How do we understand formal languages?

Consider a sentence from some formal language; for example, a sentence of quantified modal logic:

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  “There is an F which could have been a non-F.”

What fixes the meaning of this sentence? How do we make sense of it? And then, what is the status of our judgments about truth conditions, validity, consequence, etc. for formal sentences?

A candidate answer. (Peter van Inwagen defends this view in “Meta-Ontology”, 1998.) The symbols in a formal language are defined in terms of some natural language, like English. For instance, is defined to mean “there is”, to mean “possibly”, and so on. We understand the formal sentence by replacing each symbol with its English definiens, and we understand the English sentence directly. On this view, formal languages are just handy abbreviations for the natural languages we have already mastered, perhaps with some extra syntactic markers to remove ambiguity.

Suppose A, an English speaker, claims that is intuitively valid. If B wants to argue that is in fact invalid, she has only three options. (1) Use a different English translation from A. In this case, though, B would merely be talking past A. (2) Deny that A correctly understands the English sentence—so B is controverting a datum of natural language semantics. (3) Deny A’s logical intuition. So B’s only options are pretty drastic: to deny a native speaker’s authority on the meaning of her own language, or to deny a (let’s say pretty strong) logical intuition.

I’m pretty sure the candidate answer is wrong. First, because the obvious English translations for a logical symbol often turn out to wrong—witness the logician’s conditional, or the rigid “actually” operator—and we can go on understanding the symbol even before we have found an adequate translation. Also, we don’t typically explain the use of a symbol by giving a direct English translation: rather, we describe (in English, or another formal language) generally how the symbol is to be used. Furthermore, we can have non-trivial arguments over whether a certain English gloss of a formal sentence is the right one.

Here’s an alternative picture. In order to do some theoretical work, we introduce a regimented language as a tool. What we need for the job is some sentences that satisfy certain semantic constraints. should mean that snow is white. should be valid. should have as a consequence. We generally won’t have codified these constraints, but we internalize them in our capacity as theorists using a particular language; someone who doesn’t use the language in accordance with the constraints doesn’t really understand it. (This view is like conceptual role semantics, except that constraints that specify the meaning directly, in other languages, are allowed.)

In using the language, we assume that some interpretation satisfies our constraints—to use a formal language is, in effect, to make a theoretical posit. Insofar as our constraints underdetermine what such an interpretation would be, our language’s interpretation is in fact underdetermined. If no language satisfies all the constraints, then we’ve been speaking incoherently; we need to relax some constraints. The constraints are partly a matter of convention, but also partly a matter of theory: internally, in using the language, we commit ourselves to its coherence, and thus to the existence of an interpretation that satisfies the constraints; and externally, the constraints are determined by the theoretical requirements we have of the language.

Say A judges to be valid. What this involves is A’s judgment that “ is valid” is a consequence of a certain set of implicit semantic constraints on the language. Again suppose that B denies A’s validity intuition. Now there are two ways to go. (1) Deny A’s logic: B might agree on the relevant constraints, but disagree that they have “ is valid” as a consequence. (2) Deny A’s constraints: B might say that some of the constraints A imposes are not appropriate for the language in question. This might be based on an internal criticism—some of A’s constraints are inconsistent—or, more likely, external criticism: some of A’s constraints don’t adequately characterize the role the language is intended to play. The important upshot is that, unlike on van Inwagen’s view, B can disagree not only on linguistic grounds or logical grounds, but also on theoretical grounds. (Of course, since on my view the constraints also fix the meaning of the language, there is no bright line between the linguistic and theoretical grounds for disagreement—this is Quine’s point.)