"Plato loves Socrates" says that Plato loves Socrates

To analyse those forms of propositions in psychology such as 'A believes that p is the case' and 'A has the thought p', Wittgenstein says at 5.542 of Tractatus that:

It is clear, however, that 'A believes that p', 'A has the thought p', and 'A says p' are of the form '"p" says p': and this does not involve a correlation of a fact with an object, but rather the correlation of facts by means of the correlation of their objects.

Wittgenstein points exactly against the superficial thought of such forms of propositions - it looks as if the proposition p stood in some kind of relation to an object A (5.541).

First of all, how to understand / explain '"p" says p'? Let us consider a proposition of the form '"q" says p':

"Wittgenstein was taller than Russell" says that Socrates was fatter than Plato.

Everyone, as long as he / she understands this English sentence, can tell immediately without any thoughtful logical analysis that it is a false statement. Furthermore, he / she can also tell immediately that the following proposition is true (no matter whether it is the case that Wittgenstein was really taller than Russell):

"Wittgenstein was taller than Russell" says that Russell was shorter than Wittgenstein.

As Wittgenstein says at 3.14, propositional sign (such as "q") is a fact. Logically, to justify if such propositions of the form '"q" says p' is true, we can examine whether it is the case that q if and only if p, which is just a truth-operation of the p and q as Wittgenstein asserts at 5.54. Besides, to perceive a complex means to perceive that its constituents are related to one another in such and such a way (5.5423). Of course not all propositions of the form '"q" says p' can be understood in this way. Clearly, the following statement is not true:

"If it is raining, then it is raining" says that if it is hot, then it is hot.

Although it is of the form '"q" says p', where q if and only if p, both p and q are in fact tautologies. As Wittgenstein says at 4.461, tautologies say nothing. In the first place, "q" says nothing, so that we cannot even assert the following:

"If it is raining, then it is raining" says that if it is raining, then it is raining.

We cannot assert the following statement either:

"Wattginstein was taller than Sucrotis" says that Sucrotis was shorter than Wattginstein.

As long as the symbols "Wattginstein" and "Sucrotis" do not signify anything, "Wattginstein was taller than Sucrotis" is nonsensical. We cannot compare nonsensical propositions in the first place.

Then how to understand / explain / justify 'A says p'? We have to examine what A said on the objects of the fact p (if A has said something about them) to determine the correlation of this fact (what A said) and the fact p, as Wittgenstein asserts at 5.542. Logically, "p" should say something in the first place, and we compare the fact p with each of the facts represented by the finite set of sensical statements of A. A composite soul would no longer be a soul (5.5421). What about if A says nonsense? We can just tell that A says nonsense, but we cannot have a statement like:

A says that Sucrotis Wattginstein was tham shorter.

It is nothing but just another piece of nonsense! Hence, Wittgenstein says at 5.5422 that

The correct explanation of the form of the proposition, 'A makes the judgement p', must show that it is impossible for a judgement to be a piece of nonsense.

Thus, if I tell you, "I make the judgement that Sucrotis was shorter than Wattginstein," and if I do not tell you a piece of nonsense, it means that I really know two guys called Sucrotis and Wattginstein, and maybe Sucrotis was in fact taller than Wattginstein so that my judgement may be wrong, but at least my judgement is not a piece of nonsense. Besides, propositions occur in such propositions still as bases of truth-operations as Wittgenstein asserts at 5.54.

Nevertheless, Russell says in his introduction to Tractatus that

This problem is simply one of a relation of two facts, namely, the relation between the series of words used by the believer and the fact which makes these words true or false.

But on the other hand, he concludes that the proposition does not occur at all in the same sense in which it occurs in a truth-function. I think that is why Wittgenstein believed that Russell did not really understand the Tractatus.

Anyway, is the heading "proposition" a good example? Is "Plato loves Socrates" really a sensical proposition (it is an example in Russell's introduction to Tractatus)? Is "love" what we cannot speak about? Must we pass "love" over in silence (as the conclusion at the end of Tractatus)? Er... I have such a thought, but my wife doesn't think so!

Identity in Tractatus

In Tractatus, identity of object Wittgenstein expresses by identity of sign, and not by using a sign for identity (5.53). He speaks roughly that (5.5303):

To say of two things that they are identical is nonsense, and to say of one thing that it is identical with itself is to say nothing at all.

Then he also says at 5.535 that Russell's "Axiom of Infinity", which says that there are infinitely many objects (at 4.1272, Wittgenstein says one even cannot say so), would express itself in language through the existence of infinitely many names with different meanings. In fact, as Russell says in his introduction to Tractatus, the rejection of identity removes one method of speaking of the totality of things.

To Wittgenstein, tautologies also say nothing at all (4.461), although they are not nonsensical (4.46211). Of course, everyone absolutely agrees that tautologies like "if it is raining, then it is raining" really say nothing at all. However, the "problem" of human beings (or of the world) is that the world is so complicated that we cannot determine many tautologies at first sight.

Identity is a relation in Mathematics. There are occasions at which one object may have more than one name / sign - signs are used to refer to the descriptions of some objects, and eventually it is found (proved) that these descriptions have the same reference. In the real world, we might call the murderer of some case of murder X, and eventually we found that he was the man called A: A = X. The introduction of the identity-sign simplifies our deduction.

Negative & Positive Propositions in Tractatus

As Wittgenstein says at the very beginning of Tractatus that the world is all that is the case. What is the case is a fact. For the totality of facts determines what is the case, and also whatever is not the case (1.12) - if we have the collection of all facts, then we can tell what is not in this collection. Maybe we can call what is not the case a negative fact. A negative fact is just a fact that tells you something that is not the case.

If p is the sign of a proposition that asserts something is the case, then the sign of the corresponding negative proposition can be constructed (by truth-operations) as ~p. However (5.5151), it is also possible to express the negative proposition by means of a negative fact: p is not the case. Functions f(x) like "x is the case", "x is not the case", "x is true", and "x is false" are not truth-functions (in the sense of Tractatus, e.g., 5). But really even in this case the negative proposition is constructed by an indirect use of the positive (5.5151).

Consider a simple example with the notation of Mathematics: 0 > 1. We can simply say that "0 > 1" is not the case. In fact, this negative proposition can also be expressed by 0 ≦ 1. We could define first the sign > for all real numbers, and then just define for any real numbers that a ≦ b whenever "a > b" is not the case if we want such a sign ≦ for the sake of simplicity. Such a sign ≦ is in fact not necessary. Nevertheless, it is clear that "a > b" presupposes the existence of "a ≦ b" and vice versa - if we know when "a" stands to "b" in a certain relation, then we can tell when "a" does not stand to "b" in that relation.

As Wittgenstein concludes at 5.5151, the positive proposition necessarily presupposes the existence of the negative proposition and vice versa.

The General Propositional Form in Wittgenstein's Tractatus

As it was mentioned in the last article that we use the perceptible sign of a proposition as a projection of a possible situation (3.11) such as spatial relations. Wittgenstein emphasizes at 3.13 that:

A proposition, therefore, does not actually contain its sense, but does contain the possibility of expressing it. ("The content of a proposition" means the content of a proposition that has sense.) A proposition contains the form, but not the content, of its sense.

In addition, the general propositional form is the essence of a proposition (5.471). For all that are possible in logic, Wittgenstein says at 5.473 and 5.4733 that

If a sign is possible, then it is also capable of signifying. Whatever is possible in logic is also permitted. Any possible proposition is legitimately constructed.

He also gives the sentence "Socrates is identical" as an example of a possible proposition. This proposition is, therefore, not only legitimately constructed and permitted, but also capable of signifying. It has no sense just because we have failed to give a meaning to the sign "identical", even if we think that we have done so (5.4733). It is we who failed to use the perceptible sign of this proposition as a projection of a possible situation (3.11).

Relations in Wittgenstein's Tractatus

Wittgenstein says at 3.1432 of Tractatus that

Instead of, 'The complex sign "aRb" says that a stands to b in the relation R' we ought to put, 'That "a" stands to "b" in a certain relation says that aRb.'

It seems scarcely comprehensible at first sight, especially for those who are familiar with set theory. In mathematics "aRb" really says that a stands to b in the relation R. It is because the world of mathematics is the totality of "such" propositions, while the (real) world is all that is the case as Wittgenstein says at the very beginning of Tractatus. We use the perceptible sign of a proposition (spoken or written, etc.) as a projection of a possible situation (3.11). The distinction is more clear if we consider spatial relation as in 3.1431. All because of such projection, we can say, for example, geometry is a study of spatial relations.

Nevertheless, no matter relations in mathematics or those in Tractatus, they should be well-defined, in the sense that we can tell at the same time what objects are in such relations (which pairs of objects are / are not in such binary relations), and assigns truth values to the objects (to the pairs of objects for binary relations).

Therefore Wittgenstein says at 5.42 that Frege's and Russell's "primitive signs" of logic such as ∨ (disjunction) and → (implication) are not signs for relations. In the first place, such logical signs did not give truth values. A proposition like p→q just tells you something (if p then q), but does not tell you that something (some relation) is true. They belongs to syntax. However, we can actually define relations on all propositions such that, for example, pRq if it is the case that p→q. These relations belongs to semantics.

Besides, Wittgenstein also says at 5.42 that Frege's and Russell's "primitive signs" of logic are not even primitive signs. Primitive signs are names that cannot be analysed further by any definition (3.26), while Frege and Russell even tried to define these logical signs. In fact, "well-definedness" is also a requirement of logical signs in Tractatus (5.46). If a sign is not primitive, we should be able to analyse it further by definition at the same time for all combinations with other well-defined signs (including brackets). If a sign was primitive, we should have introduced (primitive signs cannot be defined) the sense of all combinations with other signs. Therefore, there are no primitive logical sign in Tractatus. The real general primitive signs are the most general form of their combinations (5.46). At 5.461, Wittgenstein further comments that such pseudo-relations of logic need brackets, which is an indication that they are not primitive signs. For example, (p→q)→r and p→(q→r) are 2 different propositions, in which brackets are necessary. More precisely, it is an indication that they alone are "sometimes" even not logical signs, instead of an indication that the logical signs are not primitive. In fact, we can always write (p→q) instead of p→q in Frege's and Russell's notation. In this case, the pair of brackets and → form the logical sign of implication. However, is such a logical sign primitive?

Jan Łukasiewicz (1878 – 1956) has introduced a bracketless notation, in which, for example, Cpq is written instead of p→q. In that case, (p→q)→r and p→(q→r) will be expressed as CCpqr and CpCqr respectively. C is not a sign of relation yet because it does not give any truth value. Nevertheless, because of 5.461, we still wonder if C is a primitive logical sign. It depends.

Many systems of classical logic developed after Wittgenstein's Tractatus really introduced, but not defined, two (primitive) logical signs such as the negation and implication, and then defined all other logical signs such as disjunction, conjunction and biconditional. The syntax of the propositions (formulas) were studied.

Comparing (p→q) with Cpq in Łukasiewicz's notation, we can see immediately that the "aggregation" done by brackets is done by the rule for reading an expression containing the sign C. No matter which notation is used, the logical signs are just used to clarify the meaning. They belongs to syntax. As Wittgenstein concludes at 5.461, signs for logical operations (including those of Łukasiewicz) are just punctuation-marks.

Functions and Operations in Wittgenstein's Tractatus

Functions in Wittgenstein's Tractatus Logico-Philosophicus do not have exactly the same meaning as those in modern Mathematics. From its context in Tractatus, a function needs not have a well-defined domain (the set of arguments at which the function is well-defined) as in Mathematics, as long as it is logico-syntactical.

If all propositions form a set (the general form is given in 6 of Tractatus), then operations in Tractatus have the same meaning as those in Mathematics -- they are just functions (in the mathematical sense) of one or more arguments (propositions) to the set of all propositions. They are well-defined on all propositions.

As states in 5.251, a function cannot be its own argument because it is not logico-syntactical (3.333), whereas an operation can take one of its own results as its base because an operation is well-defined on all propositions as its bases and produces a proposition as its result.

If all propositions form a set, then what is the meaning of a function of Wittgenstein's with these propositions as arguments? Such a Wittgenstein's function is in fact the value of a mathematical function. In other words, if we denote the function with one argument (for example) by F in Mathematics, then Wittgenstein's function is just F(p) for some proposition p. Wittgenstein calls such F an operation. As states in 5.234, truth-functions of elementary propositions are results of operations with elementary propositions as bases. In Mathematics, we can have a composite function - a function of a function. That's why operations and functions must not be confused with each other (5.25); and a function cannot be its own argument, whereas an operation can take one of its own results as its base.

(Written on 22 September 2008; revised on 12 May 2009)

Wittgenstein's Tractatus Logico-Philosophicus & Russell's Paradox

"No proposition can make a statement about itself, because a propositional sign cannot be contained in itself," Wittgenstein says at 3.332 of Tractatus Logico-Philosophicus (the decimal figures hereafter refers to the numbers in the Tractatus). He also "disposes" Russell's paradox at 3.333.

Let us consider the following proposition p:

This statement is false.

Clearly, if p is true, then from the content of p, p is false. On the contrary, if p is false, then what p says is true, so that p is true. It is the liar paradox.

At first glance, the problem of the liar paradox is due to self-reference of the proposition. Let us consider a more general version of the paradox. Let P and Q denote the following proposition respectively:

Q is true; P is false.

We see that the paradoxes arise from the actually meaningless definitions of the objects (that is, the propositions) concerned. To define an object, we must use already well-defined terms. A self-referential proposition is meaningless because it is defined by itself. For the last version of the paradox, both P and Q are defined by what are being defined! Let us return to the self-referential liar paradox and rewrite p as

p is false.

Consider a function F(q):

q is false.

Then F(p) is the proposition:

p is false.

Besides, F(F(p)) is the proposition:

F(p) is false.

Here, as mentioned in 3.333, p, F(p) and F(F(p)) are in fact different propositions, and they should not be signified by the same sign such as p (as what liar paradox does), as states in 3.325:

In order to avoid such errors we must make use of a sign-language that excludes them by not using the same sign for different symbols and by not using in a superficially similar way signs that have different modes of signification: that is to say, a sign-language that is governed by logical grammar--by logical syntax. (The conceptual notation of Frege and Russell is such a language, though, it is true, it fails to exclude all mistakes.)

Wittgenstein does not mention the liar paradox but that of Russell. Russell's paradox is a set-theoretical paradox that asks whether the set R defined by

R = {x: x is not an element of x}

is an element of itself. If we use f(x) to denote the proposition

x is not an element of x,

then

R = {x: f(x)}.

However, is the definition of R meaningful? If it was, it had to be able to tell if any given object was an element of R. In particular, could it tell if the object R (if it could be defined) itself was an element? In other words, is f(R) true (or false)? Actually, if R was well-defined, f(R) could be rewritten as f({x: f(x)}), which was also a meaningless self-referential proposition (covered by the definition)!

Russell's solution to his paradox was his theory of types. He arranged all propositions into a hierarchy. The lowest level of the hierarchy consisted of propositions about individuals, not sets. The next lowest level consisted of propositions about sets of individuals. The next lowest level consisted of propositions about sets of sets of individuals, and so on. Thereby the definition (one of these "valid" propositions) of any set only referred to objects of the same "type" (at the same level).

It seems that Russell's theory of types is just an ad hoc solution to the problem of self-reference of his paradox (3.332). However, we see that the rules in his theory are not just syntactic, but semantic - he had to mention the meaning of signs when establishing the rules for them (3.331). On the other hand, Wittgenstein provided a syntactic solution to these paradoxes. More precisely, Wittgenstein did not provide any solution (because a solution is more or less ad hoc), but it is just that there is no such paradox in the system of his Tractatus.