Formal Languages and Natural Languages
Formal Languages and Natural Languages
Philosophers of language have argued over the relationship between so-called formal languages and natural languages (the kind people regularly speak). Some say formal languages supply the underlying logical form of the sentences of natural languages; some say formal languages improve on natural languages; some say formal languages are technical inventions that distort natural languages; some say they reflect the innate language of thought unlike natural languages; some say they are pointless abstractions bearing no meaningful relationship to natural languages. The correct answer, however, is this: they are part of natural language. Every formula of a formal language can be read as a sentence of English (or any other natural language), though the resulting utterance may sound stilted (“there is something x such that for everything y…”). A so-called formal language is really a formal notation, where “formal” just means, “looks like mathematics”; the language is good old ordinary language, suitably configured. All we are doing is translating some strings of English words into other strings of English words, as when we replace (say) “Everyone loves someone” with “For every person there exists another person such that the first person loves the second person”. The symbols of the formal language are just invented signs for words we already have–for example, using a backwards upper case “E” to stand in for “there exists”. I take it this is completely obvious. 
But it makes a difference to how we view certain kinds of proposal. The theory of descriptions, say, is just the proposal that one kind of sentence of natural language containing “the” can be translated into another kind of sentence of natural language containing only words like “there is”, “for all”, and “uniquely” (itself translated by the word “identical” suitably positioned). We are using one part of natural language to paraphrase another part, invidiously preferring some sentences to others. This cannot be an improvement on natural language, since it isnatural language; we might just prefer one part of it to another for philosophical or other reasons. Nor can the preferred part be the underlying form of the other part: for how could one part of natural language be the underlying form of another part? Both are overt sentences of the language, neither “deep” nor “superficial”. We may describe one sentence as the analysis of another, but how could one sentence literally contain another? By the same token, the “formal” part could not be inferior to natural language, though it might not exhaust the full resources of natural language. There is no such thing as an opposition between “logical language” and “ordinary language”: both are just versions of natural language. All we can really talk about is whether one or other part of natural language is preferable for certain reasons or purposes. We can argue about the utility or perspicuity of certain notations, which are just abbreviations for natural language expressions; but we cannot argue about the relationship between natural languages and some other type of non-natural language.  Everything we can say (or write) is part of natural language, which is why we can always convert a logical formula into familiar words of the vernacular. Thus a theory of truth for a formal language is in reality a theory of truth for one section of a natural language. A so-called formal language is not some sort of transcendent symbolic system standing outside of natural language. What logicians have done is simply invent a code for a certain part of the language they already speak.
The innate and universal language of thought can be externalized in different kinds of notational system—including English and Japanese, predicate calculus and modal logic—but these are all systems that express the same underlying cognitive structure. If we call this structure LANGUAGE, then what are called formal and natural languages are just different ways of externalizing LANGUAGE. And the formal (mathematical-looking) mode of externalization is really just a part of the natural-language mode of externalization. There is no opposition here, no rivalry, no better or worse. Principia Mathematica is actually a piece of ordinary English. We might say that logicians speak a certain dialect of their native language. A formal language is just so much (stilted) informal language; it is not something standing magnificently apart from the common language we all speak. We use different parts of our native language for different purposes, altering our vocabulary and style; a so-called logical language is just one such variation.
 There exists the theoretical possibility of a formal language not expressible in the sentences of natural language: then the standard range of options regarding its relationship to natural language would be available. But that is not the situation in which we find ourselves, which is that logic texts simply consist of ordinary language written in novel orthography (variables, brackets, etc).
 In the same way we can meaningfully talk about the relative merits of different human languages, say English and French, but this is clearly an intra natural language issue, not an issue about a natural language versus an unnatural language. In what way is a logical language unnatural? It is so only in the sense that it is visually unfamiliar and awkward to speak. We use the same linguistic competence in the logic classroom that we use in daily life.
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