Manolo P-O’s reply to Williamson’s argument for the validity of the Barcan Formula (BF) in his contribution to the LOGOS Seminar last Wednesday appears to depend on taking a sort of instrumentalistic attitude deflating the ontological commitments incurred in building an account of relations of logical consequence (in his language L4). However, he grants that (i) L4 does have explanatory commitments, in that it is intended to account for consequence relations among modal statements. Here I would like to raise some concerns that, prima facie, (i) is inconsistent with (ii) Manolo’s rejection of the validity of BF, together with (iii) his crucial “material” contention in section 7, that there are objects not in the domain of D(w*). Let me elaborate.
(i) Firstly, consider Manolo’s analogy with the first-order case (section 4). Granted that “there is a model with respect to which Socrates is human” is a very misleading way of putting the L4 way of stating the non-validity of ‘Socrates is human’. However, unconstrained talk of functions and interpretations will not do to properly reflect the explanatory commitments of the relevant L4 statements. For, appealing to that unconstrained talk, we could also establish that ‘Socrates is identical to himself’ is not valid. The functions and models of which we talk in L4 must be restricted by a correct theory capturing that, whatever it is, in virtue of which some statements and arguments are first-order valid and some others are not. According to many, this means that they must preserve the meanings of the first-order “logical constants”; in the case of ‘Socrates is human’, the semantic categories to which ‘Socrates’ and ‘is human’ belong, i.e., that they must be assigned, respectively, object-like meanings and monadic-property-like meanings. The same applies, mutatis mutandis, when we move to the modal case; and here we must assume that the modal operators, ‘possibly’ and ‘necessarily’, count among the logical constants.
(ii) Now, Manolo rejects BF, i.e., he thinks that while, e.g., ‘it is possible that there is an object that LW fathered’ is true, ‘there is an object such that it is possible that LW fathered him’ is false; while (iii) he accepts that there are objects not in the domain of D(w*), WHICH ARE PRECISELY THOSE WE NEED AS WITNESSES for the non-validity of BF, i.e., for ascriptions of truth and falsity exactly like the previous ones. Thus, (*) THERE IS an object that contributes to making ‘it is possible that there is an object that LW fathered’ is true. My main concern is now this: Given the explanatory commitments granted for L4 in (i), how is it that (*) is consistent with counting ‘there is an object such that it is possible that LW fathered him’ as false? Given those explanatory commitments (particularly, that of respecting the meanings that the modal operators like ‘possibly’ do have), how is it that (*) does not commit us to the existence of a-possible-entity-fathered-by-LW? But this appears to be to grant that ‘there is an object such that it is possible that LW fathered him’ is, after all, true.
I think that M P-O needs to say more about the commitments and lack thereof derived from the explanatory endevours associated to L4; merely gesturing towards a distinction between “structural” and “material” issues is not enough for a reply to Williamson’s argument for possibilia.