List,
Now we come to the crux of Lowell 2, and perhaps the crucial issue involved in existential graphs - an issue that Peirce himself did not fully recognize until years after the Lowell lectures, and may never have fully resolved. The roots of the problem can be traced back to the beginning of Lowell 2, where Peirce introduced the concept of a cut. He did this by appealing to the imagination of his audience, asking them to contemplate an important logical phenomenon, namely the "conditional de inesse," diagrammatically - as a "scroll" made up of two cuts, one inside the other: [[ . in order to assert of the universe of discourse that if it rains then a pear is ripe I must put on the blackboard this: I draw the two ovals which I call a scroll in blue because I do not want you to regard them as ordinary lines. I want you to join me in making believe that they are cuts through the surface, and that inside the outer one the skin of the board has been stripped off disclosing another surface below. This I call the bottom or area. Therefore "It rains" is not scribed on the blackboard or, as I say, is not scribed on the sheet of assertion. For what is scribed on that sheet is asserted to be true of the universe of discourse; while the statement "It rains" is a mere supposition. Let us say that that bottom inside the outer cut represents another universe, a universe of supposition, and that it is only in that universe that it is said to rain. ]] The cut, then, is so named because it cuts off everything inside it from the universe of discourse represented by the blackboard or "sheet of assertion." Then the "bottom" or "area" inside this cut "represents another universe, a universe of supposition" - a suitable way of representing the significance of the if-clause of a conditional proposition. But what about the then--clause? Since it does not belong to a universe of supposition, it has to be enclosed in another cut; and what does the area inside this cut represent? Yet another universe, this time doubly cut off from the universe of discourse (the sheet of assertion)? Apparently not, because the entire scroll is supposed to make an assertion about that universe of discourse, and to represent it as simply as possible; and the imagination balks at picturing a universe within a universe within a universe (and so on, indefinitely). So Peirce recoils from the "separate universe" significance of the cut and asks us to imagine that the universe inside the inner cut is "like" the universe outside the outer cut: [[ the bottom of the outer cut contains the inner cut which interrupts its surface; and inside the inner we will make believe that a patch is put in with a surface like that of the blackboard, although cut off from it. I use the word area for any part of the surface unbounded or bounded by cuts, never extending through a cut. ]] This leaves the concept of "area" somewhat ambiguous: is it just "another universe" cut off from the one on the other side of the cut - and thus one of many possible universes - or is it one of just two universes, one "positive" and the other "negative", one "true" and the other "false"? Peirce evidently chooses the latter route, and to me the reason is clear enough: Peirce at this point (in 1903) is investigating the logic of mathematics, starting with the simplest mathematics - and this requires a two-valued logic. He says as much in one of the Lowell drafts, MS 466: "The simplest possible kind of mathematics would be the mathematics of a system of only two values. Do we in experience meet with any case in which such mathematics could have any application? I reply that we do. There are just two values that any assertion can have. It is either true when it has all the value it can have, or it is false, and has no value at all, It follows that our system of Existential Graphs is precisely an application of the mathematics of two values." In Lowell 2, having started by saying that the first (outer) cut represents "a universe of supposition," Peirce tries to show that it must thereby represent negation. He uses the rather awkward contrivance of the "blot" to show this, and thus to render the existential graph system "an application of the mathematics of two values," where the negative of the negative is simply the positive. But as he observed several times in the Lowell drafts, this is a "Pure Mathematical" rendition of EGs, and compromises its value as a tool for the analysis of logic. It glosses over the essential difference between mathematics and logic which Peirce made very clear in MS 466: [[ The only reason I do not agree with Dedekind in making mathematics a branch of logic is that logic is not a science of pure assumptions but is a study of positive truth. The mathematician seeks only to trace out the consequences of his assumptions in the readiest and speediest way. The logician does not care much what the conclusions from this or that system of assumptions may be. What he is interested in is in dissecting reasonings, in finding out what their elementary steps are, and in showing what positive facts about the real universe of things and of thoughts it is from which the necessity of the mathematician's reasonings and the validity of other kinds of reasonings depend, and exactly what the nature of that dependence is. I have already pointed out that the characters that make a system of symbols good for mathematical purposes are quite contrary to those which would make it good for logical purposes. ]] Peirce had also made it very clear at the very beginning of Lowell 2 that, as a logician, his interest in developing EGs was "in dissecting reasonings, in finding out what their elementary steps are, and in showing what positive facts about the real universe of things and of thoughts it is from which the necessity of the mathematician's reasonings and the validity of other kinds of reasonings depend, and exactly what the nature of that dependence is." But once he had committed to interpreting EGs according to a two-valued logic, he found it difficult, if not impossible, to use EGs to represent some "positive facts about the real universe of things and of thoughts." The ambiguity lurking in the concept of the cut turned out to cause problems for the system of gamma graphs, but already in Lowell 2 there are symptoms of the root problem, especially in 2.17 with Peirce's dictum that "a line of identity is, as I have said, a graph and as such it cannot be part on one side of a cut and part on the other side." In short, a graph can only be meaningful in one universe of discourse, and if different areas of a diagram represent different universes, then a graph cannot cross the cut which divides one universe from another. Peirce had to invent the "ligature" defined in 2.17 to work around this problem - but it never quite went away. In 1906, Peirce apparently thought he had solved the problem by re-imagining the area inside a cut as the verso of the sheet of assertion, and giving the cut a dimension of depth. Thus the cut or verso signified not merely negation, but another universe, this time a "universe of possibility" rather than a "universe of supposition" as in Lowell 2. This universe also had a new dimension of depth: "The cut may be imagined to extend down to one or another depth into the paper, so that the overturning of the piece cut out may expose one stratum or another, these being distinguished by their tints; the different tints representing different kinds of possibility" CP 4.578). Peirce announced this in his April 1906 address to the National Academy of Science (CP 4.579), referring to his "Syllabus" which accompanied the Lowell Lectures: [[ In a certain partly printed but unpublished "Syllabus of Logic," which contains the only formal or full description of Existential Graphs that I have ever undertaken to give, I laid it down, as a rule, that no graph could be partly in one area and partly in another; and this I said simply because I could attach no interpretation to a graph which should cross a cut. As soon, however, as I discovered that the verso of the sheet represents a universe of possibility, I saw clearly that such a graph was not only interpretable, but that it fills the great lacuna in all my previous developments of the logic of relatives. For although I have always recognized that a possibility may be real, that it is sheer insanity to deny the reality of the possibility of my raising my arm, even if, when the time comes, I do not raise it; and although, in all my attempts to classify relations, I have invariably recognized, as one great class of relations, the class of references, as I have called them, where one correlate is an existent, and another is a mere possibility; yet whenever I have undertaken to develop the logic of relations, I have always left these references out of account, notwithstanding their manifest importance, simply because the algebras or other forms of diagrammatization which I employed did not seem to afford me any means of representing them. I need hardly say that the moment I discovered in the verso of the sheet of Existential Graphs a representation of a universe of possibility, I perceived that a reference would be represented by a graph which should cross a cut, thus subduing a vast field of thought to the governance and control of exact logic. ]] In this way Peirce got back in touch with certain "facts about the real universe of things and of thoughts" which he had swept under the rug by reducing the meaning of the cut to simple negation. But it seems he never found a way to reconcile the more logical side with the mathematical side of existential graphs, which to a large extent depended on treating the cut (or later, the shaded area) as negation. In the long run, as Roberts ruefully notes (p. 89), the cost of "making the cut do double duty" was too high. This may explain why Peirce abandoned his attempt to reconcile the two purposes of EGs in gamma graphs, and in his later years reverted to the simpler two-valued version of EGs that he had developed in 1903. For instance in MS L 231 (1911), p.13-14, he wrote that "Every indivisible graph instance must be wholly contained in a single area. The line of identity may be regarded as a graph composed of any number of dyads "- is -" or as a single dyad. But it must be wholly in one area. Yet it may abut upon another line of identity in another area." However, Peirce did manage to convey some of the vital insights gained from the more complex logical side of EGs, in his 1906 "Prolegomena to an Apology for Pragmaticism" (published in the Monist); and perhaps after that there was nothing more to be gleaned from his decade-long development of existential graphs - at least nothing that would accomplish the purpose assigned to them in Lowell 2, which was "to enable us to separate reasoning into its smallest steps so that each one may be examined by itself. Observe, then, that it is not the purpose of this system of expression to facilitate reasoning and to enable one to reach his conclusions in the speediest manner. Were that our object, we should seek a system of expression which should reduce many steps to one; while our object is to subdivide one step into as many as possible. Our system is intended to facilitate the study of reasoning but not to facilitate reasoning itself" (Lowell 2.1). To go more deeply into the phenomena of reasoning, rather than seeking a new notation which is at best mathematically equivalent to other notations of formal logic operations, we need to venture into the positive science of phenomenology, and that is exactly what Peirce did in Lowell Lecture 3. And that's where we will go, after dealing with the closing pages of Lecture 2 this weekend. From: [email protected] [mailto:[email protected]] Sent: 30-Nov-17 18:55 Continuing from Lowell 2.16, https://fromthepage.com/jeffdown1/c-s-peirce-manuscripts/ms-455-456-1903-low ell-lecture-ii/display/13626 Lines of identity bring only one new rule of illative transformation. (Illative transformation, by the way, is transformation of the nature of necessary inference.) But lines of identity require some slight changes to be made in the three primary rules already given. Under the rule of omission and insertion, it is to be noted that a line of identity may be broken within an even number of cuts or on the sheet of assertion, while two lines may be joined within an odd number of cuts. We also have the curious fact that as far as this rule is concerned a point on a cut is to be treated as being within the cut, although in other respects it is to be treated as being outside the cut. The cause of this is that we take it for granted that something exists. How this cause produces this effect I shall leave it for you to make out. The rule about the double enclosure receives an extension since not only are two cuts of no effect when no graph is between them, but they are equally so when nothing is between them except lines of identity that traverse the space between the two cuts. The rule of iteration and deiteration takes a form which cannot easily be expressed without defining a new term, ligature. Namely, a line of identity is, as I have said, a graph and as such it cannot be part on one side of a cut and part on the other side. In the graph there are two lines of identity. But their having a common point on the cut identifies the individuals they denote. By a ligature is meant a line of identity together with all other lines of identity that have points in common with it. For example means any man loves himself. It has four lines of identity, one attached to the monad spot "is a man," two attached to the dyad spot loves and one joining the triple point to the inner cut. But all those make a single "ligature." http://gnusystems.ca/Lowell2.htm }{ Peirce's Lowell Lectures of 1903 https://fromthepage.com/jeffdown1/c-s-peirce-manuscripts/ms-455-456-1903-low ell-lecture-ii
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