Wittgenstein on the Arbitrariness of Grammar

Chapter 1
WITTGENSTEIN'S CONCEPTION OF GRAMMAR
We should begin by considering what Wittgenstein means by "grammar." For, although, as we shall see later, he himself sometimes in fact implies otherwise, he at least seems to employ this word as a term of art with a meaning which bears only a rather remote resemblance to that which it has in everyday usage.
Wittgenstein's most basic conception of grammar is that it consists in rules which govern the use of words and which thereby constitute meanings or concepts.1 Thus, he identifies grammar in general with the "rules for use of a word" (PG, I, #133; cf. BT, p. 136); or to cite a more specific example, he says of mathematics, which he understands to be an important part of grammar, that "in mathematics we are convinced of grammatical propositions; so the expression, the result, of our being convinced is that we accept a rule" (RFM, III, #26). And since, famously, he believes that a word's use may (generally) be equated with its meaning, he holds that the rules for use of words which make up grammar "determine meaning (constitute it)" (PG, I, #133), that "the meaning of a sign lies . . . in the rules in accordance with which it is used/in the rules which prescribe its use" (BT, p. 84); or to cite a more specific example, the mathematical part of grammar again, he says that "mathematics forms concepts" (RFM, VII, #67).2
Wittgenstein maintains, in an important and persistent analogy, that "grammar . . . has somewhat the same relation to the language as . . . the rules of a game have to the game" (PG, I, #23; cf. WWK, pp. 103-5; LC, pp. 48 ff.; BT, pp. 138-39, 168; WLC, pp. 3-4; OC, #95). Hence in his notorious characterization of linguistic practices as "language-games," grammar plays the role of the rules which govern these "games" in contrast to the moves that are made within them.3 To pursue some of the more central implications of this analogy:
(1) Just as the rules of a game constitute the game and first make possible the moves which occur within it, likewise grammar constitutes an area of language and first makes possible the linguistic moves which occur within it.4
(2) More specifically, just as in a game such as chess the rules prescribe or permit certain moves and proscribe others for the pieces (for example, the bishop may move diagonally but not orthogonally), and thereby also constitute the identity of the pieces required for making particular moves within the game (for example, the bishop in essential part simply is the piece subject to the rule just mentioned), likewise grammar prescribes or permits certain linguistic moves and proscribes others (for example, it prescribes or permits that in a context where one has counted 2 items and another 2 items one judge there to be a total of 4 items, and it proscribes that one judge there to be a total of 5 items), and thereby also constitutes the identity of the concepts required for making particular linguistic moves (for example, the concept "2" in essential part simply is the concept subject to the mathematical rule just mentioned).5
(3) Just as the rules of games not only govern and essentially constitute the particular moves made within games but also provide a standard for adjudicating these moves' success or failure, so the rules of grammar in addition to governing and essentially constituting particular linguistic moves also provide a standard for adjudicating their success or failure.6
(4) Just as the rules of a game are not assertions but instead more like commands or imperatives, similarly grammatical rules are not assertions but more like commands, commandments, or categorical imperatives (RFM, V, #13, #17; VI, #30; VII, #72).
(5) Like the rules of games, grammatical rules are in some sense conventions: "Grammar consists of conventions" (PG, I, #138; cf. BT, p. 167; PI, #354-55).7
(6) Just as the rules of games may be either explicitly formulated in language (as in most commercial board games, for example) or else implicit (as in some young children's games, for instance), and in the latter case they may subsequently achieve explicit formulation (see PI, #54), likewise the rules of grammar may either be explicitly formulated in language (as they are in the case of the principles of mathematics, for example) or else implicit, and if they are implicit they may subsequently achieve explicit formulation.8
(7) Again, just as the rules of games may in some cases be definite but in others vague or fluctuating, so the rules of grammar may in some cases be definite but in others vague or fluctuating (PI, #79-83; Z, #438-41).

The above gives what one might perhaps call Wittgenstein's generic conception of "grammar." However, he usually employs this term in more specific applications, and it is especially important to focus on one of these in particular.