Functional Logic • Inquiry and Analogy

[Jon Awbrey]

Functional Logic • Inquiry and Analogy • Preliminaries
• https://inquiryintoinquiry.com/2023/06/20/functional-logic-inquiry-and-analogy-preliminaries-2/

Functional Logic • Inquiry and Analogy
• https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy

This report discusses C.S. Peirce's treatment of analogy,
placing it in relation to his overall theory of inquiry.
We begin by introducing three basic types of reasoning
Peirce adopted from classical logic. In Peirce's analysis
both inquiry and analogy are complex programs of logical
inference which develop through stages of these three types,
though normally in different orders.

Note on notation. The discussion to follow uses logical conjunctions,
expressed in the form of concatenated tuples e₁…eₖ, and minimal negation
operations, expressed in the form of bracketed tuples (e₁,…,eₖ), as the
principal expression-forming operations of a calculus for boolean-valued
functions, that is, for propositions. The expressions of this calculus
parse into data structures whose underlying graphs are called “cacti” by
graph theorists. Hence the name “cactus language” for this dialect of
propositional calculus.

Resources —

Logic Syllabus
• https://oeis.org/wiki/Logic_Syllabus

Boolean Function
• https://oeis.org/wiki/Boolean_function

Boolean-Valued Function
• https://oeis.org/wiki/Boolean-valued_function

Logical Conjunction
• https://oeis.org/wiki/Logical_conjunction

Minimal Negation Operator
• https://oeis.org/wiki/Minimal_negation_operator

Regards,

Jon

[Jon Awbrey]

Functional Logic • Inquiry and Analogy • 1
• http://inquiryintoinquiry.com/2023/06/21/functional-logic-inquiry-and-analogy-1-2/

All,

Figure 1 gives a quick overview of traditional terminology
I'll have occasion to refer to as discussion proceeds.

Figure 1. Types of Reasoning in Aristotle
• https://inquiryintoinquiry.files.wordpress.com/2022/04/types-of-reasoning-in-aristotle.jpg

Regards,

Jon

[Jon Awbrey]

Functional Logic • Inquiry and Analogy • 2
• https://inquiryintoinquiry.com/2023/06/22/functional-logic-inquiry-and-analogy-2-2/

Types of Reasoning in C.S. Peirce
• https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy#Types_of_Reasoning_in_C.S._Peirce

Peirce gives one of his earliest treatments of the three types of
reasoning in his Harvard Lectures of 1865 “On the Logic of Science”.
There he shows how the same proposition may be reached from three
directions, as the result of an inference in each of the three modes.

<QUOTE CSP:>
We have then three different kinds of inference.

• Deduction or inference à priori,
• Induction or inference à particularis,
• Hypothesis or inference à posteriori.

(Peirce, CE 1, 267).

• If I reason that certain conduct is wise because it has
a character which belongs only to wise things, I reason à priori.

• If I think it is wise because it once turned out to be wise,
that is, if I infer that it is wise on this occasion because
it was wise on that occasion, I reason inductively [à particularis].

• But if I think it is wise because a wise man does it, I then make
the pure hypothesis that he does it because he is wise, and I reason
à posteriori.

(Peirce, CE 1, 180).
</QUOTE>

Suppose we make the following assignments.

• A = Wisdom
• B = a certain character
• C = a certain conduct
• D = done by a wise man
• E = a certain occasion

Recognizing a little more concreteness will aid understanding,
let us make the following substitutions in Peirce’s example.

• B = Benevolence = a certain character
• C = Contributes to Charity = a certain conduct
• E = Earlier today = a certain occasion

The converging operation of all three reasonings is shown in Figure 2.

Figure 2. A Triply Wise Act
• https://inquiryintoinquiry.files.wordpress.com/2022/04/triply-wise-act.jpg

The common proposition concluding each argument is AC,
contributing to charity is wise.

• Deduction could have obtained the Fact AC from
the Rule AB, benevolence is wisdom, along with
the Case BC, contributing to charity is benevolent.

• Induction could have gathered the Rule AC,
contributing to charity is exemplary of wisdom, from
the Fact AE, the act of earlier today is wise, along with
the Case CE, the act of earlier today was an instance of
contributing to charity.

• Abduction could have guessed the Case AC,
contributing to charity is explained by wisdom, from
the Fact DC, contributing to charity is done by this wise man, and
the Rule DA, everything wise is done by this wise man.

Thus, a wise man, who does all the wise things there are to do,
may nonetheless contribute to charity for no good reason and even
be charitable to a fault. But on seeing the wise man contribute to
charity it is natural to think charity may well be the “mark” of his
wisdom, in essence, that wisdom is the “reason” he contributes to
charity.

Regards,

Jon

[Jon Awbrey]

Functional Logic • Inquiry and Analogy • 3
• https://inquiryintoinquiry.com/2023/06/23/functional-logic-inquiry-and-analogy-3-2/

Inquiry and Analogy • Comparison of the Analyses
• https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy#Comparison_of_the_Analyses

The next two Figures will be of use when we turn to comparing
the three types of inference as they appear in the respective
analyses of Aristotle and Peirce.

Figure 3. Types of Reasoning in Transition
• https://inquiryintoinquiry.files.wordpress.com/2022/04/types-of-reasoning-in-transition.jpg

Figure 4. Types of Reasoning in Peirce
• https://inquiryintoinquiry.files.wordpress.com/2022/04/types-of-reasoning-in-peirce.jpg

Regards,

Jon

[Jon Awbrey]

Functional Logic • Inquiry and Analogy • 4 (Part 1)
• https://inquiryintoinquiry.com/2023/06/24/functional-logic-inquiry-and-analogy-4-2/

Inquiry and Analogy • Aristotle’s “Apagogy” • Abductive Reasoning
•
https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy#Aristotle.27s_.E2.80.9CApagogy.E2.80.9D_.E2.80.A2_Abductive_Reasoning_as_Problem_Reduction

Peirce's notion of abductive reasoning is derived from Aristotle's
treatment of it in the “Prior Analytics”. Aristotle's discussion
begins with an example which may seem incidental but the question
and its analysis are echoes of the investigation pursued in one of
Plato's Dialogue, the “Meno”. It concerns nothing less than the
possibility of knowledge and the relationship between knowledge and
virtue, or between their objects, the true and the good. It is not
just because it forms a recurring question in philosophy, but because
it preserves a close correspondence between its form and its content,
that we shall find this example increasingly relevant to our study.

<QUOTE Aristotle:>
We have Reduction (απαγωγη, abduction): (1) when it is obvious that
the first term applies to the middle, but that the middle applies to
the last term is not obvious, yet nevertheless is more probable or
not less probable than the conclusion; or (2) if there are not many
intermediate terms between the last and the middle; for in all such
cases the effect is to bring us nearer to knowledge.

(1) E.g., let A stand for “that which can be taught”, B for “knowledge”,
and C for “morality”. Then that knowledge can be taught is evident; but
whether virtue is knowledge is not clear. Then if BC is not less probable or
is more probable than AC, we have reduction; for we are nearer to knowledge
for having introduced an additional term, whereas before we had no knowledge
that AC is true.

(2) Or again we have reduction if there are not many intermediate terms
between B and C; for in this case too we are brought nearer to knowledge.
E.g., suppose that D is “to square”, E “rectilinear figure”, and F “circle”.
Assuming that between E and F there is only one intermediate term — that the
circle becomes equal to a rectilinear figure by means of lunules — we should
approximate to knowledge.

Aristotle, “Prior Analytics” 2.25, Hugh Tredennick (trans.)
</QUOTE>

A few notes on the reading may be helpful. The Greek text seems to imply
a geometric diagram, in which directed line segments AB, BC, AC indicate
logical relations between pairs of terms taken from A, B, C. We have two
options for reading the line labels, either as implications or as subsumptions,
as in the following two paradigms for interpretation.

Table of Implications
• https://inquiryintoinquiry.files.wordpress.com/2022/04/table-of-implications-tuv.png

Table of Subsumptions
• https://inquiryintoinquiry.files.wordpress.com/2022/04/table-of-subsumptions-tuv.png

In the latter case, P ⩾ Q is read as “P subsumes Q”, that is,
“P applies to all Q”, or “P is predicated of all Q”.

Regards,

Jon

[Jon Awbrey]

Functional Logic • Inquiry and Analogy • 4 (Part 2)

• https://inquiryintoinquiry.com/2023/06/24/functional-logic-inquiry-and-analogy-4-2/

Inquiry and Analogy • Aristotle’s “Apagogy” • Abductive Reasoning
•
https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy#Aristotle.27s_.E2.80.9CApagogy.E2.80.9D_.E2.80.A2_Abductive_Reasoning_as_Problem_Reduction

The method of abductive reasoning bears a close relation to the
sense of reduction in which we speak of one question reducing to
another. The question being asked is “Can virtue be taught?”
The type of answer which develops is as follows.

If virtue is a form of understanding, and if we are willing to
grant that understanding can be taught, then virtue can be taught.
In this way of approaching the problem, by detour and indirection,
the form of abductive reasoning is used to shift the attack from the
original question, whether virtue can be taught, to the hopefully
easier question, whether virtue is a form of understanding.

The logical structure of the process of hypothesis formation in
the first example follows the pattern of “abduction to a case”,
whose abstract form is diagrammed and schematized in Figure 5.

Figure 5. Teachability, Understanding, Virtue
• https://inquiryintoinquiry.files.wordpress.com/2022/04/teachability-understanding-virtue-3.0.png

The sense of the Figure is explained by the following assignments.

Term, Position, Interpretation
https://inquiryintoinquiry.files.wordpress.com/2022/04/term-position-interpretation-tuv.png

Premiss, Predication, Inference Role
• https://inquiryintoinquiry.files.wordpress.com/2022/04/premiss-predication-inference-role-tuv.png

Abduction from a Fact to a Case proceeds according to the following schema.

Fact: V ⇒ T?
Rule: U ⇒ T.
──────────────
Case: V ⇒ U?

Regards,

Jon

[Jon Awbrey]

Functional Logic • Inquiry and Analogy • 5
• https://inquiryintoinquiry.com/2023/06/25/functional-logic-inquiry-and-analogy-5-2/

Inquiry and Analogy • Aristotle’s “Paradigm” • Reasoning by Analogy
•
https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy#Aristotle.27s_.E2.80.9CParadigm.E2.80.9D_.E2.80.A2_Reasoning_by_Analogy_or_Example

Aristotle examines the subject of analogical inference or “reasoning by example” under the heading of the Greek word
παραδειγμα, from which comes the English word “paradigm”. In its original sense the word suggests a kind of
“side‑show”, or a parallel comparison of cases.

<QUOTE Aristotle:>
We have an Example (παραδειγμα, or analogy) when the major extreme is shown to be applicable to the middle term by means
of a term similar to the third. It must be known both that the middle applies to the third term and that the first
applies to the term similar to the third.

E.g., let A be “bad”, B “to make war on neighbors”, C “Athens against Thebes”, and D “Thebes against Phocis”. Then if
we require to prove that war against Thebes is bad, we must be satisfied that war against neighbors is bad. Evidence of
this can be drawn from similar examples, e.g., that war by Thebes against Phocis is bad. Then since war against
neighbors is bad, and war against Thebes is against neighbors, it is evident that war against Thebes is bad.

Aristotle, “Prior Analytics” 2.24, Hugh Tredennick (trans.)
</QUOTE>

Figure 6 shows the logical relationships involved in Aristotle’s example of analogy.

Figure 6. Aristotle's “Paradigm”
• https://inquiryintoinquiry.files.wordpress.com/2013/11/aristotles-paradigm.jpg

Regards,

Jon

[Jon Awbrey]

Functional Logic • Inquiry and Analogy • 6
• https://inquiryintoinquiry.com/2023/06/26/functional-logic-inquiry-and-analogy-6-2/

Inquiry and Analogy • Peirce's Formulation of Analogy • Version 1
• https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy#Peirce_on_Analogy_1

All,

Next we look at a couple of ways Peirce analyzed analogical inferences.

Version 1 —

<QUOTE CSP:>

C.S. Peirce • “On the Natural Classification of Arguments” (1867)

The formula of analogy is as follows:

S′, S″, and S‴ are taken at random from such a class
that their characters at random are such as P′, P″, P‴.

T is P′, P″, P‴,

S′, S″, S‴ are Q;

∴ T is Q.

Such an argument is double. It combines the two following:

1.

S′, S″, S‴ are taken as being P′, P″, P‴,

S′, S″, S‴ are Q;

∴ (By induction) P′, P″, P‴ is Q,

T is P′, P″, P‴;

∴ (Deductively) T is Q.

2.

S′, S″, S‴ are, for instance, P′, P″, P‴,

T is P′, P″, P‴;

∴ (By hypothesis) T has the common characters of S′, S″, S‴,

S′, S″, S‴ are Q;

∴ (Deductively) T is Q.

Owing to its double character, analogy is very strong with only a moderate number of instances.

(Peirce, CP 2.513, CE 2, 46–47)
</QUOTE>

Figure 7 shows the logical relationships involved in the above analysis.

Figure 7. Peirce's Formulation of Analogy (Version 1)
• https://inquiryintoinquiry.files.wordpress.com/2022/04/peirces-formulation-of-analogy-version-1.jpg

Regards,

Jon

[Jon Awbrey]

Functional Logic • Inquiry and Analogy • 7
• https://inquiryintoinquiry.com/2023/06/27/functional-logic-inquiry-and-analogy-7-2/

Inquiry and Analogy • Peirce’s Formulation of Analogy • Version 2
• https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy#Peirce_on_Analogy_2

All,

Here's another formulation of analogical inference Peirce gave some years later.

<QUOTE CSP:>

C.S. Peirce • “A Theory of Probable Inference” (1883)

The formula of the analogical inference
presents, therefore, three premisses, thus:

S′, S″, S‴, are a random sample of some undefined class X,
of whose characters P′, P″, P‴, are samples,

T is P′, P″, P‴;

S′, S″, S‴, are Q's;

Hence, T is a Q.

We have evidently here an induction and an hypothesis
followed by a deduction; thus:

[Parallel Column Display]
• https://inquiryintoinquiry.files.wordpress.com/2022/04/peirce-on-analogy-e280a2-cp-2.733.png

Hence, deductively, T is a Q.

(Peirce, CP 2.733, with a few changes in Peirce's notation to facilitate comparison between the two versions)
</QUOTE>

Figure 8 shows the logical relationships involved in the above analysis.

Figure 8. Peirce's Formulation of Analogy (Version 2)
• https://inquiryintoinquiry.files.wordpress.com/2022/04/peirces-formulation-of-analogy-version-2.jpg

Regards,

Jon

[Jon Awbrey]

[Jon Awbrey]

Functional Logic • Inquiry and Analogy • 8
• https://inquiryintoinquiry.com/2023/06/28/functional-logic-inquiry-and-analogy-8-2/

Inquiry and Analogy • Dewey's “Sign of Rain” • An Example of Inquiry
• https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy#Dewey_Rain

All,

To illustrate the place of the sign relation in inquiry
we begin with Dewey's elegant and simple example of
reflective thinking in everyday life.

<QUOTE Dewey:>

A man is walking on a warm day. The sky was clear the last time
he observed it; but presently he notes, while occupied primarily
with other things, that the air is cooler. It occurs to him that
it is probably going to rain; looking up, he sees a dark cloud
between him and the sun, and he then quickens his steps. What, if
anything, in such a situation can be called thought? Neither the
act of walking nor the noting of the cold is a thought. Walking is
one direction of activity; looking and noting are other modes of
activity. The likelihood that it will rain is, however, something
“suggested”. The pedestrian “feels” the cold; he “thinks of”
clouds and a coming shower.

(John Dewey, How We Think, 6–7)
</QUOTE>

Inquiry and Interpretation
• https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy#Dewey_Rain_1

In Dewey's narrative we can see the components of a sign relation
laid out in the following fashion. “Coolness” is a Sign of the
Object “rain” and “the thought of the rain's likelihood” is the
Interpretant of that sign with respect to that object. In the
present description of reflective thinking Dewey distinguishes
two phases, “a state of perplexity, hesitation, doubt” and “an
act of search or investigation” (p. 9), comprehensive stages
which are further refined in his later model of inquiry.

Reflection is the action the interpreter takes to establish
a fund of connections between the sensory shock of coolness
and the objective danger of rain by way of the impression
rain is likely. But reflection is more than irresponsible
speculation. In reflection the interpreter acts to charge or
defuse the thought of rain (the probability of rain in thought)
by seeking other signs the thought implies and evaluating the
thought according to the results of that search.

Figure 9 shows the semiotic relationships involved in Dewey's
story, tracing the structure and function of the sign relation
as it informs the activity of inquiry, including the movements
of surprise explanation and intentional action. The labels on
the outer edges of the semiotic triple suggest the “significance”
of signs for eventual occurrences and the “correspondence” of ideas
with external orientations. But there is nothing essential about
the dyadic role distinctions they imply, as it is only in special
or degenerate cases that their shadowy projections preserve enough
information to determine the original sign relation.

Figure 9. Dewey's “Sign of Rain” Example
• https://inquiryintoinquiry.files.wordpress.com/2022/04/deweys-sign-of-rain-example.jpg

Regards,

Jon

[Jon Awbrey]

Functional Logic • Inquiry and Analogy • 9
• https://inquiryintoinquiry.com/2023/07/01/functional-logic-inquiry-and-analogy-9-2/

Inquiry and Analogy • Dewey's “Sign of Rain” • An Example of Inquiry
• https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy#Dewey_Rain

All,

We turn again to Dewey's vignette, tracing figures of logic on grounds of semiotic.

Inquiry and Inference
• https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy#Dewey_Rain_2

If we follow Dewey's “Sign of Rain” example far enough to consider
the import of thought for action, we realize the subsequent conduct
of the interpreter, progressing up through the natural conclusion of
the episode — the quickening steps, seeking shelter in time to escape
the rain — all those acts form a series of further interpretants,
contingent on the active causes of the individual, for the originally
recognized signs of rain and the first impressions of the actual case.

Just as critical reflection develops the associated and alternative
signs which gather about an idea, pragmatic interpretation explores
the consequential and contrasting actions which give effective and
testable meaning to a person’s belief in it.

Figure 10 charts the progress of inquiry in Dewey's Sign of Rain example
according to the stages of reasoning identified by Peirce, focusing on
the compound or mixed form of inference formed by the first two steps.

Figure 10. Cycle of Inquiry
• https://inquiryintoinquiry.files.wordpress.com/2022/04/cycle-of-inquiry-grayscale.jpg

Step 1 is Abductive,
abstracting a Case from the consideration of a Fact and a Rule.
• Fact : C ⇒ A, In the Current situation the Air is cool.
• Rule : B ⇒ A, Just Before it rains, the Air is cool.
• Case : C ⇒ B, The Current situation is just Before it rains.

Step 2 is Deductive,
admitting the Case to another Rule and arriving at a novel Fact.
• Case : C ⇒ B, The Current situation is just Before it rains.
• Rule : B ⇒ D, Just Before it rains, a Dark cloud will appear.
• Fact : C ⇒ D, In the Current situation, a Dark cloud will appear.

What precedes is nowhere near a complete analysis of Dewey's example,
even so far as it might be carried out within the constraints of the
syllogistic framework, and it covers only the first two steps of the
inquiry process, but perhaps it will do for a start.

Regards,

Jon

[Jon Awbrey]

Functional Logic • Inquiry and Analogy • 10
• http://inquiryintoinquiry.com/2023/07/07/functional-logic-inquiry-and-analogy-10-2/

Functional Conception of Quantification Theory
• https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy#Quantification

All,

Up till now quantification theory has been based on the assumption of
individual variables ranging over universal collections of perfectly
determinate elements. The mere act of writing quantified formulas
like (∀x∈X)f(x) and (∃x∈X)f(x) involves a subscription to such notions,
as shown by the membership relations invoked in their indices.

As we reflect more critically on the conventional assumptions in the
light of pragmatic and constructive principles, however, they begin
to appear as problematic hypotheses whose warrants are not beyond
question, as projects of exhaustive determination overreaching
the powers of finite information and control to manage.

Thus it is worth considering how the scene of quantification theory
might be shifted nearer to familiar ground, toward the predicates
themselves which represent our continuing acquaintance with phenomena.

Regards,

Jon

[Jon Awbrey]

Functional Logic • Inquiry and Analogy • 11
• https://inquiryintoinquiry.com/2023/07/08/functional-logic-inquiry-and-analogy-11-2/

Inquiry and Analogy • Higher Order Propositional Expressions
• https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy#HOPE

Higher Order Propositions and Logical Operators (n = 1)
• https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy#HOPE_1

All,

A “higher order proposition” is, roughly speaking,
a proposition about propositions. If the original
order of propositions is a set of indicator functions
f : X → B then the next higher order of propositions
consists of maps of the type m : (X → B) → B.

For example, consider the case where X = B. There are
exactly four propositions one can make about the elements
of X. Each proposition has the concrete type f : X → B and
the abstract type f : B → B. Then there are exactly sixteen
higher order propositions one can make about the initial set
of four propositions. Each higher order proposition has the
abstract type m : (B → B) → B.

Table 11 lists the 16 higher order propositions about propositions
on one boolean variable, organized in the following fashion.

• Columns 1 and 2 form a truth table for the four propositions
f : B → B, turned on its side from the way one is most likely
accustomed to see truth tables, with the row leaders in Column 1
displaying the names of the functions f_i, for i = 1 to 4, while
the entries in Column 2 give the values of each function for the
argument values listed in the corresponding column head.

• Column 3 displays one of the more usual expressions
for the proposition in question.

• The last sixteen columns are headed by a collection of
conventional names for the higher order propositions,
also known as the “measures” m_j, for j = 0 to 15,
where the entries in the body of the Table record
the values each m_j assigns to each f_i.

Table 11. Higher Order Propositions (n = 1)
• https://inquiryintoinquiry.files.wordpress.com/2022/05/higher-order-propositions-n1.png

Regards,

Jon

[Jon Awbrey]

Functional Logic • Inquiry and Analogy • 12
• https://inquiryintoinquiry.com/2022/05/07/functional-logic-inquiry-and-analogy-12/

Interpretive Categories for Higher Order Propositions (n = 1)
• https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy#HOPE_1.2

All,

Referring to “Table 11. Higher Order Propositions (n = 1)”
from the previous post:
• https://inquiryintoinquiry.files.wordpress.com/2022/05/higher-order-propositions-n1.png

Table 12 presents a series of “interpretive categories” for
the higher order propositions in Table 11. I'll leave those
for now to the reader's contemplation and discuss them when
we get two variables into the mix. The lower dimensional
cases tend to exhibit “condensed” or “degenerate” structures
and their full significance will become clearer once we get
beyond the 1‑dimensional case.

Table 12. Interpretive Categories for Higher Order Propositions (n = 1)
• https://inquiryintoinquiry.files.wordpress.com/2022/05/interpretive-categories-for-higher-order-propositions-n1.png

Regards,

Jon

[Jon Awbrey]

Functional Logic • Inquiry and Analogy • 13
• https://inquiryintoinquiry.com/2023/07/10/functional-logic-inquiry-and-analogy-13-2/

Inquiry and Analogy • Higher Order Propositional Expressions
• https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy#HOPE

Higher Order Propositions and Logical Operators (n = 2)
• https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy#HOPE_2

All,

There are 2¹⁶ = 65536 measures of type m : (B² → B) → B.
Table 13 introduces the first 24 of those measures in the
fashion of higher order truth table I used before.

The column headed m_j shows the values of the measure m_j on
each of the propositions f_i : B² → B, for i = 0 to 23, with
blank entries in the Table being optional for values of zero.
The arrangement of measures which continues according to the
plan indicated here is referred to as the “standard ordering”
of those measures. In this scheme of things, the index j of
the measure m_j is the decimal equivalent of the bit string
associated with m_j’s functional values, which are obtained
in turn by reading the j-th column of binary digits in the
Table as the corresponding range of boolean values, taking
them up in the order from bottom to top.

Table 13. Higher Order Propositions (n = 2)
• https://inquiryintoinquiry.files.wordpress.com/2022/05/higher-order-propositions-n2-2.0.png

Regards,

Jon

[Jon Awbrey]

Functional Logic • Inquiry and Analogy • 14
• https://inquiryintoinquiry.com/2023/07/11/functional-logic-inquiry-and-analogy-14-2/

Umpire Operators
• https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy#Umpire_Operators

All,

The 2¹⁶ measures of type (B × B → B) → B present a formidable array
of propositions about propositions about 2-dimensional universes
of discourse. The early entries in their standard ordering define
universes too amorphous to detain us for long on a first pass but
as we turn toward the high end of the ordering we begin to recognize
familiar structures worth examining from new angles.

Instrumental to our study we define a couple of higher order operators,

• Υ : (B × B → B)² → B

and

• Υ₁ : (B × B → B) → B,

referred to as the relative and absolute “umpire operators”,
respectively. If either operator is defined in terms of more
primitive notions then the remaining operator can be defined
in terms of the one first established.

Let X = ⟨u, v⟩ be a 2-dimensional boolean space, X ≅ B × B,
generated by 2 boolean variables or logical features u and v.

Given an ordered pair of propositions e, f : ⟨u, v⟩ → B
as arguments, the relative umpire operator reports the
value 1 if the first implies the second, otherwise it
reports the value 0.

• Υ(e, f) = 1 if and only if e ⇒ f

Expressing it another way:

• Υ(e, f) = 1 ⇔ ¬( e ¬( f )) = 1

In writing this, however, it is important to observe that
the 1 appearing on the left side and the 1 appearing on the
right side of the logical equivalence have different meanings.
Filling in the details, we have the following.

• Υ(e, f) = 1 ∈ B ⇔ ¬( e ¬( f )) = 1 : ⟨u, v⟩ → B

Writing types as subscripts and using the fact that X = ⟨u, v⟩,
it is possible to express this more succinctly as follows.

• Υ(e, f) = 1_B ⇔ ¬( e ¬( f )) = 1_{X → B}

Finally, it is often convenient to write the first argument
as a subscript. Thus we have the following equation.

• (Υ_e)(f) = Υ(e, f).

The “absolute umpire operator”, also called the “umpire measure”,
is a higher order proposition Υ₁ : (B × B → B) → B defined by the
equation Υ₁(f) = Υ(1, f). In this case the subscript 1 on the left
and the argument 1 on the right both refer to the constant proposition
1 : B × B → B. In most settings where Υ₁ is applied to arguments it
is safe to omit the subscript 1 since the number of arguments indicates
which type of operator is meant. Thus, we have the following identities
and equivalents.

• Υf = Υ₁(f) = 1_B ⇔ ¬( 1 ¬( f )) = 1 ⇔ f = 1_{B × B → B}

The umpire measure Υ₁ is defined at the level of boolean functions
regarded as mathematical objects but it can be understood also in
terms of the judgments it induces on the syntactic level. In that
interpretation Υ₁ recognizes theorems of the propositional calculus
over [u, v], giving a score of 1 to tautologies and a score of 0 to
everything else, counting all contingent statements as no better than
falsehoods.

Regards,

Jon

[Jon Awbrey]

Functional Logic • Inquiry and Analogy • 15
• https://inquiryintoinquiry.com/2023/07/12/functional-logic-inquiry-and-analogy-15-2/

Inquiry and Analogy • Measure for Measure
• https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy#Measure_for_Measure

All,

Let us define two families of measures,

• α_i, β_i : (B × B → B) → B for i = 0 to 15,

by means of the following equations:

• α_i f = Υ(f_i, f) = Υ(f_i ⇒ f),

• β_i f = Υ(f, f_i) = Υ(f ⇒ f_i).

Table 14 shows the value of each α_i on each of the
16 boolean functions f : B × B → B.

In terms of the implication ordering on the 16 functions,
α_i f = 1 says f is “above or identical to” f_i in the
implication lattice, that is, f ≥ f_i in the implication
ordering.

Table 14. Qualifiers of the Implication Ordering α_i f = Υ(f_i, f)
• https://inquiryintoinquiry.files.wordpress.com/2022/05/qualifiers-of-implication-ordering-ceb1-2.0.png

Table 15 shows the value of each β_i on each of the
16 boolean functions f : B × B → B.

In terms of the implication ordering on the 16 functions,
β_i f = 1 says f is “below or identical to” f_i in the
implication lattice, that is, f ≤ f_i in the implication
ordering.

Table 15. Qualifiers of the Implication Ordering β_i f = Υ(f, f_i)
• https://inquiryintoinquiry.files.wordpress.com/2022/05/qualifiers-of-implication-ordering-ceb2-2.0.png

Applied to a given proposition f, the qualifiers α_i and β_i
tell whether f is above f_i or below f_i, respectively, in the
implication ordering.

By way of example, let us trace the effects of several such measures,
namely, those which occupy the limiting positions in the Tables.

• α₀f = 1 iff f₀ ⇒ f iff 0 ⇒ f, hence α₀f = 1 for all f.

• α₁₅f = 1 iff f₁₅ ⇒ f iff 1 ⇒ f, hence α₁₅f = 1 iff f = 1.

• β₀f = 1 iff f ⇒ f₀ iff f ⇒ 0, hence β₀f = 1 iff f = 0.

• β₁₅f = 1 iff f ⇒ f₁₅ iff f ⇒ 1, hence β₁₅f = 1 for all f.

Expressed in terms of the propositional forms they value positively,
α₀ = β₁₅ is a wholly indifferent or indiscriminate measure, accepting
every proposition f : B × B → B, whereas the measures α₁₅ and β₀ value
the constant propositions 1 : B × B → B and 0 : B × B → B, respectively,
above all others.

Regards,

Jon

[Jon Awbrey]

Functional Logic • Inquiry and Analogy • 16
• https://inquiryintoinquiry.com/2023/07/13/functional-logic-inquiry-and-analogy-16-2/

Extending the Existential Interpretation to Quantificational Logic
• https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy#Ex_Quant

All,

One of the resources we have for this investigation is
a formal calculus based on C.S. Peirce's logical graphs.
For now we'll adopt the “existential interpretation” of
that calculus, fixing the meanings of logical constants
and connectives at the core level of propositional logic.
To build on that core we'll need to extend the existential
interpretation to encompass the analysis of quantified
propositions, or “quantifications”. That in turn will
take developing two further capacities of our calculus.

On the formal side we'll need to consider higher order
functional types, continuing our earlier venture above.
In terms of content we'll need to consider new species
of “elemental” or “singular“ propositions.

Let us return to the 2‑dimensional universe X• = [u, v].
A bridge between propositions and quantifications can be
built by defining a foursome of measures or “qualifiers”
ℓ_ij : (B × B → B) → B defined by the following equations.

Display 1. Qualifiers ℓ_ij
• https://inquiryintoinquiry.files.wordpress.com/2022/05/qualifiers-lij-2.0.png

A higher order proposition ℓ_ij : (B × B → B) → B tells us
something about the proposition f : B × B → B, specifically,
which elements in B × B are assigned a positive value by f.

Taken together, the ℓ_ij operators give us a way to express
many useful observations about the propositions in X• = [u, v].
Figure 16 summarizes the action of the ℓ_ij operators on the
propositions of type f : B × B → B.

Figure 16. Higher Order Universe of Discourse
[ℓ_00, ℓ_01, ℓ_10, ℓ_11] ⊆ [[u, v]]
• https://inquiryintoinquiry.files.wordpress.com/2022/05/venn-diagram-4-dimensions-uv-cacti-8-inch.png

Regards,

Jon

[Jon Awbrey]

Functional Logic • Inquiry and Analogy • 17
• https://inquiryintoinquiry.com/2023/07/15/functional-logic-inquiry-and-analogy-17-2/

Inquiry and Analogy • Application of Higher Order Propositions to Quantification Theory (Part 1)
• https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy#App_Quant_1

All,

Our excursion into the expanding landscape of higher order propositions
has come round to the point where we can begin to open up new perspectives
on quantificational logic.

Though it may be all the same from a purely formal point of view,
it does serve intuition to adopt a slightly different interpretation
for the two‑valued space we take as the target of our basic indicator
functions. In that spirit we declare a novel type of “existence‑valued
functions” f : Bⁿ → E where E = {-e, +e} = {empty, existent} is a pair of
values indicating whether anything exists in the cells of the underlying
universe of discourse. As usual, we won't be too picky about the coding
of those functions, reverting to binary codes whenever the intended
interpretation is clear enough.

With that interpretation in mind we observe the following correspondence
between classical quantifications and higher order indicator functions.

Table 17. Syllogistic Premisses as Higher Order Indicator Functions
• https://inquiryintoinquiry.files.wordpress.com/2022/05/syllogistic-premisses-as-higher-order-indicator-functions-4.0.png

Regards,

Jon

[Jon Awbrey]

Functional Logic • Inquiry and Analogy • 18
• https://inquiryintoinquiry.com/2023/07/16/functional-logic-inquiry-and-analogy-18-2/

Inquiry and Analogy • Application of Higher Order Propositions to Quantification Theory (Part 2)
• https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy#App_Quant_2

All,

Last time we took up a fourfold scheme of quantified propositional forms
traditionally known as a “Square of Opposition”, relating it to a quartet
of higher order propositions which, depending on context, are also known
as “measures”, “qualifiers”, or “higher order indicator functions”.

Table 18 develops the above ideas in further detail, expressing a larger set
of quantified propositional forms by means of propositions about propositions.

Table 18. Simple Qualifiers of Propositions (Version 1)
• https://inquiryintoinquiry.files.wordpress.com/2022/05/simple-qualifiers-of-propositions-version-1.png

Regards,

Jon

[Jon Awbrey]

Functional Logic • Inquiry and Analogy • 19
• https://inquiryintoinquiry.com/2023/07/17/functional-logic-inquiry-and-analogy-19-2/

Inquiry and Analogy • Application of Higher Order Propositions to Quantification Theory (Part 3)
• https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy#App_Quant_3

<QUOTE John Dewey>
Reflection is turning a topic over in various aspects and in
various lights so that nothing significant about it shall be
overlooked — almost as one might turn a stone over to see
what its hidden side is like or what is covered by it.

John Dewey • How We Think
</QUOTE>

All,

Tables 19 and 20 present the same information as Table 18, sorting
the rows in different orders to reveal other symmetries in the arrays.

Table 18. Simple Qualifiers of Propositions (Version 1)
• https://inquiryintoinquiry.files.wordpress.com/2022/05/simple-qualifiers-of-propositions-version-1.png

Table 19. Simple Qualifiers of Propositions (Version 2)
• https://inquiryintoinquiry.files.wordpress.com/2022/05/simple-qualifiers-of-propositions-version-2.png

Table 20. Simple Qualifiers of Propositions (Version 3)
• https://inquiryintoinquiry.files.wordpress.com/2022/05/simple-qualifiers-of-propositions-version-3.png

Regards,

Jon

[Jon Awbrey]

Functional Logic • Inquiry and Analogy • 20
• https://inquiryintoinquiry.com/2023/07/18/functional-logic-inquiry-and-analogy-20-2/

Inquiry and Analogy • Application of Higher Order Propositions to Quantification Theory (Part 4)
• https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy#App_Quant_4

All,

Table 21 provides a thumbnail sketch of the relationships discussed in this section.

Table 21. Relation of Quantifiers to Higher Order Propositions
• https://inquiryintoinquiry.files.wordpress.com/2022/06/relation-of-quantifiers-to-higher-order-propositions-1.0.png

Regards,

Jon

[Jon Awbrey]

Functional Logic • Inquiry and Analogy • 21
• https://inquiryintoinquiry.com/2023/07/19/functional-logic-inquiry-and-analogy-21-2/

Inquiry and Analogy • Generalized Umpire Operators
• https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy#Gen_Ump_Ops

All,

To get a better handle on the space of higher order propositions
and continue developing our functional approach to quantification
theory, we'll need a number of specialized tools.

To begin, we define a higher order operator Υ, called the
“umpire operator”, which takes 1, 2, or 3 propositions as
arguments and returns a single truth value as the result.
Operators with optional numbers of arguments are called
“multigrade operators”, typically defined as unions over
function types. Expressing Υ in that form gives the
following formula.

UMP 1. https://inquiryintoinquiry.files.wordpress.com/2023/07/ump-1-1.png

In contexts of application, that is, where a multigrade operator
is actually being applied to arguments, the number of arguments in
the argument list tells which of the optional types is “operative”.

In the case of Υ, the first and last arguments appear as indices,
the one in the middle serving as the main argument while the other
two arguments serve to modify the sense of the operation in question.
Thus, we have the following forms.

UMP 2. https://inquiryintoinquiry.files.wordpress.com/2023/07/ump-2.png

The operation Υₚ^r q = Υ(p, q, r) evaluates the proposition q
on each model of the proposition p and combines the results
according to the method indicated by the connective parameter r.

In principle, the index r may specify any logical connective on
as many as 2^k arguments but in practice we usually have a much
simpler form of combination in mind, typically either products
or sums.

By convention, each of the accessory indices p, r is assigned
a default value understood to be in force when the corresponding
argument place is left blank, specifically, the constant proposition
1 : B^k → B for the lower index p and the continued conjunction or
continued product operation ∏ for the upper index r. Taking the
upper default value gives license to the following readings.

UMP 3. https://inquiryintoinquiry.files.wordpress.com/2023/07/ump-3.png

This means Υₚ(q) = 1 if and only if q holds for all models of p.
In propositional terms, this is tantamount to the assertion that
p ⇒ q, or that ¬(p ¬(q)) = 1.

Throwing in the lower default value permits the following abbreviations.

UMP 4. https://inquiryintoinquiry.files.wordpress.com/2023/07/ump-4.png

This means Υq = 1 if and only if q holds for the whole universe
of discourse in question, that is, if and only q is the constantly
true proposition 1 : B^k → B. The ambiguities of this usage are
not a problem so long as we distinguish the context of definition
from the context of application and restrict all shorthand notations
to the latter.

Regards,

Jon