Indefinite Extensibility in Natural Language

Abstract

ABSTRACTThe Monist's call for papers for this issue ended: If formalism is true, then it must be possible in principle to mechanize meaning in a conscious thinking and language-using machine; if intentionalism is true, no such project is intelligible. We use the Grelling-Nelson paradox to show that natural language is indefinitely extensible, which has two important consequences: it cannot be formalized and model theoretic semantics, standard for formal languages, is not suitable for it. We also point out that object-object mapping theories of semantics, the usual account for the possibility of nonintentional semantics, do not seem able to account for the indefinitely extensible productivity of natural language.1. IntroductionThere exist at least three published papers (Cook 2007, 2009; Schlenker 2010) arguing that natural language is indefinitely extensible in some sense. More concretely, all three papers start from the Liar paradox and by means of Strengthened and Strengthened Strengthened Liars construct a hierarchy of truth values with as many of them as there are ordinal numbers. Both authors renounce the principle of Bivalence in order to solve the Liar and its revenge forms.Bivalence is, however, a fundamental law of classical logic with a considerable intuitive appeal; it is the logical counterpart of the ontological principle that any well-defined situation either obtains or does not obtain. As we see it, trading such a fundamental principle for a solution of the Liar is no bargain. The price could be too high for many people. So, we offer a path to the indefinite extensibility of natural language that is compatible with all principles commonly held as laws of logic. Instead of the Liar, we will use the Grelling-Nelson paradox.We will first address the issue of formalizability, just to show that natural language is not formalizable. Then we will interpret the result in terms of indefinite extensibility as defined below.2. VaguenessSome readers may contemplate natural language as trivially nonformalizable because of its inherent vagueness. We do not wish to go into the discussion whether vagueness can be formalized or not. We just wish to show mat, leaving vagueness aside, natural language turns out to be nonformalizable on quite another account, namely, indefinite extensibility.So we ask the reader to assume that we are dealing with a definite segment of English - which we will call ?-English - so that all metalinguistic predicates here used are definite, and the corresponding sets, when they exist, are ordinary, nonfuzzy sets. The reader may well not believe such a crisp core of English to exist, but this should not prevent him from following us: we only purport to show that on the assumption that ?English exists we can all the same prove that it cannot be formalized; this will reveal at least that vagueness is not the unique feature of natural language that makes it incapable of being formalized.3. Natural Language Cannot Be FormalizedIn order to prove the proposition in the title we first introduce some definitions.An interpreted language L is a set of ordered pairs , where a signifier is a syntactic object and a signified is the meaning or semantic value of the signifier (the terms are borrowed from Saussure [ 1 972]). For present purposes, we need not commit ourselves to any particular theory of meaning. It is clear that natural language is intrinsically interpreted.It seems reasonable to require for L to be formalizable that there be an effective procedure enumerating all of L's signifiers and assigning them the corresponding signified, which amounts to the requirement that L be recursively enumerable. So we say that an interpreted language L can be formalized if and only if there is an algorithm ALG that enumerates L.We will establish the following proposition: there is no ALG providing a formalization of the predicates of ? …