1 Wittgenstein’s Tractatus in Brief: Remarks on the Scaffolding
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"Mr. Wittgenstein's Tractatus Logico-Philosophicus, whether or not it proves to give the ultimate truth on the matters with which it deals, certainly deserves, by its breadth and scope and profundity, to be considered an important event in the philosophical world." |
"The book deals with the problems of philosophy, and shows, I believe that the reason why these problems are posed is that the logic of our language is misunderstood. The whole sense of the book might be summed up in the following words: what can be said at all can be said clearly, and what we cannot talk about we must pass over in silence." |
| —Bertrand Russell, Introduction to the Tractatus | —Ludwig Wittgenstein, Preface to the Tractatus |
Wittgenstein’s philosophy of logic in the Tractatus was the result of his study of Gottlob Frege’s writings and study with Bertrand Russell on the philosophy of mathematics. Frege’s and Russell’s work arose from their logicism—an attempt to construct and secure the foundations of mathematics, especially arithmetic, on the basis of logic and set theory. The thesis of logicism can be set forth as follows:
Wittgenstein’s Tractatus is inspired, not so much by logical or mathematical rigor, but rather by a sense of the austere and aesthetic beauty of formal logic. Wittgenstein’s goal was to construct a philosophy of logic that could be reconciled with mathematics, science, probability, morality, the self, and his belief in a mystical worldview.
Wittgenstein’s Kantian aim in the Tractatus is to understand the structure and limits of thought by examining the structure and limits of language. The limits of language are determined by its internal logical structure. Using his theory of logic, Wittgenstein places the propositions of science as within the limits of sense but places religion and morality beyond those limits. The logical positivists use this demarcation line to abandon the claims of religion, morality, and aesthetics as cognitively meaningless, but for Wittgenstein, this demarcation line protected the realms of religion, morality and aesthetics from intrusion of scientific verificationism. According to Wittgenstein, philosophical troubles arise from the desire to transcend the world of human thought and experience and trying to adopt an Archimedean standpoint. The over-arching architectonic or formal structure of the Tractatus can be seen from its whole numbered propositions:
| 1 | The world is all that is the case. | |
| 2 | What is the case — a fact — is the existence of states of affairs. | |
| 3 | A logical picture of facts is a thought. | |
| 4 | A thought is a proposition with a sense. | |
| 5 | A proposition is a truth-function of elementary propositions. | |
| 6 | The general form of a proposition is \([\bar{p},\bar{\xi},N(\bar{\xi})]\). This is the general form of a proposition. |
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| 6.54 | My propositions serve as elucidations in the following way: anyone who understands me eventually recognizes them as nonsensical, when he has used them—as steps—to climb up beyond them. (He must, so to speak, throw away the ladder after he has climbed up it.) | |
| 7 | What we cannot speak about we must pass over in silence. |
The genesis of the Tractatus is the grand statement that the world is all that is the case. And then it states that what is the case, a fact, is characterized as the existence of a state of affairs. A proposition asserts the existence of a state of affairs. Which is a logical picture of facts or a thought. The flow of the Tractatus continues in this way by starkly elegant propositions connecting one term to the next until one reaches the famous claim that the propositions of the Tractatus themselves are only the rungs of a ladder that must be kicked away. Wittgenstein’s dichotomy that logical relations can only be “shown” not “said” casts doubt on the meaningfulness of the previous propositions, which, after all, attempt to say a great deal about logic. The Tractatus ends with the mystical admonition that “of that we cannot speak, we must consign to silence.”
Rudolf Carnap (1891 - 1970), who was described by the American logician W. V. O. Quine as the “embodiment of logical positivism, logical empiricism, the Vienna Circle,”1 gives a vivid portrait of Wittgenstein’s manner of philosophizing:
1 Quine’s 1970 Tribute to Carnap.
- His point of view and his attitude towards people and problems, even theoretical problems, were much more similar to those of a creative artist than to those of a scientist; one might almost say, similar to those of a religious prophet or a seer. When he started to formulate his view on some specific philosophical problem, we often felt the internal struggle that occurred to him at that very moment, a struggle by which he tried to penetrate from darkness to light under an intense and painful strain, which was even visible on his most expressive face. When finally, sometimes after a prolonged arduous effort, his answer came forth, his statement stood before us like a newly created piece of art or a divine revelation. Not that he asserted his views dogmatically… But the impression he made on us was as if insight came to him as through a divine inspiration, so that we could not help feeling that any sober rational comment or analysis of it would be profanation.
The goal of this short tour of the Tractatus is to give you a sense of some of its main doctrines and to point out its clashing conception of logic found in the works of Gödel and Turing. Although Gödel and Wittgenstein never met, it is clear that they were very aware of the reputation of each other work. Wittgenstein writes in his Remarks on the Foundations of Mathematics:
- Let us suppose I prove the unprovability (in Russell’s system) of P; then by this proof I have proved P. Now if this proof were one in Russell’s system — I should in this case have proved at once that it belonged and did not belong to Russell’s system. — That is what comes of making up such sentences. But there is a contradiction here! – Well, then there is a contradiction here. Does it do any harm here? p. 51e
In a letter to his friend Karl Menger, who had organized a mathematics colloquium which branches off from the Vienna Circle and which was less interested in dismissing religious claims with more of an emphasis on mathematics, Gödel wrote:
“I also read parts of it [Wittgenstein’s Remarks on the Foundations of Mathematics]. It seemed to me at the time that the benefit created by it may be mainly that it shows the falsity of the assertions set forth in it” and in a footnote added: “and in the Tractatus (the book itself contains very few assertions).” 2
2 “Gödel’s remarks about Wittgenstein are cited by Solomon Feferman, the editor-in-chief of the monumental Gödel’s Collected Works, as a”gem” in his reflections in “The Gödel Editorial Project: A Synopsis”, reprinted in Kurt Gödel: Essays for His Centennial, edited by Feferman, Parsons, and Simpson (Cambridge University Press), p. 11.
There is a growing literature on Wittgenstein’s “notorious passage” about Gödel Incompleteness addressing the question whether Wittgenstein understood Gödel’s theorems and their significance.
Juliet Floyd and Hilary Putnam, “A Note on Wittgenstein’s ‘Notorious Paragraph’ about the Gödel theorem,” Journal of Philosophy, vol. 97 (2000): 624–632. Timothy Bays, “On Floyd and Putnam on Wittgenstein on Gödel,” Journal of Philosophy, CI.4 (April 2004): 197–210.
Logical Atomism: Facts (1 – 1.21), States of Affairs (2 – 2.0141); Objects (2.02 – 2.063)
| 1 | The world is all that is the case. | |
| 1.1 | The world is the totality of facts, not things. | |
| 1.11 | For the totality of facts determines both was what is the case, and also whatever is not the case. | |
| 1.12 | The facts in logical space are the world. | |
| 1.13 | The facts in logical space are the world. | |
| 1.2 | The world divides into facts. | |
| 2 | What is the case—a fact—is the existence of states of affairs. | |
| 2.01 | A state of affairs (a state of things) is a combination of objects (things). |
Picture Theory of Meaning: Pictures (2.1 – 2.225); Thoughts (3 – 3.13); Propositions and Names (3.14 – 3.261)
| 1 | The world is all that is the case. | |
| 1.1 | The world is the totality of facts, not things. | |
| 1.11 | For the totality of facts determines both was what is the case, and also whatever is not the case. | |
| 1.12 | The facts in logical space are the world. | |
| 1.13 | The facts in logical space are the world. | |
| 1.2 | The world divides into facts. | |
| 2 | What is the case—a fact—is the existence of states of affairs. | |
| 2.01 | A state of affairs (a state of things) is a combination of objects (things). |
Wittgenstein’s theory of factual propositions depends on the fundament idea of exclusion. A factual proposition always excludes certain possibilities. The proposition that \(P\) excludes the possibility that not-P. All factual propositions are truth-functions of elementary propositions.
1.1 Philosophy of Logic: Convention (3.62 – 3.5), Philosophy (4 – 4.0031), True and False (4.01 – 4.0641); Science (4.1 – 4.116), Formal Concepts (4.12 – 4.2), Truth function (4.21 – 4.45); Tautology (4.46 – 5.101); Inference (5.11 – 5.156)
Wittgenstein’s view is that necessity is logical necessity and that the necessary truths of logic are all empty tautologies. This view is tantamount to a denial of Kant’s claim there are synthetic a priori truths.
Suppose that \(P\) and \(Q\) are atomic facts. Then one logical fact composed from these logical atoms could be expressed by something like this: \(P\) if and only if \(Q\) is equivalent to saying not both \(P\) and \(Q\) only if neither \(P\) nor \(Q\). Here the notion of equivalence is tautological. Given the truth of the biconditional \(P \iff Q\) is would be incorrect to say \(P\) is equivalent to \(Q\). The biconditional \(P \iff Q\) is true if either both \(P\) and \(Q\) are true or both \(P\) and \(Q\) are false. \((P \land Q)\) is tautologically equivalent to \((\neg P → Q)\) because \([(P \land Q) \iff (\neg P \to Q)]\) is a tautology. Similarly, given the truth of the conditional \(P \iff Q\) it would be incorrect to say P implies Q. The conditional \((P to Q)\) is true whenever \(P\) happens to be true or \(Q\) happens to be false. \(\neg P\) tautologically implies \((P \to Q)\) because the conditional \(\neg P \to (P \to Q)\) is a tautology.
Although Wittgenstein stalwartly refuses to give examples of atomic facts, let’s suppose, for the sake of definiteness that the propositional letters P and Q stand for “Socrates is a philosopher” and “Socrates asks a question”, then the above logical fact has as an instance:
Socrates is a philosopher if and only if he asks a question is tautologically equivalent to it is not the case that Socrates is both a philosopher and asks a question only if it is the case that Socrates is neither a philosopher nor asks a question.
We can construct a truth table for this “molecular fact”:
| \(P\) | Q | (P | ↔︎ | Q) | ↔︎ | [~ | (P | ∧ | Q) | → | ~ | (P | ∨ | Q)] | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| T | T | T | T | T | T | F | T | T | T | T | F | T | T | T | |
| T | F | T | F | F | T | T | T | F | F | F | F | T | T | F | |
| F | T | F | F | T | T | T | F | F | T | F | F | F | T | T | |
| F | F | F | T | F | T | T | F | F | F | T | T | F | F | F | |
| 1 | 3 | 2 | 9 | 6 | 1 | 4 | 2 | 8 | 7 | 1 | 5 | 2 |
The initial two columns on the left assign to the component propositions \(P\) and \(Q\) all the possible truth assignments. These correspond to the four logically possible worlds—the world in which both \(P\) and \(Q\) are true, the world in which \(P\) is true but \(Q\) is false, the world in which \(P\) is false but \(Q\) is true, and the world in which both \(P\) and \(Q\) are false. Filling out the truth table is a matter of computing the various columns based on the truth rules for each of the connectives:
- The negation \(\neg P\) has the opposite truth-value of \(P\).
- A conjunction \((P \land Q)\) is true if and only if both conjuncts \(P\) and \(Q\) are true.
- A disjunction \((P \lor Q)\) of sentences is false if and only if both disjuncts \(P\) and \(Q\) are false.
- A conditional \((P \to Q)\) is false if and only if its antecedent \(P\) is true and its consequent \(Q\) is false.
- A biconditional \((P \leftrightarrow Q)\) is true if and only if its constituents \(P\) and \(Q\) have the same truth value.
This particular molecular proposition has several interesting logical properties. First, it combines the five most common logical connective and operators \(\{\neg, \land, \lor, \to, \leftrightarrow\}\) to one a single propositional. Secondly, this proposition is a tautology or logical truth as can be seen from that fact that column 9, the final column to be calculated, has all T’s or the truth value true in every row of its final column. Intuitively, this means that the proposition is true in all logically possible worlds. Thirdly, the proposition implicitly contains the two, and only two, logical connectives that are adequate to express all 16 of the possible binary connectives. The symbolic sentence \(\neg(P \land Q)\), which states that not both \(P\) and \(Q\) are true, is symbolized by the Sheffer stroke \(P \mid Q\) and is known as nand. The sentences \(\neg(P \lor Q)\) which states that neither \(P\) nor \(Q\) is true, is symbolized by the dagger \(P \downarrow Q\) and is known as nor. Notice that this last fact is a fact about logic—a metalogical fact—of the sort that violates Wittgenstein’s dichotomy that truths of logic cannot be said but only shown. Nevertheless, it remains an interesting fact about the logic of logic. Wittgenstein’s dichotomy of showing and saying therefore begs the question against the very idea of meta-mathematics.
| 1 | The world is all that is the case. | |
| 1.1 | The world is the totality of facts, not things. | |
| 1.11 | For the totality of facts determines both was what is the case, and also whatever is not the case. | |
| 1.12 | The facts in logical space are the world. | |
| 1.13 | The facts in logical space are the world. | |
| 1.2 | The world divides into facts. | |
| 2 | What is the case—a fact—is the existence of states of affairs. | |
| 2.01 | A state of affairs (a state of things) is a combination of objects (things). |
Another doctrine of Wittgenstein’s was that tautologies are empty and say the same thing, that is nothing at all. From a semantic point of view, all tautologies are true in every logically possible world and hence do not say something distinctive about any possible world. However, not all tautologies are created equal. From a syntactic point of view, some tautologies have more logical depth than others insofar as they have more deductive consequences. The following propositional counterparts to Aristotle’s Three Laws of Thought, are tautologies that “say” different things:
- Law of Excluded Middle: \(\phi \lor \neg \phi\)
- Law of Non-Contradiction: \(¬(\phi \land \neg\phi)\)
- Law of Identity: \(\phi \iff \phi\)
And perhaps it is not immediately obvious what the following tautologies “say”:
- Pierce’s Law: \([(P \to Q) \to P] \to P\)
- Consequentia Mirabilis: \((\neg P \to P) \iff P\)
- Scotus’s Law: \((P ∧ \neg P) \to X\)
- \((P \to Q) ∨ (Q \to R)\)
The following set of tautological schemata form an axiomatic basis for conditional logic together with the single rule of inference modus ponens:
- \((\mathbf{L}_1)\) \(\phi \to (\psi \to \phi)\)
- \((\mathbf{L}_2)\) \([\phi \to (\psi \to \chi)] \to [(\phi \to \psi) \to (\phi \to \chi)]\)
- \((\mathbf{L}_3)\) \((\neg \psi \to \neg \phi) \to (\phi \to \psi)\)
Modus ponens: From \(\phi \to \psi\) and \(\phi\) to infer \(\psi\).
1.2 Syntax of Arithmetic:
| 1 | The world is all that is the case. | |
| 1.1 | The world is the totality of facts, not things. | |
| 1.11 | For the totality of facts determines both was what is the case, and also whatever is not the case. | |
| 1.12 | The facts in logical space are the world. | |
| 1.13 | The facts in logical space are the world. | |
| 1.2 | The world divides into facts. | |
| 2 | What is the case---a fact---is the existence of states of affairs. | |
| 2.01 | A state of affairs (a state of things) is a combination of objects (things). |
Wittgenstein’s view that numbers are the exponents of functions, while suggestive of the successor operation, does not give a rigorous analysis of numbers. Instead, Wittgenstein’s manipulation of exponents seems to presuppose, rather than provide an explanation for, the laws of arithmetic.
1.3 Solipsism: Belief (5.54 – 5.5423), Self and World (5.55 – 5.641)
| 1 | The world is all that is the case. | |
| 1.1 | The world is the totality of facts, not things. | |
| 1.11 | For the totality of facts determines both was what is the case, and also whatever is not the case. | |
| 1.12 | The facts in logical space are the world. | |
| 1.13 | The facts in logical space are the world. | |
| 1.2 | The world divides into facts. | |
| 2 | What is the case---a fact---is the existence of states of affairs. | |
| 2.01 | A state of affairs (a state of things) is a combination of objects (things). |
The problem with solipsism is that when the solipsist denies the existence of anything but himself, he uis unable to point to what it is that, according to him, does not exist.
1.4 Logic, Science, and Mathematics: Logic and Mathematics (6 – 6.241), Science (6. – 6.372)
| 1 | The world is all that is the case. | |
| 1.1 | The world is the totality of facts, not things. | |
| 1.11 | For the totality of facts determines both was what is the case, and also whatever is not the case. | |
| 1.12 | The facts in logical space are the world. | |
| 1.13 | The facts in logical space are the world. | |
| 1.2 | The world divides into facts. | |
| 2 | What is the case---a fact---is the existence of states of affairs. | |
| 2.01 | A state of affairs (a state of things) is a combination of objects (things). |
Wittgenstein believed himself “to have found, on all essential points, the final solution to the problems’ of philosophy.” The goal is to deal ‘with the problems of philosophy’ by ‘set[ting] a limit thought.’ The way to do this is to set a limit ‘not to thought, but to the expressions of thoughts,’ i.e., to language. Wittgenstein’s principles are too restrictive:
Wittgenstein’s Principle of _______________ (2.02) ____________ physics.
Wittgenstein’s Principle of _______________ (4.2211) __________ mathematics.
Wittgenstein’s Principle of ________________(5.54) ___________ psychology.
1.5 Value and Mysticism: Value (6.373 – 6.522), What cannot be said (6.53 – 7)
| 1 | The world is all that is the case. | |
| 1.1 | The world is the totality of facts, not things. | |
| 1.11 | For the totality of facts determines both was what is the case, and also whatever is not the case. | |
| 1.12 | The facts in logical space are the world. | |
| 1.13 | The facts in logical space are the world. | |
| 1.2 | The world divides into facts. | |
| 2 | What is the case---a fact---is the existence of states of affairs. | |
| 2.01 | A state of affairs (a state of things) is a combination of objects (things). |
The reason why the problems of philosophy (‘metaphysical problems in particular’) ‘are posed is that the logic of our language is misunderstood.’ Therefore, what ‘can be said at all can be said clearly, and what we cannot talk about we must consign to silence.’
Wittgenstein’s famous ladder passage is anticipated, as pointed out by Hao Wang, in the Chinese Diamond Sutra (“The dharma I am preaching is analogous to a raft [which is to be discarded after use]’ even the dharma can be discarded, a fortiori the non-dharma”. It is also anticipated by Sextus Empiricus’ Against the Logicians, II, 480-1 ( “And again, just as it is not impossible for a man who ascends to a high place to overturn the ladder with his foot after the ascent, so also it is not unlikely that the Skeptic after he has arrived at the demonstration of his thesis by means of the argument…as it were a step ladder, should then abolish this very argument”).
Wittgenstein did not follow his own advice in proposition 7. So perhaps it is fitting to conclude with the paradoxical humor of Zhūangzi:
荃者所以在魚，得魚而忘荃；蹄者所以在兔，得兔而忘蹄；言者所以在意，得意而忘言。 吾安得夫忘言之人而與之言哉！
The bamboo fish net exists for catching fish. Once the fish is caught, forget the net! The rabbit snare exists for trapping rabbits. Once the rabbit is trapped, forget the snare! Words exist because they are used for expressing meaning. Once the meaning is grasped, forget the words! Where can I meet those who have forgotten words so I can have a word with them?