Definition 2
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.
</pre>
<pre>
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.
</pre>
<pre>
Definition 4
Definition 4
If X c U,
If X c U,
then the following are identical subsets of UxB:
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]}
</pre>
<pre>
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}.
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.
</pre>
Given an indexed set of sentences, Sj for j C J, it is possible to consider the logical conjunction of the corresponding propositions. Various notations for this concept are be useful in various contexts, a sufficient sample of which are recorded in Definition 6.
Given an indexed set of sentences, Sj for j C J, it is possible to consider the logical conjunction of the corresponding propositions. Various notations for this concept are be useful in various contexts, a sufficient sample of which are recorded in Definition 6.
<pre>
Definition 6
Definition 6
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.
</pre>
<pre>
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].
</pre>
<pre>
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
</pre>
<pre>
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
</pre>
<pre>
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). :$1b
</pre>
<pre>
Rule 8
Rule 8
about things in the universe U,
about things in the universe U,
then the following are equivalent:
then the following are equivalent:
R8a. S <=> T. :D7a
R8a. S <=> T. :D7a
R8g. ConjUu (( [S] , [T] ))$(u). :R7f
R8g. ConjUu (( [S] , [T] ))$(u). :R7f
</pre>
For instance, the observation that expresses the equality of sets in terms of their indicator functions can be formalized according to the pattern in Rule 9, namely, at lines (a, b, c), and these components of Rule 9 can be cited in future uses as "R9a", "R9b", "R9c", respectively. Using Rule 7, annotated as "R7", to adduce a few properties of indicator functions to the account, it is possible to extend Rule 9 by another few steps, referenced as "R9d", "R9e", "R9f", "R9g".
For instance, the observation that expresses the equality of sets in terms of their indicator functions can be formalized according to the pattern in Rule 9, namely, at lines (a, b, c), and these components of Rule 9 can be cited in future uses as "R9a", "R9b", "R9c", respectively. Using Rule 7, annotated as "R7", to adduce a few properties of indicator functions to the account, it is possible to extend Rule 9 by another few steps, referenced as "R9d", "R9e", "R9f", "R9g".
<pre>
Rule 9
Rule 9
If X, Y c U,
If X, Y c U,
then the following are equivalent:
R9a. X = Y. :R5a
R9a. X = Y. :R5a
::
::
R9g. ConjUu (( {X} , {Y} ))$(u). :R7f
R9g. ConjUu (( {X} , {Y} ))$(u). :R7f
</pre>
<pre>
Rule 10
Rule 10
If X, Y c U,
If X, Y c U,
then the following are equivalent:
then the following are equivalent:
R10a. X = Y. :D2a
R10a. X = Y. :D2a
R10h. ConjUu (( [u C X] , [u C Y] ))$(u). :R8g
R10h. ConjUu (( [u C X] , [u C Y] ))$(u). :R8g
</pre>
<pre>
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
R11c. {X} c UxB
R11c. {X} c UxB
: {X} = {< u, v> C UxB : v = [S](u)}. :R
: {X} = {< u, v> C UxB : v = [S](u)}. :R
R11d. {X} : U -> B
R11d. {X} : U -> B
: {X}(u) = [S](u), for all u C U. :R
: {X}(u) = [S](u), for all u C U. :R
R11e. {X} = [S]. :R
R11e. {X} = [S]. :R
</pre>
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.
<pre>
Fact 1
Fact 1
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=====
=====1.3.12.2. Derived Equivalence Relations=====
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.
A different way of seeing that two sets are equal, or of seeing whether two sets are equal, is based on the following observation:
A different way of seeing that two sets are equal, or of seeing whether two sets are equal, is based on the following observation:
<pre>
Two sets are equal as sets
Two sets are equal as sets
<=> the values of these functions are equal on all domain elements.
<=> the values of these functions are equal on all domain elements.
</pre>
It is important to notice the hidden quantifier, of a universal kind, that lurks in all three equivalent statements but is only revealed in the last.
It is important to notice the hidden quantifier, of a universal kind, that lurks in all three equivalent statements but is only revealed in the last.
Recall the definition of Con(R), the connotative component of R, in the following form:
Recall the definition of Con(R), the connotative component of R, in the following form:
Con(R) = RSI = {< s, i> C SxI : <o, s, i> C R for some o C O}.
: Con(R) = RSI = {< s, i> C SxI : <o, s, i> C R for some o C O}.
Equivalent expressions for this concept are recorded in Definition 8.
Equivalent expressions for this concept are recorded in Definition 8.
<pre>
Definition 8
Definition 8
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}
</pre>
The dyadic relation RIS that constitutes the converse of the connotative relation RSI can be defined directly in the following fashion:
The dyadic relation RIS that constitutes the converse of the connotative relation RSI can be defined directly in the following fashion:
Con(R)^ = RIS = {< i, s> C IxS : <o, s, i> C R for some o C O}.
: Con(R)^ = RIS = {< i, s> C IxS : <o, s, i> C R for some o C O}.
A few of the many different expressions for this concept are recorded in Definition 9.
A few of the many different expressions for this concept are recorded in Definition 9.
<pre>
Definition 9
Definition 9
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)^
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}
</pre>
D9g. {< i, s> C IxS : <o, s, i> C R for some o C O}
Recall the definition of Den(R), the denotative component of R, in the following form:
Recall the definition of Den(R), the denotative component of R, in the following form:
Den(R) = ROS = {<o, s> C OxS : <o, s, i> C R for some i C I}.
: Den(R) = ROS = {<o, s> C OxS : <o, s, i> C R for some i C I}.
Equivalent expressions for this concept are recorded in Definition 10.
Equivalent expressions for this concept are recorded in Definition 10.
<pre>
Definition 10
Definition 10
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}
</pre>
The dyadic relation RSO that constitutes the converse of the denotative relation ROS can be defined directly in the following fashion:
The dyadic relation RSO that constitutes the converse of the denotative relation ROS can be defined directly in the following fashion:
Den(R)^ = RSO = {< s, o> C SxO : <o, s, i> C R for some i C I}.
: Den(R)^ = RSO = {< s, o> C SxO : <o, s, i> C R for some i C I}.
A few of the many different expressions for this concept are recorded in Definition 11.
A few of the many different expressions for this concept are recorded in Definition 11.
<pre>
Definition 11
Definition 11
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}
</pre>
The "denotation of x in R", written "Den(R, x)", is defined as follows:
The "denotation of x in R", written "Den(R, x)", is defined as follows:
Den(R, x) = {o C O : <o, x> C Den(R)}.
: Den(R, x) = {o C O : <o, x> C Den(R)}.
In other words:
In other words:
Den(R, x) = {o C O : <o, x, i> C R for some i C I}.
: Den(R, x) = {o C O : <o, x, i> C R for some i C I}.
Equivalent expressions for this concept are recorded in Definition 12.
Equivalent expressions for this concept are recorded in Definition 12.
<pre>
Definition 12
Definition 12
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)}
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}
</pre>
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 jumping to the conclusions that are implied by these latter terms.
To define the "equiference" of signs in terms of their denotations, one says that "x is equiferent to y under R", and writes "x =R y", to mean that Den(R, x) = Den(R, y). Taken in extension, this notion of a relation between signs induces an "equiference relation" on the syntactic domain.
To define the "equiference" of signs in terms of their denotations, one says that "x is equiferent to y under R", and writes "x =R y", to mean that Den(R, x) = Den(R, y). Taken in extension, this notion of a relation between signs induces an "equiference relation" on the syntactic domain.
For each sign relation R, this yields a binary relation Der(R) c SxI that is defined as follows:
For each sign relation R, this yields a binary relation Der(R) c SxI that is defined as follows:
Der(R) = DerR = {<x, y> C SxI : Den(R, x) = Den(R, y)}.
: Der(R) = DerR = {<x, y> C SxI : Den(R, x) = Den(R, y)}.
These definitions and notations are recorded in the following display.
These definitions and notations are recorded in the following display.
<pre>
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)}
</pre>
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.
# 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.
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.
<pre>
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
} :D12a
} :D12a
</pre>
<pre>
Fact 2.2
Fact 2.2
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 :
} :Log
} :Log
F2.2e. {DerR} = {<x, y, v> C SxIxB :
F2.2e. {DerR} = {<x, y, v> C SxIxB :
} :Log
} :Log
F2.2f. {DerR} = {<x, y, v> C SxIxB :
F2.2f. {DerR} = {<x, y, v> C SxIxB :
} :$
} :$
</pre>
<pre>
Fact 2.3
Fact 2.3
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>
=====1.3.12.3. Digression on Derived Relations=====
=====1.3.10.16 Digression on Derived 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.
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.
To that end, let the derivation Der(R) be expressed in the following way:
To that end, let the derivation Der(R) be expressed in the following way:
{DerR}(x, y) = Conj(o C O) (( {RSO}(x, o) , {ROS}(o, y) )).
: {DerR}(x, y) = Conj(o C O) (( {RSO}(x, o) , {ROS}(o, y) )).
From this abstract a form of composition, temporarily notated as "P#Q", where P c XxM and Q c MxY are otherwise arbitrary dyadic relations, and where P#Q c XxY is defined as follows:
From this abstract a form of composition, temporarily notated as "P#Q", where P c XxM and Q c MxY are otherwise arbitrary dyadic relations, and where P#Q c XxY is defined as follows:
{P#Q}(x, y) = Conj(m C M) (( {P}(x, m) , {Q}(m, y) )).
: {P#Q}(x, y) = Conj(m C M) (( {P}(x, m) , {Q}(m, y) )).
Compare this with the usual form of composition, typically notated as "P.Q" and defined as follows:
Compare this with the usual form of composition, typically notated as "P.Q" and defined as follows:
{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) ).
===1.4. Outlook of the Project : All Ways Lead to Inquiry===
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 being the shorter term, because of the following advantages:
# It allows one to broaden the scope of investigation to include any form of proceeding toward knowledge that merely aims at such a method.
# It allows one to finesse the issue, for the time being, of how much "method" there is in science.
This Subdivision and the next deal with opposite aspects of inquiry. In many ways it might have been better to interlace the opposing points of comparison, taking them up in a parallel fashion, but this plan was judged to be too distracting for a first approach. In other ways, the negative sides of each topic are prior in point of time to the positive sides of the issue, but sensible people like to see the light at the end of the tunnel before they trouble themselves with the obscurities of the intervening journey. Thus, this Subdivision of the text emphasizes the positive features of inquiry and the positive qualities of its objective, while the next Subdivision is reserved to examine the negative aspects of each question.
at least the names of enough positive concepts to propose addressing the
negative conditions of knowledge in which inquiry necessarily starts.
In this Subdivision I stand back once again from the problem of inquiry
In the order of nature, the absence of a feature naturally precedes the full development of its presence. In the order of discussion, however, positive terms must be proposed if it is desired to say anything at all.
The reader may take my apology for this style of presentation to be
The discussion in this Subdivision is placed to serve a primer, declaring at least the names of enough positive concepts to propose addressing the negative conditions of knowledge in which inquiry necessarily starts.
In this Subdivision I stand back once again from the problem of inquiry and allow myself take a more distant view of the subject, settling into what I think is a comfortable and a natural account of inquiry, the best that I have at my command, and attending to the task of describing its positive features in a positive light. I present my personal view of inquiry as I currently understand it, without stopping to justify every concept in detail or to examine every objection that might be made to this view. In the next Subdivision I discuss a few of the more obvious problems that stand in the way of this view and I try to remove a few of the more tractable obscurities that appear ready to be cleared up. The fact that I treat them as my "personal insights" does not mean that all of these ideas about inquiry originate with me, but only that I have come to adopt them for my personal use. There will be many occasions, the next time that I go over this ground, to point out the sources of these ideas, so far as I know them.
In this subdivision I stand back once again from the problem of inquiry and allow myself take a more distant view of the subject, settling into what I think is a comfortable and a natural account of inquiry, the best that I have at my command, and attending to the task of describing its positive features in a positive light. I present my personal view of inquiry as I currently understand it, without stopping to justify every concept in detail or to examine every objection that might be made to this view. In the next subdivision I discuss a few of the more obvious problems that stand in the way of this view and I try to remove a few of the more tractable obscurities that appear ready to be cleared up. The fact that I treat them as my "personal insights" does not mean that all of these ideas about inquiry originate with me, but only that I have come to adopt them for my personal use. There will be many occasions, the next time that I go over this ground, to point out the sources of these ideas, so far as I know them.
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.
====1.4.1 The Matrix of Inquiry (2)====
====1.4.1. The Matrix of Inquiry====
<blockquote>
<blockquote>
<p>Thus when mothers have chidren suffering from sleeplessness, and want to lull them to rest, the treatment they apply is to give them, not quiet, but motion, for they rock them constantly in their arms; and instead of silence, they use a kind of crooning noise; and thus they literally cast a spell upon the children (like the victims of a Bacchic frenzy) by employing the combined movements of dance and song as a remedy.</p>
<p>Thus when mothers have children suffering from sleeplessness, and want to lull them to rest, the treatment they apply is to give them, not quiet, but motion, for they rock them constantly in their arms; and instead of silence, they use a kind of crooning noise; and thus they literally cast a spell upon the children (like the victims of a Bacchic frenzy) by employing the combined movements of dance and song as a remedy.</p>
<p>(Plato, Laws, VII, 790D).</p>
<p>(Plato, ''Laws'', VII, 790D).</p>
</blockquote>
</blockquote>
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, <math>y_0 = y \cdot y,</math> 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.
I can sum all this up by saying that an inquiry requires an inquirer to suggest a hypothesis about the subject of interest and then to put that particular model of the subject to the test. This in turn requires one to devote a modicum of personal effort to the task of testing the chosen hypothesis, to put a quantum of personal interest at stake for the sake of finding out whether the model fits the subject, and, overall, to take the risk of being wrong. Any model that is feasible is also defeasible, at least, where it concerns a contingent subject of inquiry.
I can sum all this up by saying that an inquiry requires an inquirer to suggest a hypothesis about the subject of interest and then to put that particular model of the subject to the test. This in turn requires one to devote a modicum of personal effort to the task of testing the chosen hypothesis, to put a quantum of personal interest at stake for the sake of finding out whether the model fits the subject, and, overall, to take the risk of being wrong. Any model that is feasible is also defeasible, at least, where it concerns a contingent subject of inquiry.
Leaving behind the mythical conditions of pure experience and naive observation, and at least by the time that one comes to give a name to the subject of investigation, one's trek through the data is already half-shod, half-fettered by the connotations of the name, and in turn by all of the concepts that it invokes in its train. The name, the concepts that it suggests, and the tacit but vague definition of the subject that this complex of associations is already beginning to constellate, attract certain experiences to the complex and filter out other observations from having any bearing on the subject matter. By this point, one is already busy translating one's empirical acquaintance with the subject into an arrangement of concepts that is intended to define its essential nature.
Leaving behind the mythical conditions of pure experience and naive observation, and at least by the time that one comes to give a name to the subject of investigation, one's trek through the data is already half-shod, half-fettered by the connotations of the name, and in turn by all of the concepts that it invokes in its train. The name, the concepts that it suggests, and the tacit but vague definition of the subject that this complex of associations is already beginning to constellate, attract certain experiences to the complex and filter out other observations from having any bearing on the subject matter. By this point, one is already busy translating one's empirical acquaintance with the subject into an arrangement of concepts that is intended to define its essential nature.
An array of concepts that is set up to capture the essence of a subject is a provisional definition of it, an implicit model of the subject that contains the makings of an explicit theory. It amounts to a selection from the phenomenal aspects of the subject, expresses a guess about its relevant features, and constitutes a hypothesis in explanation of its experienced characteristics. This incipient order of model or theory is tantamount to a definition because it sets bounds on the "stretches" and the "holds" of a term - its extension, intension, and intention - but this is not the kind of definition that has to be taken on faith, or that constitutes the first and the last word on the subject. In other words, it is an empirical definition, one that is subject to being falsified in reference to its intended subject, by failing to indicate the necessary, the pertinent, or the relevant features that account for the presence of its phenomena or the persistence of its process.
An array of concepts that is set up to capture the essence of a subject is a provisional definition of it, an implicit model of the subject that contains the makings of an explicit theory. It amounts to a selection from the phenomenal aspects of the subject, expresses a guess about its relevant features, and constitutes a hypothesis in explanation of its experienced characteristics. This incipient order of model or theory is tantamount to a definition because it sets bounds on the "stretches" and the "holds" of a term — its extension, intension, and intention — but this is not the kind of definition that has to be taken on faith, or that constitutes the first and the last word on the subject. In other words, it is an empirical definition, one that is subject to being falsified in reference to its intended subject, by failing to indicate the necessary, the pertinent, or the relevant features that account for the presence of its phenomena or the persistence of its process.
If I reflect on the conduct of inquiry, seeking to fix it in a fitting image and trying to cast it in a positive light, the best I can do is this:
If I reflect on the conduct of inquiry, seeking to fix it in a fitting image and trying to cast it in a positive light, the best I can do is this:
Inquiry is a process that aims at achieving belief or knowledge.
: Inquiry is a process that aims at achieving belief or knowledge.
But even this simple a description already plunges the discussion deep into a number of obscurities. Most prominently, there is the disjunction between belief and knowledge that cries out to be explained or resolved. Stirring beneath the surface, and not quite fading into the background, many of the other terms that are invoked in the description are capable of hiding the entire contents of the original ignorance that the image as a whole is aimed to dispell. And yet, there is nothing that I can do in this avowedly positive context but to mark these points down as topics for future discussion.
But even this simple a description already plunges the discussion deep into a number of obscurities. Most prominently, there is the disjunction between belief and knowledge that cries out to be explained or resolved. Stirring beneath the surface, and not quite fading into the background, many of the other terms that are invoked in the description are capable of hiding the entire contents of the original ignorance that the image as a whole is aimed to dispel. And yet, there is nothing that I can do in this avowedly positive context but to mark these points down as topics for future discussion.
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.
=====1.4.1.1 Inquiry as Conduct=====
=====1.4.1.1. Inquiry as Conduct=====
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 species of prototypically human action that is both dynamic and deliberate, or conceived to fall under a form of purposeful control, usually conscious but possibly not. For the sake of clarity, it helps to seek a more formal definition of conduct, one that expresses the concept in terms of abstract features rather than trying to suggest it by means of typical examples.
This invokes the technical term "conduct", referring to the
species of prototypically human action that is both dynamic
and deliberate, or conceived to fall under a form of purposeful
control, usually conscious but possibly not. For the sake of
clarity, it helps to seek a more formal definition of conduct,
one that expresses the concept in terms of abstract features
rather than trying to suggest it by means of typical examples.
Conduct is action with respect to an object. The distinction between action and conduct, reduced to the level of the most abstract formal relations that are involved, can be described in the following manner.
Conduct is action with respect to an object. The distinction between action and conduct, reduced to the level of the most abstract formal relations that are involved, can be described in the following manner.
What does this definition of conduct say about the temporal ordering of the object with respect to the states? The states are conceived to be ordered in time, but so far nothing has been said to pin down where in relation to these states the object must be conceived to fall in time. Nor does the definition make any particular specification necessary. This makes the question of relative time a secular parameter of the definition, allowing the consideration of the following options:
What does this definition of conduct say about the temporal ordering of the object with respect to the states? The states are conceived to be ordered in time, but so far nothing has been said to pin down where in relation to these states the object must be conceived to fall in time. Nor does the definition make any particular specification necessary. This makes the question of relative time a secular parameter of the definition, allowing the consideration of the following options:
# If the object is thought to precede the action of the conduct, then it tends to be regarded as a creative act, an initial intention, an original stimulus, a principal cause, or a prime mover.
# If the object is thought to succeed the action of the conduct, then it tends to be regarded as an end, a goal, or a purpose, in other words, a state envisioned to be fulfilled.
# If the object is thought to be concurrent, immanent, or transcendent throughout the action of the conduct, then it tends to be regarded as falling under one of the following possibilities: a prevailing value, a controlling parameter, a universal system of effective forces, a pervasive field of potentials, a ruling law, or a governing principle.
A prevailing value or a controlling parameter, which guides the temporal development of a system, is a term that fits into a law or a principle, which governs the system at a higher level. The existence of a value or a law that rules a system, and the information that an agent of the system has about its parameters and its principles, are two different matters. Indeed, a major task of development for an inquiring agent is to inform itself about the values and the laws that form its own system. Thus, one of the objects of the conduct of inquiry is a description in terms of laws and values of the rules that govern and guide inquiry.
A prevailing value or a controlling parameter, which guides the temporal development of a system, is a term that fits into a law or a principle, which governs the system at a higher level. The existence of a value or a law that rules a system, and the information that an agent of the system has about its parameters and its principles, are two different matters. Indeed, a major task of development for an inquiring agent is to inform itself about the values and the laws that form its own system. Thus, one of the objects of the conduct of inquiry is a description in terms of laws and values of the rules that govern and guide inquiry.
The elaboration of an object in terms of this rich vocabulary -- as a cause, end, field, force, goal, intention, law, parameter, principle, purpose, system, or value -- adds colorful detail and concrete sensation to the account, and it helps to establish connections with the arrays of terminology that are widely used to discuss these issues. From a formal and relational point of view, however, all of these concepts are simply different ways of describing, at possibly different levels of generality, the object of a form of conduct. With that in mind, I find it useful to return to the simpler form of description as often as possible.
The elaboration of an object in terms of this rich vocabulary — as a cause, end, field, force, goal, intention, law, parameter, principle, purpose, system, or value — adds colorful detail and concrete sensation to the account, and it helps to establish connections with the arrays of terminology that are widely used to discuss these issues. From a formal and relational point of view, however, all of these concepts are simply different ways of describing, at possibly different levels of generality, the object of a form of conduct. With that in mind, I find it useful to return to the simpler form of description as often as possible.
This account of conduct brings to the fore a number of issues, some of them new and some of them familiar, but each of them allowing itself to be approached from a fresh direction by treating it as an implication of a critical thesis just laid down. I next examine these issues in accord with the tenets from which they stem.
This account of conduct brings to the fore a number of issues, some of them new and some of them familiar, but each of them allowing itself to be approached from a fresh direction by treating it as an implication of a critical thesis just laid down. I next examine these issues in accord with the tenets from which they stem.
Once again, there is a distinction between (a) the properties of an action, agent, conduct, or system, as expressible by the agent that is engaged in the conduct, or as representable within the system that is undergoing the action, and (b) the properties of the same entities, as evident from an "external viewpoint", or as statable by the equivalent of an "outside observer".
Once again, there is a distinction between (a) the properties of an action, agent, conduct, or system, as expressible by the agent that is engaged in the conduct, or as representable within the system that is undergoing the action, and (b) the properties of the same entities, as evident from an "external viewpoint", or as statable by the equivalent of an "outside observer".
3. Reflection is a part of inquiry.
3. Reflection is a part of inquiry. Reflection is a form of conduct.
The task of reflection on conduct is to pass from a purely interior view of one's own conduct to an outlook that is, effectively, an exterior view.
The task of reflection on conduct is to pass from a purely interior view of one's own conduct to an outlook that is, effectively, an exterior view.
What does this definition of conduct say about the temporal ordering of the object with respect to the states? The states are conceived to be ordered in time, but so far nothing has been said to pin down where in relation to these states the object must be conceived to fall in time. Nor does the definition make any particular specification necessary. This makes the question of relative time a secular parameter of the definition, allowing the consideration of the following options:
What does this definition of conduct say about the temporal ordering of the object with respect to the states? The states are conceived to be ordered in time, but so far nothing has been said to pin down where in relation to these states the object must be conceived to fall in time. Nor does the definition make any particular specification necessary. This makes the question of relative time a secular parameter of the definition, allowing the consideration of the following options:
# If the object is thought to precede the action of the conduct, then it tends to be regarded as a creative act, an initial intention, an original stimulus, a principal cause, or a prime mover.
# If the object is thought to succeed the action of the conduct, then it tends to be regarded as an end, a goal, or a purpose, in other words, a state envisioned to be fulfilled.
# If the object is thought to be concurrent, immanent, or transcendent throughout the action of the conduct, then it tends to be regarded as falling under one of the following possibilities: a prevailing value, a controlling parameter, a universal system of effective forces, a pervasive field of potentials, a ruling law, or a governing principle.
A prevailing value or a controlling parameter, which guides the temporal development of a system, is a term that fits into a law or a principle, which governs the system at a higher level. The existence of a value or a law that rules a system, and the information that an agent of the system has about its parameters and its principles, are two different matters. Indeed, a major task of development for an inquiring agent is to inform itself about the values and the laws that form its own system. Thus, one of the objects of the conduct of inquiry is a description in terms of laws and values of the rules that govern and guide inquiry.
A prevailing value or a controlling parameter, which guides the temporal development of a system, is a term that fits into a law or a principle, which governs the system at a higher level. The existence of a value or a law that rules a system, and the information that an agent of the system has about its parameters and its principles, are two different matters. Indeed, a major task of development for an inquiring agent is to inform itself about the values and the laws that form its own system. Thus, one of the objects of the conduct of inquiry is a description in terms of laws and values of the rules that govern and guide inquiry.
The elaboration of an object in terms of this rich vocabulary - as a cause, end, field, force, goal, intention, law, parameter, principle, purpose, system, or value - adds colorful detail and concrete sensation to the account, and it helps to establish connections with the arrays of terminology that are widely used to discuss these issues. From a formal and relational point of view, however, all of these concepts are simply different ways of describing, at possibly different levels of generality, the object of a form of conduct. With that in mind, I find it useful to return to the simpler form of description as often as possible.
The elaboration of an object in terms of this rich vocabulary — as a cause, end, field, force, goal, intention, law, parameter, principle, purpose, system, or value — adds colorful detail and concrete sensation to the account, and it helps to establish connections with the arrays of terminology that are widely used to discuss these issues. From a formal and relational point of view, however, all of these concepts are simply different ways of describing, at possibly different levels of generality, the object of a form of conduct. With that in mind, I find it useful to return to the simpler form of description as often as possible.
This account of conduct brings to the fore a number of issues, some of them new and some of them familiar, but each of them allowing itself to be approached from a fresh direction by treating it as an implication of a critical thesis just laid down. I next examine these issues in accord with the tenets from which they stem.
This account of conduct brings to the fore a number of issues, some of them new and some of them familiar, but each of them allowing itself to be approached from a fresh direction by treating it as an implication of a critical thesis just laid down. I next examine these issues in accord with the tenets from which they stem.
1. Inquiry is a form of conduct.
1. Inquiry is a form of conduct.
This makes inquiry into inquiry a special case of inquiry into conduct. Certainly, it must be possible to reason about conduct in general, especially if forms of conduct need to be learned, examined, modified, and improved.
This makes inquiry into inquiry a special case of inquiry into conduct. Certainly, it must be possible to reason about conduct in general, especially if forms of conduct need to be learned, examined, modified, and improved.
Placing the subject of inquiry within the subject of conduct and making the inquiry into inquiry a subordinate part of the inquiry into conduct does not automatically further the investigation, especially if it turns out that the general subject of conduct is more difficult to understand than the specialized subject of inquiry. But in those realms of inquiry where it is feasible to proceed hypothetically and recursively, stretching the appropriate sort of hypothesis over a wider subject area can act to prime the pump of mathematical induction all the more generously, and actually increase the power of the recursion. Of course, the use of a recursive strategy comes at the expense of having to establish a more extended result at the base.
Placing the subject of inquiry within the subject of conduct and making the inquiry into inquiry a subordinate part of the inquiry into conduct does not automatically further the investigation, especially if it turns out that the general subject of conduct is more difficult to understand than the specialized subject of inquiry. But in those realms of inquiry where it is feasible to proceed hypothetically and recursively, stretching the appropriate sort of hypothesis over a wider subject area can act to prime the pump of mathematical induction all the more generously, and actually increase the power of the recursion. Of course, the use of a recursive strategy comes at the expense of having to establish a more extended result at the base.
2. The existence of an object that rules a form of conduct and the information that an agent of the conduct has about the object are two different matters.
2. The existence of an object that rules a form of conduct and the information that an agent of the conduct has about the object are two different matters.
This means that the exact specification of the object can require an order of information that the agent does not have available, at least, not for use in reflective action, or even an amount of information that the agent lacks the capacity to store. No matter how true it is that the actual course of the agent's conduct exactly reflects the influence of the object, and thus, in a sense, represents the object exactly, the question is whether the agent possesses the equivalent of this information in any kind of accessible, exploitable, reflective, surveyable, or usable form of representation, in effect, any mode of information that the agent can use to forsee, to modify, or to temper its own temporal course.
This means that the exact specification of the object can require an order of information that the agent does not have available, at least, not for use in reflective action, or even an amount of information that the agent lacks the capacity to store. No matter how true it is that the actual course of the agent's conduct exactly reflects the influence of the object, and thus, in a sense, represents the object exactly, the question is whether the agent possesses the equivalent of this information in any kind of accessible, exploitable, reflective, surveyable, or usable form of representation, in effect, any mode of information that the agent can use to forsee, to modify, or to temper its own temporal course.
This issue may seem familiar as a repetition of the "meta" question. Once again, there is a distinction between (a) the properties of an action, agent, conduct, or system, as expressible by the agent that is engaged in the conduct, or as representable within the system that is undergoing the action, and (b) the properties of the same entities, as evident from an "external viewpoint", or as statable by the equivalent of an "outside observer".
This issue may seem familiar as a repetition of the "meta" question. Once again, there is a distinction between (a) the properties of an action, agent, conduct, or system, as expressible by the agent that is engaged in the conduct, or as representable within the system that is undergoing the action, and (b) the properties of the same entities, as evident from an "external viewpoint", or as statable by the equivalent of an "outside observer".
3. Reflection is a part of inquiry. Reflection is a form of conduct.
3. Reflection is a part of inquiry. Reflection is a form of conduct.
The task of reflection on conduct is to pass from a purely interior view of one's own conduct to an outlook that is, effectively, an exterior view. What is sought is a wider perspective, one that is able to incorporate the sort of information that might be available to an outside observer, that ought to be evident from an external vantage point, or that one reasonably imagines might be obvious from an independent viewpoint. I am tempted to refer to such a view as a "quasi-objective perspective", but only so long as it possible to keep in mind that there is no such thing as a "completely outside perspective", at least, not one that a finite and mortal agent can hope to achieve, nor one that a reasonably socialized member of a community can wish to take up as a permanent station in life.
The task of reflection on conduct is to pass from a purely interior view of one's own conduct to an outlook that is, effectively, an exterior view. What is sought is a wider perspective, one that is able to incorporate the sort of information that might be available to an outside observer, that ought to be evident from an external vantage point, or that one reasonably imagines might be obvious from an independent viewpoint. I am tempted to refer to such a view as a "quasi-objective perspective", but only so long as it possible to keep in mind that there is no such thing as a "completely outside perspective", at least, not one that a finite and mortal agent can hope to achieve, nor one that a reasonably socialized member of a community can wish to take up as a permanent station in life.
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.
=====1.4.1.2 Types of Conduct=====
=====1.4.1.2. Types of Conduct=====
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?
=====1.4.1.3 Perils of Inquiry=====
=====1.4.1.3. Perils of Inquiry=====
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?
In making a hypothesis or choosing a model, one appears to select from a vaster number of conceivable possibilities than a finite agent could ever enumerate in complete detail or consider as an articulate totality. As the very nature of a contingent description and the very character of a discriminate action is to apply in some cases but not in others, there is no escaping the making of a risky hypothesis or a speculative interpretation, even in the realm of a purely mental action. Thus, all significant thought, even thinking to any purpose about thought itself, demands a guess at the subject or a grasp of the situation that is contingent, dubious, fallible, and uncertain.
In making a hypothesis or choosing a model, one appears to select from a vaster number of conceivable possibilities than a finite agent could ever enumerate in complete detail or consider as an articulate totality. As the very nature of a contingent description and the very character of a discriminate action is to apply in some cases but not in others, there is no escaping the making of a risky hypothesis or a speculative interpretation, even in the realm of a purely mental action. Thus, all significant thought, even thinking to any purpose about thought itself, demands a guess at the subject or a grasp of the situation that is contingent, dubious, fallible, and uncertain.
If all this is true - if inquiry begins with doubt, if every significant hypothesis is itself a dubious proposition, if the making and the testing of a hypothesis are instances of equally doubtful actions, and if every action has the potential to alter the very situation and the very subject matter that are being addressed - then it leads to the critical question: How is the conduct of inquiry, that begins by making a hypothesis and that continues by testing this description in action, supposed to help with the situation of uncertainty that incites it in the first place and that is supposed to maintain its motivation until the end is reached? The danger is that the posing of a hypothesis may literally introduce an irreversible change in the situation or the subject matter in question. The fear is that this change might be one that too conveniently fulfills or too perversely subverts the very hypothesis that engenders it, that it may obstruct the hypothesis from ever being viewed with equanimity again, and thus prevent the order of reflection that is needed to amend or discard the hypothesis when the occasion to do so arises.
If all this is true — if inquiry begins with doubt, if every significant hypothesis is itself a dubious proposition, if the making and the testing of a hypothesis are instances of equally doubtful actions, and if every action has the potential to alter the very situation and the very subject matter that are being addressed — then it leads to the critical question: How is the conduct of inquiry, that begins by making a hypothesis and that continues by testing this description in action, supposed to help with the situation of uncertainty that incites it in the first place and that is supposed to maintain its motivation until the end is reached? The danger is that the posing of a hypothesis may literally introduce an irreversible change in the situation or the subject matter in question. The fear is that this change might be one that too conveniently fulfills or too perversely subverts the very hypothesis that engenders it, that it may obstruct the hypothesis from ever being viewed with equanimity again, and thus prevent the order of reflection that is needed to amend or discard the hypothesis when the occasion to do so arises.
If one fears that merely contemplating a special hypothesis is enough to admit a spurious demonstration into the foundations of one's reasoning, even to allow a specious demon to subvert all one's hopes of a future rationality and to destroy all one's chances of a reasonable share of knowledge, then one is hardly in a state of mind that can tolerate the tensions of a full-fledged, genuine inquiry. If one is beset with such radical doubts, then all inquiry is no more comfort than pure enchoiry. Sometimes it seems like the best you can do is sing yourself a song that soothes your doubts. Perhaps it is even quite literally true that all inquiry comes back at last to a form of "enchoiry", the invocation of a nomos, a way of life, or a song and a dance. But even if this is the ultimate case, it does no harm and it does not seem like a bad idea to store up in this song one or two bits of useful lore, and to weave into its lyric a few suggestions of a practical character.
If one fears that merely contemplating a special hypothesis is enough to admit a spurious demonstration into the foundations of one's reasoning, even to allow a specious demon to subvert all one's hopes of a future rationality and to destroy all one's chances of a reasonable share of knowledge, then one is hardly in a state of mind that can tolerate the tensions of a full-fledged, genuine inquiry. If one is beset with such radical doubts, then all inquiry is no more comfort than pure enchoiry. Sometimes it seems like the best you can do is sing yourself a song that soothes your doubts. Perhaps it is even quite literally true that all inquiry comes back at last to a form of "enchoiry", the invocation of a nomos, a way of life, or a song and a dance. But even if this is the ultimate case, it does no harm and it does not seem like a bad idea to store up in this song one or two bits of useful lore, and to weave into its lyric a few suggestions of a practical character.
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.
=====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:
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 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.
In general, a conduct relation serves as a "model of conduct" (MOC), not always the kind of model that is meant to be emulated, but the type of model that captures an aspect of structure in a form of conduct.
In general, a conduct relation serves as a ''model of conduct'' (MOC), not always the kind of model that is meant to be emulated, but the type of model that captures an aspect of structure in a form of conduct.
The question arises: What is the relationship between signs and states? On the assumption that signs and states are comparable in their levels of generality, consider the following possibilities:
The question arises: What is the relationship between signs and states? On the assumption that signs and states are comparable in their levels of generality, consider the following possibilities:
# Signs are special cases of states.
# Signs and states are the same sorts of things.
# States are special cases of signs.
Depending on how one answers this question, one is also choosing among the following options:
Depending on how one answers this question, one is also choosing among the following options:
# Sign relations are special cases of conduct relations.
# Sign relations and conduct relations are the same sorts of things.
# Conduct relations are special cases of sign relations.
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.
=====1.4.1.5 Models of Inquiry=====
=====1.4.1.5. Models of Inquiry=====
The value of a hypothesis, or the worth of a model, is not to be given a prior justification, as by a deductive proof, but has to be examined in practice, as by an empirical probation. It is not intended to be taken for granted or to go untested, but its meaning in practice has to be articulated before its usefulness can be judged. This means that the conceivable practical import of the hypothesis or the model has to be developed in terms of its predicted and its promised consequences, after which it is judged by the comparison of these speculative consequences with the actual results. But this is not the end of the matter, for it can be a useful piece of information to discover that a particular kind of conception fails a particular kind of comparison. Thus, the final justification for a hypothesis or a model is contained in the order of work that it leads one to do, and the value of this work is often the same whether or not its premiss is true. Indeed, the fruitfulness of a suggestion can lie in the work that proves it untrue.
The value of a hypothesis, or the worth of a model, is not to be given a prior justification, as by a deductive proof, but has to be examined in practice, as by an empirical probation. It is not intended to be taken for granted or to go untested, but its meaning in practice has to be articulated before its usefulness can be judged. This means that the conceivable practical import of the hypothesis or the model has to be developed in terms of its predicted and its promised consequences, after which it is judged by the comparison of these speculative consequences with the actual results. But this is not the end of the matter, for it can be a useful piece of information to discover that a particular kind of conception fails a particular kind of comparison. Thus, the final justification for a hypothesis or a model is contained in the order of work that it leads one to do, and the value of this work is often the same whether or not its premiss is true. Indeed, the fruitfulness of a suggestion can lie in the work that proves it untrue.
My plan then has to be, rather than trying to derive a model of inquiry in a deductive fashion from a number of conditions like y0 = y.y, only to propose a plausible model, and then to test it under such conditions. Each of these tests is a "two-edged sword", and the result of applying a particular test to a proposed model can have either one of two effects. If one believes that a particular test is a hard and fast rule of inquiry, or a condition that any inquiry is required to satisfy, then the failure of a model to live up to its standard tends only to rule out that model. If one has reason to believe that a particular model of inquiry covers a significant number of genuine examples, then the failure of these models to follow the prescribed rule can reflect badly on the test itself.
My plan then has to be, rather than trying to derive a model of inquiry in a deductive fashion from a number of conditions like <math>y_0 = y \cdot y,</math> only to propose a plausible model, and then to test it under such conditions. Each of these tests is a two-edged sword, and the result of applying a particular test to a proposed model can have either one of two effects. If one believes that a particular test is a hard and fast rule of inquiry, or a condition that any inquiry is required to satisfy, then the failure of a model to live up to its standard tends only to rule out that model. If one has reason to believe that a particular model of inquiry covers a significant number of genuine examples, then the failure of these models to follow the prescribed rule can reflect badly on the test itself.
In order to prime the pump, therefore, let me offer the following account of inquiry in general, the whole of which can be taken as a plausible hypothesis about the nature of inquiry in general.
In order to prime the pump, therefore, let me offer the following account of inquiry in general, the whole of which can be taken as a plausible hypothesis about the nature of inquiry in general.
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".
# 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.
# 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
Does the inquiry into inquiry begin with a surprise or a problem concerning the process or the conduct of inquiry? In other words, does the inquiry into inquiry start with one of the following forms of departure: (1) a surprising difference between what is expected of inquiry and what is observed about it, or (2) a problematic difference between what is observed about inquiry and what is intended for it?
Does the inquiry into inquiry begin with a surprise or a problem concerning the process or the conduct of inquiry? In other words, does the inquiry into inquiry start with one of the following forms of departure: (1) a surprising difference between what is expected of inquiry and what is observed about it, or (2) a problematic difference between what is observed about inquiry and what is intended for it?
====1.4.2 The Moment of Inquiry====
====1.4.2. The Moment of Inquiry====
<blockquote>
<blockquote>
<p>Every young man - not to speak of old men - on hearing or seeing anything unusual and strange, is likely to avoid jumping to a hasty and impulsive solution of his doubts about it, and to stand still; just as a man who has come to a crossroads and is not quite sure of his way, if he be travelling alone, will question himself, or if travelling with others, will question them too about the matter in doubt, and refuse to proceed until he has made sure by investigation of the direction of his path.</p>
<p>Every young man — not to speak of old men — on hearing or seeing anything unusual and strange, is likely to avoid jumping to a hasty and impulsive solution of his doubts about it, and to stand still; just as a man who has come to a crossroads and is not quite sure of his way, if he be travelling alone, will question himself, or if travelling with others, will question them too about the matter in doubt, and refuse to proceed until he has made sure by investigation of the direction of his path.</p>
<p>(Plato, Laws, VII, 799C).</p>
<p>(Plato, ''Laws'', VII, 799C).</p>
</blockquote>
</blockquote>
In any case, belief or knowledge is the feature of state that an agent of inquiry lacks at the moment of setting out. Inquiry begins in a state of impoverishment, need, or privation, a state that is absent the quality of certainty. It is due to this feature that the agent is motivated, and it is on account of its continuing absence that the agent keeps on striving to achieve it, at least, with respect to the subject in question, and, at any rate, in sufficient measure to make action possible.
In any case, belief or knowledge is the feature of state that an agent of inquiry lacks at the moment of setting out. Inquiry begins in a state of impoverishment, need, or privation, a state that is absent the quality of certainty. It is due to this feature that the agent is motivated, and it is on account of its continuing absence that the agent keeps on striving to achieve it, at least, with respect to the subject in question, and, at any rate, in sufficient measure to make action possible.
====1.4.3 The Modes of Inquiry====
====1.4.3. The Modes of Inquiry====
<blockquote>
<blockquote>
<p>Let the strange fact be granted, we say, that our hymns are now made into "nomes" (laws), just as the men of old, it would seem, gave this name to harp-tunes, - so that they, too, perhaps, would not wholly disagree with our present suggestion, but one of them may have divined it vaguely, as in a dream by night or a waking vision: anyhow, let this be the decree on the matter:- In violation of public tunes and sacred songs and the whole choristry of the young, just as in violation of any other "nome" (law), no person shall utter a note or move a limb in the dance.</p>
<p>Let the strange fact be granted, we say, that our hymns are now made into "nomes" (laws), just as the men of old, it would seem, gave this name to harp-tunes, — so that they, too, perhaps, would not wholly disagree with our present suggestion, but one of them may have divined it vaguely, as in a dream by night or a waking vision: anyhow, let this be the decree on the matter: — In violation of public tunes and sacred songs and the whole choristry of the young, just as in violation of any other "nome" (law), no person shall utter a note or move a limb in the dance.</p>
<p>(Plato, Laws, VII, 799E-800A).</p>
<p>(Plato, ''Laws'', VII, 799E–800A).</p>
</blockquote>
</blockquote>
Other names for descriptive laws are "declarative" or "empirical" laws. Other names for prescriptive laws are "procedural" or "normative" laws.
Other names for descriptive laws are "declarative" or "empirical" laws. Other names for prescriptive laws are "procedural" or "normative" laws.
Implicit in a descriptive law is the connection to be found or made, discovered or created, between past experience and present expectation. What one knows about these connections is kept in a descrptive model.
Implicit in a descriptive law is the connection to be found or made, discovered or created, between past experience and present expectation. What one knows about these connections is kept in a descriptive model.
Implicit in a prescriptive law is the connection to be found or made, discovered or created, between current conduct and future experience. What one knows about these connections is kept in a prescriptive model.
Implicit in a prescriptive law is the connection to be found or made, discovered or created, between current conduct and future experience. What one knows about these connections is kept in a prescriptive model.
If it were only a matter of doing propositional reasoning as efficiently as possible, I would simply use the cactus language and be done with it, but there are several other reasons for revisiting the syllogistic model. Treating the discipline that is commonly called "logic" as a cultural subject with a rich and varied history of development, and attending to the thread of tradition in which I currently find myself, I observe what looks like a critical transition that occurs between the classical and the modern ages. Aside from supplying the barest essentials of a historical approach to the subject, a consideration of this elder standard makes it easier to appreciate the nature and the character of this transformation. In addition, and surprisingly enough to warrant further attention, there appear to be a number of cryptic relationships that exist between the syllogistic patterns of reasoning and the ostensibly more advanced forms of analysis and synthesis that are involved in the logic of relations.
If it were only a matter of doing propositional reasoning as efficiently as possible, I would simply use the cactus language and be done with it, but there are several other reasons for revisiting the syllogistic model. Treating the discipline that is commonly called "logic" as a cultural subject with a rich and varied history of development, and attending to the thread of tradition in which I currently find myself, I observe what looks like a critical transition that occurs between the classical and the modern ages. Aside from supplying the barest essentials of a historical approach to the subject, a consideration of this elder standard makes it easier to appreciate the nature and the character of this transformation. In addition, and surprisingly enough to warrant further attention, there appear to be a number of cryptic relationships that exist between the syllogistic patterns of reasoning and the ostensibly more advanced forms of analysis and synthesis that are involved in the logic of relations.
=====1.4.3.1 Deductive Reasoning=====
=====1.4.3.1. Deductive Reasoning=====
In this subsection, I present a trimmed-down version of deductive reasoning in Aristotle, limiting the account to universal syllogisms, in effect, keeping to the level of propositional reasoning. Within these constraints, there are three basic "figures" of the syllogism.
In this subsection, I present a trimmed-down version of deductive reasoning in Aristotle, limiting the account to universal syllogisms, in effect, keeping to the level of propositional reasoning. Within these constraints, there are three basic "figures" of the syllogism.
In addition to this terminology, it is convenient to make use of the following nomenclature:
In addition to this terminology, it is convenient to make use of the following nomenclature:
# The ''Fact'' is the proposition that applies the term in the first position to the term in the third or last position.
# The ''Case'' is the proposition that applies the term in the second or intermediate position to the term in the third or last position.
# The ''Rule'' is the proposition that applies the term in the first position to the term in the second or intermediate position.
Because the roles of Fact, Case, and Rule are defined with regard to positions rather than magnitudes they are insensitive to whether the proposition in question is being used as a premiss or is being drawn as a conclusion.
Because the roles of Fact, Case, and Rule are defined with regard to positions rather than magnitudes they are insensitive to whether the proposition in question is being used as a premiss or is being drawn as a conclusion.
The "first figure" of the syllogism is explained as follows:
The ''first figure'' of the syllogism is explained as follows:
<blockquote>
<blockquote>
<p>When three terms are so related to one another that the last is wholly contained in the middle and the middle is wholly contained in or excluded from the first, the extremes must admit of perfect syllogism. By "middle term" I mean that which both is contained in another and contains another in itself, and which is the middle by its position also; and by "extremes" (a) that which is contained in another, and (b) that in which another is contained. For if A is predicated of all B, and B of all C, A must necessarily be predicated of all C. ... I call this kind of figure the First.</p>
<p>When three terms are so related to one another that the last is wholly contained in the middle and the middle is wholly contained in or excluded from the first, the extremes must admit of perfect syllogism. By "middle term" I mean that which both is contained in another and contains another in itself, and which is the middle by its position also; and by "extremes" (a) that which is contained in another, and (b) that in which another is contained. For if A is predicated of all B, and B of all C, A must necessarily be predicated of all C. ... I call this kind of figure the First.</p>
<p>(Aristotle, Prior Analytics, 1.4).</p>
<p>(Aristotle, ''Prior Analytics'', 1.4).</p>
</blockquote>
</blockquote>
There is the Case:
There is the Case:
"All canaries are birds." (C => B)
: "All canaries are birds." (C => B)
There is the Rule:
There is the Rule:
"All birds are animals." (B => A)
: "All birds are animals." (B => A)
One deduces the Fact:
One deduces the Fact:
"All canaries are animals." (C => A)
: "All canaries are animals." (C => A)
The propositional content of this deduction is summarized on the right. Taken at this level of detail, deductive reasoning is nothing more than an application of the transitive rule for logical implications.
The propositional content of this deduction is summarized on the right. Taken at this level of detail, deductive reasoning is nothing more than an application of the transitive rule for logical implications.
The "second figure" of the syllogism is explained as follows:
The ''second figure'' of the syllogism is explained as follows:
<blockquote>
<blockquote>
When the same term applies to all of one subject and to none of the other, or to all or none of both, I call this kind of figure the Second; and in it by the middle term I mean that which is predicated of both subjects; by the extreme terms, the subjects of which the middle is predicated; by the major term, that which comes next to the middle; and by the minor that which is more distant from it. The middle is placed outside the extreme terms, and is first by position. (Aristotle, Prior Analytics, 1.5).
<p>When the same term applies to all of one subject and to none of the other, or to all or none of both, I call this kind of figure the Second; and in it by the middle term I mean that which is predicated of both subjects; by the extreme terms, the subjects of which the middle is predicated; by the major term, that which comes next to the middle; and by the minor that which is more distant from it. The middle is placed outside the extreme terms, and is first by position.</p>
<p>(Aristotle, ''Prior Analytics'', 1.5).</p>
</blockquote>
</blockquote>
There is the Fact:
There is the Fact:
"All opossums are mammals." (O => M)
: "All opossums are mammals." (O => M)
There is the Rule:
There is the Rule:
"No newts are mammals." (N.M = 0)
: "No newts are mammals." (N.M = 0)
One deduces the Case:
One deduces the Case:
"No newts are opossums." (N.O = 0)
: "No newts are opossums." (N.O = 0)
The propositional content of this deduction is summarized on the right. Expressed in terms of the corresponding classes, it says that if O c M and if N intersects M trivially, then N must also intersect O trivially. Here, I use a raised dot "." to indicate either the conjunction of two propositions or the intersection of two classes, and I use a zero "0" to indicate either the identically false proposition or the empty class, leaving the choice of interpretation to the option of the reader.
The propositional content of this deduction is summarized on the right. Expressed in terms of the corresponding classes, it says that if O c M and if N intersects M trivially, then N must also intersect O trivially. Here, I use a raised dot "." to indicate either the conjunction of two propositions or the intersection of two classes, and I use a zero "0" to indicate either the identically false proposition or the empty class, leaving the choice of interpretation to the option of the reader.
The "third figure" of the syllogism is explained as follows:
The ''third figure'' of the syllogism is explained as follows:
<blockquote>
<blockquote>
If one of the terms applies to all and the other to none of the same subject, or if both terms apply to all or none of it, I call this kind of figure the Third; and in it by the middle I mean that of which both the predications are made; by extremes the predicates; by the major term that which is [further from] the middle; and by the minor that which is nearer to it. The middle is placed outside the extremes, and is last by position. Aristotle, Prior Analytics, 1.6).
<p>If one of the terms applies to all and the other to none of the same subject, or if both terms apply to all or none of it, I call this kind of figure the Third; and in it by the middle I mean that of which both the predications are made; by extremes the predicates; by the major term that which is [further from] the middle; and by the minor that which is nearer to it. The middle is placed outside the extremes, and is last by position.</p>
<p>(Aristotle, ''Prior Analytics'', 1.6).</p>
</blockquote>
</blockquote>
There is the Fact:
There is the Fact:
"All sonnets are poems." (S => P)
: "All sonnets are poems." (S => P)
There is the Case:
There is the Case:
"Some sonnets are rhapsodies." (S.R > 0)
: "Some sonnets are rhapsodies." (S.R > 0)
One deduces the Rule:
One deduces the Rule:
"Some rhapsodies are poems." (R.P > 0)
: "Some rhapsodies are poems." (R.P > 0)
The propositional content of this deduction is summarized on the right. Expressed in terms of the corresponding classes, it says that if S c P and if R intersects S non-trivially then R must intersect P non-trivially.
The propositional content of this deduction is summarized on the right. Expressed in terms of the corresponding classes, it says that if S c P and if R intersects S non-trivially then R must intersect P non-trivially.
=====1.4.3.2 Inductive Reasoning=====
=====1.4.3.2. Inductive Reasoning=====
(Aristotle, ''Prior Analytics'', 2.23).
=====1.4.3.3 Abductive Reasoning=====
=====1.4.3.3. Abductive Reasoning=====
A choice of method cannot be justified by deduction or by induction, at least, not wholly, but involves an element of hypothesis. In ancient times, this mode of inference to an explanatory hypothesis was described by the Greek word "apagoge", articulating an action or a process that "carries", "drives", or "leads" in a direction "away", "from", or "off". This was later translated into the Latin "abductio", and that is the source of what is today called "abduction" or "abductive reasoning". Another residue of this sense survives today in the terminology for "abductor muscles", those that "draw away (say, a limb or an eye) from a position near or parallel to the median axis of the body" (Webster's).
A choice of method cannot be justified by deduction or by induction, at least, not wholly, but involves an element of hypothesis. In ancient times, this mode of inference to an explanatory hypothesis was described by the Greek word "apagoge", articulating an action or a process that "carries", "drives", or "leads" in a direction "away", "from", or "off". This was later translated into the Latin "abductio", and that is the source of what is today called "abduction" or "abductive reasoning". Another residue of this sense survives today in the terminology for "abductor muscles", those that "draw away (say, a limb or an eye) from a position near or parallel to the median axis of the body" (Webster's).
Suppose I have occasion to reason as follows:
Suppose I have occasion to reason as follows:
"It looks like a duck, so I guess it is a duck."
: "It looks like a duck, so I guess it is a duck."
Or even more simply:
Or even more simply:
"It looks blue, therefore it is blue."
: "It looks blue, therefore it is blue."
These are instances in which I am using abductive reasoning, according to the pattern of the following schema:
These are instances in which I am using abductive reasoning, according to the pattern of the following schema:
I observe a Fact:
I observe a Fact:
"It looks like X." (X')
: "It looks like X." (X')
I have in the back of my mind a general Rule:
I have in the back of my mind a general Rule:
"If it is X, then it looks like X." (X => X')
: "If it is X, then it looks like X." (X => X')
I reason my way back from the observed Fact and the assumed Rule to assert what I guess to be the Case:
I reason my way back from the observed Fact and the assumed Rule to assert what I guess to be the Case:
"It is X." (X)
: "It is X." (X)
The abduction is a hypothetical inference that results in a diagnostic conclusion, that is, a statement of opinion as to what is conjectured to be the case. In each case the operation of abductive reasoning starts from a complex configuration, involving a number of explicit observations in the foreground and a class of implicit assumptions in the background, and it offers a provisional statement about certain possibility, one that is typically less conspicuous, obvious, or prominent, but still potentially present in the situation, and hopefully serving to explain the surprising or the problematic aspects of the whole state of affairs.
The abduction is a hypothetical inference that results in a diagnostic conclusion, that is, a statement of opinion as to what is conjectured to be the case. In each case the operation of abductive reasoning starts from a complex configuration, involving a number of explicit observations in the foreground and a class of implicit assumptions in the background, and it offers a provisional statement about certain possibility, one that is typically less conspicuous, obvious, or prominent, but still potentially present in the situation, and hopefully serving to explain the surprising or the problematic aspects of the whole state of affairs.
But what if an example of a good method is already known to exist, one that has all of the commonly accepted properties that appear to define what a good method ought to be? In this case, the abductive argument acquires the additional strength of an argument from analogy.
But what if an example of a good method is already known to exist, one that has all of the commonly accepted properties that appear to define what a good method ought to be? In this case, the abductive argument acquires the additional strength of an argument from analogy.
=====1.4.3.4 Analogical Reasoning=====
=====1.4.3.4. Analogical Reasoning=====
The classical treatment of analogical reasoning by Aristotle explains it as a combination of induction and deduction. More recently, C.S. Peirce gave two different ways of viewing the use of analogy, analyzing it into complex patterns of reasoning that involve all three types of inference. In the appropriate place, it will be useful to consider these alternative accounts of analogy in detail. At the present point, it is more useful to illustrate the different versions of analogical reasoning as they bear on the topic of choosing a method.
The classical treatment of analogical reasoning by Aristotle explains it as a combination of induction and deduction. More recently, C.S. Peirce gave two different ways of viewing the use of analogy, analyzing it into complex patterns of reasoning that involve all three types of inference. In the appropriate place, it will be useful to consider these alternative accounts of analogy in detail. At the present point, it is more useful to illustrate the different versions of analogical reasoning as they bear on the topic of choosing a method.
==Notes==
==Notes==
===Critique Of Functional Reason : Note 78===
<pre>
<pre>
MW = Matthew West:
MW = Matthew West:
Therefore, "((x_1),(x_2), ...,(x_k))" partitions the
Therefore, "((x_1),(x_2), ...,(x_k))" partitions the
universe of discourse, saying "Just one x_j is true".
universe of discourse, saying "Just one x_j is true".
</pre>
===Critique Of Functional Reason : Note 83===
<pre>
| Tantum ergo sacramentum
| Tantum ergo sacramentum
| veneremur cernui,
| veneremur cernui,
But advise you solely of its transformed instantiations,
But advise you solely of its transformed instantiations,
And fix my faith on imagination to sense the supplement.
And fix my faith on imagination to sense the supplement.
</pre>
===Critique Of Functional Reason : Note 92===
<pre>
I need to try and say some things at his point about
I need to try and say some things at his point about
why formal language theory is interesting and useful,
why formal language theory is interesting and useful,
but maybe it will supply a grub-stake of motivational
but maybe it will supply a grub-stake of motivational
victuals for the grueling labors of exploration ahead.
victuals for the grueling labors of exploration ahead.
</pre>
===IDS. Incitatory Note 1===
IDS. Incitatory Note 1
<pre>
| Each ground-principle must be proved entirely
| Each ground-principle must be proved entirely
| by that same kind of inference which it supports.
| by that same kind of inference which it supports.
|'Volume 1, 1857-1866', Peirce Edition Project,
|'Volume 1, 1857-1866', Peirce Edition Project,
| Indiana University Press, Bloomington, IN, 1982.
| Indiana University Press, Bloomington, IN, 1982.
</pre>
===IDS. Meditative Note 1===
IDS. Meditative Note 1
<pre>
I would like to start from a "common sense practical" (CSP) point of view,
I would like to start from a "common sense practical" (CSP) point of view,
and, indeed, never to lose sight of what appears evident from that station,
and, indeed, never to lose sight of what appears evident from that station,
Or for a rest ...
Or for a rest ...
</pre>
===IDS. Meditative Note 2===
IDS. Meditative Note 2
<pre>
So I may begin with an object and a sign in a tenuous relation,
So I may begin with an object and a sign in a tenuous relation,
with the subject matter indexed under the topic name "inquiry",
with the subject matter indexed under the topic name "inquiry",
["sign-ficant" [stet]]
["sign-ficant" [stet]]
</pre>
===IDS. Obligatory Note 1===
While I remain compelled to remain silent on the status of the absolute fiat, the irrelative notion of the unmotivated motion and the disinterested stance, let me then turn to the other axes of description, descriptive vs. normative. Axes of description, indeed, you can almost hear one branch of the recursion already beginning to wind up its whine to the verge of a howl, but toss it a sop and try to persevere in the quest.
In this view, I regard the very idea of a norm as invoking its due pragma — aim, business, concern, desire, end, function, goal, intention, interest, objective, purpose, its names are legion — and the good sense of the norm is simply to suggest what one ought to do, contingent, of course, on one's motive to achieve that pragma.
If we keep in mind the kinds of ''applied research task'' (ART) that your everyday artist, designer, engineer, mathematician, scientist, or other type of technical worker has to carry out on an everyday basis, we note how these axes of description can be used to frame their activities and to depict their forms of conduct, without mistaking either the frame or the picture for the object of the picture so framed. Nor does any body imagine that the observer must flatten out into a single plane or align with a single axis, in order to make a vantage of the frame so pictured.
Common sense practical wit tells us that effective action toward the achievement of a desirable result will naturally depend on acquiring good descriptions of the lay of the land in which we hope to advance.
===IDS. Projective Note 1===
IDS. Projective Note 1
<pre>
Good morning. Thanks. I had a bad night.
Good morning. Thanks. I had a bad night.
I blame Bernard Morand, who wrote me this:
I blame Bernard Morand, who wrote me this:
Why am I asking this question?
Why am I asking this question?
</pre>
===IDS. Projective Note 2===
IDS. Projective Note 2
<pre>
So we may rest assured that we do have a "subject matter", an empirical domain,
So we may rest assured that we do have a "subject matter", an empirical domain,
or a realm of experience that is indexed, however dimly, generally, or vaguely,
or a realm of experience that is indexed, however dimly, generally, or vaguely,
I hope I have made this sound as truly and
I hope I have made this sound as truly and
as trivially obvious as it ought to be.
as trivially obvious as it ought to be.
</pre>
===IDS. Reflective Note 1===
IDS. Reflective Note 1
<pre>
In reflecting on what in the world a "Theory of Inquiry" (TOI) might be,
In reflecting on what in the world a "Theory of Inquiry" (TOI) might be,
it occurs to me that there are many different things that one might mean
it occurs to me that there are many different things that one might mean
I will think about that, and get back to you.
I will think about that, and get back to you.
</pre>
===IDS. Reflective Note 2===
IDS. Reflective Note 2
<pre>
| How will I approach this problem about the nature of inquiry?
| How will I approach this problem about the nature of inquiry?
|
|
http://members.door.net/arisbe/menu/library/bycsp/newlist/nl-main.htm
http://members.door.net/arisbe/menu/library/bycsp/newlist/nl-main.htm
</pre>
===IDS. Work Area===
IDS. Work Area
<pre>
From this point of view, inquiry is form of conduct,
From this point of view, inquiry is form of conduct,
an applied research task, like many others that we
an applied research task, like many others that we
have to carry out, and that can be done either
have to carry out, and that can be done either
better or worse.
better or worse.
</pre>
==Document History==
<pre>
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| Document History
|
| Subject: Inquiry Driven Systems: An Inquiry Into Inquiry
| Contact: Jon Awbrey <jawbrey@oakland.edu>
| Version: Draft 10.00
| Created: 23 Jun 1996
| Revised: 02 Mar 2003
| Advisor: M.A. Zohdy
| Setting: Oakland University, Rochester, Michigan, USA
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
</pre>
</pre>
===Inquiry List===
* http://stderr.org/pipermail/inquiry/2004-May/thread.html#1434
* http://stderr.org/pipermail/inquiry/2004-May/thread.html#1564
* http://stderr.org/pipermail/inquiry/2004-June/thread.html#1574