Saturday, December 16, 2006

Manolo P-O on Williamson on Possibilia

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.


Manuel Pérez Otero said...

Model-theoretic accounts versus “witness” elucidations

Manolo GC says I am commited to a certain thesis, (*): THERE IS an object that contributes to making ‘it is possible that there is an object that LW fathered’ is true, and suggests that (*) is inconsistent with another thesis that I, undoubtedly, defend: (**):‘there is an object such that it is possible that LW fathered him’ is false. My reply: there are two important independent problems about (*). I would accept (*) only if it is qualified by some provisos or clarificatory remarks in order to avoid these problems. But with these qualifications (*) is consistent with (**) (or we would need an argument to show it isn’t).
The two problems with (*):
(1) The terminology in (*) and specially the jargon about objects that are “witnesses” for the non-validity of a formula (cf. GC’s comment) are similar to L3. I can accept them. But they could be misleading. I appeal again to the analogy with first order (classical, non-modal) logic (FOL). Sentence (S), ‘Socrates is human’, is not valid because it is false relatively to some models. Let’s consider one of these models, < A , F >, where the function of interpretation F assigns Nixon and the set of green objects to ‘Socrates’ and ‘is human’, respectively. Now consider: (N): Nixon and the fact that he is not green are witnesses for the non-validity of (S); (N’): Nixon and the fact that he is not green contribute to making (S) is not valid. I think that (*) and the other claims about “witnesses” used by Manolo GC are not more acceptable than (N) or (N’). They can be used, as vivid ways to put in different words the standard precise model-theoretic accounts (expressed in L4 and FOL, respectively) (model-theoretic accounts that should include –I agree– the explanatory additional remarks mentioned in paragraph (i) of Manolo GC’s comment). But if we are asked for the exact meaning of them, then the precise model-theoretic accounts (expressed in L4 and FOL, respectively) should be invoked. Under this reading of (N), (N’) and (*) their ontological (material) commitments don’t go beyond those of the corresponding model-theoretic more precise versions.
(2) Even when we look in that way to (N), (N’) and (*), these claims are not literally true. The reason is similar fot both kinds of cases (the FOL and the L4 case). Relatively to some of the models that falsifies (S), Nixon “contributes” to making it not valid; but relatively to some other models that falsifies (S) (e.g. models whose domain doesn’t contain Nixon) Nixon doesn’t “contributes” at all. It would be absurd to sustain that (N) and (N’) are true independently of the model we are considering. In possible worlds semantics the constraints required to select an intended model don’t select a single model. In fact, if we talk of functions and interpretations in the unconstrained sense Manolo GC mentions in his paragraph (i), there are two kind of constraints which are relevant to select intended models. I agree with what GC says in (i) (except for a detail which is not relevant to the present discussion). There he mentions the constraints required to preserve the meanings of the “logical constants” (I was pressuposing these constraints in my talk). Further constraints are required if we want, for instance, that ‘water is H2O’ and (assuming that Nixon was fathered by William) ‘if Nixon exists, then he was fathered by William’ are necessary (true relatively to every possible world in the intended models). Anyway, I see no reason to suppose that the required constraints select just a single model, with its corresponding single w* and D(w*). Relatively to some model I, whose “actual world” is wI*, number 7 is an entity not contained in D(wI*) but belonging to the denotation of ‘fathered by LW’ in some other world, m, of that model I; therefore, relatively to model I, number 7 is one of the “witnesses” (in Manolo GC’s words) of the non-validity of the Barcan Formula (BF). But relatively to some other intended models that falsifies (BF) number 7 is not such a “witness”; for instance, a model J, whose “actual world” wJ* has a domain, D(wJ*), containing number 7 (so, other objects, not contained in D(wJ*), are the “witnesses” relatively to J of the non-validity of the Barcan Formula). Consequently, I am not commited to (*), but just to something like (#): For any intended model, I, that falsifies (BF), THERE IS an object that contributes to making ‘it is possible that there is an object that LW fathered’ is true (i. e., an actual object not contained in D(wI*)).
But I turn to the considerations in (1). I am commited to (#) only in the innocuous sense in which I (and Manolo GC, for instance) am commited to claims like (##): For any model, I, that falsifies (S), THERE IS an object that contributes to making (S) is not valid. When Nixon is one of these “witnesses” of the non-validity of (S), that doesn’t imply any kind of ontologically interesting “dependence” of the non-validity of (S) on the existence of Nixon. Mutatis mutandis for (#). We would need from Manolo GC an argument to sustain that (#), in the innocuous sense, is not consistent with (**).
Regarding Williamson argument and what I think that my discussion would achieve: Williamson argument crucially rely on (W): For any intended model, I, the domain of the actual world is identical to D(wI*). Thesis (W) is far from being obvious. My discussion aims to explain why we don’t need to assume (W). In that sense, my reflection can contribute at least to change the perception about the onus of the proof: it is Williamson who should adduce a proper justification for (W), a justification that could overcome the drawbacks in his counter-intuitive ontological theory of possibilia.

Manuel Pérez Otero

manolo g-c said...

There is an interesting discussion of highly related matters in the first part of Karen Bennett's recently published "Two Axes of Actualism", Phil. Review 2005, 297-326, about her "first axis". I assume that, putting it in her terms, P-O wants to hold actualism, the slogan that "everything is actual" (where the domain of the quantifier is not trivially understood as restricted to the actual world), and the view that the domain of the actual world does not include the domain of every other world. She argues that, given the former view, the latter "cannot be stated within Kripke's semantics. Either the semantics must be modified by the addition of a representation function, or else we must accept that the [actualist] slogan is a piece of presemantic stage direction" (309). I guess that P-O instrumentalistic suggestions go some way in the direction of elaborating the latter option.

My first post was just intended to challenge the defender of a view along these lines to try and clarify the proposal. I still think that the challenge has not been met, and that merely returning the burden of the proof back to Williamson and others similarly skeptic about the tenability of the two claims is not satisfactory; although I also think that P-O contentions go much further into meeting it than mere invocation of the metaphor of "presemantic stage directions".