Cf: Functional Logic • Inquiry and Analogy • 9
https://inquiryintoinquiry.com/2022/05/02/functional-logic-inquiry-and-anal…
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_…
Inquiry and Inference
https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy#Dewey_…
If we follow Dewey’s “Sign of Rain” example far enough to consider
the import of thought for action, we realize the subsequent conduct
of the interpreter, progressing up through the natural conclusion of
the episode — the quickening steps, seeking shelter in time to escape
the rain — all those acts form a series of further interpretants,
contingent on the active causes of the individual, for the originally
recognized signs of rain and the first impressions of the actual case.
Just as critical reflection develops the associated and alternative
signs which gather about an idea, pragmatic interpretation explores
the consequential and contrasting actions which give effective and
testable meaning to a person’s belief in it.
Figure 10 charts the progress of inquiry in Dewey’s Sign of Rain example
according to the stages of reasoning identified by Peirce, focusing on
the compound or mixed form of inference formed by the first two steps.
Figure 10. Cycle of Inquiry
https://inquiryintoinquiry.files.wordpress.com/2022/04/cycle-of-inquiry-gra…
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
There is a movement afoot to develop GQL as a kind of SQL for graphs. It's
just starting up, and I'm afraid that they may create YAK (Yet Another
Kluge). But since they want to use SQL conventions, there is a possibility
for making it compatible with DOL. If they do that, it could be a winner.
Otherwise, it could be a disaster.
The header of SQL is neutral: SELECT ... WHERE /* A statement in some
version of logic */
My recommendation is to base the WHERE-clause on DOL, which is an OMG
standard that can relate all the logics of the Semantic Web and all the
diagrams of Formal UML to Common Logic.
In March of 2020, I presented a talk at the Knowledge Graph conference,
for which I got the best presentation award. I updated it for a keynote
speech at the European Semantic Web Conference in June 2020. In both
talks, I emphasized the importance of interoperability and the use of DOL
for relating all the logics to one another via the freely available
software that was available. See http://jfsowa.com/talks/eswc.pdf .
In eswc.pdf, I emphasized the relationship of graph logics to Common Logic
and the OMG standard for DOL. I believe that approach could be used to
allow any logic supported by DOL as a candidate for the WHERE clause. That
would include a broad range of logics that are already being used plus any
new logics that could be mapped to and from graphs. As examples, I used
existential graphs and conceptual graphs. They are general enough to
include all the current knowledge graphs as proper subsets -- and both EGs
and CGs can support the same version of Common Logic as DOL they can also
be extended to support IKL -- but that would not be in the first version of
GQL.
Some people might complain that Common Logic is too powerful. But DOL
supports mappings among a very wide range of logics from the simplest up to
some very rich versions. A standard based on DOL would allow and encourage
implementers to choose any level of expressive power that any DOL logic
supports.
That would allow implementers to start small and add as much expressive
power as they find useful at any time they wish And it would show them an
open-ended growth path for the future.
Bottom Line: The standards organizations have a motto: "Standards should
be built on other standards." By building GQL on DOL, they would create a
bridge to all the logics of the Semantic Web, Formal UML, and SQL, which
supports a subset of first-order logic.
John
Cf: Survey of Differential Logic • 3
https://inquiryintoinquiry.com/2021/05/15/survey-of-differential-logic-3/
All,
Linked above is a Survey of blog and wiki posts on Differential Logic,
material I plan to develop toward a more compact and systematic account.
Note. One effect of the pandemic has been been to blot out my memory
of much work I blogged over the year and many group discussions I have
in mind as “recent” and “I’ll get back to it” actually occurred several
months ago. Thinking it will serve memory to recycle the more eddifying
currents of water under the bridge, here’s an update of my Survey page on
Differential Logic.
There's a lot of links, so I'll leave Readers on their own recognizance
to follow what they will from the linked blog post.
Regards,
Jon
Cf: Survey of Differential Logic • 3
https://inquiryintoinquiry.com/2021/05/15/survey-of-differential-logic-3/
All,
Linked above is a Survey of blog and wiki posts on Differential Logic,
material I plan to develop toward a more compact and systematic account.
Note. One effect of the pandemic has been been to blot out my memory
of much work I blogged over the year and many group discussions I have
in mind as “recent” and “I’ll get back to it” actually occurred several
months ago. Thinking it will serve memory to recycle the more eddifying
currents of water under the bridge, here’s an update of my Survey page on
Differential Logic.
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
