Abstract
Trial and error classifiers, corresponding to concepts which change their extensions over time, are introduced and briefly philosophically motivated. A fragment of the language of classical first-order logic is given a new semantics, using \(\omega \)-sequences of classical models, in order to interpret the basic predicates as classifiers of this kind. It turns out that we can use a natural deduction proof system which differs from classical logic only in the conditions for application of existential elimination. Soundness and completeness theorems are proved for this system.
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Notes
There is related contemporary work on models with variable interpretations and truth predicates, see T. Achourioti. Intensionality and Truth: A Study Beyond Deflationism. Dissertation, Amsterdam University (forthcoming).
Two short comments on what is not handled here: (i) Over time, we may subdivide, or even give up, the concept term P, but such events are not really modelled in this paper; (ii) Vagueness of terms is also deferred to other theories. Here we just assume our predicates to be precise at discourse level, whether by stipulation or through other kinds of “semantic negotiations”.
Basically, an object intuitively “coming into existence” at time s later than t will be part of the formal timless domain even at t, and if it, counterfactually, would have been classified as an animal (say) according to the conceptual and experimental schemes at t, our formal representation will consider it so classified already at t (though the situation may of course have changed when s finally arrives, if the ‘animal’ classifier works differently then). Conversely, an animal that dies at t will still be classified at s, though not necessarliy as an animal, if the concept has changed by then.
So, describing the dynamics of the classification process in this external language will result in an (intentional) massive “loss of information”. This is as it should be. Compare the loss of information when a relational structure is described using the basic modal language, instead of a first order language.
In this paper we do not treat generalized quantifiers, but only the ordinary universal and existential ones. This is not to say that there aren’t interesting trial and error aspects of, say, quantifiers such as ‘most’ or ‘infinitely many’.
One might also consider enriching the language with several (e.g. weaker and stronger) universal quantifiers, etc.
In fact, from a technical point of view, we don’t really need (in the completeness proof) models with any change at all in the interpretation of individual constants, cf. footnote 15. On the other hand, demanding constants to “converge immediately” would not fit the philosophical intuition.
We will use \(\vDash _{\mathrm{te}}\) for trial-and-error truth/consequence and \(\vDash _{\mathrm{c}}\) for classical truth/consequence.
So, \(m \in [\![\beta ]\!]_{{\mathcal {M}}_j}\) is just a convenient way to write \({\mathcal {M}}_j \vDash _{\mathrm{c}} \beta [m]\).
And similarly for the denotation of constants.
In the proposition and its corollary, \(\varGamma \) and \(\varphi \) are of course in the restricted language of trial and error logic.
Refer back to the examples in Sect. 3.3, and note that classical proofs of these consequences all use existential elimination.
The system would still be complete if we in (ii) further restrict the form of \(\gamma \) to \(\exists x \alpha \) or \(\bot \). But such extreme parsimony is not really a virtue here. (Cf. footnote 14.) On the other hand, we could also keep soundness with slight liberalizations, so as to avoid some “detours” in derivations.
If the rule is restricted as in footnote 13, we use the following derivation:
Note that the construction would still work if we in all models interpret c as c.
References
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Acknowledgments
As usual, I am deep in intellectual debt to my collegues in the logic group at my department at the University of Gothenburg, and special mention goes to Christian Bennet and Fredrik Engström for outstanding generosity with their time. Valuable comments from two anonymous reviewers have been most helpful in revising the paper. Most ideas which underlie, or eventually became part of, this paper have previously been presented in different contexts, and I would in particular like to thank the Logic and Language group at ILLC, Amsterdam, for inviting me to give a talk at their colloquium on this subject. The work on this paper was partly funded by the foundation Kungliga och Hvitfeldtska stiftelsen.
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Kaså, M. A Logic for Trial and Error Classifiers. J of Log Lang and Inf 24, 307–322 (2015). https://doi.org/10.1007/s10849-015-9222-7
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DOI: https://doi.org/10.1007/s10849-015-9222-7