Re: [PEIRCE-L] Higher-Order Logics (was Problems in mixing quantifiers with modal logic)
Jerry, Jon, List, Peirce never used the term "graphic object". In his classification of the sciences, pure mathematics does not depend on anything else. Phaneroscopy is free to use any imaginable mathematical patterns to analyze, classify, and interpret anything in the phaneron, no matter where it came from -- by any of the external or internal senses or by some sort of dreaming or imagining, JLRC: Given a graphic object, how does one decipher the logical content of it? By phaneroscopy, whatever is in the phaneron is interpreted as a pattern or diagram that consists of a pieces or parts that may resemble patterns previously observed and classified images together with unfamiliar parts that may be classified according to degrees of resemblance to previously observed and classified patterns. JLRC: What types of semantics can be associated with what types of visual distinctions? That partially interpreted phaneron is further interpreted evaluated according to esthetical, ethical, and semeiotic methods. Semeiotic is a more detailed analysis, which includes anything that may be call semantics. JLRC: How many distinctions are to be associated with a graphic object? And how are these distinctions associated with the forms embodied in the objects with logical premises OF ANY ORDER? There is no limit to the amount or depth of analysis that may be applied to anything in the phaneron. Different individuals with different kinds of background and experience may analyze anything at any level of detail. The kind and amount of logic that anybody may use depends on their knowledge, interests, and experience. JAS: Your questions [by JLRC] as posed are extremely general, and their answers depend heavily on the particular context of interest. I agree. As for Delta graphs, I'll send a partial draft in a few days that goes into more detail. John From: "Jerry LR Chandler" Jon, John, List: The attempts to interpret the on going discussions leads to simple questions about meaning of symbols and logics. Given a graphic object, how does one decipher the logical content of it? What types of semantics can be associated with what types of visual distinctions? How many distinctions are to be associated with a graphic object? And how are these distinctions associated with the forms embodied in the objects with logical premises OF ANY ORDER? I pose these questions because as the discussion unfolds into the vast richness of modal logics within the modern forms of symbolic logics, the roles of individual minds in expressing semes appears to become dominant. In other words, the boundaries between symbols and icons seems to disappearing... Cheers Jerry On Mar 8, 2024, at 9:45 PM, Jon Alan Schmidt wrote: 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-Graphs-of-Charles-S.-Peirce.pdf, 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
Re: [PEIRCE-L] Higher-Order Logics (was Problems in mixing quantifiers with modal logic)
Jon, John, List: The attempts to interpret the on going discussions leads to simple questions about meaning of symbols and logics. Given a graphic object, how does one decipher the logical content of it? What types of semantics can be associated with what types of visual distinctions? How many distinctions are to be associated with a graphic object? And how are these distinctions associated with the forms embodied in the objects with logical premises OF ANY ORDER? I pose these questions because as the discussion unfolds into the vast richness of modal logics within the modern forms of symbolic logics, the roles of individual minds in expressing semes appears to become dominant. In other words, the boundaries between symbols and icons seems to disappearing... Cheers Jerry > On Mar 8, 2024, at 9:45 PM, Jon Alan Schmidt wrote: > > 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-Graphs-of-Charles-S.-Peirce.pdf, > 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.  > > 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
[PEIRCE-L] Higher-Order Logics (was Problems in mixing quantifiers with modal logic)
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" Sent: 3/9/24 1:02 PM To: "Jon Alan Schmidt" , "Peirce-L" 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
Re: [PEIRCE-L] Higher-Order Logics (was Problems in mixing quantifiers with modal logic)
John, List: JFS: 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. When using Beta EGs to implement first-order predicate logic (FOPL), the fixed domain of quantification consists of *indefinite individuals *within the universe of discourse, such that heavy lines of identity (LoIs) correspond to variables in the standard notation. General concepts are attributed to those individuals by attaching names to LoIs, thereby making those individuals more definite and those concepts more determinate. Your position seems to be that modal logic is not just *analogous *to FOPL, but *formally equivalent* to FOPL--the fixed domain of quantification now consists of *possible worlds*, and the "predicates" being attributed to them are propositions that would be true in them. Accordingly, what Peirce scribes on R 339:[340r] could be interpreted as a new application of Beta EGs instead of a plausible candidate for the new Delta EGs. I continue to be skeptical of this suggestion because there is an obvious and fundamental *semiotic *difference between describing *things *with names (rhemes/semes) and describing *states of things* with propositions (dicisigns/phemes). As a simple test, we can examine whether the usual modal axioms can be translated into theorems of FOPL as implemented using Beta EGs. Distributive axiom K = □(*p* → *q*) → (□*p* → □*q*) becomes ∀*x*(P*x* → Q*x*) → (∀*x*P*x* → ∀*x*Q*x*), which is valid. However, serial axiom D = □*p* → ◇ *p* becomes ∀*x*(P*x*) → ∃*x*(P*x*), which is invalid--as Peirce himself emphasizes repeatedly, an oddly enclosed (shaded) LoI does not assert the *existence *of anything. Moreover, reflexive axiom T = □*p* → *p* cannot be translated into a well-formed formula (WFF) at all because there is no counterpart for asserting a proposition to be true in the *actual *world--a name must *always *be attached to at least one LoI. Symmetric axiom B = ◇□*p* → *p*, transitive axiom 4 = □*p* → □□*p*, and euclidean axiom 5 = ◇□*p* → □*p* likewise cannot be translated into WFFs because there is no counterpart for iterated modalities. The first two limitations might be overcome by implementing a different version of FOPL that includes existential import (serial) and singular terms (reflexive), with corresponding modifications to Beta EGs, but the others would require stipulated axioms just like the corresponding modal systems. JFS: 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. ... 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. I would caution against making such sweeping and dismissive pronouncements. After all, there might very well be applications of logics beyond second-order in science, engineering, or computer systems that have not yet come to your attention or that get discovered in the future, perhaps by one of those graduate students. In any case, I remind you again that according to Peirce, "True science is distinctively the study of useless things. For the useful things will get studied without the aid of scientific men" (CP 1.76, c. 1896). JFS: 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. On the contrary, Zeman's statement is *not *mistaken. In the Gamma EG that he is discussing (CP 4.470, LF 2/1:165, 1903), a special LoI with dotted lines on either side of it facilitates quantification over general concepts (predicates) in addition to the usual quantification over indefinite individuals corresponding to ordinary LoIs in Beta EGs. In your own words, "For second order logic, the domain D2 consists of all possible functions and/or predicates that range over entities in D1." JFS: 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. I am not disputing this. I just see no evidence in R 514, R L376, or elsewhere to support the *specific *claim that what Peirce has in mind for Delta EGs is adding metalanguage to Beta EGs, since his only stated reason for needing "a *Delta *part" at all is "in order to deal with modals." It seems
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" 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-Graphs-of-Charles-S.-Peirce.pdf, 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
[PEIRCE-L] Higher-Order Logics (was Problems in mixing quantifiers with modal logic)
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-Graphs-of-Charles-S.-Peirce.pdf, 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: 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 <
