Cf: Sign Relations, Triadic Relations, Relation Theory • Discussion 6
http://inquiryintoinquiry.com/2022/03/01/sign-relations-triadic-relations-r…
Re: FB | Charles S. Peirce Society
https://www.facebook.com/groups/peircesociety/posts/2551077815028195/
::: Alain Létourneau
https://www.facebook.com/groups/peircesociety/posts/2551077815028195?commen…
All,
Alain Létourneau asks if I have any thoughts
on Peirce's Rhetoric. I venture the following.
Classically speaking, rhetoric (as distinguished from dialectic)
treats forms of argument which “consider the audience” — which
take the condition of the addressee into account. But that is
just what Peirce's semiotic does in extending our theories of
signs from dyadic to triadic sign relations.
We often begin our approach to Peirce's semiotics by saying he puts the
interpreter back into the relation of signs to their objects. But even
Aristotle had already done that much. Peirce's innovation was to apply
the pragmatic maxim, clarifying the characters of interpreters in terms
of their effects — their interpretants — in the flow of semiosis.
Some reading —
Awbrey, J.L., and Awbrey, S.M. (1995),
“Interpretation as Action • The Risk of Inquiry”,
Inquiry : Critical Thinking Across the Disciplines 15(1), 40–52.
https://www.academia.edu/57812482/Interpretation_as_Action_The_Risk_of_Inqu…
Regards,
Jon
Cf: Functional Logic • Inquiry and Analogy • 7
https://inquiryintoinquiry.com/2022/04/29/functional-logic-inquiry-and-anal…
Inquiry and Analogy • Peirce’s Formulation of Analogy • Version 2
https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy#Peirce…
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-e2…
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-…
Regards,
Jon
Cf: Functional Logic • Inquiry and Analogy • 5
https://inquiryintoinquiry.com/2022/04/26/functional-logic-inquiry-and-anal…
Inquiry and Analogy • Aristotle’s “Paradigm” • Reasoning by Analogy
https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy#Aristo…
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 “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)
</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.…
Regards,
Jon
Cf: Functional Logic • Inquiry and Analogy • 4 (Part 1)
https://inquiryintoinquiry.com/2022/04/24/functional-logic-inquiry-and-anal…
Inquiry and Analogy • Aristotle’s “Apagogy” • Abductive Reasoning
https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy#Aristo…
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/table-of-implication…
Table of Subsumptions
https://inquiryintoinquiry.files.wordpress.com/2022/04/table-of-subsumption…
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 1)
https://inquiryintoinquiry.com/2022/04/24/functional-logic-inquiry-and-anal…
Inquiry and Analogy • Aristotle’s “Apagogy” • Abductive Reasoning
https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy#Aristo…
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/table-of-implication…
Table of Subsumptions
https://inquiryintoinquiry.files.wordpress.com/2022/04/table-of-subsumption…
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 • Preliminaries
http://inquiryintoinquiry.com/2021/11/14/functional-logic-inquiry-and-analo…
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 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 )
• Cactus Language ( https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Overview )
Regards,
Jon
Cf: Peirce’s 1870 “Logic of Relatives” • Preliminaries
https://inquiryintoinquiry.com/2014/01/27/peirces-1870-logic-of-relatives-p…
All,
I need to return to my study of Peirce’s 1870 Logic of Relatives,
and I thought it might be more pleasant to do that on my blog than
to hermit away on the wiki where I last left off.
Peirce’s 1870 “Logic of Relatives” • Part 1
===========================================
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1
Peirce’s text employs lower case letters for logical terms of general reference
and upper case letters for logical terms of individual reference. General terms
fall into types, namely, absolute terms, dyadic relative terms, and higher adic
relative terms, and Peirce employs different typefaces to distinguish these.
The following Tables indicate the typefaces used in the text below for Peirce’s
examples of general terms.
Table 1. Absolute Terms (Monadic Relatives)
https://inquiryintoinquiry.files.wordpress.com/2021/11/peirces-1870-lor-e28…
Table 2. Simple Relative Terms (Dyadic Relatives)
https://inquiryintoinquiry.files.wordpress.com/2021/11/peirces-1870-lor-e28…
Table 3. Conjugative Terms (Higher Adic Relatives)
https://inquiryintoinquiry.files.wordpress.com/2021/11/peirces-1870-lor-e28…
Individual terms are taken to denote individual entities falling under
a general term. Peirce uses upper case Roman letters for individual terms,
for example, the individual horses H, H′, H″ falling under the general term h
for horse.
The path to understanding Peirce’s system and its wider implications
for logic can be smoothed by paraphrasing his notations in a variety
of contemporary mathematical formalisms, while preserving the semantics
as much as possible. Remaining faithful to Peirce’s orthography while
adding parallel sets of stylistic conventions will, however, demand close
attention to typography-in-context. Current style sheets for mathematical
texts specify italics for mathematical variables, with upper case letters
for sets and lower case letters for individuals. So we need to keep an
eye out for the difference between the individual X of the genus x and
the element x of the set X as we pass between the two styles of text.
References
==========
• Peirce, C.S. (1870), “Description of a Notation for the Logic of Relatives,
Resulting from an Amplification of the Conceptions of Boole’s Calculus of Logic”,
Memoirs of the American Academy of Arts and Sciences 9, 317–378, 26 January 1870.
Reprinted, Collected Papers (CP 3.45–149), Chronological Edition (CE 2, 359–429).
Online:
• https://www.jstor.org/stable/25058006
• https://archive.org/details/jstor-25058006
• https://books.google.com/books?id=fFnWmf5oLaoC
• 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 volume.paragraph).
• Peirce, C.S., Writings of Charles S. Peirce : A Chronological Edition,
Peirce Edition Project (eds.), Indiana University Press, Bloomington and
Indianapolis, IN, 1981–. Cited as (CE volume, page).
Resources
=========
• Peirce’s 1870 Logic of Relatives
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Overview
Regards,
Jon
Eric B> ! avoid 3 Dim because unstable ... Four is perfect, unless not
"commutative" ... For "dimensions", infinity is quite too much
For any specific application, it's usually best to use whatever is
appropriate. And I agree that most people who use an ontology never think
about or worry about proofs in any dimension.
But please read the article I cited. It shows that the general case
(infinity) includes all the special cases as subtypes. Furthermore, it
also shows that proofs for the general case (infinity) are often simpler
than proofs for any of the special cases.
This fact is important for the people who are defining general principles
for ontology. The people who just work on specific problems never need to
worry about the proofs. But they do need assurance that the people who
define the general case know what they're doing.
John
ACM has now made all their publications from 1951 to 2000 freely available.
They plan to make more available in the next 5 years.
John
----------------------------------
ACM has opened the articles published during the first 50 years of its publishing program. These articles, published between 1951 and the end of 2000, are now open and freely available to view and download via the ACM Digital Library.
ACM's first 50 years backfile contains more than 117,500 articles on a wide range of computing topics. In addition to articles published between 1951 and 2000, ACM has also opened related and supplemental materials including data sets, software, slides, audio recordings, and videos.
"We at ACM are especially proud to make this announcement now as we celebrate the 75th anniversary of our organization," said ACM President Gabriel Kotsis. "ACM has published many of the foundational works by pioneers of the computing field, and we are delighted to share this treasure trove with the world. And in doing so, we take another large step in our evolution to become a fully open access publisher."
Making the first 50 years of its publications and related content freely available expresses ACM's commitment to open access publication and represents another milestone in our transition to full open access within the next five years.
Alex, Matteo, Igor, Lists,
A one-dimensional structure is often an awkward approximation to some
n-dimensional structure. For example, C. S. Peirce invented the
one-dimensional notation for predicate calculus (which Peano modified by
introducing letters drawn upside-down and backwards). But he later
simplified and generalized the notation with graphs, which can be drawn in
two dimensions. But they are even simpler in three or more dimensions,
when they avoid issues about cross-overs.
More recently, category theory has been generalized to "infinity
categories". Infinity may sound complex, but it is actually simpler
because it removes many details that depend on some specific number N. The
proofs are often simpler when many of the details can be ignored.
For an introduction to infinity mathematics, see the recent article in
the Scientific American:
https://www.scientificamerican.com/article/infinity-category-theory-offers-a
-birds-eye-view-of-mathematics1/
Application to ontology: Nearly everything we deal with in our daily
lives involves processes in three dimensions plus time. The linear
definitions in language and logic are usually drastically oversimplified
approximations. The logic may seem to be precise, but that precision is
often an illusion.
As Lord Kelvin said, "Better a rough answer to the right question than an
exact answer to the wrong question."
John
