Monday, 20 June 2016

Forthcoming in Disputatio


My paper 'On Identity Statements: In Defense of a Sui Generis View' is forthcoming in Disputatio. The final draft is available at PhilPapers. It's the longest, most substantial paper I've gotten accepted for publication so far, and its argument is quite difficult and controversial. The first version of it was written in 2009 for a course taught by Adrian Heathcote. I presented its main arguments at the 2010 Australasian Association for Logic Conference and received a mostly encouraging response. I put it aside for a few years after a few failed attempts to publish it, and only recently began to try again. The Disputatio referee report was really good and has caused the paper to be miles better than it was.

Thursday, 16 June 2016

Modal Realism and Counterpossibles: A Tension in Lewis

David Lewis held that possible worlds are worlds as concrete as our own (cf. Lewis (1986)). He also held, in his work on counterfactuals (cf. Lewis (1973)), that all counterfactuals with impossible antecedents - 'counterpossibles' - are vacuously true. These two views do not fit together well. Embracing modal realism leads to especially compelling counterexamples - counterexamples given modal realism, that is - to the thesis that counterpossibles are all true. These take the form of conditionals whose antecedents are not intuitively impossible, but which are impossible given modal realism.

These arise because, according to modal realism, reality as a whole – that is, the totality of the posited worlds – is necessarily the way it is. Lewis is very upfront about this. Witness:

There is but one totality of worlds; it is not a world; it could not have been different. (Lewis 1986: 80.)

So, for example:

'If there had been two fewer men in reality as a whole than there actually are, there would have been fewer women.'

There is no reason to think this is true. And yet Lewis's thesis about counterpossibles, together with modal realism, implies that it is vacuously true.

Perhaps worse:

'If there had been fewer men in reality as a whole than there actually are, there would have been just as many men in reality as a whole as there actually are.'

This seems positively false.

Postscript

Thanks to Quentin Ruyant for pointing out that the last counterfactual, given modal realism and the thesis that there are infinitely many worlds with men in them, actually seems to come out true in a funny way: if there had been two fewer men, there still would have been infinitely many. So this was a bad example. Consider instead:

'If there had been no Model-T Fords in reality as a whole, there still would have been some Model-T Fords in reality as a whole'.

References

Lewis, David K. (1973). Counterfactuals. Blackwell Publishers.

Lewis, David K. (1986). On the Plurality of Worlds. Blackwell Publishers.

Tuesday, 17 May 2016

Sufficient Conditions for Some Inference-Patterns Involving Conditionals

This is a continuation of the last post, which was a continuation of this article. The following three theorems show that, given a plausible assumption about the semantics of conditionals:

(1) You can't go wrong in applying hypothetical syllogism (transitivity for conditionals) so long as the set of relevant scenarios for the output conditional is a subset of that for the second input and the set of relevant scenarios for the second input is a subset of that for the first.

(2) You can't go wrong in applying contraposition so long as the set of relevant scenarios for the output conditional is a subset of that for the input.

(3) You can't go wrong in applying strengthening the antecedent so long as the same condition as in (2) is fulfilled.

Let us assume that a conditional A > C is true iff in all relevant scenarios the corresponding material conditional is true. 


I will use Sc(X) to denote the set of relevant scenarios for a conditional X .


Theorem 1. For any three conditionals A > B, B > C and A > C such that  Sc(B > C) ⊆ Sc(A > B) and Sc(A > C) ⊆ Sc(B > C), A > C will be true if A > B and B > C are true.


Proof: Take any three conditionals A > B, B > C and A > C such that Sc(B > C) ⊆ Sc(A > B) and Sc(A > C) ⊆ Sc(B > C).  Now suppose that A > B and A > C are true. Since all the relevant scenarios for A > C are also relevant for A > B and B > C, and A > B and B > C are true, the material conditionals A ⊃ B and ⊃ C will both be true at all the relevant scenarios for A > C (by the assumed semantics). Since transitivity holds for material conditionals, ⊃ C will be true at all the relevant scenarios for A > C, making A > C true (by the assumed semantics). Therefore, if A > B and A > C are true, A > C will be true.


IllustrationIf I had spoken to a cat then I would have spoken to an animal. If I had spoken to an animal I would have been happy. Therefore, if I had spoken to a cat then I would have been happy. (It is natural to think of the set of relevant scenarios for the first sentence as larger than that for the second. This could be further brought out by adding something like 'no matter what' to the first sentence.)

Theorem 2. For any two conditionals A > B and ~B > ~A such that Sc(~B > ~A) ⊆ Sc(A > B), ~B > ~A will be true if A > B is true.


Proof: Take any two conditionals A > B and ~B > ~A such that Sc(~B > ~A) ⊆ Sc(A > B). Now suppose that A > B is true. Since all the relevant scenarios for ~B > ~A are also relevant for A > B, and A > B is true, the material conditional A ⊃ B will be true at all the relevant scenarios for ~B > ~A (by the assumed semantics). Since contraposition holds for material conditionals, ~B  ~A will be true at all the relevant scenarios for ~B > ~A, making ~B > ~A true (by the assumed semantics). Therefore, if A > B is true, ~B > ~A will be true.


Illustration: (Assume for the following Q & A that it is analytic that all bachelors are men.)

Q: Do you think that, in view of the fact that we get energy from shooting men, if we don't shoot a man tonight, we won't shoot a bachelor? 

A: Of course: If we will shoot a bachelor tonight then, no matter what, we will shoot a man. And what you're asking about follows from that.

Theorem 3. For any two conditionals A > B and (A & C) > B such that Sc((A & C) > B⊆ Sc(A > B), (A & C) > B will be true if A > B is true.

Proof: (Same as for Theorem 2 but with (A & C) > B in place of ~B > ~A, (A & C)  B in place of ~B  ~A, and an appeal to the fact that strengthening the antecedent holds for material conditionals in place of the appeal to the fact that contraposition holds for material conditionals.)


Illustration:
(As with Theorem 2, assume for the following Q & A that it is analytic that all bachelors are men.)

Q: Do you think that we would have shot a man today if we had gone out and shot five bachelors and taken care to go for the masculine-looking ones?


A: If we had gone out and shot five bachelors, then, no matter what, we would have shot a man. So yes, of course we would have 
 shot a man today if we had gone out and shot five bachelors and taken care to go for the masculine-looking ones.


Thursday, 10 March 2016

Transitivity and Conditionals

Followup:

Let us assume that a conditional A > C is true iff in all relevant scenarios the corresponding material conditional is true.

Let's leave it completely open here what makes a scenario relevant for a conditional. Let's also leave it open what scenarios are like.

(That something like the above is true for counterfactual or subjunctive conditionals seems more widely accepted than that something like it is true for indicatives, so the following will be most widely acceptable as an observation about the logic of counterfactuals. I think it probably applies to indicatives too. That it holds on the assumption of the above schematic semantics seems to me to be almost beyond dispute.)

In their 2008 paper 'Counterfactuals and Context', Brogaard and Salerno attempt to block a famous counterexample to transitivity for counterfactuals (cf. Lewis , p. 33) with the proposal that to have conditionals for which different scenarios are relevant figuring in the same argument is illicit.

But an inference from A > B and B > C to A > C will be truth-preserving as long as the set of relevant scenarios for the second is a subset of that for the first, and the set of relevant scenarios for the third is a subset of that for the second. (Note I don't say 'proper subset': they could all be the same set, but that's a special case.)

Illustration: If I had spoken to a cat then I would have spoken to an animal. If I had spoken to an animal I would have been happy. Therefore, if I had spoken to a cat then I would have been happy. (It is natural to think of the set of relevant scenarios for the first sentence as larger than that for the second. This could be further brought out by adding something like 'no matter what' to the first sentence.)

(This post builds on this.)

References

Brogaard, Berit & Salerno, Joe (2008). Counterfactuals and context. Analysis 68 (297):39–46.

Lewis, David K. (1973). Counterfactuals. Blackwell Publishers.

Wednesday, 9 March 2016

Five Objections to Sider's Quasi-Conventionalism About Modality

In a recent post I described Sider's quasi-conventionalism about modality, which in my view takes an important step forward with respsect to necessity de dicto but is mistaken in other ways. (My account of necessity de dicto shares a structure with it.) Here I give five objections to Sider's view.

None of these take the form of counterexamples. As Merricks (2013) observes:
[...] Sider’s general approach—as opposed to specific instances of that approach—is immune to counterexample. For suppose that Sider lists the “certain sorts.” You then come up with an absolutely compelling example of a proposition that is necessarily true and not of a sort on the list. Sider need not abandon his overall approach to reducing necessity. Instead, he could just add a new sort to the list to accommodate that example. Or suppose you come up with an absolutely compelling example of a true proposition that is not necessarily true and is of a sort on the list. Sider could just expunge that sort from the list.
1. Necessity does not seem disjunctive or arbitrary (at least, not to this extent).

This is an objection centering on our intuitive grasp of the concept of necessity de dicto. It seems like this is a notion we can grasp, with the help of Kripke’s characterizations as supplemented in this post. Now, when we grasp this idea, it seems we are grasping a single, unified concept: necessary truths could not have been otherwise, no matter how things had turned out. This just doesn’t seem like a disjunctive matter, and nor does it seem like the sort of thing we make one way or the other with any kind of arbitration - although of course there are unclear or borderline cases, which we may perhaps make stipulations about to some extent.

This is not a knock-down objection, of course. Sometimes philosophy can reveal things to be other than they might seem. But I think it is hard to deny, if we are willing and able to grasp the concept of necessity de dicto and careful to hold in abeyance any of our pet theoretical proclivities which may suggest otherwise, that the notion does seem more unitary and less arbitrary than Sider’s theory would have us believe. And I propose that that should count as a mark against Sider’s theory.

Furthermore, insofar as appearance really is different from what Sider says the reality is when it comes to necessity, there is some explanatory work for Sider, or more generally the would-be quasi-conventionalist, to do here: why the discrepancy? As far as I know, no answer has yet been given.

2. The ersatz substitute worry.

A starting point for this worry is the unapologetically ad hoc nature of Sider’s successive extensions of the toy version of his approach that he begins with (where the “certain sort” of propositions he takes as “modal axioms” are just the mathematical truths). This process seems to be one of going back and forth between a growing list of types of propositions, the list at the heart of an increasingly disjunctive account, and our grasp of the real modal notion of necessity. This gives rise to the worry that all we are doing is building an ersatz substitute for the real notion, by looking at the extensional behaviour of the latter and stipulating this behaviour into the account. No matter how far we pursue this strategy, the disjunctive notion we are building will remain fundamentally different in character from the notion whose behaviour we are modelling with it. Supposing that what we want from an ‘if and only if’ style account of necessity de dicto is not some substitute for that notion, but a biconditional which gives us insight into the notion itself, Sider’s approach will never satisfy.

Something of this worry is even suggested by what Sider says about family resemblances, rehearsed in the previous post as point (6). The quasi-conventionalist could simply insist that each of the items on their list of the types of propositions which count as modal axioms is there as a brute fact - that’s just how the notion of necessity works. But, Sider says, the quasi-conventionalist ‘need not be quite so flat-footed’, and is ‘free to exhibit similarities between various modal axioms, just as one might exhibit similarities between things that fall under our concept of a game, to use Wittgenstein’s example’. This move, offered as an optional extra for the quasi-conventionalist, is plausibly in tension with the way Sider’s successively extended accounts are formulated. Just as the concept of a game - allowing for the sake of argument that it is a family resemblance concept - is plausibly not actually captured by any particular disjunction, but is as we might say inherently open-ended, it is also plausible that we should admit that the real “certain sort” or “modal axiom” notion doing the all-important work in Sider’s account - allowing for the sake of argument that it is a family resemblance concept - is not captured by any particular disjunction either.

This of course suggests a variant of Sider’s approach, where it is held that the “modal axiom” notion is a family resemblance concept, and admitted that any definite, disjunctive list of types of propositions could only yield, when plugged into the overall account, an ersatz substitute for the notion of necessity de dicto. This variant is not, or at any rate less, vulnerable to the the ersatz substitute worry. But it is not clear whether it could really satisfy a philosopher who wants insight into the notion of necessity de dicto, let alone a philosopher with Sider’s motivations. For instance, can it really claim to be modally reductive? It might on the contrary seem that the family resemblance notion in question should be counted as thoroughly modal. Furthermore, it may seem to yield an account which is insufficiently insightful - essentially all we are now getting is (Schema) itself, together with the pronouncement that the condition C is given by a family resemblance concept. Is there nothing more which can be said? Relatedly, the question now arises: is it after all true that the notion in question is a family resemblance concept? What reason have we to believe that? (I will suggest, somewhat ironically given that I am on the whole much more admiring of Wittgenstein’s philosophy than Sider is, that it isn’t true. The notion playing this ‘condition C’ role, i.e. the notion which when combined with the notion of truth yields a notion playing Sider’s “modal axiom” role, can be defined in terms of a single necessary and sufficient condition.)

3. No iteration?

When Sider says early on in the modality chapter of his (2011) that the account he offers will be partial, there is a footnote to this remark which runs as follows:
(16) For example, the account defines a property of propositions that do not themselves concern modality, and thus is insufficient to interpret iterable modal operators.
This raises the question: how come, faced with this failure of coverage, Sider doesn’t simply make the same move with modal statements as he does with analyticities, “metaphysical” statements, and natural kind statements - namely include them expressly in the account?

Perhaps the answer is that this would threaten the account’s claim of reductiveness. For it seems that in order to include modal statements on the list, we need the concept ‘modal’.
The question then becomes: is ‘modal’ modal? If it is, Sider’s account is in serious trouble: it cannot, as a matter of principle, handle iterated modality. For remember, it is supposed to be modally reductive. And if iterated modality is a real, legitimate thing, then what use is a theory which gives us - by design - some extensionally correct answers but cannot handle this whole class of cases? It seems such a theory could give us an ersatz substitute for modality at best (to recall the above objection by that name). Its failure, if it is a failure, to be extendable to a salient class of cases should perhaps suggest to us that it is on the wrong track.

So, is ‘modal’ modal? It is an interesting question, and suggests interesting analogous questions about other kinds of concepts. One reason to think it is, is that we don’t seem to have a general way of saying what ‘modal’ means which doesn’t work by way of example. We seem to need examples of modal notions - necessity, or contingency, or possibility, or impossibility, or some combination of them - to do the job. To be sure, we could be said to be mentioning rather than using these notions in our explanations of ‘modal’, but is that any help? Don’t we need to use them in some broad sense in order to mention them in the appropriate way?

Another way out which may occur to the reader is to somehow delineate the modal statements using notions which are distinct from ‘modal’ and the like, but which fortuitously give the right extension. I am pessimistic about this. For a start, I can’t think of any good candidate notions. Furthermore, even if there were notions around which could do the job, wouldn’t using them for this purpose play further into the ersatz substitute worry described above? In particular, it seems like this strategy, while it may help Sider’s account deliver extensionally correct answers, would take the account (even) further from the real meaning of modal expressions, or the real nature of modal notions.

One possible strategy remains to be considered: accepting that ‘modal’ is modal and simply giving up the claim to full modal reductivity. From one angle, this seems not unreasonable; the way that ‘modal’ introduces modality, assuming it does, into the account, seems quite special and different from the way modality would be introduced if a notion of possibility or necessity were directly used. So perhaps there is room to claim that a broadly Siderian quasi-conventionalist account involving the notion of ‘modal’ as an unreduced modal element could still constitute a theoretical advance. I have no knockdown objection to this, but I do want to suggest that once this concession is made, other objections, such as the first two considered here - (i) that necessity does not seem as disjunctive or arbitrary as quasi-conventionalism would have us believe and (ii) the ersatz substitute worry - become all the more acute; I am not sympathetic with the following sort of move, but you might try to argue that biting those bullets is worth it if we get in return a complete reduction of modality, with its attendant payoff in eliminating puzzlement and vindicating certain sorts of metaphysical visions, but you can’t do that anymore under the present strategy. Indeed, the whole spirit of the quasi-conventionalist approach seems to be in tension with allowing such a modal element into the mix.

In sum, there is reason to suspect that iterated modality, and the failure of any existing version of Sider’s approach to cover it, poses a serious threat to Sider’s approach in general.

4. Reductivity a bug, not a feature.

Essentially this objection is raised against Sider’s theory by Merricks (2013). The objection is simply that, if we have reason to think that a modal notion like that of necessity de dicto cannot be reduced to non-modal notions, or if we just intuitively feel that to be right, then we should on that score alone be suspicious of Sider’s theory, since it purports to give a reduction. In making this objection, Merricks cites an argument he gives elsewhere (namely in Chapter 5 of his (2007)) for the conclusion that such modal notions indeed cannot be reduced to non-modal ones.

5. Questionable motivation.

As we said at the outset, Sider’s account is partly motivated by general puzzlement about modality, and partly by a metaphysical vision. Both these facets of the motivation can be made the focus of criticism. The following is not supposed to constitute a sharp, incisive objection, but rather to cast some doubt on these general features of Sider’s approach.

Regarding general puzzlement: yes, modality is puzzling to philosophers. But perhaps this puzzlement is not to be treated exclusively by means of reduction (or, for that matter, by ‘if and only if’ analysis whether modally reductive or not). Indeed, pursuing reduction can even be seen as pursuing an easy way out - albeit one which may be impossible in principle. Perhaps the only real way forward, with parts of our puzzlement at least, is, rather than trying to reduce modality to non-modal terms, to work on our way of looking at modal concepts themselves, using philosophical methods other than reductive analysis. (One method which comes to mind is the method, due to Wittgenstein, of imagining simplified language games and comparing and contrasting them to ours. In the Brown Book some steps are taken towards doing this with modality, but only cursorily. I mention this to give a particularly concrete and well-known example of a possibly helpful method, but this is just one among many - I do not mean to suggest it could suffice all by itself.) Non-modally-reductive accounts of necessity de dicto such as mine do not face this criticism, since they do not aim to clear up all of our puzzlement about modality, or even just some core of it, by means of an ‘if and only if’ style analysis. Nor do they aim even to point the way to such a clearing up. By being less ambitious on that front, they offer a more realistic hope of genuine theoretical progress on our understanding of de dicto modal notions - how they relate to other notions both modal and non-modal.

Regarding the metaphysical vision: it is beyond the scope of this post to criticize Sider’s Hume-influenced, Lewis-influenced metaphysical vision head-on. But we may note that, insofar as there may be grave problems with this sort of metaphysics for all we know given the present state of philosophical inquiry - nothing of the sort may be tenable, ultimately - there may also be problems with a highly ambitious approach to modality which is in service of this sort of metaphysics. More generally, perhaps there is reason to be dubious of any approach to modality based upon a metaphysical vision. One reason may be that the vision is, so to speak, too antecedent to modal considerations: perhaps one should let modal considerations shape one’s approach to metaphysical questions, rather than trying to explain modality (away, if you like) in terms of an approach to metaphysical questions which had its appeal quite apart from, or even in spite of, modal considerations. Another reason may be that the best way to make theoretical progress on the notion of necessity de dicto is to keep sweeping metaphysical visions out of it. We may do better to instead treat our topic along broadly logical lines. One way this may help is that it might free us up to throw a wider variety of conceptual resources at the problem - for instance, semantic notions or other modal notions which may seem problematic against some special metaphysical backdrop but are actually quite in order.

That concludes our list of objections or worries. For two further objections, see Merricks (2013).

I think the cumulative effect of the objections canvassed above should be for us to regard Sider’s theory as highly problematic. But note that none of these objections threaten (Schema). This raises the question: what if these were a more soberly motivated, more theoretically satisfying (Schema)-embodying account available? Some other candidate for the condition C in (Schema) which avoids these objections?

References
Merricks, Trenton (2007). Truth and Ontology. Oxford University Press.
Merricks, Trenton (2013). Three Comments on Writing the Book of the World. Analysis 73 (4):722-736.
Sider, Theodore (2003). Reductive theories of modality. In Michael J. Loux & Dean W. Zimmerman (eds.), The Oxford Handbook of Metaphysics. Oxford University Press 180-208.
Sider, Theodore (2011). Writing the Book of the World. Oxford University Press.
Sider, Theodore (2013). Symposium on Writing the Book of the World. Analysis 73 (4):751-770.

Saturday, 5 March 2016

Forthcoming in The Reasoner


My paper 'Against the Brogaard-Salerno Stricture' is forthcoming in The Reasoner. It is about the logic of conditionals, and derives from this blog post. The final draft is available at PhilPapers. Here is a post building on it.

Friday, 26 February 2016

Fifth Anniversary of Beginning Sprachlogik

February 24 this year marks the fifth anniversary of the first Sprachlogik post.

Thanks to everyone who has been supportive of the blog!