Sentences, sense, and logical truth

In the Tractatus Wittgenstein does not take up the example from Notebooks for explaining how we form sentences. In Notebooks he compares the construction of a sentence to the way a judge views the evidence and constructs the case (Wittgenstein, Ludwig. Notebooks 1914–1916. September 29, 1914). In the Tractatus Wittgenstein considers the construction of the structure of a sentence in relation to reality and compares the sentence to a picture. We read in Tractatus 4.01: “The sentence is a picture of reality. The sentence is a model of the reality as we think it is.” We must not consider the “model of reality as we think it is” as a hypothesis of natural science that is ready for falsification or affirmation. Tractatus 4.01 cannot be seen as the proof of Wittgenstein being a positivist. The positivist fallacy is the most popular misunderstanding of Wittgenstein’s Tractatus. We must read and interpret Tractatus 4.01 in relation to Tractatus 4.0141, where Wittgenstein compares the speaker of a sentence to a musician who “is able to read the symphony out of the score”. The musician is capable of reading music out of the score because she or he has learned the “general rule” that is necessary for writing a musical score and therefore she or he understands the score (Tractatus 4.0141). Wittgenstein constructs his picture theory with the help of the art of music without forgetting about empirical science. The “general rule” that enables the musician to understand the musical score applies also to the gramophone record. From the lines of the gramophone record one can reconstruct the music and the same “general rule” allows to write the musical score for the music. The following of the “general rule” founds an “internal relation” between the musician reading a score, a gramophone record reproducing the music of the score and writing down the score of the music (ibid.). Wittgenstein understands the “general rule” as the “law of projection” “which projects the symphony into the language of the musical score” (ibid.) The projection of the musical score into the gramophone record Wittgenstein calls “translation” (ibid.). The law of projection of the symphony into the gramophone record follows the laws of physics, but Wittgenstein does not speak of physics. He speaks of the translation of the musical score into the gramophone record and of the projection of the symphony into the musical score. I need a musician for projecting the symphony into the musical score and I need a set of musicians, an orchestra, to translate the musical score into music which will be projected into the gramophone record. The hermeneutics of translating tells us that every translation is an interpretation. Interpreting is not a science, it is an art. Wittgenstein speaks of the “language of the musical score” and of the “language of the gramophone record”, he speaks of the art of music, of producing and of reproducing music. The construction of sentences can be compared to the art of composing, of projecting music into scores or into old and new sound carriers. The performance of a symphony, the musical score of the symphony and the operation of the sound carrier reproducing the symphony are pictures of music. Analogically, Wittgenstein understand sentences of language as being pictures, pictures of reality. Tractatus 4.0141 shows that we must consider reality as inclusive reality of all forms of culture, of art, of science, and above all of language.

The differentiation between “to show” and “to say” is one of the fundamental ideas of the Tractatus. A sentence shows something and says something. In order to understand this differentiation, we have to remember the construction of the case, that is the facts that form a state of affairs. A description of a state of affairs is called a sentence. In 4.023 Wittgenstein speaks of facts andsentences: “A sentence is the description of a fact”, or the sentence is a description of a state of affairs. The sense of a sentence is the description of the facts that form a state of affairs. The picture theory considers the facts that form a state of affairs as elements of a picture. In Tractatus 2.141 we read “The picture is a fact.” Tractatus 2.15 explains the structure of a picture: “That the elements of the picture are combined with one another in a definite way, represents that the things are so combined with one another.” The fundamental differentiation between “to show” and “to say” Wittgenstein affirms in Tractatus 4.021: “The sentence shows its sense. The sentence shows how things stand, if it is true. And it says, that they do so stand.”

If I understand a sentence, I know the state of affairs presented by it, I know the facts presented by it. Another way of expressing that I know the state of affairs presented by the sentence is the expression that I understand the sense of the sentence. If I understand the sense of the sentence and if I know therefore “how things stand”, I do not yet know if things actually “do so stand”. The sentence says that what it shows is the case. Whether what it says is true, the sentence itself cannot decide. The sentence only says what the case is if it is true. We still must find the truth of the sentence, that is logical truth of the sentence. Wittgenstein assesses in the Tractatus the two truth-possibilities “true” and “false,” we speak of a two-valued logic. Computers, databases, the internet, and social media still work based on a two-valued logic. Our contemporary world relies on a two-valued truth for the possibilities and realizations of globalized communication.

Wittgenstein uses the two-valued logic and invents the truth-tables as a method for the logical analysis of sentences and complexes of sentences. With the help of Wittgenstein’s two-valued logic we can logically analyze sentences that show at least one logical operator. The realization of this logical analysis is possible independently of the conviction that elementary sentences must exist. When explaining truth-tables and truth-functions Wittgenstein uses the term “elementary sentence”. Although it looks like his logical analysis of sentences applies only to elementary sentences, this is not the case. Already in Tractatus 4.023 Wittgenstein affirms determination of reality to the truth values “true” and “false” for all sentences and not exclusively for elementary sentences: “The sentence determines reality to this extent, that one only needs to say “Yes” or “No” to it to make it agree with reality. Reality must therefore be completely described by the sentence. A sentence is the description of a fact.” Concerning the truth-tables Wittgenstein is clear in Tractatus 5.31 that we do not necessarily deal with elementary propositions: “The Schemata No. 4.31 are also significant, if ‘p’, ‘q’, ‘r’, etc. are not elementary sentences.” The schemes in Tractatus 4.31 fix the sequence of the truth-possibilities of three sentences “p”, “q”, and “r”, of two sentences “p” and “q”, and of “p”. The sequence of truth-possibilities determines the order of operating the truth-functions of sentences. We are speaking of logical truth and logical operators. Logical constants or operators like the words “and,” “or,” “not,” “some,” “all,” “identical,” “if … then,” etc signify logical operations.

Our sentences very often show logical operators, or they can at least be formulated in a way that shows the logical operators. The logical analysis of sentences is of interest for demonstrating the logical coherence of our arguments that we present with sentences or series of sentences. It is a possibility-condition for this logical analysis to accept that the logical operators do not represent objects. We follow Wittgenstein’s operative understanding of the logical construction of the sentence: the logical operators signify operations and not logical objects. The sense of a sentence is shown by the sentence. The sentence is able to show the circumstances, that is the truth-possibilities, if the sentence is true or false. Together with the given sentence “s” we are also given all the “circumstances” that make the sentence “s” true or false. With the help of the logical operator, we can see that the sense of the sentence operates a function. This function is a logical truth-function in a two-valued logic, whose logical operators show how to operate the function. The truth-values are the results of this operation. In Tractatus 5.2341 we read: “The sense of a truth-function of p is a function of the sense of p. Denial, logical addition, logical multiplication, etc. etc., are operations. (Denial reverses the sense of a proposition.)”.

Tractatus 4.442 is an example for a truth-table, that is a scheme for the logical analysis of the sentences p and q, a sign for the logical function of a sentence, or as Wittgenstein says, a “propositional sign”. Tractatus 4.442:

Thus e.g. “

p

q

T

T

T

F

T

T

T

F

F

F

T

“

Is a propositional sign.

The sign shows in the first row the sentences p and q. In the following rows we see in the first two columns the combination of the truth-possibilities true (T) and false (F) that are also called “truth-arguments”. The third column shows the truth-conditions of one combination of truth-possibilities. In the same number 4.442 of Tractatus Wittgenstein allows writing the third column as the row “(TTFT) (p,q)”. “Those truth-possibilities of its truth-arguments, which verify the proposition, I shall call its truth-grounds.” (Tractatus 5.101).

The truth-conditions of the above truth-table correspond to the truth-function “logical implication”. Wittgenstein sees the truth-conditions as the result of operations. Proceeding along the series of the truth-conditions of the two sentences p and q, we can repeat the operation of co-ordinating truth-conditions to the truth-possibilities. It is the same function we are repeatedly operating. It is the repeated operation that operates the same function that makes us go from one line to the next. Wittgenstein wants to analyse every series of sentences with the help of this repeated operation of truth-functions.

In Tractatus 5.101 Wittgenstein identifies 16 truth-functions. Number 6 in the count is the truth-function “not-p”. Number 7 in the count is “not-q”). In the following sentences of Tractatus Wittgenstein struggles to be able to present a kind of primal sign, a basic logical sign or concept that explains the logic of all possible sentences. The task to identify the one logical truth-function that constitutes the basis for all possible truth-functions of sentences takes Wittgenstein more than half of the whole Tractatus. In Tractatus 6 and 6.1 he presents the general form of the truth-function as successive negation:

“6 The general form of truth-function is: [p, ξ, N(ξ)]. This is the general form of proposition.

6.001 This says nothing else than that every proposition is the result of successive applications of the operation N’(ξ) to the elementary propositions.”

Russel complements Wittgenstein’s explanation of the symbol [p, ξ, N(ξ)]: “p stands for all atomic propositions. ξ stands for any set of propositions. N’(ξ) stands for the negation of all the propositions making up ξ” (Russel. 1989. “Introduction”. In Ludwig Wittgenstein. Logisch-philosophische Abhandlung. Tractatus logico-philosophicus, edited by Brian McGuinnes and Joachim Schulte. 258-287. Frankfurt am Main: Suhrkamp. 272). In Tractatus 5.101 Wittgenstein constructs sentences from the truth-functions of conjunction (using the logical operator “and”), and of disjunction (using the logical operator “or”)and derives from conjunction and disjunction the truth-functions logical implication (using the operator “if … then” in the function “if p then q”), and of equality (using the operator “=” as in the function “p = q”) and other 12 truth-functions based on the truth values true and false for the sentences p and q. With a two-valued logic for sentences, the truth-values of truth-functions can be used again as basic points for other truth-functions. For the process described by the symbol [p, ξ, N(ξ)] Wittgenstein makes use of Sheffer’s stroke. Sheffer has shown “that all truth-functions of a given set of propositions can be constructed out of the two functions “not-p or not-q” or “not-p and not-q” (Russel 270).

We know that around 1930 Wittgenstein abandoned this theory of the general form of sentences and the two-valued logic with the truth-possibilities true and false. In March 1928 Wittgenstein went to Vienna to listen to L. E. Brouwer (1881-1966). The mathematician from the Netherlands spoke on mathematics and logic to voice his criticism of the principle “tertium non datur” (Richter. Untersuchungen zur operativen Logik der Gegenwart. Freiburg: Karl Alber.1965. 42). The two-valued logic is an expression of the classical principle of the excluded third: “Tertium non datur”.

Around 1900 mathematicians still held true the axiom and believed that all problems of mathematics can be solved (Richter 1965: 43). Brouwer recognized that this axiom is the equivalent of the axiom of the excluded third and criticised the principle tertium non datur: The reason for the erroneous assumption that the axiom tertium non datur was true lies in the application of the logic of finite sets of numbers for the logic of infinite sets of numbers (ibid.). The finite numbers as a potential set of infinite numbers depend in the operative concept of mathematics on a rule for constructing them. The rule for constructing finite numbers is given in Tractatus 6.03. To obtain numbers we have to operate the function given in Tractatus 6.03 (Richter 1965: 42). Richter demonstrates that this kind of operation does not help, if we ask for the possibility of a set of infinite numbers, for example: Are odd numbers greater than 1 perfect numbers? Perfect numbers are natural numbers that equal the sum of their proper divisors. The number 6 is, for example, a perfect number because 6 = 1 + 2 + 3. The number 28 is also a perfect number: 28 = 1 + 2 + 4 + 7 + 14 (ibid.). To answer the question whether there are perfect odd numbers we would have to construct a potentially infinite sequence of single operations asking: Is the number 3 a perfect number? Is the number 5 a perfect number? ect. (Richter 1965: 43). We will obtain answers to these individual questions, but we do not have at our disposal some rule or algorithm for answering the question in general. The lack of this kind of rule or decision procedure for answering our question leads to the recognition of insoluble questions or questions that we cannot decide and Richter documents that in 1931 Gödel presented the general proof for the existence of insoluble problems in mathematics (ibid.). Wittgenstein accepted in 1930 Brouwer’s criticism of the axiom of the excluded third (Richter 1965: 47) and opens the a priori of the sense of the sentence and the speech-acts to the investigation of the great variety of language games (Richter 1965: 49).

The epistemological turn from the two-valued logic of logical truth to the criticism of the axiom of the excluded third and the acceptance of a third possibility that includes the truth-value “I do not know” gives rise to significant consequences for philosophical and theological argumentation. Tractatus 7, the last sentence of the Tractatus, says: “Whereof one cannot speak, thereof one must be silent.” Tractatus 6.522 clearly identifies the inexpressible that one cannot put into words and speak of: “There is indeed the inexpressible. This shows itself; it is the mystical.” We learned about the mystical from Tractatus 6. 44: “Not how the world is, is the mystical, but that it is.” Wittgenstein talks about the fact that the world exists and his reaction to this fact. We recognize that the consideration of the mystical is a central preoccupation of women, men and queer of many religions and theologies and equally of women, men and queer, who do not profess any religion or theological belief.

The turn from a two-valued logic to the criticism of the principle of the excluded third was important for speaking about themes like the mystical that the Tractatus thought one must be silent on. In his theological and philosophical discussions with Karl Rahner (1904-1984), Vladimir Richter learned from his Jesuit brother that it was fundamental to be able to theologize on the basis of a reflected logic for theological knowledge and insight (Richter, Vladimir. “Logik und Geheimnis.” In Philosophische Grundlagen, theologische Grundfragen, biblische Themen. Vol. 1 of Gott in Welt. Freiburg: Herder. 1964. 188-207. 189). Richter discovered in his investigations of a possible logic of the mystical that the logic of the mystical resembles the logic of the so-called insoluble problems of mathematics (ibid.). The logical structure is similar if we compare the sentences of theology and the sentences of mathematics, for which we do not have at our disposal a method with which we can decide exclusively with the truth-value true or the truth-value false. For example, we have in mathematics no procedure to decide the question whether all perfect numbers are even or whether there are perfect numbers greater than one that are odd numbers. The impossibility of deciding on the basis of a two-valued logic that would be able either to prove right or to prove wrong the sentence that there is an odd number greater than one that is a perfect number leads to a third possibility. This third possibility consists of a logic that accepts not being able to positively prove a sentence right or wrong and therefore turns to a logic of proving wrong the principle of the excluded third. This kind of logic would be capable of proving wrong the refutation of the truth-value true for theological sentences and accepts not being able to prove right the theological sentence in question. Today it is no longer a scandal to theologize as a Catholic Christian on the basis that accepts that sentences of religious beliefs such as expressed by words like “the mystical, creation or creator”, cannot positively be proven to be the case and cannot be attributed to the truth-value true or false of the two-valued logic of empirical science. Richter insists on the necessity of using this kind of logic in theology; theology needs to demonstrate its awareness of the difference between the refutation of the refutation and positive demonstrability (Richter 1964: 196).

Richter wanted theology to generally be able to not only demonstrate the logical coherence of its sentences with the help of the principle of non-contradiction, but he wanted to also develop some kind of formalized procedure for demonstrating that the theological sentences of a theological thesis do not contradict each other. At the same time, Richter insists that it is logically possible to show that the refutation of a theological sentence can be logically refuted. This formalized procedure Richter develops in the discussion with the dialogical interpretation of the intuitionist logic by Paul Lorenzen (1915-1994), who founded together with Wilhelm Kamlah (1905-1976) the philosophical School of Erlangen (Richter 1964: 197). Intuitionist logic is the philosophical term for a logic that accepts and operates on the basis of the refutation of the principle of the excluded third. Richter points at the dialogical beginning of Greek logic in dialectic and rhetoric and recalls the technique of disputation in medieval logic in order to defend Lorenzen’s dialogical logic that compares dialogue to a game between two parties, the proponent and the opponent (ibid.). The rules of the game leave room in the dialogue between the two truth-possibilities of the affirmation yes and the negation no, following a concept of negation that accepts the possibility that no decision is possible and therefore turns to the negation of the refutation and the negation of the affirmation as a third truth-possibility (Richter 1964: 206). It was very interesting for me to discover that this intensive discussion of the intuitionist logic of Lorenzen led Richter not only to the assurance of the logical legitimacy of faith-sentences, but also provoked an answer by Lorenzen concerning ethical and theological matters. I write about the correspondence between Lorenzen and Richter in my next post.