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Arbitrariness of Grammar in Wittgenstein's Transitory Period: Zuzana Fořtová


Even though the theme of the arbitrariness of grammar and its rules is discussed primarily in the connection with Wittgenstein's later philosophy, i.e. with the Philosophical Grammar and even with the Philosophical Investigations[1], the most important argumentative grounds for the thesis of the arbitrariness of grammar are offered by Wittgenstein already in the period of his return to philosophy and of the debates with the Vienna Circle, i.e. between 1929-32. In this paper, I would like to focus on the theme of the arbitrariness of grammar as we can find it in this period. Firstly, I shall discuss in some length Wittgenstein's essential arguments (from the Philosophical Remarks[2] and the 1930-32 Lectures[3]) in which he argues that grammatical rules are unjustifiable, necessary and non-superfluous, these characteristics of rules creating the grounds for the possibility of calling these rules arbitrary. And secondly, I will further investigate the arbitrariness of grammar with the aid of the distinction between calculus and its application from Wittgenstein's discussions with the Vienna Circle.[4]


1. Arguments against External Justification of Grammar

According to Medina[5] (Medina, p. 51), throughout 1929-1932 Wittgenstein developed three different and independent arguments designed to prove that grammatical rules are unjustifiable, primarily in the sense of the impossibility of justification or refutation of rules by reference to what these rules describe, i.e. reality. These arguments are: (1) a reductio ad absurdum, (2) a charge of circularity and (3) a regress argument.


(1)   A reductio ad absurdum

With reference to Wittgenstein´s Lectures (p. 47), Medina shows that, according to Witttgenstein, if rules were susceptible to justification, then they also could be susceptible to refutation, i.e. in case I can justify rules, I can also refute them (as wrong or superfluous), i.e. I can show that reality is otherwise. But Wittgenstein emphasizes the fact that we cannot refute rules with the aid of a description of reality, because “if we were able to describe a reality that supposedly contradicted the grammar of our language, then, by definition, that reality would not refute our grammar: the very act of our description would put that reality in harmony with grammar” (Medina, 51). In this description, Medina refers to the Lectures and PR, article 6: “in order to describe a reality in which grammar was otherwise I should have to use the very combinations which grammar forbids” (Lectures, p. 47), “[b]ut I cannot use language to get outside language” (PR, §6). To make the matter more perspicuous, the core of this reductio argument is as follows:

If rules can be justified (by reference to reality), then they can be also refuted (by the same reference).

But the conclusion (refutability of rules) is invalid since in being able to describe a reality which contradicts rules, we have to use these rules already and therefore the reality does not refute these rules.

The consequence (refutability of rules) is not valid and, thus, the presupposition (the possibility of justifying grammar) is not valid either. The argument is called reductio ad absurdum, because in order to describe reality which is different from what our rules suggested, I would already have to use these rules to express this fact of non-accordance.


(2)   Argument from Circularity

Knowing the reductio argument, the argument from circularity is easy to follow. In this case, Medina cites this place from the Philosophical Remarks: “Grammatical conventions cannot be justified by describing what is represented. Any such description already presupposes the grammatical rules” (PR, §7). Rules cannot be justified by what they describe (reality), because every description already presupposes the rules. As such, this argumentation can be conceived also as a reductio argument, because it again emphasizes the fact that in order to describe reality, I already have to employ grammatical rules. Circularity lies then in the fact that while I can describe reality by the aid of rules, I cannot, reversely, describe rules by reference to reality, “on pain of circularity” (Medina, 52). Thus, even though Medina speaks about this argument as about an argument from circularity, we can as well understand this argument as a reductio ad absurdum, because it shows the same thing as the first argumentation while being only enriched by the addition that we cannot take as valid the reverse case (rules as presupposing reality in their justification).


(3)   A Regress Argument

In this kind of argumentation, Wittgenstein employs his well-known reluctance towards meta-constructs[6]. We could suppose that we would be able to justify one grammar and its rules by reference to another, meta-grammar in which we could explain the rules of the first-level grammar. But, irrespective of the fact that Wittgenstein simply denies the existence of meta-grammars, this presupposition leads to a regress ad infinitum, since we can suppose that this meta-grammar is again justified by another, i.e. meta-meta-grammar. But, as Wittgenstein puts it in his Lectures, “grammar is not something higher, with another grammar beyond it” (Lectures, p. 87), i.e. “there is no such thing as a hierarchy of grammars” (Medina, p. 52).

To summarize Medina's interpretation, rules can be neither justified nor refuted by reference to reality, because we have to use rules to describe reality (and not conversely reality to describe rules - argument from circularity) and, therefore, reality cannot – if it is to be expressed – refute these rules (reductio ad absurdum). Plus, we can neither justify nor refute rules by some meta-grammar (regress argument). Even though I believe Medina is right in his summarizing of Wittgenstein's arguments, I also believe we can charge him with avoiding the difficulty of interpreting one essential argument from the Philosophical Remarks. We should focus on it because it shows new features of grammar, i.e. that grammar is necessary and non-superfluous. Let us now investigate this argument beginning with its full-length citation and continuing with its analysis:

 [1] If I could describe the point of grammatical conventions by saying they are made necessary by certain properties of the colours (say), then that would make the conventions superfluous, since in that case I would be able to say precisely that which the conventions exclude my saying.

[2] Conversely, if the conventions were necessary, i.e. if certain combinations of words had to be excluded as nonsensical, then for that very reason I cannot cite a property of colours that makes the conventions necessary, since it would then be conceivable that the colours should not have this property, and I could only express that by violating the conventions. (PR, 53)

Strictly speaking, the proof consists of two proofs, the first of them showing that rules are not superfluous, the other that rules are necessary. In the citation, I marked these two parts by numbers in square brackets (not to be found in the original text).

(1) Proof by contradiction manifesting that rules are not superfluous:
If I could describe rules with reference to reality then rules would be superfluous
I would be able to say precisely that which the rules exclude my saying.
But I cannot say this,
(1) rules cannot be described with reference to reality and (therefore) (2) rules are not superfluous.

(2) Proof manifesting that rules are necessary:
If rules were necessary then I cannot refer to reality as a basis which makes the rules necessary
in that case reality could also be otherwise which I could express only by violating rules.

In this second proof, even though Wittgenstein also speaks in an unreal conditional (if rules were necessary …) and thus we anticipate again a proof by contradiction, both premises are correct (i.e. rules are really necessary and I cannot refer to reality as a basis which makes them necessary), because the explanation (since it would then be conceivable ...)  is also correct, i.e. I cannot cite reality as a basis for my rules, because in that case I would have an immediate access toward reality which I could see also otherwise than it is in my description. But in that case, I would have to describe this reality only by violating my rules to which it would contradict. But then rules would be superfluous (because we would have unmediated access to reality) which is the opposite of the result of the first argument. For being in accordance with the first argument, we thus have to admit that we cannot violate (contradict) rules, because (from the first argument) they represent the only access to reality and in that sense are necessary (in transmitting this access).

To summarize, while the arguments Medina follows were designed to show that rules are not externally justifiable, in this complex argument Wittgenstein also tries to prove rules are (not only unjustifiable, but also) non-superfluous (argument 1) and necessary (argument 2). Thus, not only can't I justify grammar with its rules; but neither can I refer to reality as to something which explains/describes these rules.


2. Arbitrariness of Grammar (Calculus and Its Application)

From the arguments we have been focusing on, we have seen that according to Wittgenstein rules of grammar are (1) unjustificatory, (2) necessary and (3) non-superfluous. These three characteristics are just other words for the arbitrariness of grammatical rules, since the arbitrariness of rules lies primarily in the fact that they cannot be either justified or explained or described from without the language; and in case we cannot justify grammar by reference to a non-linguistic reality, than we have to use this grammar in the attempt to elucidate it on its own and in that sense, grammar is necessary and irreducible. The question I shall deal with in this section is how explicitly Wittgenstein further develops the thesis of the arbitrariness of grammar, employing an important distinction between calculus and its application.

While interpreting Wittgenstein's conception of the arbitrariness of grammar in the Vienna-Circle period, Coffa[7] is confused by the fact that we can find theses concerning this arbitrariness that (at least seemingly) contradict each other. On the one side, “Wittgenstein often talked of rules of grammar in a conventionalist tone of voice, as if to change them were no more difficult than to stop playing chess and start playing checkers or to replace a meterstick by a yardstick” (Coffa, p. 270-271, my emphasis). There, it seems that there are no reasons whatsoever on the basis of which we should choose one set of rules and not another (expect for our taste or personal preference). On the other side, Wittgenstein emphasizes exactly the opposite, saying that it is almost unimaginable that (and how) we could change the rules we actually employ. He even compares the difficulty of this vision of different sets of rules with the vision that we cease to fear fire or to believe in inductive reasoning.[8] Therefore, on the one hand, it seems that for Wittgenstein rules are easily changeable, because there is no reason why to prefer one set of rules, while on the other hand he also emphasizes the fact that rules cannot be so easily changed.[9]

Coffa not only points at this inconsistency, but also offers a solution explaining why this inconsistency is only illusive, his solution lying in the fact that in speaking about changeability and unchangeability of rules and grammar, Wittgenstein is employing a different concept of grammar. Specifically stated, there is a difference between “grammar regarded as a calculus” and between “an applied grammar” (Coffa, p. 271). While in the case of calculus, we can, Coffa asserts, quite easily imagine the changeability, in the case of applied calculus, the changeability is not so easily attainable since we are deeply accustomed to the usage of one calculus (in our case – one grammar with its rules). In order to understand this, we have to elucidate Wittgenstein's essential notion of calculus and its application whereas our main attention will be devoted to the notes from the debates with the Vienna Circle.

Calculus” is for Wittgenstein in this period a general notion for all types of structures which are governed by some rules, i.e. the notion of “calculus” can be applied to mathematical or geometrical systems (WVC, 149), the game of chess (WVC, 133) or to our language (in which case another word for “calculus” is “grammar”) (WVC, 168). In his discussion of mathematical formalism, Wittgenstein stresses exactly the fact that that in which formalism is right is that “every syntax can be conceived of as a system of rules of a game”[10], i.e. both game (e.g. chess) and language have something in common which is, as Wittgenstein adds at the same place, that they are both calculi. And right thereafter, Wittgenstein further adds that these calculi are arbitrary (WVC, 103).[11] Wittgenstein then sees as an essential question “what it is that distinguishes the syntax of a language from the game of chess”, while the answer is that “it is its [i.e. of syntax] application and nothing else” (WVC, p. 104). Thus, for Wittgenstein, there is an essential difference between calculi – some are here for fun (chess), some are presumably not only for fun, but can be quite easily changed if necessary, e.g. if some contradictions within them occur (mathematics, geometry) and some are used in everyday life (language) and cannot be so easily changed. The essential point is that from the calculus itself, i.e. somehow a priori (without taking into account its usage), I cannot say whether it is only a game or whether it is “serious”, “the calculus itself implies no relation either to seriousness or to fun”:[12] “Detached from its application and considered by itself it [language] is a game just like chess”. On the other hand, if we used chess for predicting the moves in wars (WVC, p.  104, 163), it would not be an innocent game anymore.



To summarize, Coffa is right in distinguishing two notions of grammar, i.e. “grammar regarded as a calculus” and “applied grammar”. In the last section I tried to show that this distinction is grounded in Wittgenstein's notions of calculi and their applications, applications being that which distinguishes calculi which would be all the same if taken a priori, i.e. calculi from which themselves I cannot decide whether they serve for fun or are a serious matter. But still, it is important to bear in mind that the arguments for arbitrariness focused on in the first section hold both for grammar as a calculus and for applied grammar, because according to Wittgenstein, even applied grammar is unjustifiable, necessary and non-superfluous. The only thing that the distinction “calculus-application” tells us about the grammar of language is that it is not a mere game, i.e. it is serious. But still the question remains of why we cannot change grammar of our language easily, i.e. why exactly language is not only a game, why it is “serious”. To find the answer to this question, I believe we have to wait for the later Wittgenstein who will tell us that there are at least some “general facts of nature”[13] which are expressed by our language.

[1]              e.g. Baker, G. P. and Hacker P. M. S., Wittgenstein - Rules, Grammar and Necessity, Cambridge University Press, Cambridge, 1991, Forster, Michael N., Wittgenstein on the Arbitrariness of Grammar, Princeton University Press, 2004, Princeton, especially the chapter The Sense in Which Grammar is Arbitrary, p. 21-65, Anscombe, G. E. M., From Parmenides to Wittgenstein, Blackwell, Oxford, 1981 etc.

[2]              Wittgenstein, Ludwig, Philosophical Remarks, Blackwell, Oxford, 1975; thereafter PR

[3]              Witthenstein, Ludwig, Lectures, Cambridge, 1920-23, University of Chicago Press, 1989

[4]              Ludwig Wittgenstein and the Vienna Circle, Conversations Recorded by Friedrich Waismann, Basil Blackwell, 1983,  thereafter WVC

[5]              Medina, Chosé, The Unity of Wittgenstein's Philosophy: Necessity, Intelligibility, and Normativity, State University of New York Press, Albany, 2002; thereafter Medina

[6]              In WVC this reluctance is expressed especially in the case of meta-mathematics: p. 121, 133, 136.

[7]              Coffa, J. Alberto, The Semantic Tradition from Kant to Carnap – to the Vienna Station, Cambridge University Press, New York, 1993; thereafter Coffa

[8]              “At other times Wittgenstein talked as if it were difficult to imagine circumstances under which one might be inclined to adopt a grammatical system different from the one we actually use (e.g., Lectures, 1930-32, p. 49; Philosophical Grammar, p. 110). The preference for a grammatical system, he appears to say on these occasions, is no less justified than the fear of fire or our belief in inductive reasoning.,” Coffa, 271

[9]              Although he does not offer us an explanation of why the change of rules should be so difficult, we can respond that this explanation is as unnecessary as in the case of fear of fire, i.e. we simply do not need grounds for this fear as well as for the usage of our rules. The grounds and explanation of this difficulty of change began to be interesting for Wittgenstein in his later philosophy when he became to be concerned with customs, institutions etc.

[10]             WVC, p. 103

[11]             “I have been thinking about what Weil may mean when he says that a formalist conceives of the axioms of mathematics as like chess rules. I want to say that not only the  axioms of mathematics but all syntax is arbitrary.,” WVC, p. 103

[12]             “For we call things serious if we apply the results of the calculus to everyday life. I apply e.g. the calculation 8x7=56 thousands of times, and this is why we take it seriously. But, at bottom, this multiplication is not the least bit different from one I merely do for fun. The difference does not lie in the calculation itself, and that is why you cannot tell by looking at the calculus whether it is a serious business or merely serves to entertain us.[my emphasis] Hence, I cannot say: If a calculus amuses me, it is a game, but only: If I can take a calculus to be something that amuses me, then it is a game. The calculus itself implies no relation either to seriousness or to fun.,” WVC, 170

[13]            “Wittgenstein appears to suggest that what ´corresponds to´ our concept of colour is not (as we might have wished to say) a distinct feature of things, but rather the ´very general fact of nature´ that colour and shape are independent”,  in:  Anscombe, G. E. M., From Parmenides to Wittgenstein, Blackwell, Oxford, 1981, p. 113.  In a similar sense Hacker also interprets these “general facts of nature”: “But we apply our grammars, use our languages against a context of normality conditions consisting of very general regularities of nature. Like a legal system (Z, §§129, 350) our concepts can be said to rest upon such normality conditions, not in the sense of being made true or correct by them, but in the sense of having a point only in such contexts.“, Hacker, P. M. S., Insight and Illusion, Themes in the Philosophy of Wittgenstein, Revised edition, Clarendon Press, Oxford, 1986, p. 190

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