Functional Logic • Inquiry and Analogy
Cf: Functional Logic • Inquiry and Analogy • Preliminaries
http://inquiryintoinquiry.com/2021/11/14/functionallogicinquiryandanalogypreliminaries/ All, 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 expressionforming operations of a calculus for booleanvalued 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 ) • BooleanValued Function ( https://oeis.org/wiki/Booleanvalued_function ) • Logical Conjunction ( https://oeis.org/wiki/Logical_conjunction ) • Minimal Negation Operator ( https://oeis.org/wiki/Minimal_negation_operator ) • Cactus Language ( https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Overview ) Regards, Jon


Cf: Functional Logic • Inquiry and Analogy • 1
https://inquiryintoinquiry.com/2022/04/17/functionallogicinquiryandanalogy1/ All, 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 https://inquiryintoinquiry.files.wordpress.com/2022/04/typesofreasoninginaristotle.jpg Regards, Jon


Cf: Functional Logic • Inquiry and Analogy • 2
https://inquiryintoinquiry.com/2022/04/19/functionallogicinquiryandanalogy2/ 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 All, 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/triplywiseact.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


Cf: Functional Logic • Inquiry and Analogy • 3
https://inquiryintoinquiry.com/2022/04/20/functionallogicinquiryandanalogy3/ Inquiry and Analogy • Comparison of the Analyses ================================================ https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy#Comparison_of_the_Analyses All, 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/typesofreasoningintransition.jpg Figure 4. Types of Reasoning in Peirce https://inquiryintoinquiry.files.wordpress.com/2022/04/typesofreasoninginpeirce.jpg Regards, Jon


Cf: Functional Logic • Inquiry and Analogy • 4 (Part 1)
https://inquiryintoinquiry.com/2022/04/24/functionallogicinquiryandanalogy4/ 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 All, 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) </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/tableofimplicationstuv.png Table of Subsumptions https://inquiryintoinquiry.files.wordpress.com/2022/04/tableofsubsumptionstuv.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


Cf: Functional Logic • Inquiry and Analogy • 4 (Part 2)
https://inquiryintoinquiry.com/2022/04/24/functionallogicinquiryandanalogy4/ 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 All, 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/teachabilityunderstandingvirtue3.0.png The sense of the Figure is explained by the following assignments. Term, Position, Interpretation https://inquiryintoinquiry.files.wordpress.com/2022/04/termpositioninterpretationtuv.png Premiss, Predication, Inference Role https://inquiryintoinquiry.files.wordpress.com/2022/04/premisspredicationinferenceroletuv.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


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, Lyle


Cf: Functional Logic • Inquiry and Analogy • 5
https://inquiryintoinquiry.com/2022/04/26/functionallogicinquiryandanalogy5/ 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 All, 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 “sideshow”, 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) </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/aristotlesparadigm.jpg Regards, Jon


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 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 careerlong 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: Relation Arithmetic Revived


Cf: Functional Logic • Inquiry and Analogy • 6
https://inquiryintoinquiry.com/2022/04/28/functionallogicinquiryandanalogy6/ 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/peircesformulationofanalogyversion1.jpg Regards, Jon


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, Lyle


Cf: Functional Logic • Inquiry and Analogy • 7
https://inquiryintoinquiry.com/2022/04/29/functionallogicinquiryandanalogy7/ 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/peirceonanalogye280a2cp2.733.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/peircesformulationofanalogyversion2.jpg Regards, Jon


Cf: Functional Logic • Inquiry and Analogy • 8
https://inquiryintoinquiry.com/2022/04/30/functionallogicinquiryandanalogy8/ 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 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 https://inquiryintoinquiry.files.wordpress.com/2022/04/deweyssignofrainexample.jpg Regards, Jon


Cf: Functional Logic • Inquiry and Analogy • 9
https://inquiryintoinquiry.com/2022/05/02/functionallogicinquiryandanalogy9/ 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 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/cycleofinquirygrayscale.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


Cf: Functional Logic • Inquiry and Analogy • 10
https://inquiryintoinquiry.com/2022/05/04/functionallogicinquiryandanalogy10/ Inquiry and Analogy • 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


Cf: Functional Logic • Inquiry and Analogy • 11
https://inquiryintoinquiry.com/2022/05/05/functionallogicinquiryandanalogy11/ 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 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) https://inquiryintoinquiry.files.wordpress.com/2022/05/higherorderpropositionsn1.png Regards, Jon


Cf: Functional Logic • Inquiry and Analogy • 12
https://inquiryintoinquiry.com/2022/05/07/functionallogicinquiryandanalogy12/ 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/higherorderpropositionsn1.png 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) https://inquiryintoinquiry.files.wordpress.com/2022/05/interpretivecategoriesforhigherorderpropositionsn1.png Regards, Jon


Cf: Functional Logic • Inquiry and Analogy • 13
https://inquiryintoinquiry.com/2022/05/09/functionallogicinquiryandanalogy13/ 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 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 jth 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/higherorderpropositionsn22.0.png Regards, Jon


Cf: Functional Logic • Inquiry and Analogy • 14
https://inquiryintoinquiry.com/2022/05/17/functionallogicinquiryandanalogy14/ Umpire Operators (Part 1 of 2) https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy#Umpire_Operators All, [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 2dimensional 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 2dimensional 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). Regards, Jon


Cf: Functional Logic • Inquiry and Analogy • 14 (Part 2)
https://inquiryintoinquiry.com/2022/05/17/functionallogicinquiryandanalogy14/ Inquiry and Analogy • Umpire Operators (Part 2) https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy#Ump_Abs All, 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. Regards, Jon

