The Substitution Interpretation of the Quantifiers

Quine and His Place in History

“…It is fairly easy to see what Barcan Marcus' canonical language should look like; one proposal of such a language was sketched by Dunn and Belnap (Dunn and Belnap, 1968). Its lexicon would at minimum contain a finite or denumerable set of constants, a finite or denumerable set of predicates, a denumerable set of variables, the usual truth-functional operators (for instance, '¬' and '&'), the quantifiers ('С', 'Ф'), and an identity predicate.…”

“…All other atomic sentences, including those with non-tag singular terms, are assigned truth values by the interpretation. All connectives as well as the substitutional quantifiers can be assigned true or false in terms of truth alone: '¬p' is true iff 'p' is false, 'p & q' is true iff 'p' is true and 'q' is true, and СxFx' is true iff 'Ft' is true for all terms 't'; 'ФxFx' is true iff 'Ft' is true for at least one term 't' (Dunn and Belnap, 1968). All substitution instances which contain non-tag singular terms as substituends -for Barcan Marcus, this would include fictional terms, higherorder terms, and terms for mere possibilia, for example -will have been assigned truth values by the interpretation quite independently of any ontological considerations, and have no bearing on the ontology.…”

On Horwich's way out

“…One problem derives from the fact that the w -rule is intimately connected to the substitutional interpretation of quantifiers. Namely, the w -rule is valid only if the relevant quantifier can be interpreted substitutionally (see Dunn andBelnap 1968, Hazen 1998). Substitutional quantification is standardly explained in terms of the truth of the substitution instances of quantified sentences.…”

Classical Logic and the Liar

“…The incoherence is independent of (T) and of any willingness to infer from a sentence to calling it 'true'. 4 In order to tackle the liar paradox, we need to deviate either from schemas like (T), by refusing to apply them to paradoxical sentences, or from classical logic, by refusing to follow all its rules when our reasoning contains such sentences. If we deviate from classical logic, we will continue to accept some of its rules even when reasoning with a paradoxical sentence, but we will not accept all of them.…”

“…Say we are facing a theoretical difficulty and, in order to overcome it, we need either to reject that rule or to reject (T). It is not clearly preferable to 4 Traditional emotivists and prescriptivists were happy to say things like 'Murder is evil' but deny that such sentences are true. Were they incoherent?…”