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@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-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 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
On Fri, Mar 8, 2024 at 5:11 PM Jeffrey Brian Downard <peirce-l@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/#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