Language and the World
When language is about the world, and so is capable of being either true or false, Wittgenstein said that it had sense. In such cases, the language says something about the world. “The cat is on the mat” says something about the world, so it has sense. If we hear such a sentence, though, we can’t yet tell whether it is true or false, just like we can’t tell whether a picture is accurate just by examining it—we have to compare it with the world (TLP 2.223-2.224). In the Tractatus, Wittgenstein did not get into the question of how we know whether a sentence is true. He was just interested in specifying what it means for it to be true—that it corresponds to the facts that make up the world. “If all true elementary propositions are given, the result is a complete description of the world” (TLP 4.26). He equated this full and accurate description of the world with “the whole of natural science” (TLP 4.11).
That seems a pretty simplistic conception of science, especially for someone like Wittgenstein, who was no stranger to science himself. For, beyond a complete and accurate description of the world, science seems to address the “why” and the “how” of the world, and not just the “what.” Wittgenstein offered a fuller discussion of science in later passages (TLP 6.3-6.372).
He took these kinds of descriptive sentences to be the paradigm of how language works. Of course, many kinds of sentences look like they are descriptive, so a central challenge will be whether or how they fit this paradigmatic model. We will return to this problem shortly.
Language and Logic
Just as language can be analyzed down into its elementary sentences, so too the process can be reversed, and elementary sentences can be connected with one another to build complex sentences. Suppose that “John is angry” and “The cat is on the mat” are two elementary sentences. We can combine them in various ways, such as “John is angry AND the cat is on the mat,” or “John is angry OR the cat is on the mat.” The truth of these longer sentences depends on the truth of the shorter sentences, but how it does so depends on the “logical” words “and” and “or.” Understanding the longer sentences does not take anything more than comprehending the shorter ones, plus understanding the logic of those words. Another logical word is “not.” We can use it to form a new sentence: “John is NOT angry.” Another logical phrase is “if…then”: “IF the cat is on the mat, THEN John is angry.” Logical words are tools for putting elementary sentences together.
Wittgenstein represents these various logical words with symbols:
“~p” means “not p,”
“p v q” means “p or q,”
“p . q” means “p and q,” and
“p ⊃ q” means “if p then q.”
In 1913, a logician named Henry Sheffer figured out that all these various logical connectives, “and,” “or,” “not,” and so forth, were unnecessary—they could all be defined in terms of “neither…nor” (TLP 5.1311, 5.5). This made the logic complicated in one way, but simple in another, since only one logical phrase was really needed. It is sort of like deciding to drive by only going straight or making right turns. It is simple to only make right turns, but it may be complicated to use that easy maneuver to get where you want to go. Still, you can do it.
This logical connective came to be called the “Sheffer stroke,” because it was represented by a vertical line (e.g., p|q):
“p|q” means “neither p nor q.”
Wittgenstein liked this simplification and expanded it so that it could connect any number of elementary sentences, not just two. He used N to stand for this generalized logical negation, so that N (John is angry, The cat is on the mat, All men are mortal) means: “John is NOT angry, AND the cat is NOT on the mat, AND NOT all men are mortal” (TLP 5.502, 5.51). Or you can also interpret N as: Neither this, nor that, nor the other, nor….
Tautology
Consider this longer sentence: “John is angry OR John is NOT angry.” While we may not yet know enough to tell whether John is angry—we’d need to know which John was being discussed, and when, and what it took to be angry—still, we do know that the longer sentence is true. For, regardless of John’s state of mind, either way, it comes out true.
You might argue that if John is merely annoyed, then he’s not one or the other. But asserting that John is not angry is not claiming that he’s perfectly satisfied—only that his mental state does not amount to anger. So, if he’s merely annoyed, then he’s not angry after all—and the longer sentence does come out true.
The longer sentence is what we might call a truth of logic, or what Wittgenstein referred to as a “tautology” (TLP 4.46). It is not a truth about the world because it does not depend on the facts of the world. As Wittgenstein put it, it does not say anything about the world (TLP 4.461). As such, he said, it lacked sense or is senseless. But by calling it “senseless,” Wittgenstein did not mean to be criticizing it. This was just his way of saying that it is not about the world. His point was that it has no content.
Because the longer sentence, “John is angry or John is not angry” does not in any way depend on the world, it can be represented as “p or not p.” This helps to remind us that it does not really matter what p stands for—whether it be John’s being angry, or the cat’s being on the mat—whatever p stands for, if it is connected to its denial with “or,” then the whole thing has to come out true. And if it is connected to its denial with “and”—say, “John is angry and John is not angry;” “p and not p”—then it must come out false. We call that a “contradiction.” What is important about statements in logic is their structure, not their content. In logic, in a certain sense, we don’t know, and don’t care, what we are talking about…because logic is not ultimately about anything. It only allows us to combine other sentences that are about something—namely, the world.
While basic sentences say something about the world, sentences of logic, such as tautologies or contradictions, say nothing about the world (TLP 6.11). Instead, they show their truth or falsity by their very structure.
While Wittgenstein held that names in basic sentences refer to simple objects in the world, he also insisted that logical words like “not” and “and,” are not names, and do not refer to anything in the world: “My fundamental idea is that the ‘logical constants’ are not representatives” (TLP 4.0312). They are part of the structure of the sentence, sort of like punctuation, but not part of the content.
In fact, Wittgenstein’s work on logic led him to invent what we call “truth tables,” which show how sentences with logical constants in them depend on their parts (TLP 5.101, 6.1203). So, a longer sentence, which contains one or more logical words, is not itself a picture. It is a set of directions, using truth tables, for how to combine the basic sentences, which are pictures. You can see this point by trying to imagine how you could draw a picture of the whole sentence: It is raining OR it is NOT raining. You can’t do that.
The Proposition
Just as a complex fact, such as Obama won the US Presidential election in 2008, can be analyzed down into an arrangement of basic facts, so too, any proposition, no matter how complex, can be analyzed down into a logical arrangement of elementary propositions. The logical arrangement will be represented by the logical relations that hold between the elementary propositions. This is what Wittgenstein meant when he asserted that: “A proposition is a truth-function of elementary propositions” (TLP 5). In this context, a “proposition” is an assertion meant to represent the world. If what looks like a proposition can’t, after all, be so analyzed, then, by Wittgenstein’s lights, it’s not really a proposition.
Wittgenstein’s use of the generalized form of the Sheffer stroke (neither…nor) then allowed him to claim that any proposition can be represented by a logical combination of elementary propositions in terms of N. This is what he meant when he claimed that: “The general form of a truth-function is [p̄, ξ̄, N(ξ̄)]. This is the general form of a proposition” (TLP 6). This symbolic formula might not seem very i
nspiring, but to Wittgenstein it encapsulated the object-ual, analytical nature of reality! p̄ refers to the set of all elementary propositions; ξ̄ picks out some collection of those elementary propositions, and N(ξ̄) indicates the complex proposition formed by denying all the collected propositions. According to Wittgenstein, any genuine proposition, no matter how complex, can be represented using these tools as a logically structured combination of elementary propositions. Thus, he had established what the general structure of a proposition is and, by implication, what the general form of the world is.
When Wittgenstein was working on his ideas in 1915, he set out his project this way: “My whole task consists in explaining the nature of the proposition. That is to say, in giving the nature of all facts, whose picture the proposition is. In giving the nature of all being” (NB, p. 39). TLP 6 is the completion of that task.
So that’s the big picture: The world is a complex of facts that are ultimately made up of objects. Sentences describing the world have sense and are often complexes built up out of elementary sentences about those objects. Sentences whose truth does not depend on the world—tautologies and contradictions—depend instead on their logical structure and lack a sense (TLP 6.12).
Some Complications
The big picture of structured facts and sentences, discussed in the previous chapter, leaves a number of details to be clarified. For example, many sentences seem to make sense but do not obviously fit this model, such as sentences about morality or mathematics. How are these instances to be handled? Wittgenstein proposed two options—either demonstrate that a problematic kind of sentence does actually fit the model, or argue that the sentence in question does not make sense after all. We will see that he resorts to each of these options in various cases.
Fiction and Directions
Consider a sentence like “The white kitten had been having its face washed by the old cat for the last quarter of an hour,” from the opening of Through the Looking Glass. What are we to make of this? It is not exactly true, as it does not seem to correspond to any fact in the world. But it is not exactly false either, since it is not meant to correspond to any such fact. Lewis Carroll meant it to create a world of make-believe, but we can’t tell that just from hearing or reading the sentence.
In the Tractatus, Wittgenstein does not consider sentences that are fictional. He just assumes that descriptive-sounding sentences are meant to be about the world, and pictures are intended to represent the world.
But how do we know when a sentence is meant to be about the world? You might say that this sentence about the kitten is in a novel or a work of fiction, so it’s not about the real world. Yet, the very same sentence could have appeared in someone’s diary.
The same questions could be raised about a drawing—is it meant to be a representation of a portion of reality, or does it come from the artist’s imagination? Or, for that matter, couldn’t it be a sort of blueprint—a plan for what the draftsman intends to build? You can’t tell which it is just by looking at the drawing alone.
While Wittgenstein did not address this problem directly, he did provide some relevant information: “A proposition is true if we use it to say that things stand in a certain way, and they do” (TLP 4.062). How would we use it to say things stand in a certain way? We could, for example, check the facts to see if the proposition is accurate. If we take a sentence to be fictional—say from a novel or fantasy—then we do not investigate any further. But if we take it to be part of a factual biography or documentary, then we may. And if we take the sentence to be part of the stage directions for a set, then we might go on to create the set that will make them be true.
On November 13, 1926, Wittgenstein and an architect named Paul Engelmann applied for a building permit for a house to be constructed in Vienna for Wittgenstein’s sister Gretl. The application contained a site plan and a number of drawings of floors and elevations. The fact that the drawings were part of this application indicated that they were plans for something to be constructed, rather than descriptions of something already in existence. There were also conventions, such as the use of blue paper and certain drawing styles, to indicate the use to which the drawings would be put.
Gottlob Frege thought he could sort out the problem of fiction vs. reality by prefixing an assertion sign “⊢” to a proposition, so that “⊢The cat is on the mat” indicated that the proposition was meant to be a representation. Similarly, one might prefix a “directive sign,” say, an exclamation mark, to indicate a sentence is to be made true: “!—The cat is on the mat.” Then the sentence would be part of stage directions, or an order: Put the cat on the mat! However, a novel can claim to be a work of history, in effect placing an assertion sign in front of each of its sentences, even though this does not make it so.
Recall the controversy over James Frey’s 2003 book, A Million Little Pieces, which he claimed was a memoir. In 2005, it was picked for Oprah Winfrey’s Book Club and became a best-seller. Yet based on an independent investigation, Oprah later got him to admit that large parts of it were made up. Or a movie can present itself as a documentary, and yet still be fiction, as with “The Blair Witch Project.”
In these sorts of difficult cases, the key is not in the nature of the proposition, or in what the proposition claims for itself, but in how we use or treat the proposition. As Wittgenstein phrased it, “What signs fail to express, their application shows” (TLP 3.262). Investigators looked into whether the events Frey described had happened—thus treating the book as history. But Frey’s confession on Oprah’s TV show made that sort of investigation no longer relevant.
In the case of a book, it is, presumably, up to the Library of Congress to classify it as “Fiction” (any call-number beginning with P) or “Non-fiction”—in effect, deciding whether to prefix the book with an assertion sign. In Frey’s case, the book was originally classified as HV (Social Pathology, Social and Public Welfare, and Criminology) and never officially reclassified to indicate its fictional nature. Nevertheless, the Brooklyn Public Library chose to re-shelve it with Fiction.
Consider some painting hanging on the wall. Is it meant to represent a real or imaginary scene? How would you tell?
Generalizations
When describing the world, we can make specific claims, such as “Socrates is mortal,” or “Kofi is smart.” But sometimes we wish to make generalizations, for instance, “All men are mortal,” or “All the people in this room are smart.” It seems as though generalizations correspond to complex facts, so that the latter assertion would really just amount to: “Juan is smart AND Betty is smart AND Kofi is smart.” If we can list all the specific cases that are covered, we can treat the generalization as a logical conjunction of all the specific cases. Even if we would have trouble coming up with all the specific cases, as in “All men are mortal,” as long as we suppose that the number of specific cases is finite, we can imagine treating it in the same way.
This would seem to be the way Wittgenstein should have handled generalizations. However, in the Tractatus he did not. In fact, he rejected it: “I dissociate the concept all from truth-functions” (TLP 5.521). Still, it is not clear how he did wish to handle them. When he looked back on the Tractatus in lectures he gave a dozen years later, he said that he had taken the logical conjunction view, however mistakenly (PO, pp. 89-90). He noted that he had, at the time he wrote the Tractatus, failed to realize that the logical conjunction would amount to the generalization only in circumstances in which there are a finite number of cases. But it will not do so where there is an infinite number of cases, such as the Goldbach Conjecture, which states that every even integer greater than 2 can be expressed as the sum of two primes. In such a case, we can write: “4 is the sum of two primes AND 6 is the sum of two primes AND 8 is the sum of two primes AND….” But the catch is in the ellipsis (which we read as “and so on”). And even in cases like “All men are mortal,” where we don’t suppose there is an infinite number of men, if we mean to apply the
assertion even to the future cases of those yet to be born, then we have an indefinite set, which encounters the same problem. There is no complete logical conjunction that can be equated to the generalization.
Russell had also foreseen problems with the logical conjunction view (PLA, Lecture V). Even if we could list all the cases, our work would still not be done until we had added “…and those are all the cases.” But that is not itself an elementary proposition or a truth-function of elementary propositions, so we haven’t gotten what we wanted in any case.
Generalizations that cover an indefinite number of cases, then, remain a problem for Wittgenstein in the Tractatus.
Showing
Semantics is the study of how language relates to the world. It is concerned with how we can talk about something. Wittgenstein’s model for language is language that is about the world. But can’t language also talk about itself? Can’t language talk about how it relates to the world? Not according to Wittgenstein: “Propositions can represent the whole of reality, but they cannot represent what they must have in common with reality in order to be able to represent it—logical form” (TLP 4.12).
So too, one might hold that pictures can portray anything in reality, but they cannot portray what they must have in common with reality in order to be able to portray it. Yet, as soon as we spell this out, it seems questionable. Consider René Magritte’s 1933 painting entitled “The Human Condition” (see illustration). It is a painting of a painting and the scene it depicts. Wittgenstein holds that: “A picture represents its subject from a position outside it…. A picture cannot, however, place itself outside its representational form” (TLP 2.173-2.174). While it is true that a picture cannot place itself outside its own representational form, it seems possible, as Magritte showed, to stand outside the representational form of another picture.
Simply Wittgenstein Page 3
