Abstract
This paper provides finitary jointly necessary and sufficient acceptance and rejection conditions for the logical constants of a first order quantificational language. By introducing the notion of making an assignment as a distinct object level practice—something you do with a sentence—(as opposed to a meta-level semantic notion) and combining this with the practice of (hypothetical and categorical) acceptance and rejection and the practice of making suppositions one gains a structure that is sufficiently rich to fully characterize the class of classical first order theories. The analysis thus provides a way of characterizing classical first order quantification by expressivist means.
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Notes
The literature here is extensive. Carnap (1937) and Stevenson (1937) are early proponents of the view that moral language lack standards of objective correctness; examples of influential modern proponents are Blackburn (1993) and Gibbard (1990). Ramsey’s (1931) (see also Adams 1975; Edgington 1995) treatment of conditionals can likewise be seen as expressivist. Price (1983a) and Yalcin (2007) discuss probabilistic and modal expressivism.
This link between sentences with context dependent free variables and the corresponding closed decontextualized counterpart has an obvious model theoretic analysis: a sentence q(x) is true (relative to a model) within the context of a premise p(x) if and only if q(x) is true in every assignment to x that makes p(x) true in the model if and only if \(\forall x (p(x) \supset q(x))\) is true in the model. While ‘context dependent indeterminate reference’ may have the air of a mysterious concept, its model-theoretic explication is straight-forward.
Without such dependencies we get the decidable fragment of first order logic where sentences are of the form ∃X∀Y A (with A quantifier-free).
Here one might specially mention Restall’s (2012) bilateral framework as he—much as in the present approach—takes sentences with ‘arbitrary names’ (e.g. free variables) to be meaningful and employs such sentences in his interpretation of the quantifiers. Restall’s framework differs from the present in other respects, most importantly he does not seek to give expressivist requirements of use (in the present sense of ‘expressive’: necessary and sufficient conditions for acceptance and rejection).
The claim is that if it is possible for any given speaker and sentence to decide whether the speaker actually accepts/rejects that sentence (there is no assumption that one can decide whether the speaker should accept/reject the sentence), then it is possible to decide, for any given sentence, whether the speaker makes linguistically competent use of the logical constants as used in that sentence. Note, however, that linguistic competence does not entail or presuppose general logical or inferential competence: it does not follow that it is in general decidable whether a speaker satisfies the structural requirements that govern general inferential practices.
The use of dynamic assignments also creates an analogy to computer programs and so, as an anonymous referee pointed out, raises questions about the relationship between the present framework and issues raised in the substantial literature on the Curry–Howard correspondence between systems of natural deduction and the typed lambda calculus. However, such comparisons are beyond the scope of this paper.
Quine calls such sentences ‘predicates’ but acknowledges that sentences with free variables are used and asserted.
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Cantwell, J. First Order Expressivist Logic. Erkenn 78, 1381–1403 (2013). https://doi.org/10.1007/s10670-012-9421-4
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DOI: https://doi.org/10.1007/s10670-012-9421-4