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" <[email protected]> Sent: 3/9/24 1:02 PM To: "Jon Alan Schmidt" <[email protected]>, "Peirce-L" <[email protected]> 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" <[email protected]> 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.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 <[email protected]> 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/#HighOrdeLogiVisVisTypeTheo 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
L 376, December 6, 1911. Houghton Library. My dear Risteen: A Diagrammatic Syntax. I mentioned to you, while you were [here] last year, that I have a diagrammatic syntax which analyzes the syllogism into no less than six inferential steps. I now describe its latest state of development for the first time. I am glad to think that my account of it will have one such a reader as you. C.S.P. This syntax, which I have hitherto called the "system of Existential Graphs", was suggested to me in reading the proof sheets of an article by me that was published in the Monist of Jan. 1897; and I at once wrote a full account of it for the same journal. But Dr. Carus would not print it. I gave an oral account of it, soon after, to the National Academy of Sciences; and in 1903 for my audience of a course of Lectures before the Lowell Institute, I printed a brief account of it. An account of slightly further development of it was given in the Monist of Oct. 1906. In this I made an attempt to make the syntax cover Modals; but it has not satisfied me. The description was, on the whole, as bad as it well could be, in great contrast to the one Dr. Carus rejected. For although the system itself is marked by extreme simplicity, the description fills 55 pages, and defines over a hundred technical terms applying to it. The necessity for these was chiefly due to the lines called "cuts" which simply appear in the present description as the boundaries of shadings, or shaded parts of the sheet. 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. 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. The Conventions. The ultimate purpose of contriving this diagrammatic syntax, is to enable one with facility to divide any necessary, or mathematical, reasoning into its ultimate logical steps. It is more accurate to call such reasoning "necessary" than to call it "mathematical"; but the latter designation will give a person who recognizes that not all mathematical reasonings have to do with quantity, but who has happily not received any of the sort of instruction in Logic that is now usually given, a better notion of what is meant than if the reasoning of the kind meant were described as "necessary". For example, all sound reasoning in applying the doctrine of chances belongs to the class of reasonings meant; and all such reasoning is truly "necessary" reasoning,|that is to say, what it concludes must necessarily (i.e. would always) be true, provided its "premisses", or the hypotheses upon which it is based, be true. But because what it reasons about is probability,|concluding that such and such an event would always, under such and such circumstances, have a stated probability, a person unfamiliar with the theory of reasoning might very naturally take it for probable reasoning. But I usually designate the kind of reasoning in question as Deductive Reasoning, or Deduction, so giving this word a broader meaning than many logicians do. If I am asked how a person can, in deductive reasoning, be absolutely certain that his conclusion is true, if his premisses are so, I answer that in point of fact he is not. One can never be absolutely certain of anything. All men make mistakes in addition and multiplication, for example; and they may repeat the same mistakes in going over the computation a second time. Consequently, however improbable it may be, it is possible, strictly speaking, that the same mistake should be repeated a million of million of times. This may have happened,|ridiculous as it would be really to suppose so,|every time that anybody undertook to say how many twice two came too. Absolute accuracy is beyond human powers. But a phrase often used in reference to deductive reasoning expresses correctly the nature of such approach to certainty as we can attain. Namely, it is often said that the reasoner must admit the deductive conclusion from his own premisses, or "fall into self-contradiction". That is precisely the essence of Deduction: the fact which its conclusion asserts, if it is a logically perfect deduction, was already asserted in its premisses,|not, usually, in either one, but in the totality of the premisses taken together. By means of the accurate expression of the meanings of premisses and conclusions in this diagrammatic syntax, it will be made clear whether or not the nature of deductive [reasoning] really be such as I say it is. One obvious reason for doubting the theory is that it would seem to render any valid deductive conclusion instantaneously evident upon an examination of the premisses; so that mathematics would be almost too simple a thing to be dignified by the name of a science, in strange conflict with the known reputation of that science as being the most difficult of all the sciences. But the theory ought not to be condemned upon this ground alone, without careful analysis of the reasonings of mathematics, since there are two reasons for believing that a science where reasonings should be of the sort that the theory supposes might have all the difficulty that mathematics has. The first of these reasons is that if certain propositions of such a science were of such extreme complexity, owing to the great number of individual objects almost exactly alike of which the premisses might make assertions, these assertions at the same time each of them dealing with many such objects, the result might be, if there were many such premisses, that almost any mind would become confused in considering them; so that it might readily happen that it should become far from an easy matter for a person to determine offhand whether a given state of things, only describable in a similarly confusing way, had been asserted to exist or not. Now it cannot be denied that one is frequently confronted in mathematics with situations of the character of that which has just been described. Whoever has read or tried to read Jordan's Traite des Substitutions, or Gauss's six demonstrations of the fundamental theorem of algebra, or a great deal of the Mecanique celeste must acknowledge it. But this is not all. My second reason is found in the peculiar character of mathematical postulates. These pronounce that certain things are possible. But these possibles are not, of course, single things, for a single thing must be more or less than possible: they embrace whole infinite series of infinite series of objects in each postulate; and it is upon the statement of the possibility of one single one of those objects or single one for each set of certain others, that some essential part of the conclusion is founded. How many demonstrations, for example, and very simple ones too, as mathematics goes, depend, each of them, upon the possibility of a single straight line; while this possibility is only asserted in the postulate that there is, or may be, a straight line through any two points of space. In that statement the possibility of every single straight line in space is asserted, including the single one whose existence is pertinent and concerning which a similar postulate directly or mediately asserts something which is an essential ingredient of the conclusion. The Phemic Sheet. Since the sole purpose of the Syntax I am describing is to facilitate the anatomy, and thereby the physiology of deductive reasonings, the reader will have anticipated the fact that no occasion has been found for supplying it with any means of expressing mere feelings or complexes of mere feelings, such as abound in the arts of music and of painting. Nor has any need been found for furnishing it with means of expressing commands,|not even such as take the softened forms of requests and inquiries. We need only a mode of indicating that what is "scribed",|i.e. is marked, whether by coveting or drawing or by a mixture of these two arts, is meant, and is not scribed for some other purpose, as, for example, to show how it might be asserted. Moreover, however minutely we may analyze our assertions, there will never be the slightest need of any such fragment of meaning as that of a noun or that of an English verb. The simplest part of speech which this syntax contemplates, which, as scribed, I shall term a blot (a vocable I choose because it is cognate with Greek [Gr.] floid, for which see L.& S. p. 1692, under [Gr.] flao, where many words containing this root are given; and there are many others in all European languages] is itself an assertion. Ought it to be an affirmation or a denial? A denial is logically the simpler, because it implies merely that the utterer recognizes, however vaguely, some discrepancy between the fact and the speech, while an affirmation implies that he has examined all the implications of the latter and finds no discrepancy with the fact. This is a circumstance to be borne in mind; but since the denial implies recognition of the affirmation, while the affirmation is so far from implying recognition of the denial, that one might imagine a paradisaic state of innocence in which men never had the idea of falsity, and yet might reason, we must admit that affirmation is psychically the simpler. Now I think that upon this point we must prefer psychical to logical simplicity. I therefore make the blot an affirmation. The subject of it must for simplicity be completely indefinite. The simplest blots, which are not relative, are such as `it snows', `it thunders', etc. All thought, which is the process of forming, under self-control, an intellectual habit, requires two functionaries; an utterer and an interpreter, and though these two functionaries may live in one brain, they are nevertheless two. In order to distinguish the actual performance of an assertion, though it be altogether a mental act, from a mere representation or appearance, the difference between a mere idea jotted down on a bit of paper, from an affidavit made before a notary, for which the utterer is substantially responsible,|I provide my system with a phemic sheet, which is a surface upon which the utterer and interpreter will, by force of a voluntary and actually contracted habit, recognize that whatever is scribed upon it and is interpretable as an assertion is to be recognized as an assertion, although it may refer to a mere idea as its subject. If "snows" is scribed upon the Phemic Sheet, it asserts that in the universe to which a special understanding between utterer and interpreter has made the special part of the phemic sheet on which it is scribed to relate, it sometime does snow. For they two 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. Any visible form which, if it were scribed on the phemic sheet would be an assertion is called a graph. If it actually be so scribed, it would be incorrect to say that the graph itself is put upon the sheet. For that would be an impossibility, since the graph itself [is] a mere form, an abstraction, a "general", or as I call it a "might be", i.e. something which might be if conditions were otherwise than they are; and in that respect it [is] just like a "word", any word, say camel. (Not that it is right to pronounce a might-be to be a fiction, or to deny its reality without other reason than that it is a mere might-be. For what do we mean by real? The legal sense, that in which we speak of "real" estate, is a good deal older than the ordinary sense. This latter, though I have once or twice met with it in older writings,|it is such a natural formation from res,|may be said with substantial truth to have been introduced by Duns Scotus, who died, if I recollect the exact date (it can only be a few years out) in 1309. It only came slowly into use, outside of the Scotists, because "in re" etc. were used instead. It would be no use quoting Scotus in order to fix the meaning: that would but render confusion worse confounded. The best way is to analyze our own meaning in some of the many common phrases in which we use it. What would a man have in his mind who should inquire whether Sancho Panza was a real person? Observe that his phrase may be "Is Sancho Panza a real character". He knows well enough that he cannot now be living or existent. But that does not affect his Reality. Is it really true, or, in other words, "is it a Real fact", that two brothers founded Rome? If the nature of it is such that its being depends upon how ancient historians have thought, that comes pretty nearly to the same thing as its not being "Real", or "in re", as the scholastics would have said.) Yet this would not hold in all cases; since it is certainly a Real fact that I had a dream last night,|a pretty rare occurrence with me. Yet the truth of this consisted in my thinking, in a general sense (that is, imagining) something. On the other hand, Kant's holding that such concepts as cause etc. are merely due to the nature of the human mind is not considered by him, and certainly ought to be considered any reason for pronouncing "cause" etc. not to be Real. Most philosophers, nowadays, would, on the contrary, think, as I certainly do, that if it were absolutely certain that anything,|Time, for example,|were a mental phenomenon of mental origin due to the structure of the human mind in such a way that no man could ever by any possibility escape from the idea of events owing in time, that would constitute the reality of time. For the real universe in which man lives is the system of those ideas which research, carried far enough, would ultimately establish. If, on the other hand, all that is true concerning a given object is due to how an individual man, or an individual collection of men happened to think, and that not about other things, as in the case of the dream, but how they happened to think about the very fact or thing whose reality is in question, then plainly the very being and truth of that object is a mere accident of thought; and all we have to do is to stop thinking about it, not only to cause the cessation of its being, but to cause it to cease from ever having been; and that sort of thing we all call unreal. Now a great many people believe the laws of nature are little more than that. They think that Sir Isaac Newton not merely discovered gravitation but that he created it. No less a person than Dr. Karl Pearson says so in plain terms in his Grammar of Science. They do not think, and recognize themselves that they do not believe, that there really are any laws, or regularities in Nature. I do not agree with them. John Stuart Mill who thought nearly as Pearson does, so far differed on that point that he thought the whole truth of Inductive reasoning,|not the fact that there are such truths, but that our observations afford any good reason for believing Inductive conclusions, to be likely to be more or less true,|is that it is a fact, whether the reasoner knows it or not, that under exactly the same circumstances, the same thing always occurs. I think that is no sound reason for believing in induction as a sound way of reasoning, because, in so far as we have any reason for such a pronouncement (and as a general proposition nothing can be more false), we only know it as an inductive conclusion. It is as the veracity of a witness were in question, and in his defence a lot of others were put on the stand, and on cross-examination it were found that all they knew of the suspected witness's veracity was what he himself had told them. But if I could have had an opportunity of tackling Mill, who had in him the stuff to make a philosopher, but who had taken his philosophic creed ready made from Bentham and from his father, James Mill, having no reading in philosophy to speak of, and who, while he was [end]
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