Answers to this rephrased question, some of which Slade lists, with all the attendant symbols and terminology, fill the next six notes; presumably these and like notes form the bulk of the calculus. How he arrived at some of these solutions would, presumably, have been discussed in the two, undelivered lectures. Fortunately, Slade’s students from BPR-57-c have been able to fill in much here, as this is exactly what Slade had been wrestling with during the original work sessions for three years. Some of their papers will appear in future issues.
Since Liebniz, or even Aristotle, the boundaries between mathematics and logic, and between logic and philosophy, have always been strangely fuzzy. Try to define them too carefully, and they disappear. Change your position only a fraction of a degree, and they seem clearly present once more. From this new angle we begin to define them again—and the process repeats. Is it, then, just the maverick statements that our third critic would claim Slade has simply scattered through his discussion of the logic of models that tempt us to take what seems essentially a discussion of the foundations of a limited, mathematical discipline and call it a philosophy? Your editor does not think so; we feel that for all its eccentricity of presentation, Slade’s work is philosophically significant—though already (a situation which has existed about Slade’s wofk since the publication of the Summa) articles have appeared which claim otherwise. The emblem of a philosophy is not that it contains a set of specific thoughts, but that it generates a way of thinking. Because a way of thinking is just that, it cannot be completely defined. And because Slade’s lecture is incomplete, we cannot know if he would have attempted even a partial description. Your editor feels that the parameters for a way of thinking have, in the extant notes of Shadows, been at least partially generated. Rather than try to describe it, we think it is best to close this limited exegesis with an example of it from Slade’s lecture. The note we end on—note seven—along with note twenty-two, completes the clearest nonmathematical explanation of the calculus Slade was trying to describe. (In note six, Slade talks about the efficiency of multiple modeling systems, or parallel models, over linear, or series models: his use of pictures, in note seven, to distinguish between words about reality and the real itself is a self-evident example of what he discusses in six. Slade drew the pictures hastily on his blackboard with blue chalk and pointed to them when they came up again in the flow of his talk.) Here is note seven:
There are situations in the world. And there are words—which are, to put it circularly, what we use to talk about them with. What makes it circular is that the existence of words, and their relationship to meanings, and the interrelationships among them all, are also situations. When we talk about how words do what they do, we are apt to get into trouble because we are maneuvering through a complex house of mirrors, and there is almost no way to avoid that trouble, short of resorting to pictures—which I am not above doing.
Many situations in the world have aspects that can be talked of as directed binary relationships. Some examples of talk about these situations which highlight the directed binary relationship are:
“Vivian loves the Taj Mahal.”
“Alicia built a house.”
“Chang threw the ball.”
“Sad means unhappy.”
“The hammer hit a nail.”
Let us take the last sentence, “The hammer hit a nail,” and consider it and the situation it might commonly be used in, and explore the modeling process that is occurring in some detail. First, we have a thing, the phrase the hammer, standing for a thing, c=^ . In that phrase, we have a thing, the word the, standing for an attitude toward •=4 , and we have another thing, the word hammer, standing for the object n^ itself. Next, we have a thing, the verb hit, standing for a relationship. After that, we have still another thing, the phrase a nail, standing for another thing, T . As in the first phrase, in the second we have a thing, the word a, standing for an attitude toward the object T different from the attitude modeled by the word the. And, as in the phrase the hammer, we have a thing, the word nail, which stands for the object T itself. Also, we have a relationship, composed of which thing (i.e., word) is put before the verb and which thing (i.e., word) is put after it, that stands for an aspect of the relationship c==A not completely subsumed by the verb hit alone, i.e., which object is the comparatively active one and which is the comparatively receptive one—or what can be talked of as “the direction of the binary relationship.” Now the direction of the relationship is, itself, a relationship; so here we have a relationship, between noun, verb, and noun, standing for an aspect of the relationship c^A .
Now there are other notable relationships in the sentence “The hammer hit a nail,” to attract our attention. In the phrase the hammer, for instance, which we have said consists of two things, the word the and the word hammer, it is necessary that the things appear in just that order. Likewise, the phrase a nail must preserve its order, if the sentence is to strike us as proper. What are these particular relationships necessary for? What would be wrong with the sentence “Hammer the hit nail a,” or “Hammer hit nail a the,” or “Hammer a the hit nail,” or “The a hammer hit nail”? In all of these, we still have the things in the sentence which stand for the things in the situation, and in all of them the relation between hammer, hit, and nail, which models the direction of the relation in the situation, is preserved. Is the relation between the and hammer, or a and nail, modeling anything in the situation which is suddenly lost or obscured if these relations are lost?
To the extent that our attitudes toward the objects in a relationship are not in that relationship, the simple answer is no. The relationship between the and hammer and between a and nail are necessary to preserve the integrity of the model itself; they are necessary if we are to recognize the model as a proper thing for modeling in the first place. But these relationships, between the and hammer and a and nail, do not model anything in the situation talked about by the sentence. To destroy them, however, may prevent other relationships (that may be modeling something in the situation, or may be preserving the integrity of the model) from standing forth clearly. This simple answer is, however, rather oversimplified.