Jon,
I'll go into much more detail in the preview article, which I am now working on. I'll just respond to the following point:
JAS: Your quotation here omits the crucial first part of the only sentence in R L376 that mentions Delta--"The better exposition of 1903 divided the system into three parts, distinguished as the Alpha, the Beta, and the Gamma, parts; a division I shall here adhere to, although I shall now have to add a Delta part in order to deal with modals."
I answered that before: Peirce is not saying that he is preserving the details of the 1903 logics. He is saying that he is preserving that DIVISION into Alpha (propositional logic), Beta (predicate logic), and Gamma (something beyond Alpha and Beta).
The most significant Gamma graphs are the the ones that represent the second-order version of his 1885 Algebra of Logic. He had reviewed Russell's logic of 1903, and he must have heard about (but didn't have time to study) Whitehead & Russell's logic of 1910. Both of them discussed higher-order logic (second order and higher), but not modal logic.
During the years after 1903, Peirce mentioned the modal words in English many, many times. And he experimented with new notations for modality, but he never used or even mentioned his 1903 modal logic for any purpose. In fact, he had only used it for a few examples in 1903.
But the most important evidence is to look at the developments in the years after Peirce. C. I. Lewis introduced a new version of propositional modal logic in 1932, which had been inspired by Peirce's 1903 modal logic. It was different from Peirce's version, but equivalent in expressive power to the propositional subset of his modal logic of 1903. During the 30 or 40 years after 1932, many logicians built on that logic. But many others (Quine among them) rejected it. Quine correctly said that modal logic was just a version of metalanguage about logic. Other logicians criticized it or ignored it altogether. Very few did much with it after the 1960s. From the 1970s and later, new versions of logic were developed to handle modal issues, but (a) they did not use the box and diamond operators for modality; (b) they used different words. such as contexts, situations, or domains; and (c) they combined predicate calculus with metalanguage, as Peirce did in L376.
In my preview of the Delta graph article, I'll explain these issues in more detail and discuss the directions taken in 1973 and later. Short summary: All the useful applications are based on some version of metalanguage, along the lines of the December 1911 article. Logics that use the two operators for necessary and possible, have no practical applications of any kind.
Peirce had good taste and good insights into the kind of logic required for problems in philosophy, science, and engineering. Metalanguage is the foundation for all useful modal reasoning in the 21st C. Textbooks still mention the Lewis-style of modal logic, but there are no applications to any kind of practical applications.
Summary: Any version of mathematics and/or logic that has no applications is. literally, useless. There are many such versions in the many years of published tomes. And most of them have few or no citations.
On rare occasions, something from the distant past is revived and becomes a big success. Peirce has an unusually large percentage of successful revivals. His Delta graphs are among them. I recognized their importance, because I have used and worked with similar logics from the late 20th and early 21st C.
John
----------------------------------------
From: "Jon Alan Schmidt" <jonalanschmidt(a)gmail.com>
Sent: 3/11/24 9:07 PM
To: Peirce-L <peirce-l(a)list.iupui.edu>
Subject: [PEIRCE-L] Peirce's Letter to Risteen (was Higher-Order Logics)
John, List:
JFS: For convenience, see the attached Delta376.txt.
I appreciate the complete transcription, although it would still be very helpful if you could quote specific sentences that you interpret as supporting each of your claims.
JFS: I believe that there is no way to interpret that text without acknowledging the fact that it is the beginning of a specification of Delta graphs. Note the ending of the second paragraph: "I shall now have to add a Delta part in order to deal with modals. A cross division of the description which here, as in that of 1903, is given precedence over the other is into the Conventions, the Rules, and the working of the System."
Your quotation here omits the crucial first part of the only sentence in R L376 that mentions Delta--"The better exposition of 1903 divided the system into three parts, distinguished as the Alpha, the Beta, and the Gamma, parts; a division I shall here adhere to, although I shall now have to add a Delta part in order to deal with modals." Accordingly, I believe that there is no way to interpret the 19 extant pages of this letter as the beginning of a specification of (only) Delta graphs. On the contrary, Peirce plainly states his intention to describe all four parts of EGs, but he never gets around to explaining their differences, let alone dealing with modals or discussing anything else that is unique to the new Delta part.
JFS: Then the paragraph immediately after that begins "The Conventions." And it continues with a specification of the conventions for something. I cannot imagine that the "something" is anything other than Delta graphs.
Your failure of imagination is not dispositive. Can you identify even one sentence from the entire section on "The Conventions"--or, for that matter, the rest of the letter--that is about EGs but not applicable to Alpha, Beta, and Gamma?
JFS: Note the later discussion about different "parts" of the phemic sheet, which may be asserted and interpreted in different ways.
Peirce states, "For they two [utterer and interpreter] may conceive that the 'phemic sheet' embraces many papers, so that one part of it is before the common attention at one time and another part at another, and that actual conventions between them equivalent to scribed graphs make some of those pieces relate to one subject and part to another." As I have noted before, the different parts relate to different subjects to which both parties pay attention at different times--equivalent what Peirce describes in previous texts such as R 280 (c. 1905) and CP 4.561n (1908), both of which I quoted last week (https://list.iupui.edu/sympa/arc/peirce-l/2024-03/msg00004.html). As such, this concept is fully applicable to Alpha, Beta, and Gamma EGs--it does not exclusively "deal with modals," and thus is not unique to Delta EGs.
JFS: That is why metalanguage must be used to state the many kinds of modality that Peirce discusses in the attached text.
Peirce indeed briefly discusses modality in R L376, but he does not address how to represent and reason about modal propositions using EGs, which is his only stated reason for needing a Delta part.
JFS: But the original MS, a copy of which you included in your note, had a thin line that connected the oval to the word 'is'. I suspect that who drew that diagram thought that the thin line between the oval and the word 'is' was just part of the word 'is'. But in his handwiriting, Peirce never drew a line in front of an initial letter 'i'. Therefore, that graph was mistakenly drawn.
Thank you for correcting my mistake. I noticed that line, drawn even more lightly than the one in the first EG on RLT 151, but assumed that it was part of Peirce's cursive "i"--just as Ketner evidently did. However, after looking at a few other manuscripts, I agree that Peirce generally did not include such a line when handwriting "i" as the first letter of a word, so there is indeed a lightly drawn line connecting the oval (containing a proposition) to the rheme (whose blank that proposition fills). Of course, I already brought to your attention his similar notation in a later manuscript--R 492 (1903), erroneously reproduced in CP 4.471 but corrected by both Roberts and Pietarinen--where the oval and line are dotted instead of lightly drawn (https://list.iupui.edu/sympa/arc/peirce-l/2024-02/msg00141.html). This notation in Gamma EGs asserts a proposition about a proposition, but there is no hint of anything like it in R L376 (nor R 514).
Regards,
Jon Alan Schmidt - Olathe, Kansas, USA
Structural Engineer, Synechist Philosopher, Lutheran Christian
www.LinkedIn.com/in/JonAlanSchmidt / twitter.com/JonAlanSchmidt
On Mon, Mar 11, 2024 at 3:56 PM John F Sowa <sowa(a)bestweb.net> wrote:
Jon, Jeff, Gary, List,
I am now writing the article on Delta graphs. In a few days, I'll send a preview. For convenience, see the attached Delta376.txt. (Since Peirce's paragraphs tend to be very long, I added some additional paragraph breaks,)
I believe that there is no way to interpret that text without acknowledging the fact that it is the beginning of a specification of Delta graphs. Note the ending of the second paragraph:
"I shall now have to add a Delta part in order to deal with modals. A cross division of the description which here, as in that of 1903, is given precedence over the other is into the Conventions, the Rules, and the working of the System."
Then the paragraph immediately after that begins "The Conventions." And it continues with a specification of he conventions for something. I cannot imagine that the "something" is anything other than Delta graphs. (That paragraph break, by the way, is Peirce's.)
Note the later discussion about different "parts" of the phemic sheet, which may be asserted and interpreted in different ways. That is why metalanguage must be used to state the many kinds of modality that Peirce discusses in the attached text.
John
I'll also mention that three people misinterpreted the two diagrams on p. 151 of RLT -- you, me, and Ken Ketner. I misinterpreted the first diagram as having a line of identity between an oval that encloses the sentence "You are a good girl". With that interpretation, it would assert "There exists a proposition that you are a good girl, and that proposition is much to be wished." But you correctly noticed that the line is so thin that it cannot be interpreted as a line of identity. Peirce did not state any reading for that complete EG. Therefore, I read it as asserting a complete grammatical sentence "That you are a good girl is much to be wished. That assertion is correct. It is logically equivalent to the above reading, but it is not syntactically equivalent to it.
Then Ken Ketner (or somebody else who drew the second EG) did not show an attached line between the oval and the verb phrase "is false." But the original MS, a copy of which you included in your note, had a thin line that connected the oval to the word 'is'. I suspect that who drew that diagram thought that the thin line between the oval and the word 'is' was just part of the word 'is'. But in his handwiriting, Peirce never drew a line in front of an initial letter 'i'. Therefore, that graph was mistakenly drawn.
Neither you nor Ken noticed that error. You did mention that Peirce had not introduced the convention of using an oval for negation until the next example. That is true, but it does not excuse the mistake of not noticing the thin line that connects the previous oval to the word 'is.
There is much more to say, and I'll include it in the preview, which I plan to send in the next few days.
John
In that case, I believe that the thin line implies that the proposition in the oval is a THING that is the subject of the verb phrase "is much to be wished."
Jon, Jeff, Gary, List,
I am now writing the article on Delta graphs. In a few days, I'll send a preview. For convenience, see the attached Delta376.txt. (Since Peirce's paragraphs tend to be very long, I added some additional paragraph breaks,)
I believe that there is no way to interpret that text without acknowledging the fact that it is the beginning of a specification of Delta graphs. Note the ending of the second paragraph:
"I shall now have to add a Delta part in order to deal with modals. A cross division of the description which here, as in that of 1903, is given precedence over the other is into the Conventions, the Rules, and the working of the System."
Then the paragraph immediately after that begins "The Conventions." And it continues with a specification of he conventions for something. I cannot imagine that the "something" is anything other than Delta graphs. (That paragraph break, by the way, is Peirce's.)
Note the later discussion about different "parts" of the phemic sheet, which may be asserted and interpreted in different ways. That is why metalanguage must be used to state the many kinds of modality that Peirce discusses in the attached text.
John
I'll also mention that three people misinterpreted the two diagrams on p. 151 of RLT -- you, me, and Ken Ketner. I misinterpreted the first diagram as having a line of identity between an oval that encloses the sentence "You are a good girl". With that interpretation, it would assert "There exists a proposition that you are a good girl, and that proposition is much to be wished." But you correctly noticed that the line is so thin that it cannot be interpreted as a line of identity. Peirce did not state any reading for that complete EG. Therefore, I read it as asserting a complete grammatical sentence "That you are a good girl is much to be wished. That assertion is correct. It is logically equivalent to the above reading, but it is not syntactically equivalent to it.
Then Ken Ketner (or somebody else who drew the second EG) did not show an attached line between the oval and the verb phrase "is false." But the original MS, a copy of which you included in your note, had a thin line that connected the oval to the word 'is'. I suspect that who drew that diagram thought that the thin line between the oval and the word 'is' was just part of the word 'is'. But in his handwiriting, Peirce never drew a line in front of an initial letter 'i'. Therefore, that graph was mistakenly drawn.
Neither you nor Ken noticed that error. You did mention that Peirce had not introduced the convention of using an oval for negation until the next example. That is true, but it does not excuse the mistake of not noticing the thin line that connects the previous oval to the word 'is.
There is much more to say, and I'll include it in the preview, which I plan to send in the next few days.
John
In that case, I believe that the thin line implies that the proposition in the oval is a THING that is the subject of the verb phrase "is much to be wished."
----------------------------------------
From: "John F Sowa" <sowa(a)bestweb.net>
Sent: 3/9/24 1:02 PM
To: "Jon Alan Schmidt" <jonalanschmidt(a)gmail.com>, "Peirce-L" <peirce-l(a)list.iupui.edu>
Subject: RE: [PEIRCE-L] Higher-Order Logics (was Problems in mixing quantifiers with modal logic)
Jeff, Jon, List,
In his 1885 Algebra of Logic, Peirce presented the modern versions of both first-order and second-order predicate logic. The only difference between his notation and the modern versions is the choice of symbols. Since Peano wanted to make his logic publishable by ordinary type setters, he had to avoid Peirce's Greek letters and subscripts. Therefore, he invented the practice of turning letters upside-down or backwards, which type setters could do very easily.
For every version of first-order logic, there is a fixed domain D1 of entities in the domain of quantification. Those entities could be anything of any kind -- that includes abstractions, fictions, imaginary beasts, and even hypothetical or possible worlds. For second order logic, the domain D2 consists of all possible functions and/or predicates that range over entities in D1.
Second order logic is the only kind of higher order logic that anybody uses for any practical applications in any version of science, engineering, or computer systems. When they use the term HOL, they actually mean some kind of second order logic, which may be the one described above or something with a different way of specifying D2.
The first (and most widely cited or defined) version of higher order logic that goes beyond second was developed by Whitehead and Russell (1910). It goes beyond second order logic by introducing domains D3, D4,..., which are so huge that nobody has ever found a use for them in any practical application.
Given D1 and D2 as above, W & R specified D3 as the set of all possible functions or predicates that may be defined over the union of D1 and D2. Then D4 is defined over the union of D1, D2, D3. And so on. Logicians (usually graduate students who need to find a thesis topic) publish papers about such things in the Journal of Symbolic Logic. And the only people who read them are graduate students who need to find a thesis topic.
Peirce never went beyond second order logic. But any statement in any language or logic about any language or logic is metalanguage. Since that word was coined over 20 years after Peirce, he never used it. But there are many uses of metalanguage in Peirce's publications and MSS. But he never chose or coined a word that would relate all the instances.
In the example that Jon copied below, "the line of identity denoting the ens rationis", Peirce used the term 'ens rationis' for that example of metalanguage. But he described other examples with other words.
In the passage below by Jay Zeman, "a different kind of line of identity, one which expresses the identity of spots rather than of individuals. This is an intriguing move, since it strongly suggests at least the second order predicate calculus, with spots now acquiring quantifications. Peirce did very little with this idea, so far as I am able to determine", Jay mistakenly used the term "second order PC". There is no quantified variable for some kind of logic. It is just another example of metalanguage that makes an assertion about the EG.
There is much more to say about metalanguage, which I'll discuss in a separate reply to Jon. But these examples are a small fraction of the many instances of metalanguage throughout Peirce's publications and MSS. Once you start looking for them, you'll find them throughout his writings. Unfortunately, Peirce had no standard terminology for talking about them.
I hate to say it, but this is one time when I wish Peirce had found a Greek word for it.
John
----------------------------------------
From: "Jon Alan Schmidt" <jonalanschmidt(a)gmail.com>
Jeff, List:
Indeed, as Don Roberts summarizes, "The Gamma part of EG corresponds, roughly, to second (and higher) order functional calculi, and to modal logic. ... By means of this new section of EG Peirce wanted to take account of abstractions, including qualities and relations and graphs themselves as subjects to be reasoned about" (https://www.felsemiotica.com/descargas/Roberts-Don-D.-The-Existential-Graph…, 1973, p. 64). Likewise, according to Ahti-Veikko Pietarinen, "In the Gamma part Peirce proposes a bouquet of logics beyond the extensional, propositional and first-order systems. Those concern systems of modal logics, second-order (higher-order) logics, abstractions, and logic of multitudes and collections, among others" (LF 2/1:28). Jay Zeman says a bit more about Gamma EGs for second-order logic in his dissertation.
JZ: There is also another suggestion, in 4.470, which is interesting but to which Peirce devotes very little time. Here he shows us a different kind of line of identity, one which expresses the identity of spots rather than of individuals. This is an intriguing move, since it strongly suggests at least the second order predicate calculus, with spots now acquiring quantifications. Peirce did very little with this idea, so far as I am able to determine, but it seems to me that there would not be too much of a problem in working it into a graphical system which would stand to the higher order calculi as beta stands to the first-order calculus. The continuity interpretation of the "spot line of identity" is fairly clear; it maps the continuity of a property or a relation. The redness of an apple is the same, in a sense, as the redness of my face if I am wrong; the continuity of the special line of identity introduced in 4.470 represents graphically this sameness. This sameness or continuity is not the same as the identity of individuals; although its representation is scribed upon the beta sheet of assertion, its "second intentional" nature seems to cause Peirce to classify it with the gamma signs. (https://isidore.co/calibre/get/pdf/4481, 1964, pp. 31-32)
The CP reference here is to the paragraph right before the one where Peirce suggests the notation of a dotted oval and dotted line to assert a proposition about a proposition (CP 4.471, 1903), similar to the first EG on RLT 151 (1898), as John and I discussed recently (https://list.iupui.edu/sympa/arc/peirce-l/2024-02/msg00141.html). Here is what Peirce says (and scribes) in that text; the image is from LF 2/1:165, with Peirce's handiwork on the right and Pietarinen's reproduction on the left.
CSP: Convention No. 13. The letters ρ0, ρ1, ρ2, ρ3, etc. each with a number of hooks greater by one than the subscript number, may be taken as rhemata signifying that the individuals joined to the hooks, other than the one vertically above the ρ taken in their order clockwise are capable of being asserted of the rhema indicated by the line of identity joined vertically to the ρ.
Thus, Fig. 57 expresses that there is a relation in which every man stands to some woman to whom no other man stands in the same relation; that is, there is a woman corresponding to every man or, in other words, there are at least as many women as men. The dotted lines between which, in Fig. 57, the line of identity denoting the ens rationis is placed, are by no means necessary.
[image.png]
On the other hand, as I keep pointing out, Peirce's only stated purpose for needing to add a new Delta part was "in order to deal with modals" (R L376, 1911 Dec 6), so I doubt that it would have had anything to do with higher-order logics. John Sowa seems to be convinced that Peirce had in mind a more generalized situation/context logic using metalanguage, but so far, I see no evidence for this in the extant 19 pages of that letter to Risteen. Pietarinen speculates, "Perhaps he planned the Delta part on quantificational multi-modal logics as can be discerned in his theory of tinctured graphs that was fledgling since 1905" (LF 1:21), but that also seems unlikely to me since Peirce ultimately describes the tinctures as "nonsensical" (R 477, 1913 Nov 8).
As far as I know, the only new notation that Peirce ever proposes for representing modal propositions with EGs after abandoning broken cuts (1903) and tinctures (1906) is the one in his Logic Notebook that I have been advocating (R 339:[340r], 1909 Jan 7). Echoing Zeman's remark in the quotation above, the sameness or continuity of a possible state of things (PST) as represented by a heavy line of compossibility (LoC) in my candidate for Delta EGs is not the same as the identity of individuals as represented by a heavy line of identity in Beta EGs.
Regards,
Jon Alan Schmidt - Olathe, Kansas, USA
Structural Engineer, Synechist Philosopher, Lutheran Christian
www.LinkedIn.com/in/JonAlanSchmidt / twitter.com/JonAlanSchmidt
On Fri, Mar 8, 2024 at 5:11 PM Jeffrey Brian Downard <peirce-l(a)list.iupui.edu> wrote:
Hello John, Jon, List,
Peirce examines both first and second intentional logics. The distinction appears to be similar, in some respects, to the contemporary distinction between first and second order logics. Here, for instance, is an SEP entry on higher order logics: https://seop.illc.uva.nl/entries/logic-higher-order/#HighOrdeLogiVisVisType…
Does Peirce’s explorations in the Gamma system of the EG, and his contemplation of a possible Delta system, bear some similarities to contemporary discussions of higher order logics, such as third order, or fourth order, etc.?
--Jeff D
I have made some revisions in my slides for the Ontology Summit talk on March 6. Most of the changes are minor clarifications. The most important additions are about a quotation by Zellig Harris.
Z. Harris was the inventor of Transformational Grammar and the thesis adviser to Noam Chomsky, who went much further on the syntactic features. But Harris had put more emphasis on the semantics, which many linguists believe is much more important than syntax.
The single most important quotation by Harris is one that William Frank used to quote at the end of all his notes to Ontolog Forum. I mention that quotation on slide 40 and discuss its implications on slide 41:
"We understand what other people say through empathy — imagining ourselves to be in the situation they were in, including imaging wanting to say what they wanted to say." Zellig Harris
In January 2023, I sent a note to Frank to ask for the full citation for that quotation. I got a response from his son, which said that William Frank had died in November 2022. That's why he stopped sending notes to Ontolog Forum. His notes were usually insightful, and it's sad that we no longer have his comments.
Other important issues, related to AGI, include the Central Executive, which is introduced in slide 35 in a discussion of neuroscience. Slide 36 shows how a simulation of a Central Executive could be added to an intelligent computer system.
That addition won't immediately make the system more intelligent, but it can provide a place where issues of relevance, moral, and ethical behavior can be addressed. That is important for designing systems that can evaluate and modify proposed actions that might cause dangerous or irresponsible behavior.
Slide 39 is important for addressing "wicked problems" that involve “complex interdependences between the systems involved, and incomplete, inconsistent, information about the problem context. Wicked problems are dynamic and span many domains with complex legal, ethical, and societal aspects.”
New projects of any kind open up unexplored territory, and old solutions that LLMs can dig up are almost never adequate to address, much less solve them. This is another area where new technology (or variations of older symbolic technology) are necessary. And the Central Executive is the kind of system where such issues should be addressed -- usually in discussions with human executives and advisers.
Some people working with LLMs mention the problems, but I have not seen any discussions that could address them. I believe that a Central Executive is an important step toward a solution. And I also believe that the Central Executive should have some kind of empathy with humans -- artificial empathy is better than no empathy at all.
Important addition: Whenever the Central Executive encounters a wicked problem or a proposal for a dangerous or unethical action, it should notify a human executive who could take action or call a committee meeting to consider some major actions that may be required.
John
Survey of Pragmatic Semiotic Information • 8
• https://inquiryintoinquiry.com/2024/03/01/survey-of-pragmatic-semiotic-info…
All,
This is a Survey of blog and wiki posts on a theory of information
which grows out of pragmatic semiotic ideas. All my projects are
exploratory in character but this line of inquiry is more open‑ended
than most. The question is —
• What is information and how does it impact the spectrum of activities
answering to the name of inquiry?
Setting out on what would become his lifelong quest to explore and explain
the “Logic of Science”, C.S. Peirce pierced the veil of historical confusions
obscuring the issue and fixed on what he called the “laws of information” as
the key to solving the puzzle.
The first hints of the Information Revolution in our understanding
of scientific inquiry may be traced to Peirce’s lectures of 1865–1866
at Harvard University and the Lowell Institute. There Peirce took up
“the puzzle of the validity of scientific inference” and claimed it
was “entirely removed by a consideration of the laws of information”.
Fast forward to the present and I see the Big Question as follows.
Having gone through the exercise of comparing and contrasting Peirce's
theory of information, however much it yet remains in a rough‑hewn state,
with Shannon’s paradigm so pervasively informing the ongoing revolution
in our understanding and use of information, I have reason to believe
Peirce’s idea is root and branch more general and has the potential,
with due development, to resolve many mysteries still bedeviling
our grasp of inference, information, and inquiry.
Please follow the above link for the full set of resources.
Articles and blog posts on the core ideas are linked below.
Information = Comprehension × Extension
• https://oeis.org/wiki/Information_%3D_Comprehension_%C3%97_Extension
{ Information = Comprehension × Extension }
• https://inquiryintoinquiry.com/2016/05/18/information-comprehension-x-exten…
{ Information = Comprehension × Extension } • Revisited
• https://inquiryintoinquiry.com/2019/01/23/information-comprehension-x-exten…
Pragmatic Semiotic Information • Ψ
• https://inquiryintoinquiry.com/2023/07/22/pragmatic-semiotic-information-%c…
Pragmatic Semiotic Information
• https://oeis.org/wiki/Pragmatic_Semiotic_Information
Peirce's Logic Of Information
• https://oeis.org/wiki/User:Jon_Awbrey/Peirce%27s_Logic_Of_Information
Peirce, C.S. (1867), “Upon Logical Comprehension and Extension”
• https://peirce.sitehost.iu.edu/writings/v2/w2/w2_06/v2_06.htm
Regards,
Jon
cc: https://www.academia.edu/community/5wodkP
cc: https://mathstodon.xyz/@Inquiry/112022087852233415
