Saturday, 2 November 2019

‘Two Recent Theories of Conditionals’ vs. Two Recent Theories of Conditionals

The tide is beginning to turn against counterintuitive theories of indicative conditionals which either deny them truth-values or give them apparently wrong ones, but a deductive argument in Gibbard’s 1981 paper ‘Two Recent Theories of Conditionals’ appears to show that those unhappy options are the only viable ones. Here I summarise some fascinating recent technical work on an escape route and argue that Gibbard’s reason for not taking that route stemmed from a (forgivable) failure of theoretical imagination and a too-narrow view of the motivation for granting truth-values to indicatives.
Indicative conditionals seem to have truth-values. Just as ‘I will not eat a grapefruit tomorrow’ and ‘You are a horse’ are true and false respectively, so it seems that ‘If I have breakfast tomorrow, it won’t be a grapefruit’ and ‘If tomorrow someone tells you you’re a horse, you’ll become a horse’ are true and false respectively.
It also seems that an indicative conditional does not always have the same truth-value as the corresponding material conditional (which is true in all cases except when the antecedent is true and the consequent false). For example, both ‘If you die tonight, you’ll be alive tomorrow’ and ‘If you die tonight, the French Government will collapse tomorrow’ seem false - the first due to the nature of life and death, the second due to the way the world is organized - even though ‘You will die tonight ⊃ you’ll be alive tomorrow’ and ‘You will die tonight ⊃ the French Government will collapse tomorrow’ are both true provided that you don’t die tonight.
An ingenious deductive argument from Allan Gibbard’s 1981 paper ‘Two Recent Theories of Conditionals’ appears to show that these two seemings cannot both be right. Gibbard’s collapse argument is so called because threatens to collapse any truth-conditions that an indicative conditional might have down to those of the corresponding material conditional.
But Gibbard’s argument does not by itself demonstrate collapse, and the larger context of Gibbard (1981) shows that he was aware of that fact. Indeed, he identified an escape route - one which takes some noticing, and may not be noticed by many who encounter this much-discussed argument outside the context of Gibbard’s paper. However, upon identifying the escape route Gibbard gave what may seem like a compelling reason not to take it. Having also rejected the material conditional account of indicatives, Gibbard ends up adopting the NTV thesis - the view that indicative conditionals lack truth-values.
Subsequently, theories which take the escape route Gibbard identified have been pursued in earnest anyway, with impressive results. After a period in which the NTV thesis was beginning to look like the dominant view, the tide is finally beginning to turn.
The purpose of this article is to contribute to turning the tide by confronting and neutralizing Gibbard’s reason for not taking the escape route, and along the way to provide a high-level summary of some recent relevant work (some of which can be highly technical). We will see that by drawing on this work we can uphold, in a principled way, the intuitive view that indicative conditionals do indeed have truth-values, and ones which can differ from those of the corresponding material conditionals.
1. Gibbard’s Collapse Argument
I begin with a reader-friendly reconstruction of Gibbard’s collapse argument.
If Implies Hook: An indicative conditional ‘If A then C’ always implies ‘A ⊃ C’, i.e. indicatives are at least as strong as material conditionals.
Conditional Conjunction Elimination: All indicative conditionals of the form ‘If (A & C) then C’ are logical truths.[1]
Import-Export: In any arbitrary context, all pairs of indicative conditionals of the forms ‘If A then (if B then C)’ and ‘If (A & B) then C’ are logically equivalent.
Equivalent Antecedents: In any arbitrary context, all pairs of indicative conditionals which share the same consequent, and whose antecedents are logically equivalent, are themselves logically equivalent.
Consider any arbitrary indicative conditional ‘If A then C’ in an arbitrary context and its corresponding ‘A ⊃ C’.
By Conditional Conjunction Elimination, ‘If (A & C) then C’ is a logical truth.
By Equivalent Antecedents, ‘If ((A ⊃ C) & A) then C’ is then also a logical truth, since ‘A & C’ is logically equivalent to ‘(A ⊃ C) & A’ by propositional logic.
By Import-Export, ‘If (A ⊃ C) then (if A then C)’ is then also a logical truth. (Here ‘(A ⊃ C)’ plays the ‘A’ role in Import-Export as stated above, ‘A’ plays the ‘B’ role, and ‘C’ plays the ‘C’ role.)
By If Implies Hook, ‘(A ⊃ C) ⊃ (if A then C)’ is then also a logical truth, since the implications of logical truths are logical truths.
By If Implies Hook again, ‘(If A then C) ⊃ (A ⊃ C)’ is a logical truth.
By propositional logic applied to the last two sentences, ‘(If A then C) ≡ (A ⊃ C)’ is a logical truth. Hence, any arbitrary indicative conditional in any arbitrary context is logically equivalent to its corresponding material conditional. QED.
If we accept the reasoning and want to maintain that indicatives have truth-values that don’t always agree with the corresponding material conditional, we need to reject one of the assumptions of the argument - either one of the explicit ones listed above, or some auxiliary assumption.
2. The State of the Art of Resisting Collapse
Some have suspected Import-Export. For instance, a detailed axiomatic analysis of Gibbard’s proof leads Fitelson to conclude as follows:
The only axioms that seem plausibly deniable (to me — in the context of a sentential logic containing only conditionals and conjunctions) are [...] the import-export laws, and they seem to be the most suspect of the bunch. I find it difficult to see how any of the other axioms could (plausibly) be denied (but I won’t argue for that claim here). (Fitelson (2013), p. 184.)
However, Import-Export has proven difficult to reject. It strikes many as plausible, and counterexamples have been elusive. Edgington, for instance, finds them plausible in the abstract, and suggests that any example one tries seems to obey the principle:
Here are two sentence forms instances of which are, intuitively, equivalent:
(i) If (A&B), C.
(ii) If A, then if B, C.
(Following Vann McGee (1985) I'll call the principle that (i) and (ii) are equivalent the Import-Export Principle, or “Import-Export” for short.) Try any example: “If Mary comes then if John doesn't have to leave early we will play Bridge”; “If Mary comes and John doesn't have to leave early we will play Bridge”. “If they were outside and it rained, they got wet”; “If they were outside, then if it rained, they got wet”. (Edgington (2014), Sec. 2.5.)
There is one notable attempt at a counterexample in the literature, due to Kaufmann (2005, pp. 213 - 214). In Fitelson’s (2016) presentation:
Suppose that the probability that a given match ignites if struck is low, and consider a situation in which it is very likely that the match is not struck but instead is tossed into a campfire, where it ignites without being struck. Now, consider the following two indicative conditionals.
(a) If the match will ignite, then it will ignite if struck.
(b) If the match is struck and it will ignite, then it will ignite.
It seems like it is possible to understand (a) and (b) in such a way that (a) expresses a logical truth and (b) does not, suggesting that they may not be equivalent, making for a counterexample to Import-Export. But this has been challenged. Khoo and Mandelkern (forthcoming) write:
However, we suspect the intuitive grip of this example rests on an equivocation in ‘will’ between a broadly dispositional meaning and a temporal meaning. We can disambiguate these readings by replacing ‘will ignite’ with ‘is ignitable’, to select for the dispositional meaning, and by replacing ‘will ignite’ with ‘will ignite at t’, to select for the temporal meaning. (We also replace ‘struck’ with ‘struck at t0’, to thoroughly regiment the readings.) We suspect that the reading on which (a) and (b) strike us as inequivalent is:
(a’) If the match is ignitable, then it will ignite at t if struck at t0.
(b’) If the match is struck at t0 and it will ignite at t, then it will ignite at t.
(b’) does indeed strike us as a logical truth, while (a’) certainly does not. But this pair is of course no longer a counterexample to the pattern we are exploring; we would only get a counterexample if we were to disambiguate (a) and (b) in a uniform way. But no matter how we do this, the resulting sentences strike us as equivalent. (Khoo & Mandelkern (forthcoming), pp. 8 - 9 in online version).
In view of the fact that even the most suspect of Gibbard’s explicitly stated principles has proven difficult to reject, it is not surprising that some have rejected auxiliary assumptions not directly appealed to in the derivation. According to Kratzer (1986, 2012, p. 105 in latter) - whose syntactically distinctive theory of indicatives was inspired by Lewis (1975) - the problem with Gibbard’s argument is that it relies on the assumption that indicative conditionals are propositions formed by an operator, ‘if’, which takes two propositions and yields a proposition. If instead we follow Kratzer and treat ‘if’ as a restrictor, and regard ordinary indicative conditionals as containing an unvoiced necessity operator restricted by ‘if’, the conclusion of Gibbard’s argument no longer leads to the result that indicative conditionals, if they have truth-conditions at all, are truth-functional. For Gibbard’s conclusion is that if indicative conditionals are propositions in which a two-place propositional operator is applied to two propositions, then their truth-conditions collapse to those of the material conditional.
However, as Khoo (2013) has shown in detail, an analogous argument can be given directly in terms of the semantic values of sentence-schemas, without assuming that ‘if’ is a two-place propositional operator. But it turns out that Kratzer’s theory is nevertheless able, in another way, to block both the original and the analogous argument. Kratzer’s theory predicts subtle counterexamples to the principle that whenever an indicative conditional is true, so is the corresponding material conditional, thus invalidating the If Implies Hook assumption of Gibbard’s argument. So too does Gillies’ (2009) theory, on which ‘if’ is a two-place operator, but one which is able to shift the index and context[2] against which the consequent of an indicative conditional is evaluated (in the course of the evaluation of the conditional containing it).
On Khoo’s analysis, Kratzer’s alternative view of the syntax of indicative conditionals is orthogonal to the collapse issue. Both her theory, on which ‘if’ is a restrictor, and Gillies’ theory, on which ‘if’ is a “shifty” two-place propositional operator, avoid Gibbard’s conclusion. But in consequence of how they avoid Gibbard’s conclusion - by invalidating If Implies Hook - both theories predict counterexamples to modus ponens construed as a semantic thesis according to which ‘C’ is true whenever ‘A’ and ‘If A then C’ are both true.
Completely invalidating modus ponens would be a serious issue and would naturally cast doubt on these theories. But, like McGee’s (1985) independently-motivated counterexample to modus ponens, the main conditionals in the predicted counterexamples feature indicative conditionals in their consequents. That the predicted counterexamples are in this way similar to independently-motivated ones suggests that they are not mere artefacts of faulty theories. Furthermore, while modus ponens construed as a semantic thesis as explained above turns out to be invalid on these theories, modus ponens as a practical inference rule remains unaffected, insofar as asserting or supposing something has the effect of restricting the range of possibilities against which conditionals are evaluated to ones in which that thing holds. In this way, both theories are compatible with modus ponens being “dynamically valid” (for details see Khoo (2013)). It seems reasonable to suppose that this is all the modus ponens we need.
Although the whole of this intricate story could not have been imagined by Gibbard, he certainly was aware in the abstract that theories which, like Kratzer’s and Gillies’, allow embedded indicative conditionals’ semantic values to differ from the semantic values they would get if taken alone, have the resources to avoid his conclusion.
This possibility, now realised in detail by existing theories, was the very escape route that Gibbard identified and gave reason not to take. The assumption that a given indicative conditional sentence in a given context always gets the same semantic value, regardless of whether it is embedded in a larger conditional, is thus an auxiliary assumption of Gibbard’s proof.
3. Why Gibbard Wouldn’t Take the Escape Route
Gibbard’s identification of this auxiliary assumption and his argument against rejecting it are contained in the following passage:
One other possibility remains: that → always represents a propositional function, but that what that function is depends not only on the utterer's epistemic state, but on the place of the connective in the sentence. In a → (b → c), for instance, we might suppose that the two different arrows represent two different propositional functions. Nothing we have seen rules that out.
The pursuit of such a theory, though, has now lost its advantage. A theory of indicative conditionals as propositions was supposed to give, at no extra cost, a general theory of sentences with indicative conditional components: simply add the theory of conditionals to our extant theory of the ways truth-conditions of sentences depend on the truth-conditions of their components. The alternative was to develop a new theory to account for each way indicative conditionals might be embedded in longer sentences, and that seemed costly. Now it turns out that for each way indicative conditionals might be embedded in longer sentences, a propositional theory will have to account for their propositional content, and do so in a way that is sensitive to the place of each indicative conditional in its sentence. In a → (b → c), the right and left arrows must be treated separately. What must be done with the left and right arrow in (a → b) → c or with the arrows in a & (b → c) and a ∨ (b → c) we do not yet know. Thus, for instance, no account of sentences of the form (a → b) → c will fall out of a simple general account of indicative conditionals as propositions; rather the account of indicative conditionals itself will have to confront separately the way left-embedded arrows work. A propositional theory would not save labor; instead it would demand all the labor that would have to be done without it. (Gibbard (1981), pp. 236-237)
The way Gibbard puts it, the assumption at issue is that ‘→ is a fixed propositional function’ (Gibbard (1981, p. 236)), but for present purposes it is the ‘fixed’ part that is relevant, and in view of the possibility of a Kratzerian treatment of the syntax of indicatives, we should separate the ‘fixed’ part out and state it in a way that does not presuppose that ‘→’ is syntactically a two-place propositional operator. Hence our statement of it at the end of the previous section: a given indicative conditional sentence in a given context always gets the same semantic value, regardless of whether it is embedded in a larger conditional. Or in other words again, the assumption is that in a given context, there is no more than one indicative conditional with one set of truth-conditions per pair of antecedent and consequent. Henceforth let’s call this the fixity assumption.
4. The Escape Route is Open
I will give a four-pronged argument against Gibbard’s defense of the fixity assumption. If it is successful, we are left free to abandon the fixity assumption and thus to resist the collapse of indicative conditionals into material conditionals while maintaining a truth-conditional approach to indivatives.
Prong 1. Gibbard’s description of the extra work we must do if we abandon the fixity assumption in the pursuit of truth-conditions for indicatives overplays the amount of extra work required, due to what appears to be a (forgivable) failure of theoretical imagination on his part.
Gibbard says that if we give up fixity, then ‘for each way indicative conditionals might be embedded in longer sentences, a propositional theory will have to account for their propositional content, and do so in a way that is sensitive to the place of each indicative conditional in its sentence’. This may be strictly correct, but it doesn’t follow that such a theory has to confront each form of embedding separately, or that this sensitivity to place cannot come about in an elegant, systematic way.
Indeed, the sensitivity to place of indicatives-inside-indicatives that we need in order to block Gibbard’s collapse argument does come about in an elegant, systematic way on both of the theories we have been discussing. On Kratzer’s theory, it stems from the fact that ‘if’ restricts a modal and that such restriction may occur more than once in a single sentence. On Gillies’, it stems from the fact that ‘if’ shifts index and context, and that such shifting may occur multiple times in a single sentence.
Thus, when Gibbard says that ‘no account of sentences of the form (a → b) → c will fall out of a simple general account of indicative conditionals as propositions; rather the account of indicative conditionals itself will have to confront separately the way left-embedded arrows work’, this - provided that Kratzer’s and Gillies’ theories qualify as ‘simple’ - is simply false. An account of sentences of that form does fall out of both accounts.
Krazter’s and Gillies’ theories deliver, in an elegant way, different semantic values for conditionals depending on where they are in a sentence. And it seems to me that there is a good sense in which these theories are such that we can ‘simply add the theory of conditionals to our extant theory of the ways truth-conditions of sentences depend on the truth-conditions of their components’.
Prong 2. Following Gibbard in embracing the NTV thesis creates special work of its own, which does not have to be done if we hold that they have truth-values.
For one thing, there is an irony in his complaint that if we give up the fixity assumption ‘we do not know’ what to do with the arrow in a sentence of the form ‘a ∨ (b → c)’. In keeping with what we saw in the previous prong, the fixity-denying theories we now have do not encounter any special difficulty in handling such sentences, and we do not have to consider such forms of embedding on a one-by-one basis. Now we may observe further that, if anything, it is the NTV route that leads to issues with such a form; if we deny truth-values to indicative conditionals, we don’t know what to do with the wedge in such a sentence. That is, we face the extra work of making sense of, or denying sense to, embeddings of allegedly truth-valueless sentences in what appear to be truth-functional contexts. (See, however, Edgington (1995) for a classic defense of the view that such embeddings are not problematic after all.)
That is one sort of extra work the NTV theorist seems to be saddled with. And there is another, quite different sort. Namely, the work of explaining what is going on when people appear to ascribe truth-values to indicatives. A truth-value-granting view of indicatives such as Kratzer’s or Gillies’ lets us take these ascriptions at face value, and to allow that they are often correct. An NTV view must either reinterpret these ascriptions so that they aren’t all incorrect, or explain why people so often say these incorrect things. So if we want to avoid extra work, it may be that we do better to uphold truth-value-granting theories like Kratzer’s and Gillies’.
Prong 3. It’s not all about extra work! The issue is whether we should or should not respond to the collapse argument by denying that indicatives have truth-values. To proceed as though this issue turns just on whether we save labor by maintaining that indicatives have truth-values is too narrow. Labor-saving patently isn’t the only reason why we might want to maintain that indicatives have truth-values. A distinct and arguably very important reason is that they seem to have truth-values! (How compelling you find this will depend on your philosophical orientation, but if you think that what pre-theoretically seems to be the case is an important guide in philosophy, it should count for quite a bit.)
Prong 4. Gibbard’s argument against abandoning the fixity assumption obscures the fact that, when you think about it, it makes sense to expect the assumption to be false. Rejecting the assumption is presented by Gibbard as a last resort. But rejecting the fixity assumption is not, on reflection, some intuitively unpalatable thing which we get forced into doing just so that we can uphold a prejudice.
There are well-developed, intuitively motivated views which enable us to think of indicative conditionals, schematically, as saying something like ‘In all relevant possibilities in which the antecedent holds, the consequent holds’. And it is quite natural to think that what is known to be true, or what is being supposed to be true, can affect what possibilities are relevant. Furthermore, it is quite natural to think of the antecedents of conditionals, for example, as introducing a supposition. Putting these last two things together, it is quite natural to think that the possibilities relevant for the ‘if B then C’ in ‘If A, then if B then C’ may differ from the possibilities relevant for an unembedded ‘If B then C’. In particular, it is natural to think that only A-possibilities will be relevant to the embedded conditional, while not-A-possibilities may still be relevant to the unembedded one.
So, the negation of the fixity assumption is something which has quite a bit of plausibility. At the very least, it seems plausible from within the general way of looking at indicatives which the collapse argument is supposed to threaten. Namely, a perspective according to which indicatives have truth-values and in some sense deal with ranges of relevant possibilities. And obviously, such a perspective has much to recommend it besides helping us to resist Gibbard’s argument.
5. Conclusion
Starting from the intuitiveness of the view that indicative conditionals have truth-values which can differ from those of the corresponding material conditional, we looked at how Gibbard’s collapse argument threatens that view, and how Import-Export, flagged as suspicious by Fitelson, is hard to fault. Drawing on work by Khoo, we then saw that both Kratzer’s and Gillies’ independently-motivated theories of indicative conditionals block Gibbard’s argument at the cost of invalidating modus ponens construed as a general semantic thesis, but that the predicted counterexamples coincide with McGee’s independently-motivated ones and leave modus ponens unscathed as a form of dynamically valid inference. We then looked at Gibbard’s argument against truth-value-granting theories of indicatives which, like Kratzer’s and Gillies’, reject the fixity assumption, and saw that the threat is not serious. Gibbard’s refusal to abandon fixity in pursuit of truth-conditions for indicatives stemmed from a failure of theoretical imagination and a too-narrow view of the motivations for non-material, truth-value-granting accounts of indicatives. The prospects for such accounts appear to be brightening.
Edgington, Dorothy (1995). On conditionals. Mind 104 (414):235-329.
Edgington, Dorothy (2014). Indicative Conditionals. In The Stanford Encyclopedia of Philosophy (Winter 2014 Edition), ed. Edward N. Zalta.
Fitelson, Branden (2013). Gibbard's Collapse Theorem for the Indicative Conditional: An Axiomatic Approach. In Automated Reasoning and Mathematics: Essays in Memory of William W. McCune, M.P. Bonacina and M. Stickel (eds.), Springer.
Fitelson, Branden (2016). Two new(ish) triviality results for the indicative conditional. Lecture Notes.
Gibbard, Allan (1981). Two Recent Theories of Conditionals. In William Harper, Robert C. Stalnaker & Glenn Pearce (eds.), Ifs. Reidel. pp. 211-247.
Gillies, Anthony S. (2009). On truth-conditions for if (but not quite only if ). Philosophical Review 118 (3):325-349.
Kaufmann, Stefan (2005). Conditional predictions. Linguistics and Philosophy 28 (2):181 - 231.
Khoo, Justin (2013). A note on Gibbard's proof. Philosophical Studies 166 (S1):153-164.
Khoo, Justin & Mandelkern, Matthew (forthcoming). Triviality results and the relationship between logical and natural languages. Mind.
Kratzer, A. (1986). Conditionals. Chicago Linguistics Society, 22(2), 1–15.
Kratzer, A. (2012). Collected papers on modals and conditionals. Oxford: Oxford University Press.
Lewis, D. (1975). Adverbs of quantification. In: E. L. Keenan (Ed.). Formal semantics of natural
language. Cambridge, MA: Cambridge University Press.

[1] Gibbard leaves the notion of ‘logical truth’ unexplicated in his proof, but the arguments in the present article do not turn on any particular understanding of it.
[2] Contexts, whatever they are, should be thought of as determining ranges or sets of possibilities relevant to the evaluation of conditionals in that context. Cf. Gillies (2009), p. 329 (incl. f.n. 5). Note also that the use of ‘possibilities’ here should not be taken to imply that the possibilities in question are all metaphysical possibilities.

Saturday, 17 August 2019

Imaginary Foundations, the Zombie Argument, and Modal Inertness

Recently published work by Wolfgang Schwarz, to an extent I'd thought impossible, offers an explanation of why there seem to be facts of phenomenal consciousness that we can know for sure. 'Red looks like this!', one says, as one inwardly points at the sensation, and one feels there is an indubitable fact here.

The basic idea, as I understand it, is that by having a cognitive architecture that allows us to keep track of sensation by means of belief-like things that we have full credence in, we can do things more efficiently than without such a mechanism. That's only a very rough explanation, but see:

- The main paper of Schwarz's on this, 'Imaginary Foundations'.
- A second paper outlining some of the ideas from 'Imaginary Foundations' in a more basic way. 

Schwarz's explanation of the seeming existence of phenomenal facts that we know with certainty leaves room for various metaphysical views about whether there are facts of phenomenal consciousness, and whether or not they're reducible to non-phenomenal facts. But Schwarz leans towards a sophisticated sort of eliminativism; it's not wrong to call phenomenal reports 'true', but really they don't state facts about the world. Nor do they seem to state a priori necessary facts, like mathematical sentences seem to - these apparent phenomenal facts have the flavour of a posteriori contingent matters, as Schwarz clarified for me in an exchange on his blog (link below). And so we might think there really aren't any facts here.

- See here for a blog post of Schwarz's, and the comments where he clarifies why it doesn't seem right to think of them as like non-external-worldly facts, the way we might think of mathematical facts. 

Now, one thing I'd like to do is explore the prospects of sticking with Schwarz's explanation of the seeming existence of phenomenal facts, but drawing a different metaphysical moral (or perhaps being critical of the very framework in which the apparent metaphysical options appear).

But another thing I'd like to do is run with the whole Schwarz package - the explanation of the seeming as well as the sophisticated eliminativism about phenomenal facts - and see what adopting this package might enable us to say about the Zombie Argument against materialism, the relationship between conceivability and metaphysical possibility, and the like.

Schwarz has already indicated briefly, in 'Imaginary Foundations' how his explanation of the seeming can explain why a philosophical zombie (p-zombie) - a being just like us but without any inner consciousness - might be conceivable. And similarly, how it might seem like Mary, the colour expert who has never seen red, gets a new bit of knowledge when she finally sees red; the knowledge that red looks like this

What I'd like to do, on this basis, is to put the sophisticated eliminativism in the picture and see what this lets us say about the Zombie Argument. And I have a hunch that there's an interesting and, as far as I know, novel position we could take here.

The idea is basically that we could regard these phenomenal consciousness reports as inert with respect to metaphysical possibility. If you have some big description which is true of some set of metaphysically possible worlds, or even just one, then adding or removing phenomenal consciousness propositions - the ones which we have in our minds for broadly computational reasons, although (by our sophisticated eliminativism) they don't really describe substantial facts - won't affect the metaphysical modal status of the description.

This seems to open up a new way of being a physicalist (or, for that matter, being a non-physicalist who believes in God or has other commitments which makes them a non-physicalist, but is suspicious about irreducible phenomenal consciousness). Physicalism (and other metaphysical views which do not posit irreducible consciousness) is often taken to entail that p-zombies are metaphysically impossible. If you need it to be the case that p-zombies are metaphysically impossible, then in the face of the Zombie Argument (see Chalmers (1999), (2009)), it can look like you really only have two broad options:

- Deny that p-zombies are conceivable in a strong sense (a sort of conceivability which is robust with respect to getting more non-modal facts and getting more rational). 
- Deny that (strong) conceivability entails metaphysical possibility.

But, if phenomenal consciousness reports are modally inert in the way indicated above, a third option presents itself. One can accept that p-zombies are as conceivable as one likes, and that descriptions involving phenomenal fact statements, and zombie-like descriptions where these are all negative, can both count as describing metaphysically possible worlds. By adding or removing phenomenal propositions, one just doesn't change which world or range of worlds one is talking about.

This enables one to both avoid setting implausibly high bars on what we should say is conceivable, and to avoid having to reject the idea that conceivability (of the right sort, and perhaps given the right information). One doesn't have to be a pre-Kripke style modal rationalist. One can accept that there are necessary a posteriori truths, but maintain that given certain relevant empirical truths, the modal situation becomes a priori. (I have a paper on this, and some ideas in my PhD thesis and a paper in progress.)

One issue here is that there might be more than one set of notions that are being called the metaphysical modal notions in contemporary philosophy. One set is friendly to moderate modal rationalism, but the more metaphysically loaded set may not be, or may not even be in good standing at all. I've found Rosen's 'The Limits of Contingency' very illuminating on this point. You might want to be a skeptic about the second, more metaphysical set of 'metaphysical modal notions' while being a moderate rationalist about the first set.

Nevertheless, it looks to me like Schwarz's ideas about the seeming existence of phenomenal facts give us a powerful way to be skeptical about irreducible phenomenal facts (whether this is because of physicalist or otherwise naturalistic predilections or just because of more specific suspicions about ideas about consciousness), while maintaining a (moderate, Kripke-proof) modal rationalism. Could there have been p-zombies? Sure, but that's not actually a different way for things to have been!


Chalmers, David J. (1999). Materialism and the metaphysics of modality. Philosophy and Phenomenological Research 59 (2):473-96.

Chalmers, David (2009). The Two-Dimensional Argument Against Materialism. In Brian P. McLaughlin & Sven Walter (eds.), Oxford Handbook to the Philosophy of Mind. Oxford University Press.

Haze, Tristan (2019). Linking Necessity to Apriority. Acta Analytica 34 (1):1-7.

Rosen, Gideon (2006). The limits of contingency. In Fraser MacBride (ed.), Identity and Modality. Oxford University Press. pp. 13--39.

Schwarz, Wolfgang (2018). Imaginary Foundations. Ergo: An Open Access Journal of Philosophy 5.

Schwarz, Wolfgang (forthcoming). From Sensor Variables to Phenomenal Facts. Journal of Consciousness Studies.

Saturday, 25 May 2019

Logical Pluralism

In this post I will raise an issue for the logical pluralism of Beall & Restall (hereafter 'B&R') - a much-discussed, topic-revivifying view in the philosophy of logic. My study of their view was prompted by Mark Colyvan, whose course on Philosophy of Logic at Sydney Uni I'm helping to teach this year. Thanks to Mark for encouragement and discussion. 

I'll start off in this post by looking at B&R's central thesis, and arguing that it fails to capture an interesting, controversial position in the philosophy of logic. The problem is not that the view is false, but that it's "too easily" true.

I take their 2000 paper ‘Logical Pluralism’ as the starting point, but where more detail is needed, I draw on their 2005 book Logical Pluralism.

Their claim concerns the following schema:
Generalised Tarski Thesis (GTT): An argument is validx if and only if, in every casex in which the premises are true, so is the conclusion. (2005, p. 29.)
(This is more precise than the corresponding schema, (V) (for 'validity'), from their 2000 as it makes clear what is allowed to vary.)

They write:
Logical pluralism is the claim that at least two different instances of GTT provide admissible precisifications of logical consequence. (2005, p. 29.)
Being an existential, numerical claim (‘There are at least two…’), there are many ways the view could be true. Later in this series of posts I'll look at the ways they imagine it coming out true - B&R hold that 'cases' may be taken to be worlds, Tarski-style models, or situations (in the sense of Barwise & Perry's situation theory, at least in the first instance). But here I want to highlight presumably unintended ways in which it comes out true, or at least appears to me to do so. If I'm right about this, these unintended ways threaten to rob their view of its apparent bite.

According to B&R, ‘cases’ may be models ‘Tarskian style’ (2000, p. 480) or ‘along Tarskian lines’ (2005, p. 29).

But this permits differences over exactly what a model is (even for a given language L - to fix ideas, let's consider the language of first-order logic).

For instances, do models provide assignments to variables, or just to names? That depends on how you like to treat quantification when defining 'true on a model'. (Another option, taken by Tarski, eschews assignments to variables in favour of a trick involving sequences of objects.)

Some think which option you take here is philosophically significant - see for instance Smith's 'Truth via Satisfaction?'. But few I think would want to say that not all of these options lead to Tarskian style models (in a broad sense).

But this doesn't actually matter, since there other differences, which seem definitely trivial, over what exactly a model is taken to be in various presentations of first-order logic.

Is a model a tuple of the form <D, I> (where D is the domain and I contains semantic information about all non-logical terms)? Or is it a tuple of the form <D, P, N>, where predicates’ extensions are given separately from names’ referents? Or do we bundle these ingredients informally, as is often done in introductory texts? (That is, do we think that a model is just: a domain, extensions for predicates etc., without thinking of the model as a mathematical object in its own right?)

The point is that these differences, while pretty unimportant, do lead to real differences
over which objects are the ‘cases’. And so to different ‘precisifications’ of the notion of logical consequence.

And so it seems that, if you believe that there is more than one slightly different way of doing broadly Tarksi-style model theory, then you should be a logical pluralist in Beall & Restall's sense. But that seems like the wrong outcome! And in some sense, not what they mean. B&R wanted to carve out and develop a distinctive philosophical view, one which would for instance conflict with the views of someone who thinks that nothing that is not some form of classical logic counts as 'logic'.


Beall, Jc & Restall, Greg (2000). Logical pluralism. Australasian Journal of Philosophy 78 (4):475 – 493.
Beall, Jc & Restall, Greg (2005). Logical Pluralism. Oxford University Press.
Smith, Nicholas J. J. (2017). Truth via Satisfaction? In Pavel Arazim & Tomas Lavicka (eds.), The Logica Yearbook 2016. London: College Publications. pp. 273-287.

Thursday, 25 April 2019

Williamson's Metaphysical Modal Epistemology and Vacuism about Counterpossibles

Timothy Williamson has argued that our capacity for metaphysical modal judgement comes along with our capacity for counterfactual judgement. This passage gives a flavour of his view:
Humans evolved under no pressure to do philosophy. Presumably, survival and reproduction in the Stone Age depended little on philosophical prowess, dialectical skill being no more effective then than now as a seduction technique and in any case dependent on a hearer already equipped to recognize it. Any cognitive capacity we have for philosophy is a more or less accidental byproduct of other developments. Nor are psychological dispositions that are non-cognitive outside philosophy likely suddenly to become cognitive within it. We should expect cognitive capacities used in philosophy to be cases of general cognitive capacities used in ordinary life, perhaps trained, developed, and systematically applied in various special ways, just as the cognitive capacities that we use in mathematics and natural science are rooted in more primitive cognitive capacities to perceive, imagine, correlate, reason, discuss… In particular, a plausible non-skeptical epistemology of metaphysical modality should subsume our capacity to discriminate metaphysical possibilities from metaphysical impossibilities under more general cognitive capacities used in general life. I will argue that the ordinary cognitive capacity to handle counterfactual carries with it the cognitive capacity to handle metaphysical modality. (Williamson, The Philosophy of Philosophy (2007), p. 136. Found in Section 3 of the SEP article 'The Epistemology of Modality'.)
In order to argue for this, Williamson takes a schematic semantic story about counterfactual conditionals:
Where “A>B” express “If it were that A, it would be that B”, (CC) gives the truth conditions for subjunctive conditionals: A subjunctive conditional “A>C” is true at a possible world w just in case either (i) A is true at no possible world or (ii) some possible world at which both A and C are true is more similar to w than any possible world at which both A and ¬C are true.(Formulation from Sec 3 of the SEP article.)
On the basis of this, he proves the following equivalences:

(NEC) □A if and only if (¬A>⊥)
It is necessary that A if and only if were ¬A true, a contradiction would follow.

(POS) ◊A if and only if ¬(A>⊥)
It is possible that A if and only if it is not the case that were A true, a contradiction would follow.

(Renderings and spellings out from the SEP article. The box and diamond are metaphysical modal operators, and the falsum - which looks like an upside-down 'T' - represents a contradiction.)

But this only works because the schematic theory of counterfactuals Williamson adopts is understood as working against a background of a notion possible worlds, where Williamson understands this as the notion of metaphysically possible worlds. This theory deems true all counterfactuals with metaphysically impossible antecedents, and this is crucial to his demonstation of the equivalences on the basis of his assumed schematic theory.

There are lots of reasons not to adopt a theory of counterfactuals which uses the notion of metaphysical possibility in this way. One reason to move away from a theory like this is if you think that there are counterpossibles - counterfactual conditionals with metaphysically impossible antecedents - which have their truth-values non-vacuously. But you might be agnostic about that. For instance, you might be happy with metaphysical modal distinctions but doubt that there are any clear cases where countepossibles have truth-values non-vacuously. In that case you might see no good reason for the backdrop of worlds or scenarios in a theory of counterfactuals to be exactly the metaphysically possible ones. Or, you might be skeptical about the very distinction between metaphysical possibility and impossibility, in which case you won't want to understand the backdrop in a way that involves that distinction. And it seems that you can get most, or even all, of the theoretical benefits of a Stalnaker-Lewis approach to counterfactuals without using that distinction. It doesn't really seem to play a starring role in the theories. 

If a semantic theory for counterfactuals which does not draw on any bright line between metaphysically possible and metaphysically impossible scenarios is as good or better than the theory that Williamson uses to prove his equivalences, that seriously undermines the equivalences, and in turn Williamson's story about how we get metaphysical modal knowledge.

(And note that this turns on Williamson's understanding his chosen theory so that 'possible world' means 'metaphysically possible world' - even accepting an identically worded theory, but where 'possible world' is understood in a way which does not involve the distinction between metaphysical possibility and impossibility, would make Williamson's equivalences unavailable.)

Sunday, 7 April 2019

Contradictory Premises and the Notion of Validity

When evaluating arguments in philosophy, it can be tempting to call an argument 'invalid' if you determine that it has contradictory premises. For example, in an introductory philosophy course at the University of Sydney, students are taught that a particular argument for the existence of God - called the Argument from Causation - is invalid because two of its premises contradict each other. It is tempting to call such an argument invalid because we can determine a priori that it is not sound, i.e. that it isn't both valid and such that its premises are true. But on a classical conception of validity, any argument with contradictory premises counts as valid, since it is impossible for all the premises of an argument with contradictory premises to be true, and so a fortiori impossible for the argument to have true premises and false conclusion.

I have heard this anomaly explained away by appeal to the fact that, while an argument with contradictory premises may count as formally valid, we are looking at informal validity, and in an informal sense perhaps any argument with contradictory premises should count as invalid. But I don't think that's right. If 'formal' is meant to signal that we are not interested in the meanings of non-logical terms and are only interested in what can be shown on the basis of the form of the argument, then that is clearly a different issue: premises could be determined to be contradictory on the basis of form alone, or in part on the basis of the meanings of the non-logical terms. The issue of contradictory premises is similarly orthogonal to the issue of 'formality' if 'formal' is instead meant to signify something like 'in an artificial language' or 'in a precise mathematical sense'. 

In fact, it's arguable that the standard treatment of validity of arguments in classical formal logic should be supplemented, so that an argument counts as valid iff it has no countermodel and its premises are jointly satisfiable.

If we defined 'valid' that way in classical logic, then to test an argument for validity using the tree method, you might have to do two trees. First, one to see if the premises can all be true together. If the tree says No, the argument is invalid and we can stop, but if the tree says Yes, then we do another tree to see if the premises together with the negation of the conclusion can all be true together, and if the tree says No, the argument is valid.

Whether or not it's worth adopting in practise, it is worth noting that this augmented definition of 'valid' in classical logic seems to correspond more closely to the ordinary, informal notion of deductive validity than the usual definition. This even delivers at least one of the desiderata which motivate relevance logic.

However, note that while we seem pretty disposed to call an argument invalid if it has contradictory premises, there is no equally strong tendency to say corresponding things using 'follows from', 'is a consequence of', or 'implies'. This is interesting in itself. It looks like, when we're talking about implication, our focus is on the putative implier or impliers and what can be got out of them, whether or not they're true. By contrast, when we talk about arguments, we're often more focused on the conclusion and whether it is shown to be true by the argument in question, so that validity is treated as one of the things we need to verify along the way. If validity is playing that role, it makes sense to declare an argument invalid if we work out that its premises can't all be true.

Tuesday, 5 February 2019

Two Sources of Interest in Metaphysical Modality (and Kripke in Light of Them)

Source 1: We want to know the vocabulary and syntax of being, or as Rosen puts it in 'The Limits of Contingency', 'the combinatorial essence of the world'. What are the basic elements and how may they be combined?

Source 2: We want to know, as it were in advance, whether various kinds of statements that are not put in terms of the basic elements count as high-level descriptions of any possible world. 

Many of Kripke's arguments that certain kinds of statements are necessary furnish considerations which purport to show that whatever the possibilities are exactly, none of them is correctly described as one in which '...'. 

This mode of argumentation on Kripke's part can make it look like his modal notions can ultimately be explicated along conceptual or semantic lines. But, as Putnam came to appreciate in between 'The Meaning of "Meaning"' and 'Is Water Necessarily H2O?', this is not so.

In this connection it is notable that all of Kripke's distinctive modal theses are negative as regards possibility. (His claims, in the course of arguing against descriptivism about names, that well-known facts about Aristotle etc. could have been different, are an exception, admittedly - but for those arguments, I don't see that he needs these alternative possibilities to be real, metaphysical possibilities. The modality used in those arguments could be deflated to conceptual or semantic without affecting the arguments, which are after all for a semantic conclusion - that names aren't synonymous with descriptions.) As far as I know, Kripke never seriously argues that such and such really is possible, really is a way the world could have been. Rather, he just works on the assumption that there are many quite various ways things could have been, but then seeks to draw some limits in high-level vocabulary.

But these Kripkean results, if that's what they are, only satisfy interest in metaphysical modality that derives from the second source. How we might satisfy our interest that derives from the second source is largely left untackled, and this is one reason why many have found Kripke's work frustrating. He elicits epistemic hopes, someone might complain, without giving us even so much as a roadmap for how so satisfy them.

His suggestion that metaphysical possibility may coincide more closely with physical possibility than has often been supposed may however be suggestive. On the other hand, he is against physicalism, so this couldn't be the whole story from his point of view.

Thursday, 10 January 2019

Rigidity and General Terms: Two Different Analogues of the Singular Case

This post wrestles with and begins to settle on a view about the confusing issue of how Kripke's notion of rigidity may apply to general terms. 

One analogue of rigidity for general terms: how about we think of it as a rigid connection between properties (or alternatively, an additional connection between the "rigid" term and a further property).

So for example, 'water' in the first instance picks out the property of being water, which is tied rigidly to the property of being (mainly composed of) H20.

Or 'cat' is tied in the first instance to the property of being a cat, and that is tied rigidly to the property of being an animal.

But now there is another kind of thing which seems different, and comes up with sentences like 'John has the property we talked about yesterday'.

Suppose the property we talked about yesterday was the property of having the property that is discussed in Book A. And suppose Book A contains a discussion of the property of redness. Now 'has the property we talked about yesterday' rigidly designates the property of being a property we talked about yesterday, but it also non-rigidly designates the property of having the property that is discussed in Book A, as well as the property of redness.

This motivates the picture of, behind a predicate, a stack of properties, where the top one is designated rigidly, and the ones below not.

Problem: we might want to say that 'cat' rigidly designates a certain kind of animal. And I may then want to rephrase that as: 'cat' rigidly ascribes animality. 

But doesn't the 'rigid' bit here fall away? Take the phenomenological, underlying-nature-neutral counterpart of 'cat' - 'catty thing'. Now even if in our world all the catty things are cats, it doesn't sound right to say that 'catty thing' ascribes the property of being a cat at all - it's not that it ascribes that property, only non-rigidly. 

So now it is beginning to look like the distinction we are after here is between a term merely covering things with property P, vs. ascribing to them property P.

But then that seems wrong when we go back to 'John has the property we discussed yesterday', since if what we talked about yesterday was the property of redness, there is a sense in which that sentence ascribes redness to John. 

This is hell!

But this whole problem, occupying the last few paragraphs, perhaps only arises from mixing together two different analogues of rigidity that we get when we look at predicates.

It may be protested that  'John has the property ...' is not a property ascription syntactically at all, but rather a 2-place relational statement with a non rigid second term.

Be that as it may, we can still classify 'has the ...' as a predicate and can still talk about a rigid/non-rigid distinction. And so I think we need to recognise that there are at least two quite different things going on here - two different things which are a bit similar to rigidity/non-rigidity as applied to names.

One is the difference between 'has the property discussed earlier' and 'is red'. Another is the difference between 'is water' and 'is watery' (one brings being composed of H20 along with it in counterfactual scenario descriptions, and the other doesn't), or 'is a cat' and 'is a catty thing'. 

One reason the second analogue may be counterintuitive if presented as a kind of rigidity is that in the case of singular terms, rigidity is associated with simplicity (both syntactic and semantic), but in the case of predicates (let's look at 'is gold', 'is water', 'is a cat' and put aside 'has the property...') it's the opposite. The "non-rigid" predicates just don't take any further property along with them, but the rigid ones do. I.e. 'is a catty thing' or 'is catlike' just picks out one property, but 'is a cat' is tied to the further property of being an animal.

Actually, the 'any' in 'any further property' is probably wrong! Maybe all predicates rigidly take some further properties along with them. So this second sort of "rigidity" we can talk about in connection with general terms should be thought of as relative to whatever further property is in question. (For instance, 'is a pencil' is arguably counterfactually locked to 'is a physical object'. So 'is a pencil' rigidly picks out physical objects - we might want to say something like that.)

One thing that is emerging here is that the 'has the property discussed earlier' vs. 'is red' thing is one distinction which pattern-matches with Kripke's discussion of rigidity as applied to names, but there is also another thing going on - predicates dragging further properties along with them in counterfactual scenario descriptions - which actually corresponds better with Kripke's informal applications of the notion of rigidity to general terms.

Now it looks like the general-term-"rigidity" considerations in Naming and Necessity are actually closer to the "necessity of constitution" and similar considerations than they are to the "necessity of identity" considerations.

Background Reading:



- Kripke, Saul (1980). Naming and Necessity. Harvard University Press.

- Soames, Scott (2002). Beyond Rigidity: The Unfinished Semantic Agenda of Naming and Necessity. Oxford University Press

- Salmon, Nathan (2004). Are general terms rigid? Linguistics and Philosophy 28 (1):117 - 134.

- Linsky, Bernard (2006). General Terms as Rigid Designators. Philosophical Studies 128 (3):655-667.

- Martí, Genoveva & Martínez-Fernández, José (2011). General terms, rigidity and the trivialization problem. Synthese 181 (2):277 - 293.

- Schwartz, Stephen P. (2002). Kinds, general terms, and rigidity: A reply to LaPorte. Philosophical Studies 109 (3):265 - 277.

- de Sa, Dan López (2007). Rigidity, General Terms, and Trivialization. Proceedings of the Aristotelian Society 107 (1pt1):117 - 123.

- Marti, Genoveva (2004). Rigidity and General Terms. Proceedings of the Aristotelian Society 104:131-148.

- Zouhar, Marián (2009). On the Notion of Rigidity for General Terms. Grazer Philosophische Studien 78 (1):207-229.

- Orlando, Eleonora (2014). General terms and rigidity: another solution to the trivialization problem. Manuscrito 37 (1):49-80.

- Gómez-Torrente, Mario (2004). Beyond Rigidity? Essentialist Predication and the Rigidity of General Terms. Critica 36 (108):37-54.

- Kosterec, Miloš (2018). Criteria for Nontrivial General Term Rigidity. Acta Analytica 33 (2):255-270.