Abstract
This is a largely informal and increasingly speculative reflection on the implications and the importance of formal semantics and formal logic, not just because they throw light on the subject matters with which they deal, but also because of their power to make those who become acquainted with them better speakers and better agents. The earlier parts of the paper rehearse some familiar formal points about model-theoretic accounts of the semantics of natural languages and then ask in what ways treatments can be tested against the intuitions of competent speakers. It is argued that model-theoretic accounts are testable even when speakers have firm judgements only about some of the sentences of the language fragment treated by the theory; and, further, that once a speaker who has verified the treatment to his satisfaction against those judgements of which he feels certain, the theory may help him to deepen his understanding of his language and thus also to improve as a speaker. I then go on to argue that a similar dynamics governs our interaction with axiomatic theories of non-linguistic aspects of human behaviour, such as knowledge or action: We may convince ourselves that such a theory is right by observing that it agrees with the judgements about aspects of the formalized concepts about which we are confident and then let the theory guide us to a better understanding of other aspects of those concepts; and that may change us as agents. These considerations suggest that there is more in common between formal treatments of the semantics of natural language and formal logics of aspects of cognition and action than is usually assumed.
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Notes
- 1.
The Paradox of Analysis was originally formulated in connection with the notion of conceptual analysis as propagated in the work of G. E. Moore. The ‘paradox’ is this: suppose I want to clarify a concept C and propose an analysis (or ‘explanatory definition’) A for it. Then my proposal meets the following predicament: either (i) C is understood well enough to make it possible to see that A is a correct analysis of it; but in that case the analysis cannot tell anything about C that wasn’t already known; or (ii) A does tell us something about C that wasn’t yet known; but then it isn’t possible to verify that A is a correct analysis of C. See e.g. [1, 10, 22].
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More recently there have been concerted efforts to turn Boolos’ insights into explicit semantic accounts of second order logic in which the meta-language, in which the truth definition of Second Order Logic is stated, is inspired by the way plurals enable us to say things that would otherwise have to be said by referring to sets or mereological complexes. This branch of logic and semantics is usually referred to as ‘Plural Logic’. For a recent contribution see e.g. [24].
- 4.
Specific charges of these sorts have been levelled at semantic accounts of modalities in terms of possible worlds. On the one hand the notion of a ‘possible world’ is said to be ill-defined or incoherent, which disqualifies specification formalisms that refer to possible worlds. On the other hand there is the objection that specification formalisms that analyse modalities in terms of possible worlds try to capture the meaning of essentially non-extensional notions (the modalities) in extensional terms (i.e. in terms of possible worlds). Such accounts therefore cannot fail to distort the semantics of the object language.
- 5.
In the case of Plural Logic the skepticism towards formal specification formalisms comes hand in hand with a curious optimism about our competence as native speakers: that our untutored grasp of the use of plurals in the language or languages we speak is good enough to justify the use of such a natural language to explicate the semantic foundations of higher order logic. This optimism I share even less. Here is one reason (I know it is not conclusive): Boolos [11] presents a translation algorithm from second order logic into English in which second order quantification translates into constructions involving plurals. My own experience with this algorithm is that when I try to apply it to any but the simplest formulas of second order logic, my head invariably spins out of control; I cannot get a grasp of the English sentence obtained as a translation. What I feel I desperately need in order to make sense of it is a further, formal analysis of it. But if I understand Boolos correctly, that is just the wrong thing to ask for.
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- 7.
Chomsky has always been well aware of the fact that natural languages admit strings as grammatical that are too complex for human processing and makes an emphatic distinction between competence and performance to account for this fact. But the limitations that a speaker may demonstrate because of performance constraints—e.g. to fail to recognise a certain string as grammatical although it is grammatical by the very principles she has internalised, or to fail to assign the correct interpretation to a string that she has recognised as a grammatical sentence—must be sharply distinguished from the partial mastery spoken of here. Limited performance and partial mastery of a language are very different things.
- 8.
In a reaction of Johan van Benthem to an earlier draft of this contribution he expressed his doubts that intuitions about entailments are our only means of checking the empirical adequacy of model-theoretic treatments. I would like, especially in reaction to that remark, to state my own conviction here that, yes, I also believe that there are other ways of verifying such treatments. One rests on our ability to understand the descriptions that are typically provided by such treatments for the models of their model classes. Those descriptions usually enable us to tell, when we are confronted with an actual or imaginary situation and a sentence that is presented as a description of it, what model or models from the formally specified model class corresponds to this situation. In such cases we can check correctness of the treatment by comparing the truth value that the treatment assigns to the sentence in the relevant model with our own speaker’s judgement whether or not the sentence is true in the given real or imagined situation.
There may be further ways in which model-theoretic treatments can be put to the test, but for me this is the most prominent one; and also, I suspect, the one most closely connected with the semanticists’ hunch that empirically adequate model-theoretic treatments of natural language fragments have little room for manoeuvre in assigning semantic values to the expressions belonging to these fragments.
- 9.
For some this question may have become a matter of terminology: two sentences simply won’t count as standing in a relation of entailment unless their relation can be explicated in terms of form along the lines indicated. My own intuitions go against this. The truth conditions of many sentences of natural languages are fixed with enough definiteness, I believe, to make the question which of them are related by entailment a meaningful one independently of any formal reduction.
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- 11.
I say ‘often’ because I do not want to exclude cases where we compute entailments by applying some formal account of meaning that assigns logical forms to premises and conclusion, and thereby provides ways of formally deriving the conclusion from the premises, without any direct appeal to the content representations of those sentences that may be in the mind of the agent who carries out the formal deduction. More on this point later on.
- 12.
A classical reference is [23].
- 13.
‘Commute’ in the following sense: if ref\(_{lin}\) is the reference relation for expressions from the public language and ref\(_{men}\) the reference relation for mental representations, and mrep is a 1–1 correspondence relation between well-formed expressions of the public language and mental representations of their content, then ref\(_{men}= \mathrm{{ref}}_{lin} \circ \mathrm {mrep}^{-1}\) and ref\(_{lin}=\mathrm{{ref}}_{men} \circ \mathrm {mrep}\).
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- 15.
There is of course also a third way in which exposure to logical models of cognition can affect our command of language. New scientific theories come almost inevitably with new terms, or new uses of existing words, and it is part of learning what the theory has to say about its subject matter that its students must assimilate those new terms and new meanings; for only in this way will they be able to talk, and think, about what the theory has to say. In this way the theorist’s vocabulary is extended with new elements to articulate new facts and hypotheses about this particular subject matter. But in that respect theories about human behaviour and cognition don’t differ from theories on any other subject matter—astrophysics, nano-chemistry, population genetics, what have you.
It should have been clear at this point, but let it be stressed once more, that the impact of logical models that is spoken of in the text is not of this sort. It is an impact on our linguistic core competence—on our command of that part of our language which we share with all other competent adult speakers, and not just with some select handful, with whom we share a special profession or hobby (be it entomology, chemical engineering, trading in financial products, a passion for cricket or, for that matter, cognitive science).
- 16.
The history of interactions between the fields of logic and linguistics is a topic that deserves an essay in its own right. What formal semantics of natural language owes to logic is obvious and known to all: It owes its existence to a logician who saw how methods from symbolic logic could be applied to the analysis of natural languages, and his insights are with us to this very day. That semantics can be a source of inspiration to formal logicians has been demonstrated, again and again, by outstanding contributions—to the logic of quantification by van Benthem, Keenan, Westerstahl, to name some of its most important contributors, to the logic and semantics of situations in the work of Barwise and others, and in countless contributions to the logic of conditionals and the logic of vagueness. What I am arguing here is that once again the pendulum is swinging back: that there is much that semanticists can learn from results in formal logic; and that that is so because of what those results have to tell us about various aspects of ontological structure.
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Kamp, H. (2014). Logic of and for Language, and Logic of and for Mind. In: Baltag, A., Smets, S. (eds) Johan van Benthem on Logic and Information Dynamics. Outstanding Contributions to Logic, vol 5. Springer, Cham. https://doi.org/10.1007/978-3-319-06025-5_29
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