Functional Logic • Inquiry and Analogy


Cf: Functional Logic • Inquiry and Analogy • Preliminaries


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.


• Logic Syllabus ( )
• Boolean Function ( )
• Boolean-Valued Function ( )
• Logical Conjunction ( )
• Minimal Negation Operator ( )
• Cactus Language ( )




Cf: Functional Logic • Inquiry and Analogy • 1


The following Figure is posted for future reference.
It gives a quick overview of traditional terminology
I'll have occasion to refer to as discussion proceeds.

Figure 1. Types of Reasoning in Aristotle




Cf: Functional Logic • Inquiry and Analogy • 2

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.

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).

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

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




Cf: Functional Logic • Inquiry and Analogy • 3

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

Figure 4. Types of Reasoning in Peirce




Cf: Functional Logic • Inquiry and Analogy • 4 (Part 1)

Inquiry and Analogy • Aristotle’s “Apagogy” • Abductive Reasoning


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)

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

Table of Subsumptions

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”.




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

Inquiry and Analogy • Aristotle’s “Apagogy” • Abductive Reasoning


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

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

Term, Position, Interpretation

Premiss, Predication, Inference Role

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

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



Lyle Anderson

This discussion of the three modes of reasoning seems to be analogous, and could be construed as a consequence of, the conception of calculation in The Laws of Form. 

We take the step from V to T and see that it is equivalent to the steps from V to U followed by U to T.  What we seem to do in the practice of abduction is to examine more that the one case, and try to find a rule that leads to the same conclusion.

Our confidence in the validity of the rule then depends on how many cases we can find where the rule give the "correct" answer.   In the case of abduction, we become more sure of the  "perhaps" as the cases that it is true mount up.
Best regards,


Cf: Functional Logic • Inquiry and Analogy • 5

Inquiry and Analogy • Aristotle’s “Paradigm” • Reasoning by Analogy


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)

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

Figure 6. Aristotle's “Paradigm”



James Bowery

Russell attempted an arithmetic of analogic reasoning which he called "Relation Arithmetic".  Roughtly speaking, Russell defined an analogy to be a "relation number".  A "relation number" (in Russell's sense) comprises the class of all structures that are relationally similar (ie: structures that can be identified with each other by mapping their relata to each other while holding invariant their relations).  So Russell's notion of relation number was is really a class of all relationships that are analogies of each other -- they are all relationally similar.  Note that a structure can involve any number of relata and relations.

Tom Etter found and fixed a bug in Russell's Relation Arithmetic:

Despite its great promise, relation-arithmetic didn’t get very far. The problem is that the most important combining operators for relations, such as cross product and join, are not invariant under similarity. That is, if A is similar to A’ and B is similar to B’, it does not follow that the cross product or join of A and B is similar to the cross product or join of A’ and B’. This can be seen from a very simple example. Consider a unary relation R with only one tuple (its table has one row and one column). Let the value in that row and column be v. Replacing v by any other value w produces a relation R’ that is similar to R. Now consider the cross product (Cartesian join) RR; it consists of the ordered pair vv. But the cross product RR’ consists of the ordered pair vw, so RR and RR’ are not similar, despite the similarity of their components.

Tom did this work as part of two papers he wrote while working for me at HP's "Internet Chapter 2" project where I asked him to help with my career-long effort to reformulate relational programming languages in a manner that would generalize Codd's work in a manner that incorporated the Laws of Form's dynamical descriptions (ie:  i and -i as logical operators).  


Two papers came out of that:

Structure Theory

Relation Arithmetic Revived

That work continued with Richard Shoup (originator of this mailing list) under Federico Faggin at Boundary Institute.

The importance to abduction and induction is best characterized in terms of lossless compression of observed facts under Algorithmic Information Theory.


Cf: Functional Logic • Inquiry and Analogy • 6

Inquiry and Analogy • Peirce’s Formulation of Analogy • Version 1


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

Version 1 —


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:


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.


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)

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

Figure 7. Peirce's Formulation of Analogy (Version 1)



Lyle Anderson

This is very important, conceptually, to how experimental science verifies theoretical formulas.  Let P be a series of measurements of a phenomenon, say a falling body on earth, and let S be the calculation for those measurements from the theory under test.  Then Q  is a sampling from P, and T is a sampling from the theory.  The two samples must have the matching independent parameters or variables.   There is an entire Theory of Error that covers what constitutes "a match" between each instance of Q, T.

In some cases the difference between theories can be quite spectacular.  A  good example from the turn of the turn of the 20th century was the Ultraviolet Catastrophe that occurred when classical, infinitesimal  thermodynamics was applied to black body radiation.  The classical theory said that all frequencies of light were possible, that is to say that the energy spectrum of light was continuous.  This lead to most of the energy being in the ultraviolet in a  catastrophic way, hence the name of the phenomenon.   What Planck showed was that if one assumed that the energy spectrum was quantized than you got results that matched the experiment, with a simple equation between energy and frequency that involved a single constant, which was named after him.

Best regards,


Cf: Functional Logic • Inquiry and Analogy • 7

Inquiry and Analogy • Peirce’s Formulation of Analogy • Version 2


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


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]

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)


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

Figure 8. Peirce's Formulation of Analogy (Version 2)




Cf: Functional Logic • Inquiry and Analogy • 8

Inquiry and Analogy • Dewey’s “Sign of Rain” • An Example of Inquiry


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)

Inquiry and Interpretation

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 both 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




Cf: Functional Logic • Inquiry and Analogy • 9

Inquiry and Analogy • Dewey’s “Sign of Rain” • An Example of Inquiry

Inquiry and Inference

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

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.




Cf: Functional Logic • Inquiry and Analogy • 10

Inquiry and Analogy • Functional Conception of Quantification Theory


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

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.




Cf: Functional Logic • Inquiry and Analogy • 11

Inquiry and Analogy • Higher Order Propositional Expressions

Higher Order Propositions and Logical Operators (n = 1)


A “higher order proposition” is, roughly speaking,
a proposition about propositions. If the original
order of propositions is a class 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 sixteen 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)




Cf: Functional Logic • Inquiry and Analogy • 12

Interpretive Categories for Higher Order Propositions (n = 1)


Referring to “Table 11. Higher Order Propositions (n = 1)” from the previous post:

Table 12 presents a series of “interpretive categories” for the higher order propositions
in Table 11. I’ll leave these 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)




Cf: Functional Logic • Inquiry and Analogy • 13

Inquiry and Analogy • Higher Order Propositional Expressions

Higher Order Propositions and Logical Operators (n = 2)


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 value of the measure m_j on
each of the propositions f_i : B² → B, for i = 0 to 15. 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)




Cf: Functional Logic • Inquiry and Analogy • 14

Umpire Operators (Part 1 of 2)


[Note. Please follow the first link above for better math formatting.]

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


• 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 a little 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).




Cf: Functional Logic • Inquiry and Analogy • 14 (Part 2)

Inquiry and Analogy • Umpire Operators (Part 2)


The “absolute umpire operator”, also known as 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 as
mathematical objects but can also be understood 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, regarding all contingent
statements as no better than falsehoods.