“Alfred,” I said, laying my pencil across my pad and leaning back in the leather wingchair, “I know you really are trying to save me pages of semiological hair-splitting, but you are also standing in my way—interfering, if you will, with the modular context I have been trying to establish between the rain and my drawing pad. Could you be a pal and see if you can get us some coffee . . .?”
As English summers will, that one soon ended.
As happens, a year later an Italian summer replaced it. I was spending a sunny week in a villa outside Florence. The news came from my hostess, one morning over coffee in the garden, that we were to be joined shortly by—of all people! I had thought he was somewhere in Nepal; indeed, I hadn’t thought of him for six months! And who, sure enough, should come striding across the grass ten minutes later, in rather worn-out sneakers, his bald spot not noticeably larger but his shoulder-length hair definitely longer, thumbs tucked under his knapsack straps, and a Persian vest over an out-at-the-elbow American workshirt, from the pocket of which stuck the stem of what, from the bulge at the pocket’s base, I recognized as his Van Vogtian meerschaum—Alfred!
He came across the lawn, grinning hawkishly, and said: “Do you know what you left behind in England and I have carried all the way to India and back?”
“What . . .?” I asked, quite surprised at his introduction and charmed by this dispensing with phatic chatter.
“Your sketch pad! Hello, Vanessa . . .” to our hostess, and gave her a large hug. The high, aluminum rack of his backpack swayed above his shoulders.
To explain what happened that afternoon, I might mention explicitly several things implicit already about both Alfred and Vanessa. She, for instance, is very generous, a far more talented painter than I, and has several easels in her studio—the converted top floor of the villa. And Alfred, as I’m sure you’ve realized, has a rather strange mind at the best of times, which also entails a rather strange sense of humor.
At any rate, some hours later, I was walking through the white dining room, with its sparse brass and wood decoration, when I noticed, through the open iron casement, out in the sunlit Italian garden, one of Vanessa’s easels set up a few yards from the window; and set up on the easel was my sketch pad, with my drawing of last year’s rain-battered, English sycamore.
While I looked at it, Alfred came climbing in over the windowsill, dropped to the floor, spilling a few cinders onto the waxed floorboards, and, kicking at them, gave me a great grin: “There,” he said, “Go on! Make a true statement—an accurate verbal model of the situation outside the window! Quick!”
“Well,” I said, smiling and a bit puzzled, “it seems that there’s . . .” I paused, about to say ‘my picture outside,’ but I remembered our colloquy back in rainy Britain: “. . . that there’s my model outside!”
“Just what I was hoping you would say,” Alfred said. “It saves even more pages of semiological hair-splitting!”
“And,” I said, encouraged by this, “the model outside is true, too! Alfred, what have you been doing in India?”
“Amazing amounts of shit,” Alfred said warmly. “Do you know, Plato was right, after all—at least about method. As far as semiological hairsplitting is concerned, we just dispensed with practically a chapter and a half! A dialogue that you can make up as you go along really is the only way to get anything done in philosophy.”
I looked out at my picture again. “Then it is my model. And my model is true.”
“Your first statement is true.” Alfred’s smile became warmer still. ‘Your second is nonsense—no, don’t look so crestfallen. Just listen a moment: whether your model is a statement, a drawing, or even a thought, it is still a thing like any other thing: that is, it has its particular internal structure, and its various elements are undergoing their various processes, be that merely the process of enduring. Now you may have chosen any aspect of this thing—part of its material, part of its structure, or part of its process—to do the bulk of the modeling for you, while it was in the modular context. And, yes, outside that context, the model is still the same thing. But it is outside the context. Therefore, pointing out this window at that picture and calling it, or any part of it—material, structure, or process—‘true’ or ‘false’ is just as nonsensical now as it would have been for you, back in that abysmal May we spent in South Bernham, to point out the window and call some thing out there ‘true’ or ‘false’ . . . the rain, the shape of the drops, or the falling. A fine distinction has to be made. Whether the model functions as true or functions as false within the context may have something to do with the internal structure of the model. But whether the model functions (as true or false) has to do with the structure of the context. If you would like to, look at it this way: ‘true’ and ‘false’ merely model two mutually exclusive ways a given model (which is a thing) may function in a given context, depending on other things, which may, in different contextual positions, function as models. But the meaningfulness of the ascription of true or false is dependent on the context, not the thing.” Alfred took another draw on his pipe, found it was out, and frowned. “Um . . . now why don’t you take out that piece of paper you have folded up in the breast pocket of your Pendleton and look at it again—excuse me, I could have suggested you take it out of your wallet and avoided the implication that you hadn’t washed your shirt since last summer, but now I am just trying to save you pages of semiological elaboration.”
Feeling a bit strange, I fingered into my breast pocket, found the paper I had so summarily folded up a summer before, and unfolded it, while Alfred went on: “Think of it in this wise: if something is in the proper, logical position, it may be called true or false. If it moves out of that position, though it is still the same thing, you can’t call it true or false.”
And, creased through horizontally, I read:
The statement on the other side of this paper is true.
“Alfred—” I frowned—“if there is a statement on the other side of this paper (and, unless my memory plays tricks, there is) and it is meaningful to call that statement true or false—now I’m only letting the internal structure of this statement suggest a line of reasoning, I’m not accepting from it any information about its ‘truth’ or ‘falsity’, ‘meaningfulness’ or ‘meaninglessness’—that means (does it not?) that it is in the proper position in the modular context to do some modeling.”
“Even as you or I, when we stand at the window looking at what’s outside.”
“And if that statement refers to what’s on this side of the paper (and memory assures me that it does), then they are in the same context, which means they cannot both occupy the same position in it at the same time.”
“Have you ever tried to stand out in the garden and inside the sitting room all at once? It is a bit difficult.”
“So if that is the case, then this statement has to be considered just as a . . . thing, like rain, or a sycamore, or a garden . . .”
“Or a sketch of a garden. Or a statement. Or a thought. They are things too.”
“But I recall distinctly. Alfred: The statement on the other side of the paper calls this statement—this thing!—false!”
“Wouldn’t really matter if it called it true, would it—”
“Of course it wouldn’t! In the context I just outlined, I could no more call this . . . thing—” I waved the statement—“‘true’ than I could call—” I looked out the window at the easel with my sketch—“that thing true!”
“Though that does not reflect on its potential for truth if placed in another contextual position. If, for example, the statement on the other side of the paper read: ‘Your picture is in the garden,’ then it would be perfectly fine. Actually, it can work quite serially; what we’re really establishing is simply the unidirectionality of the modular context from the real. But then, all that semiological hair-splitting . . . Better turn over the paper and see if your memory isn’t playing tricks on you.”
Hastily I did. And re
ad:
The statement on the other side of this paper is false.
“Yes,” I said, “there is a statement on this side, and it does attribute truth-or-falsity to the statement on the other. Which is nonsensical. It’s standing inside the sitting room in Bernham looking out the window and calling the rain ‘true.’”
“You never really did that,” Alfred said. “We just made a model of it that we judged nonsensical—useless in a particular sort of way. Keep looking at the side of the paper you’re looking at now—that is: Set up the context in the other direction.”
I did until I had:
“It’s the same situation. If I let the other statement occupy the modeling position and this occupy the position of the modeled thing, then the fact that the other statement attributes truth or falsity to what’s on this side means it’s nonsensical too.”
Alfred nodded. “It’s like having, on either side of your paper: ‘The thing on the other side of this paper is true (or false); the thing on the other side of this paper is false (or true).’ Which is an empty situation, in the same way that if you and, say, Vanessa, both had drawing pads and pencils and were sitting where you could see each other’s paper, and I gave you the instructions: ‘Both of you draw only what the other is drawing.’ You’d both end up with empty pictures.”
“Speaking of Vanessa,” I said, “let us go see what she is doing. She is a better artist than I am, which I suspect means that on some level, she has established a more interesting modular context with reality than I have. Perhaps she will take a break from her work and have some coffee with us.”
“Splendid,” said Alfred. “Oh, you asked me what I was doing in India? Well, while I was there, I got hold of some . . .” But that is another story too.
28. Language suggests that “truth” (or “falsity”) may be an attribute of sentences much as “redness” may be an attribute of apples. The primary language model is the adjective “true,” the secondary one a noun, “truth,” derived from the adjective. This is not the place to begin the argument against the whole concept of attributes. (It goes back to Leibniz’s inseparable subject/verbs for true predicates; Quine has demonstrated how well we can get along in formal logic without attributes, as well as without the whole concept of propositions.) But I maintain that, subsumed under the noun “truth,” is a directed binary relation, running from the real to the uttered, by way of the mind. The problems we have concerning “truth” (such as the paradox in section 27) are problems that arise from having to model a directed binary relationship without a transitive verb.
It is as if, in those situations in which we now say “The hammer strikes the nail” and “The hammer misses the nail,” we were constrained by the language only to speak of “strike nails” and “miss nails,” and to discuss “strikeness” and “missness” as attributes a given nail might or might not possess, depending on the situation, at the same time seldom allowing a mention of the hammer and never the moment of impact.
What “truth” subsumes (as well as an adjective-derived noun can) is a process through which apprehension of some area of the real (either through the senses, or through the memory, or the reality of internal sensation—again, this is not the place to discuss their accuracy) generates a descriptive utterance. This process is rendered highly complex by the existence of choice and imagination and is totally entangled in what Quine and Ullian have called “the web of belief”: confronted with the real, the speaker may choose not to speak at all, or to speak of something else, or she may be mistaken (at any number of levels), or he may generate a description in a mode to which “truth” or “falsity” are simply not applicable (it may be in G. Spencer-Brown’s “imaginary” mode). But when the speaker does generate an utterance of the sort we wish to consider, the overall process structure is still binary, and directed from reality to the sentence.
When I look out the window and say “It is raining outside,” what I perceive outside the window is controlling my utterance in a way the internal apprehension of which is my apprehension of the statement’s “truth” or “falsity.” My utterance does not affect—save possibly in the realms of Heisenberg—whatever (rain or shine) is outside the window.
People have suggested that the problem of paradox sentences is that they are self-descriptive. Yes, but the emphasis should be on descriptive, not self.
“This sentence contains six words” is just as self-descriptive as “This sentence is false.” But the first sentence is not paradoxical; it is simply wrong. (It contains five words.) The second sentence is paradoxical because part of the description (specifically “This sentence . . .”) covers two things (both the sentence “This sentence is false” and the sentence that it suggests as an equivalent translation, “This sentence is true”) and does not at all refer to the relation between them. The only predicate that is visible in “This sentence is . . .” suggests they relate in a way they do not: “This sentence ‘This sentence is true’ is the sentence ‘This sentence is false.’” And, obviously, it isn’t. But the same situation exists in Grelling’s paradox, the paradox of the Spanish barber, as well as the set-of-all-normal-sets paradox—indeed, in all antinomies.
The real generates an utterance via a process that allows us to recognize it as “true” or “false.”
If we introduce verbs into the language to stand for the specific generative processes, we fill a much stumbled-over gap. By recovering what is on both sides of the interface, and the direction the relation between them runs, we clarify much that was confused because unstated. Let us coin “generyte” and “misgeneryte,” and let us make clear that these processes are specifically mental and of the particular neurocybernetic nature that produce the utterances which, through a host of overdeter- mined and partially determined reasons, we have been recognizing as “true” and “false.” If we introduce these verbs into our paradox, it stands revealed simply as two incorrect statements.
On one side of the paper instead of “The sentence on the other side of this paper is true,” we write:
“What is on this side of the paper generytes the sentence on the other side.”
And on the other side instead of “The sentence on the other side of this paper is false,” we write:
“What is on this side of the paper misgenerytes the sentence on the other side.”
Looking at either sentence, then turning the paper over to see if it does what it claims, we can simply respond, for both cases: “No, it does not.” One (among many) properties that lets us recognize a generyted (or misgeneryted) sentence is that it is in the form of a description of whatever generyted (or misgeneryted) it; neither sentence is in that form.*
A last comment on all this:
The whole problem of relating mathematics to logic is basically the problem of how, logically, to get from conjunctions like “1 + 1 = 2 and 1 +? 1 ≠ 3,” which is the sort of thing we can describe in mathematics, to the self-evident (yet all but unprovable) logical implication: “1 + 1 = 2 therefore 1 + 1 ≠3,” which is the process that propels us through all mathematical proofs.
Now consider the following sentences, one a conjunction, one an implication:
“This sentence contains ten words and it misgenerytes itself.”
“If this sentence contains ten words, then it misgenerytes itself.”
About the first sentence we can certainly say: “That sentence contains nine words, therefore it misgenerytes itself.” If that self-evident there-fore can be considered an implication, and assumed equivalent to (“to have the same truth values as” in our outmoded parlance) the implication of the second sentence, then, working from the side of language, we have, self-evidently, bridged the logical gap into mathematics!
Before making such an assumption, however, count the words in the second sentence . . .
29. Vanessa Harpington (during a period when she [not I] thought her work was going badly), shortly after Alfred’s departure for Rumania:
“What use is love?
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“It assures neither kindness, compassion, nor intelligence between the people who feel it for one another.
“The best you can say is that when good people love, they behave well. . . sometimes.
“When bad people love, they behave appallingly.
“I wonder what the brilliant Alfred will have to say about a paradox like that!’
“First of all, Vanessa,” I reminded her as we walked the cobbled streets, with the Arno, dull silver, down every block, through the Italian summer, “you simply cannot take such abstract problems so seriously. Remember, you and Alfred are both fictions: neither of you exists. The closest I’ve ever been to passing a summer in an English country house was a weekend at John and Margery Brunner’s in Somerset, and though I spent a few weeks in Venice once, I’ve never stayed in an Italian villa in my life! I’ve never even been in Florence—”
“Oh, really,” Vanessa said. “You just don’t understand at all!” and, for the rest of the walk back, stayed a step or two ahead of me, arms folded and looking mostly somewhere else, though we did eventually talk—about other things.