Jon Awbrey · @Inquiry
259 followers · 1813 posts · Server mathstodon.xyz

Logical Graphs • Discussion 6
inquiryintoinquiry.com/2023/08

Re: Logical Graphs • First Impressions
inquiryintoinquiry.com/2023/08

Logical Graphs • Figures 1 and 2
inquiryintoinquiry.files.wordp

Re: Academia.edu • Robert Appleton
academia.edu/community/lavbw5?

RA:
❝As a professional graphic designer and non-mathematician reading your two diagrams, I need to ask for a simpler statement of their purpose. What do Fig 1 and Fig 2 represent to you? And what insight do they provide us?❞

My Comment —

Figures 1 and 2 are really just a couple of “in medias res” pump‑primers or ice‑breakers. This will all be explained in the above linked blog post, where I'm revising the text and upgrading the graphics of some work I first blogged in 2008 based on work I did even further back. I'll be taking a fresh look at that as I serialize it here.

Those two Figures come from George Spencer Brown's 1969 book Laws of Form, where he called them the Law of Calling and the Law of Crossing. GSB revived and clarified central aspects of Peirce's systems of logical graphs and I find it helpful to integrate his work into my exposition of Peirce. For now you can think of those as exemplifying two core formal principles which go to the root of the mathematical forms underlying logical reasoning.


#PropositionalCalculus #BooleanFunctions #lawsofform #spencerbrown #ExistentialGraphs #EntitativeGraphs #LogicalGraphs #logic #Peirce

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Jon Awbrey · @Inquiry
254 followers · 1811 posts · Server mathstodon.xyz

Logical Graphs • First Impressions 1
inquiryintoinquiry.com/2023/08

Introduction • Moving Pictures of Thought —

A “logical graph” is a graph-theoretic structure in one of the systems of graphical syntax Charles Sanders Peirce developed for logic.

In numerous papers on “qualitative logic”, “entitative graphs”, and “existential graphs”, Peirce developed several versions of a graphical formalism, or a graph-theoretic formal language, designed to be interpreted for logic.

In the century since Peirce initiated this line of development, a variety of formal systems have branched out from what is abstractly the same formal base of graph-theoretic structures. This article examines the common basis of these formal systems from a bird's eye view, focusing on the aspects of form shared by the entire family of algebras, calculi, or languages, however they happen to be viewed in a given application.


#PropositionalCalculus #BooleanFunctions #lawsofform #spencerbrown #existensialgraphs #EntitativeGraphs #LogicalGraphs #logic #Peirce

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Alan · @alan
6 followers · 55 posts · Server piaille.fr
Jon Awbrey · @Inquiry
254 followers · 1811 posts · Server mathstodon.xyz

Differential Logic • The Logic of Change and Difference
inquiryintoinquiry.com/2023/08

Differential logic is the logic of variation — the logic of change and difference.

Differential logic is the component of logic whose object is the description of variation, for example, the aspects of change, difference, distribution, and diversity, in universes of discourse subject to qualitative logical description. In its formalization, differential logic treats the principles governing the use of a “differential logical calculus”, in other words, a formal system with the expressive capacity to describe change and diversity in logical universes of discourse.

A simple case of a differential logical calculus is furnished by a differential propositional calculus. This augments ordinary propositional calculus in the same way the differential calculus of Leibniz and Newton augments the analytic geometry of Descartes.

Resources —

Differential Logic
oeis.org/wiki/Differential_Log
• Part 1 ( oeis.org/wiki/Differential_Log )
• Part 2 ( oeis.org/wiki/Differential_Log )
• Part 3 ( oeis.org/wiki/Differential_Log )

Differential Propositional Calculus
oeis.org/wiki/Differential_Pro
• Part 1 ( oeis.org/wiki/Differential_Pro )
• Part 2 ( oeis.org/wiki/Differential_Pro )

Differential Logic and Dynamic Systems
oeis.org/wiki/Differential_Log
• Part 1 ( oeis.org/wiki/Differential_Log )
• Part 2 ( oeis.org/wiki/Differential_Log )
• Part 3 ( oeis.org/wiki/Differential_Log )
• Part 4 ( oeis.org/wiki/Differential_Log )
• Part 5 ( oeis.org/wiki/Differential_Log )



#neuralnetworksystems #MinimalNegationOperators #differentialpropositions #QualitativeDynamics #booleandifferencecalculus #BooleanFunctions #leibniz #DiscreteDynamicalSystems #DifferentialLogic #LogicalGraphs #logic #Peirce

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Jon Awbrey · @Inquiry
253 followers · 1809 posts · Server mathstodon.xyz

Pragmatic Maxim • References, Readings, Resources
inquiryintoinquiry.com/2023/08

References —

Peirce, C.S., Collected Papers of Charles Sanders Peirce, vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931–1935, 1958. Cited as CP n.m for volume n, paragraph m.

Readings —

Awbrey, Jon, and Awbrey, Susan (1995), “Interpretation as Action • The Risk of Inquiry”, Inquiry : Critical Thinking Across the Disciplines 15(1), 40–52.

Archive
web.archive.org/web/2000121016
Journal
pdcnet.org/inquiryct/content/i
Online
academia.edu/57812482/Interpre

Resources —

Bergman and Paavola (eds.) • Commens Dictionary of Peirce's Terms
helsinki.fi/science/commens/di

Pragmatic Maxim
helsinki.fi/science/commens/te
Pragmaticism
helsinki.fi/science/commens/te
Pragmatism
helsinki.fi/science/commens/te



#methodofreflection #closureprinciple #representationprinciple #representation #operationaldefinition #extensionaldefinition #effectivedefinition #pragmatism #semiotics #standardoperatingprocedure #pragmaticmaxim #inquiry #logic #Peirce

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Jon Awbrey · @Inquiry
252 followers · 1808 posts · Server mathstodon.xyz

Inquiry Into Inquiry • Discussion 9
inquiryintoinquiry.com/2023/08

Re: Milo Gardner
academia.edu/community/VBqzR5?

MG: ❝Do you agree that Peirce was limited to bivalent logic?❞

Taking classical logic as a basis for reasoning is no more limiting than taking Dedekind cuts as a basis for constructing the real number line. For Peirce's relational approach to logic as semiotics the number of dimensions in a relation is more important than the number of values in each dimension. That is where 3 makes a difference over 2.


#triadicrelationirreducibility #RelationTheory #compositionality #categorytheory #semiotics #informationfusion #information #inference #inquiry #logic #Peirce

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Jon Awbrey · @Inquiry
251 followers · 1807 posts · Server mathstodon.xyz

Inquiry Into Inquiry • Discussion 8
inquiryintoinquiry.com/2023/08

Re: Milo Gardner
academia.edu/community/Lbxjg5?

MG: ❝Peirce sensed that bivalent syntax was superceded by trivalent syntax,
but never resolved that nagging question.❞

My Comment —

The main thing is not a question of syntax but a question of the mathematical models we use to cope with object realities and real objectives (pragmata). Signs, syntax, and systems of representation can make a big difference in how well they represent the object domain and how well they serve the purpose at hand but they remain accessory to those objects and purposes.


#triadicrelationirreducibility #RelationTheory #compositionality #categorytheory #semiotics #informationfusion #information #inference #inquiry #logic #Peirce

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Jon Awbrey · @Inquiry
250 followers · 1805 posts · Server mathstodon.xyz

Inquiry Into Inquiry • Discussion 7
inquiryintoinquiry.com/2023/08

Dan Everett has prompted a number of discussions on Facebook recently which touch on core issues in Peirce's thought — but threads ravel on and fray so quickly in that medium one rarely get a chance to fill out the warp. Not exactly at random, here's a loose thread I think may be worth the candle.

Re: Facebook • Daniel Everett
facebook.com/permalink.php?sto

My Comment —

Compositionality started out as a well-defined concept, arising from the composition of mathematical functions, abstracted to the composition of arrows and functors in category theory, and generalized to the composition of binary, two-place, or dyadic relations. In terms of linguistic complexity it's associated with properly context-free languages. That all keeps compositionality on the dyadic side of the border in Peirce's universe. More lately the term has been volatilized to encompass almost any sort of information fusion, which is all well and good so long as folks make it clear what they are talking about, for which use the term “information fusion” would probably be sufficiently vague.


#triadicrelationirreducibility #RelationTheory #compositionality #categorytheory #semiotics #informationfusion #information #inference #inquiry #logic #Peirce

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Jon Awbrey · @Inquiry
250 followers · 1805 posts · Server mathstodon.xyz

Inquiry Into Inquiry • Discussion 7
inquiryintoinquiry.com/2023/08

Dan Everett has prompted a number of discussions on Facebook recently which touch on core issues in Peirce's thought — but threads ravel on and fray so quickly in that medium one rarely get a chance to fill out the warp. Not exactly at random, here's a loose thread I think may be worth the candle.

Re: Facebook • Daniel Everett
facebook.com/permalink.php?sto

My Comment —

Compositionality started out as a well-defined concept, arising from the composition of mathematical functions, abstracted to the composition of arrows and functors in category theory, and generalized to the composition of binary, two-place, or dyadic relations. In terms of linguistic complexity it's associated with properly context-free languages. That all keeps compositionality on the dyadic side of the border in Peirce's universe. More lately the term has been volatilized to encompass almost any sort of information fusion, which is all well and good so long as folks make it clear what they are talking about, for which use the term “information fusion” would probably be sufficiently vague.


#triadicrelationirreducibility #RelationTheory #compositionality #categorytheory #semiotics #informationfusion #information #inference #inquiry #logic #Peirce

Last updated 1 year ago

Jon Awbrey · @Inquiry
250 followers · 1803 posts · Server mathstodon.xyz

Seven Ways of Looking at a Pragmatic Maxim • 7
inquiryintoinquiry.com/2023/08

The seventh excerpt is a late reflection on the reception of pragmatism. With a sense of exasperation that is almost palpable, Peirce tries to justify the maxim of pragmatism and to correct its misreadings by pinpointing a number of false impressions that the intervening years have piled on it, and he attempts once more to prescribe against the deleterious effects of these mistakes. Recalling the very conception and birth of pragmatism, he reviews its initial promise and its intended lot in the light of its subsequent vicissitudes and its apparent fate. Adopting the style of a post mortem analysis, he presents a veritable autopsy of the ways that the main idea of pragmatism, for all its practicality, can be murdered by a host of misdissecting disciplinarians, by what are ostensibly its most devoted followers.

❝This employment five times over of derivates of concipere must then have had a purpose. In point of fact it had two. One was to show that I was speaking of meaning in no other sense than that of intellectual purport. The other was to avoid all danger of being understood as attempting to explain a concept by percepts, images, schemata, or by anything but concepts. I did not, therefore, mean to say that acts, which are more strictly singular than anything, could constitute the purport, or adequate proper interpretation, of any symbol. I compared action to the finale of the symphony of thought, belief being a demicadence. Nobody conceives that the few bars at the end of a musical movement are the purpose of the movement. They may be called its upshot.❞

(Peirce, CP 5.402 note 3, 1906).

#pragmaticmaxim #Peirce

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Jon Awbrey · @Inquiry
250 followers · 1803 posts · Server mathstodon.xyz

Seven Ways of Looking at a Pragmatic Maxim • 6
inquiryintoinquiry.com/2023/08

A sixth excerpt is useful in stating the bearing of the pragmatic maxim on the topic of reflection, namely, that it makes all of pragmatism boil down to nothing more or less than a method of reflection.

❝The study of philosophy consists, therefore, in reflexion, and pragmatism is that method of reflexion which is guided by constantly holding in view its purpose and the purpose of the ideas it analyzes, whether these ends be of the nature and uses of action or of thought. … It will be seen that pragmatism is not a Weltanschauung but is a method of reflexion having for its purpose to render ideas clear.❞

(Peirce, CP 5.13 note 1, 1902).



#methodofreflection #closureprinciple #representationprinciple #representation #operationaldefinition #extensionaldefinition #effectivedefinition #pragmatism #semiotics #standardoperatingprocedure #pragmaticmaxim #inquiry #logic #Peirce

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Jon Awbrey · @Inquiry
250 followers · 1803 posts · Server mathstodon.xyz

Seven Ways of Looking at a Pragmatic Maxim • 5
inquiryintoinquiry.com/2023/08

The fifth excerpt is useful by way of additional clarification, and was aimed to correct a variety of historical misunderstandings that arose over time with regard to the intended meaning of the pragmatic maxim.

❝The doctrine appears to assume that the end of man is action — a stoical axiom which, to the present writer at the age of sixty, does not recommend itself so forcibly as it did at thirty. If it be admitted, on the contrary, that action wants an end, and that that end must be something of a general description, then the spirit of the maxim itself, which is that we must look to the upshot of our concepts in order rightly to apprehend them, would direct us towards something different from practical facts, namely, to general ideas, as the true interpreters of our thought.❞

(Peirce, CP 5.3, 1902).



#upshotofconcepts #closureprinciple #representationprinciple #representation #operationaldefinition #extensionaldefinition #effectivedefinition #pragmatism #semiotics #standardoperatingprocedure #pragmaticmaxim #inquiry #logic #Peirce

Last updated 1 year ago

Jon Awbrey · @Inquiry
249 followers · 1801 posts · Server mathstodon.xyz

Seven Ways of Looking at a Pragmatic Maxim • 4
inquiryintoinquiry.com/2023/08

The fourth excerpt illustrates one of Peirce's many attempts to get the sense of the pragmatic philosophy across by rephrasing the pragmatic maxim in an alternative way. In introducing this version, he addresses an order of prospective critics who do not deem a simple heuristic maxim, much less one that concerns itself with a routine matter of logical procedure, as forming a sufficient basis for a full-grown philosophy.

❝On their side, one of the faults that I think they might find with me is that I make pragmatism to be a mere maxim of logic instead of a sublime principle of speculative philosophy. In order to be admitted to better philosophical standing I have endeavored to put pragmatism as I understand it into the same form of a philosophical theorem. I have not succeeded any better than this:

❝Pragmatism is the principle that every theoretical judgment expressible in a sentence in the indicative mood is a confused form of thought whose only meaning, if it has any, lies in its tendency to enforce a corresponding practical maxim expressible as a conditional sentence having its apodosis in the imperative mood.❞

(Peirce, CP 5.18, 1903).



#pragmaticimperative #closureprinciple #representationprinciple #representation #operationaldefinition #extensionaldefinition #effectivedefinition #pragmatism #semiotics #standardoperatingprocedure #pragmaticmaxim #inquiry #logic #Peirce

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Jon Awbrey · @Inquiry
249 followers · 1801 posts · Server mathstodon.xyz

Seven Ways of Looking at a Pragmatic Maxim • 3
inquiryintoinquiry.com/2023/08

The third excerpt puts a gloss on the meaning of a practical bearing and provides an alternative statement of the maxim.

❝Such reasonings and all reasonings turn upon the idea that if one exerts certain kinds of volition, one will undergo in return certain compulsory perceptions. Now this sort of consideration, namely, that certain lines of conduct will entail certain kinds of inevitable experiences is what is called a “practical consideration”. Hence is justified the maxim, belief in which constitutes pragmatism; namely:

❝In order to ascertain the meaning of an intellectual conception one should consider what practical consequences might conceivably result by necessity from the truth of that conception; and the sum of these consequences will constitute the entire meaning of the conception.❞

(Peirce, CP 5.9, 1905).



#sumofconsequences #closureprinciple #representationprinciple #representation #operationaldefinition #extensionaldefinition #effectivedefinition #pragmatism #semiotics #standardoperatingprocedure #pragmaticmaxim #inquiry #logic #Peirce

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Jon Awbrey · @Inquiry
250 followers · 1794 posts · Server mathstodon.xyz

Seven Ways of Looking at a Pragmatic Maxim • 2
inquiryintoinquiry.com/2023/08

The second excerpt presents another version of the pragmatic maxim, a recommendation about a way of clarifying meaning that can be taken to stake out the general philosophy of pragmatism.

❝Pragmaticism was originally enounced in the form of a maxim, as follows:

❝Consider what effects that might conceivably have practical bearings you conceive the objects of your conception to have. Then, your conception of those effects is the whole of your conception of the object.

❝I will restate this in other words, since ofttimes one can thus eliminate some unsuspected source of perplexity to the reader. This time it shall be in the indicative mood, as follows:

❝The entire intellectual purport of any symbol consists in the total of all general modes of rational conduct which, conditionally upon all the possible different circumstances and desires, would ensue upon the acceptance of the symbol.❞

(Peirce, CP 5.438, 1878/1905).



#conceptionofeffects #closureprinciple #representationprinciple #representation #operationaldefinition #extensionaldefinition #effectivedefinition #pragmatism #semiotics #standardoperatingprocedure #pragmaticmaxim #inquiry #logic #Peirce

Last updated 1 year ago

Jon Awbrey · @Inquiry
250 followers · 1793 posts · Server mathstodon.xyz

Seven Ways of Looking at a Pragmatic Maxim • 1
inquiryintoinquiry.com/2023/08

Peirce stated the pragmatic maxim in many different ways over the years, each of which adds its own bit of clarity or correction to their collective corpus.

The first excerpt appears in the form of a dictionary entry, intended as a definition of pragmatism.

Pragmatism. The opinion that metaphysics is to be largely cleared up by the application of the following maxim for attaining clearness of apprehension:

“Consider what effects, that might conceivably have practical bearings, we conceive the object of our conception to have. Then, our conception of these effects is the whole of our conception of the object.” (Peirce, CP 5.2, 1878/1902).



#conceptionofeffects #closureprinciple #representationprinciple #representation #operationaldefinition #extensionaldefinition #effectivedefinition #pragmatism #semiotics #standardoperatingprocedure #pragmaticmaxim #inquiry #logic #Peirce

Last updated 1 year ago

Jon Awbrey · @Inquiry
250 followers · 1793 posts · Server mathstodon.xyz

Pragmatic Maxim • Introduction
inquiryintoinquiry.com/2023/08

The “pragmatic maxim”, also known as the “maxim of pragmatism” or the “maxim of pragmaticism”, is a maxim of logic formulated by Charles Sanders Peirce. Serving as a practical recommendation or regulative principle in the normative science of logic, its function is to guide the conduct of thought toward the achievement of its purpose, advising the addressee on an optimal way of “attaining clearness of apprehension”.

Charles Sanders Peirce
mywikibiz.com/Charles_Sanders_

Normative Science
mywikibiz.com/Normative_scienc


#semiotics #normativescience #CharlesSandersPeirce #cspeirce #requlativeprinciple #pragmaticmaxim #inquiry #logic #Peirce

Last updated 1 year ago

Jon Awbrey · @Inquiry
250 followers · 1793 posts · Server mathstodon.xyz

Pragmatic Maxim • Et Sic Deinceps
inquiryintoinquiry.com/2023/08

The pragmatic maxim is a guideline for the practice of inquiry formulated by Charles Sanders Peirce. Serving as a practical recommendation or regulative principle in the normative science of logic, its function is to guide the conduct of thought toward the achievement of its purpose, advising the addressee on an optimal way of “attaining clearness of apprehension”.


#semiotics #normativescience #CharlesSandersPeirce #cspeirce #requlativeprinciple #pragmaticmaxim #inquiry #logic #Peirce

Last updated 1 year ago

Jon Awbrey · @Inquiry
247 followers · 1784 posts · Server mathstodon.xyz

Survey of Precursors Of Category Theory • 4
inquiryintoinquiry.com/2023/08

A few years ago I began a sketch on the “Precursors of Category Theory”, tracing the continuities of the category concept from Aristotle, to Kant and Peirce, through Hilbert and Ackermann, to contemporary mathematical practice. A Survey of resources on the topic is given below, still very rough and incomplete, but perhaps a few will find it of use.

Background —

Precursors Of Category Theory
oeis.org/wiki/Precursors_Of_Ca

Propositions As Types Analogy
oeis.org/wiki/Propositions_As_

Blog Series —

Notes On Categories
inquiryintoinquiry.com/2013/02

Precursors Of Category Theory
1. inquiryintoinquiry.com/2013/12
2. inquiryintoinquiry.com/2013/12
3. inquiryintoinquiry.com/2014/01

Precursors Of Category Theory • Discussion
1. inquiryintoinquiry.com/2020/09
2. inquiryintoinquiry.com/2020/09
3. inquiryintoinquiry.com/2020/09

Categories à la Peirce —

C.S. Peirce • A Guess at the Riddle
inquiryintoinquiry.com/2012/03

Peirce's Categories
1. inquiryintoinquiry.com/2015/10
2. inquiryintoinquiry.com/2015/10
3. inquiryintoinquiry.com/2015/11
•••
19. inquiryintoinquiry.com/2020/05
20. inquiryintoinquiry.com/2020/05
21. inquiryintoinquiry.com/2020/06




#universals #typetheory #propositionsastypes #peircescategories #hypostaticabstraction #continuouspredicate #RelationTheory #functionallogic #foundationsofmathematics #diagrams #categorytheory #analogy #abstraction #saundersmaclane #ackermann #hilbert #carnap #kant #Peirce #aristotle

Last updated 1 year ago

Jon Awbrey · @Inquiry
247 followers · 1781 posts · Server mathstodon.xyz

Survey of Abduction, Deduction, Induction, Analogy, Inquiry • 3
inquiryintoinquiry.com/2023/04

This is a Survey of blog and wiki posts on three elementary forms of inference, as recognized by a logical tradition extending from Aristotle through Charles S. Peirce. Particular attention is paid to the way these inferential rudiments combine to form the more complex patterns of analogy and inquiry.

Blog Dialogs —

• Abduction, Deduction, Induction, Analogy, Inquiry
1. inquiryintoinquiry.com/2012/08
2. inquiryintoinquiry.com/2012/11
3. inquiryintoinquiry.com/2016/02
•••
29. inquiryintoinquiry.com/2020/03
30. inquiryintoinquiry.com/2020/12
31. inquiryintoinquiry.com/2022/10

Blog Series —

Functional Logic • Inquiry and Analogy
inquiryintoinquiry.com/2023/06
1. inquiryintoinquiry.com/2023/06
2. inquiryintoinquiry.com/2023/06
3. inquiryintoinquiry.com/2023/06
•••
19. inquiryintoinquiry.com/2023/07
20. inquiryintoinquiry.com/2023/07
21. inquiryintoinquiry.com/2023/07
Discussion
inquiryintoinquiry.com/2023/07

OEIS Wiki —

Functional Logic • Inquiry and Analogy
oeis.org/wiki/Functional_Logic

Prospects for Inquiry Driven Systems
oeis.org/wiki/User:Jon_Awbrey/

Introduction to Inquiry Driven Systems
oeis.org/wiki/Introduction_to_

Information = Comprehension × Extension
oeis.org/wiki/Information_%3D_

Inquiry Driven Systems • Inquiry Into Inquiry
oeis.org/wiki/Inquiry_Driven_S



#informationequalscomprehensiontimesextension #inquirydrivensystem #semiotics #dynamicsofinquiry #DifferentialLogic #functionallogic #inquiry #analogy #induction #deduction #abduction #logic #Peirce

Last updated 1 year ago