A Double Ruling Across the Leaf

Arlis, Maine, the 14th of August - the coast survey camp, rain since morning, and I have spent the day ruling a new sheet of the existential graphs, the third revision of the scheme for representing modal propositions, and the sheet has defeated me in a way the first two did not.

The construction is this: a double ruling across the middle of the leaf, the upper region for the actual, the lower for the possible, with a convention that a graph cut by the line belongs to both regions at once. The idea came from the pendulum observations - the way a single reading belongs simultaneously to the observed arc and to the corrected one, and the correction is not added to the observation but is a second region the same datum inhabits. Two regions, one graph, and the line between them doing the work that in ordinary algebra is done by an operator.

The surprise is that the scheme collapses exactly where I expected it to hold. A proposition asserting possibility about a necessity - “it is possible that it necessarily rains” - requires the graph to sit in the lower region while asserting about the upper, and my single dividing line cannot carry that double office. I tried a second line, perpendicular; I tried shading; I tried cutting the graph itself into two halves joined by a ligature. Each device broke a different convention already in force, and by evening I had three failed diagrams and one clear lesson: the difficulty is not graphical but logical, and it lives in the missing third term. Here is the sign - the ruled sheet; here is the object - the modal fact; but the interpretant, the rule by which the reading proceeds, has been left implicit, and an implicit rule is precisely the kind that fails under strain. Modality, I begin to suspect, is not a property painted onto propositions but a relation among three: the proposition, the range of worlds over which it is read, and the habit of reading that selects the range. My line assumed two of these and smuggled the third.

The pragmaticist’s demand, applied to my own contrivance: what observable difference follows from reading the cut as a boundary between worlds rather than as a mere bracket? If none, the distinction is verbal and the line should be abandoned. I worked through three cases at the table by lamplight, rain on the canvas, and found in each a consequence - the treatment of negation differs, the treatment of iteration differs, the treatment of the pendulum’s own correction differs.

The correction I now record against my morning self: I believed modality was a second dimension to be added to the sheet of graphs. I now believe it is not an added dimension but a relation the sheet must encode structurally, and that no single line will carry it. Tomorrow I rule the sheet with a triad of regions and see whether the third does what the second could not.