List,
I have a lot to say about 2.5, so Ill insert my comments into the text below. Gary f. From: [email protected] [mailto:[email protected]] Sent: 25-Oct-17 16:35 Continuing from Lowell 2.4, https://www.fromthepage.com/jeffdown1/c-s-peirce-manuscripts/ms-455-456-1903 -lowell-lecture-ii/display/13604: [Lowell 2.5:] The question of the proper way of expressing a conditional proposition de inesse in a system of existential graphs has formed the subject of an elaborate investigation with the reasoning of which I will not trouble you. [Gf:] Those who dont mind being troubled with this reasoning can find a few clues in CP 4.435: [[ If a system of expression is to be adequate to the analysis of all necessary consequences, it is requisite that it should be able to express that an expressed consequent, C, follows necessarily from an expressed antecedent, A. . In order to form a new and reasonable convention for this purpose we must get a perfectly distinct idea of what it means to say that a consequent follows from an antecedent. It means that in adding to an assertion of the antecedent an assertion of the consequent we shall be proceeding upon a general principle whose application will never convert a true assertion into a false one. This, of course, means that so it will be in the universe of which alone we are speaking. But when we talk logic and people occasionally insert logical remarks into ordinary discourse our universe is that universe which embraces all others, namely The Truth, so that, in such a case, we mean that in no universe whatever will the addition of the assertion of the consequent to the assertion of the antecedent be a conversion of a true proposition into a false one. But before we can express any proposition referring to a general principle, or, as we say, to a range of possibility, we must first find means to express the simplest kind of conditional proposition, the conditional de inesse, in which If A is true, C is true means only that, principle or no principle, the addition to an assertion of A of an assertion of C will not be a conversion of a true assertion into a false one. This conditional de inesse has to be expressed as a graph in such a way as distinctly to express in our system both a and c, and to exhibit their relation to one another. To assert the graph thus expressing the conditional de inesse, it must be drawn upon the sheet of assertion, and in this graph the expressions of a and of c must appear; and yet neither a nor c must be drawn upon the sheet of assertion. How is this to be managed? Let us draw a closed line which we may call a sep (sæpes, a fence), which shall cut off its contents from the sheet of assertion. Let this sep together with all that is within it, considered as a whole, be called an enclosure, this close, being written on the sheet of assertion, shall assert the conditional de inesse; but that which it encloses, considered separately from the sep, shall not be considered as on the sheet of assertion. Then, obviously, the antecedent and consequent must be in separate compartments of the close. In order to make the representation of the relation between them iconic, we must ask ourselves what spatial relation is analogous to their relation. Now if it be true that If a is true, b is true and If b is true, c is true, then it is true that If a is true, c is true. This is analogous to the geometrical relation of inclusion. So naturally striking is the analogy as to be (I believe) used in all languages to express the logical relation; and even the modern mind, so dull about metaphors, employs this one frequently. It is reasonable, therefore, that one of the two compartments should be placed within the other. But which shall be made the inner one? In order to decide which is the more appropriate mode of representation, one should observe that the consequent of a conditional proposition asserts what is true, not throughout the whole universe of possibilities considered, but in a subordinate universe marked off by the antecedent. This is not a fanciful notion, but a truth. ]] [Lowell 2.5:] Suffice it to say that it is found that there is essentially but one proper mode of representing it. Namely, 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. [Gf:] In order to explain the meaning of the scroll, Peirce appeals to our topological imagination, and introduces a third dimension to the flat sheet of assertion. The cut is not a mere boundary marking off a part of one surface from the rest of it; inside the cut is another surface, representing another universe, a universe of supposition rather than assertion. But now we find another cut inside the first one, and we know that the relation between the two spaces has to represent the relation of consequence, and to do so as iconically as possible. How do we imagine this topologically? [Lowell 2.5:] Besides this graph It rains 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. [Gf:] This time the cut, instead of disclosing another surface below, is patched to resemble the surface of the sheet of assertion. This suggests that the area inside the inner cut resembles the sheet of assertion in some way, although cut off from it as if it reverses the significance of the outer cut. This turns out to have repercussions through the whole system of EGs, as they develops cut within cuts within cuts: we find that actions permitted within an odd number of cuts (counting inward from the sheet of assertion) are often the reverse of actions permitted within an even number of cuts. If this feature of EGs is visually iconic, it is only so by a convention which you have to learn in order to correctly interpret the diagrams of the system. [Lowell 2.5:] A fixed terminology is a great comfort. [Gf:] That line makes me laugh every time, as in this context it seems such an understatement. I wonder if the original audience laughed. Anyway, we now define some of the basic terms used in reference to EGs: [Lowell 2.5:] Let us term the area on which a cut stands the place of the cut, while the area or bottom of the cut is the area within the cut. The cut itself is not a graph nor the replica of a graph. No more is the scroll. But the scroll with the two graphs scribed in its two closes or areas makes up a graph, or graph-replica; and this I call an enclosure. The term may be used indifferently to mean the graph or the replica. [Gf:] The term bottom as a synonym for area soon drops out of Peirces terminology, and indeed it doesnt seem suitable for the area inside the inner cut of a scroll, given the way he asked us to imagine it above. In any case, these terms all refer to topological spaces regarded as symbols of logical relations. To avoid confusion later on, its important to remember that the enclosure is a graph while neither the cut nor the scroll is a graph in itself. http://gnusystems.ca/Lowell2.htm <http://gnusystems.ca/Lowells.htm> }{ Peirces Lowell Lectures of 1903 https://fromthepage.com/jeffdown1/c-s-peirce-manuscripts/ms-455-456-1903-low ell-lecture-ii
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