## Abstract

The main thesis of the paper is that self-reference is the fundamental principle of language, which enables the expression of very general issues concerning the essence of the world, in particular it enables naming, paradoxically, an infinity by finitely many expressions. Any endeavor to dispose of self-reference makes language lifeless. From this perspective any description of the essence of language which tends to be sufficiently general must also encompass itself. Wittgenstein’s Picture Theory motivated by the saying-showing distinction is here treated as an example of such general description. But in a negative sense: Wittgenstein’s attempt to determine and express the boundary between meaningful expressions and senselessness is so general, i.e. self-referential, that it becomes itself senseless.

## Table of contents

On the basis of reactions that Wittgenstein received from Frege and Russell after they had read the Tractatus manuscript he realized that his inspirators did not recognize Tractatus’s main idea, which according to Wittgenstein also should be counted as the cardinal problem of philosophy: “The main point is the theory of what can be expressed (gesagt) by propositions – i.e. by language – (and, which comes to the same, what can be thought) and what can not be expressed by propositions, but only shown (gezeigt).” (Letter to Russell 19.8.19, Wittgenstein 1974, p. 71)

According to the Tractatus’s picture theory the essence of language is depicting reality through the general propositional form. A proposition is a picture of how things are. The picture and what is depicted must have the same logico-mathematical multiplicity of distinct elements of which they are composed. The picture and depicted reality share “the pictorial form” that makes it possible to structure elements within the picture for the purposes of mirroring reality in a conventionally determined way. The pictorial form as a possibility of that structure can have different kinds of representation yet their base is still one logical form.

Propositional representation is possible due to a logical isomorphism between the combination of propositional signs and the possible configuration of things in a depicted situation. The proposition is false if it depicts a non-existing combination of things nevertheless it remains meaningful when this combination of things is possible.

All meaningful propositions are truth-functions of elementary propositions. These elementary propositions are concatenations of logically proper names which are further unanalysable and their meanings are simple objects they go proxy for. The sense of an elementary proposition is one possible combination of these objects, i.e. the state of affairs it depicts. One elementary proposition cannot contain sense of any other, so one cannot entail nor contradict another. An assignment of a truth-value to one elementary proposition is logically independent of an assignment of a true-value to any other. A false elementary proposition is not the negation of a true one, it depicts the state of affairs that it does not obtain.

The idea that language and reality share a definite pictorial form presupposes the metaphysical claim that both an elementary proposition and the possible state of affairs it depicts consist of ultimate elements. Although one of its motivations was to dissolve problems concerning the relationship between elements and complexes, which Russell left unsolved, the picture theory collapses just with this atomistic metaphysics. This collapse is actuated by two of Wittgenstein’s tendencies which are interconnected: first, to describe the essence and limits of language, and second, to keep this effort as general as possible. That means that, by describing the essence and limits of language also cover this description because it is a part of the examined language.

Thus, on one hand, Wittgenstein generally analyses an elementary proposition as including two kinds of relation: the relation of names, i.e. its internal structure, and the pictorial relation correlating its names and objects in reality etc. On the other hand, when Wittgenstein expresses the essence of language this way and does it in terms of this language alone, he cannot adduce any particular example of elementary proposition without giving up generality: “If I cannot say a priori what elementary propositions there are, then the attempt to do so must lead to obvious nonsense.“ (TLP 5.5571, Wittgenstein 1961)

The description of an element is always some descriptive complex. No matter that we “talk” about a name or about an object, because language is a part of reality and simultaneously to say that “something is real” is a language expression. The generality of Wittgenstein’s analysis is embedded in the self-referential power of language and seems to lead to paradox and senselessness. This is the reason why in the Tractatus Wittgenstein intends to show that posing problems of philosophy, i.e. general problems, is a consequence of misunderstanding the logic of our language.

The aim of Tractatus, which is drawing a limit to the expression of thought or, in other words, to determine a boundary between what can be said clearly and what can be only shown because attempts to express it lead to nonsense, can be interpreted as an endeavor to protect language from what we will call “the senseless flexion” that emerges when philosophical problems are posed. The flexion strengthens with the generality of expressed problems and finally becomes self-reflexion. The main point of this interpretation is that the generality of an expressed problem, the self-reference of the expression and senselessness are mutually connected. This connection is the breeding ground of which logico-semantic paradoxes are born.

The Wittgenstein’s saying-showing distinction at the same time proscribes general propositions about the world and the realm of value as well as propositions about the essence of propositional representation. Unlike bipolar meaningful propositions and their limiting cases, tautologies and contradictions, these propositions concerning transcendental preconditions of the world and its symbolic representation are just “pseudo-propositions”. They are senseless attempts to say things that could not be otherwise.

Thus, on one hand, Tractatus’s list of ineffabilia includes ethical and aesthetical values, the logic of facts, the atomistic ontology, that there are laws of nature, that the world is my world, that there is no soul, the meaning of life, everything that is higher, the mystical etc. On the other hand, we should pass over in silence that propositions and what they depict share the pictorial form, the logical form or the logico-mathematical multiplicity, that there are the meaning of signs and the sense of propositions, that something falls under a formal concept such as a name or a number, that propositions are linked together according to the rules of logical syntax and inferences, etc. In other words, we should pass over in silence everything that is stated about the Picture Theory above.

Wittgenstein’s prohibition of logico-syntactic and semantic categories was supposed to avoid difficulties encountered by Frege and Russell. Frege’s riddle of “concept of “concept””, by which we try to name the unsaturated entity, i.e. concept, although names can only refer to something which is saturated, i.e. objects, as well as Liar-like paradoxes, whose structure can be demonstrated on that of set-theoretic paradoxes, i.e. R {X | X X} ↔ R R ↔ RR, are all based on circularity and self-referential or “impredicative” definition. Russell tried to dissolve these paradoxes by means of his theory of types which was motivated by the so called “vicious-circle principle“: “Whatever involves all of a collection must not be one of the collection.” (Whitehead and Russell 1910, p. 37)

According to Wittgenstein, Russell’s theory of types is superfluous and must be done away with by a proper theory of symbolism which should show that “what seem to be different kinds of things are symbolized by different kinds of symbols which cannot possibly be substituted in one another’s place”. (Letter to Russell 1.1913, Wittgenstein 1974, pp. 19-20) From the perspective of the saying-showing distinction it is not possible to establish rules of logical syntax by mentioning the meaning of a sign as Russell does. (See TLP 3.33-3.334, Wittgenstein 1961) The sign of a propositional function already contains the “prototype” of its argument and thus a function cannot contain itself. In case some function F(fx) could be its own argument “there would be a proposition ‘F(F(fx))’, in which the outer function F and the inner function F must have different meanings, since the inner one has the form φ(fx) and the outer one has the form ψ(φ(fx)). Only the letter ‘F’ is common to the two functions, but the letter by itself signifies nothing. This immediately becomes clear if instead of ‘F(Fu)’ we write ‘(φ):F(φu). ψu=Fu’. That disposes of Russell’s paradox.” (TLP 3.333, Wittgenstein 1961)

But is really such “proper theory of symbolism” that disposes of self-reference and paradoxes arising of it conceivable? The senseless pronouncements of Tractatus itself prove that such theory is not conceivable. Tractatus is an example of a general theory of language, i.e. the picturing of theory of picturing, which flexes to the self-picturing and at the same time tries to protect itself from this senseless-flexion. Wittgenstein’s propositions ought to be only elucidatory and who understands them finally recognizes them as senseless, “when he has climbed out through them, on them, over them. (He must so to speak throw away the ladder, after he has climbed up on it.) He must transcend these propositions, and then he will see the world aright”. (TLP 6.54, Wittgenstein 1961)

Anyway, is the picture of the world devoid of this kind of propositions the right picture of the world? Indeed, elimination of the senseless flexion from a language entails the loss of its enchanting ability of expressing an infinity by finitely many expressions. The self-reference should be conceived as an essence of language, as something not to be disposed of. Various semantic paradoxes, such as Grelling's, Berry's, Richard's, Curry's etc., can be then perceived not as a menace but as a celebration of the miracle of language.

Recently Stephen Yablo has tried to prove that “self-reference is neither necessary nor sufficient for Liar-like paradox“. He has propounded the paradox that can be paraphrased as follows: Every proposition in an infinite sequence of propositions says that all subsequent propositions are untrue. (Yablo 1993) Paradox arises because it is not possible to decide the true-value of any proposition in the sequence. But is this paradox really without self-reference? One can eliminate self-reference only if one can distinguish between “all subsequent propositions” and “all propositions” within an infinite sequence of propositions. Until we are not able to determine a rule for such infinity comparison, i.e. express that rule by denumerable words, infinitely many propositions still collapse into one proposition that says about itself that is untrue. For it is just self-reference that spreads an infinity.

The ability of saying anything, or more generally, picturing anything is based on the ability of self-picturing. Whatever can be counted as language which enables to picture anything including this picturing itself. The “creativity of language”, that allows us to compositionally form an infinite number of meaningful propositions with a finite number of marks and noises, is rooted in the self-reference, among others because the rules that determine which strings of marks and noises are linguistic expressions as well as the rules of syntax and compositionality are expressible in that language.

Generality means being able to include or subsume itself. That is how a paradox arises because an infinity mirrors in something particular and vice versa. What is vexing about paradoxes is not the circularity or our inability to decide the true-values, but our insight into an infinity. We fear we would name the infinity and thus we would become a mirror depicting another mirror. We would see only the mirror instead of our picture in the mirror. The fear of circulus vitiosus has roots in horror infiniti.

## Literature

- Whitehead, Alfred North and Russell, Bertrand 1910 Principia Mathematica, 1. vol., Cambridge: Cambridge University Press.
- Wittgenstein, Ludwig 1961 Tractatus Logico-Philosophicus, London: Routledge and Kegan Paul.
- ––– 1974 Letters to Russell, Keynes and Moore, Oxford: Basil Blackwell.
- Yablo, Stephen 1993 “Paradox without self-reference”, Analysis 53, 251-252.

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