Sunday, October 01, 2006

Chapter One [draft]

[Here's a draft - minus footnotes - of the first chapter of my honours thesis, following on from the introduction. Feedback welcome, however small!]

§1.1 Kripkean Semantics and the contingent a priori

I am thinking of a particular person X. He is very well known, and there are numerous expressions in our language – some quite colourful – that could be used to refer to him. For a drab example, I can pick him out by definite description, as ‘the President of the United States’, or else by the name ‘GWB’. These terms are co-extensional: both pick out the same entity X, namely, Bush. But the two expressions differ in modal character. For consider that possible world w where Kerry was elected president instead of Bush. When we apply our terms to that world, we find that their referents diverge. ‘The President of the United States’ picks out Kerry in world w, even though it refers to Bush in our actual world @. Kripke argues that a proper name such as ‘GWB’ refers to the same individual X – Bush – in both @ and w, and indeed in all possible worlds where he exists. A term that picks out the same object in all possible worlds may be called a rigid designator. Names, such as ‘GWB’, and natural kind terms, such as ‘water’, are paradigm instances of rigid designators. They stand in contrast to non-rigid descriptions, such as ‘the President of the United States’, which may be satisfied by different individuals in different worlds.

The content of a rigid designator may depend on how the world actually turns out, as when contingent descriptions are used to fix the reference of a rigid designator. Whatever object happens to actually fit the description, that very object is then denoted by the rigid designator in all possible worlds – even those in which the object no longer satisfies the original description. Hence the reference-fixing description and the rigidly designating term might come apart in non-actual possible worlds. In such a case, we can know a priori that the designated object meets the description, even though it is metaphysically contingent. For instance, we might use a particular stick S to define (fix the reference of) the length ‘one metre’. But once this length is so fixed, we can imagine possible worlds where S is a different length, longer or shorter than its actual length of one metre. ‘The length of S’ and ‘one metre’ would no longer be co-extensional in such worlds, assuming that the latter is a rigid designator. Hence, Kripke argues ‘S is one metre long’ is a contingent yet a priori statement. We don’t need to look at the world in order to learn that it’s true, because however long it actually turns out, that is also how long one metre is. Nevertheless, when we consider a counterfactual possible world, we hold the actual world fixed. That is, when applying the rigid designator ‘one metre’ to another possible world w, it still refers to the actual length of S, rather than to the length of S in w. It does not vary from world to world the way its associated reference-fixing description (‘the length of S’) does.

Of course, Kripke allows that the people in w might use the expression ‘one metre’ to denote the length of S in w. So they could speak truly in uttering the sentence ‘S is one metre long.’ But that is because the term ‘metre’ means something different in their mouths. We are concerned with what our terms mean when applied to counterfactual worlds. This must not be confused with what sentences would be true in the counterfactual world’s language. (There is a possible language in which ‘two’ means five. That does not make our sentence ‘two plus two equals ten’ possibly true in the relevant sense.) In our language, we may stipulate, the term ‘one metre’ rigidly designates the actual length of S. So, in our language, ‘S is one metre long’ is false of a world w where S is longer or shorter than it is in our world.

To clarify this point, consider the statement:
(A) “S is as long as S actually is”
While (A) is presumably a priori, it may yet be false in other possible worlds, so long as the “actually” operator is understood as fixedly referring to our world. There are worlds where S is a different length from what it actually is, after all. So statements like (A) are contingent despite being a priori.

To make this more precise, let us understand the modal operator ‘actually’ as signifying ‘in @’, where ‘@’ names the actual world. Consider some contingent proposition P that is actually true. Because P is contingent, there will be some possible world w at which it is false. At such a world, P is false, despite it being the case that P is true at @, i.e. ‘actually P’ is true. Because ‘P’ and ‘actually P’ differ in truth-value at world w, we find that ‘P iff actually P’ is false at w, and thus not metaphysically necessary. But again, ‘P iff actually P’ is presumably a priori: we don’t need to look at our world in order to know that the things true in it are actually true. It follows trivially from the definition of ‘actually’. This semantic trick makes clear how there could be contingent a priori statements. It is trivial a priori that ours is the actual world, even though there are other possible worlds that aren’t. We can get similar results by rigidifying other indexicals: “I am here now (if I exist)” is presumably a priori – I don’t first need to check where I am before knowing it to be true – but not metaphysically necessary, since I could have been somewhere else. The key is that the referent of ‘here’ will be fixed by wherever I actually am. In this sense, rigid designators epistemically co-vary with their reference-fixing descriptions, hence making the coincidence a priori knowable, despite being metaphysically fixed once we shift our attention to counterfactual worlds.

§1.2 A posteriori necessities

Rigid designators pick out the same entity in all possible worlds. So any two such designators that are actually co-referential will thereby also co-vary across every other possible world. Since ‘Cicero’ and ‘Tully’ both name one and the same person, so that Cicero is Tully, it follows that necessarily, Cicero is Tully. There is no possible world for which our two names pick out distinct individuals. But the sentence ‘Cicero is Tully’ is certainly not knowable a priori. The thought it expresses – and so, derivatively, the sentence itself – is an a posteriori necessity. It is conceivable (in the sense of not being a priori incoherent), but metaphysically impossible, that Cicero and Tully are distinct.

Aside from identity statements, Kripke also argued that a posteriori necessities could be found in claims of natural kind membership, where the kind in question was originally picked out by ostension, as “that kind of thing”. Consider the example of cats: it takes empirical investigation to learn about their underlying nature, or what kind of thing they are. For all we know a priori, it might turn out that cats are automata or demons, rather than animals. But if they share some deep explanatory nature, then whatever it may actually be, they have that necessarily. We might call it their “essence”. Given that our local cats are animals, we would withhold the label “cat” from any (perhaps counterfactual) entity that lacked this internal nature, no matter how superficially cat-like it might seem. Hence ‘cats are animals’ is seen to be a necessary truth, albeit a posteriori. Similarly for theoretical identifications, such as that between water and H2O: whether the identification holds can only be decided by empirical inquiry; but if it does hold, then it does so of necessity. When assessing a counterfactual world, we judge that it lacks water if it lacks the stuff (H2O) that plays the water role in the actual world.

The take-home message of all this is that the connection between apriority and necessity is not nearly so straightforward as claimed in CT. There are all sorts of sentences – e.g. about water’s being other than H2O, Cicero’s not being Tully, or cats being strange demons – that are conceivably true and yet metaphysically impossible. Just because we can coherently imagine something being true, doesn’t guarantee that there is any possible world where it really is so. Modal rationalism looks to be in trouble.

§1.3 Interlude: the state of modal rationalism

Modal rationalism, recall, is the thesis that we have a priori access to the space of possible worlds. The Kripkean counterexamples to CT suggest that there are necessary truths beyond our a priori reach – statements may be true in all possible worlds without our realizing it. But there are two ways to account for this: perhaps we are ignorant of what modal space contains, or perhaps we are merely ignorant of how to describe it. Only the former poses any real threat to modal rationalism; but it is the latter account that we should accept.

It is sometimes supposed that Kripke shrank metaphysical modal space so as to exclude some “conceptually possible” worlds. On this story, we imagine a conceptually coherent scenario and then realize (a posteriori) that it could not come to be. But this is not how we proceeded in the above sections at all. Instead, we imagined a scenario involving (say) cat-like demons, and then we determined that to call those creatures ‘cats’ would misdescribe the scenario. No total possibilities – complete scenarios or “worlds” – have been newly ruled out. Rather, certain descriptions of them have been disallowed. The significance of the Kripkean necessary a posteriori is thus more semantic than metaphysical. It suggests that there are true descriptions of modal space that we are not in a position to recognize a priori. But this need not involve any ignorance of modal space itself. Our ignorance may be purely linguistic.

Let me now sketch the sort of picture proposed by the modal rationalist. Modal space is taken to be rationally transparent, by which I mean the contents of possible worlds are a priori knowable, at least in principle. An ideally rational agent could, by reason alone, come very close to omniscience. For, in a sense, the only thing the ideal agent doesn’t know a priori is which world is actual. Her only fundamental lack is this self-locating knowledge; from that one additional fact, she could know all. (Granted, if we reject Lewisian modal realism, we will think that the significance of actuality goes beyond mere self-location. There’s an important sense in which the actual world is the only one that’s real, and so this one item of ignorance really signifies near-total ignorance of reality. Nevertheless, Lewis’ picture offers a convenient heuristic for thinking about modal knowledge.)

Any truths that are not a priori knowable must, therefore, depend on this crucial unknown fact of which world is actual. Indeed, this is precisely what we find in the Kripkean cases described above. Whether a counterfactual critter counts as a cat depends on whether it shares the same underlying nature as the cat-like critters of our actual acquaintance. The ideally rational agent might know a priori all about the intrinsic natures of all the various cat-like beings across the possible worlds. But, due simply to her ignorance of which world is actual, she cannot know which of those beings are cats. As Jackson explains, “The key point is that the right way to describe a counterfactual world sometimes depends in part on how the actual world is, and not solely on how the counterfactual world is in itself.” The Kripkean cases result from an actuality-relative semantics, whereby what (some of) our words mean depends on which world is actual. The next section will spell out the details behind this, and thereby neutralize the Kripkean threat to modal rationalism.

§1.4 Two Dimensional Semantics

Two-Dimensionalists propose that there are two ways to think about other possible worlds. We can hold the actual world fixed and consider another world as counterfactual, yielding the familiar notion of ‘metaphysical’, or as I will call it here, subjunctive possibility. Alternatively, we may consider a world as actual, which leads to a comparatively neglected notion that I will call indicative possibility. This duality complicates the platitude that S is necessary iff S is true in all possible worlds, for now we have two ways of assessing the truth of S at each world w. Let us say that w satisfies S iff the subjunctive conditional “If w had been the case, S would have been the case” is true. On the other hand, w verifies S according to the indicative conditional: “If w is the case, S is the case.” That is, the truth of S follows from the hypothesis that w is actual, or, roughly, the conditional probability P(S | w is actual) is high. Note that S is subjunctively necessary just in case it is satisfied by all worlds, and indicatively necessary just in case it is verified by all worlds. But these two types of necessity can diverge, as the Kripkean examples show.

Consider a possible world w that is much like ours but that the cat-like creatures in w are secretly demons. If we presuppose the actual animality of cats, and consider w as counterfactual in light of this assumption, we should conclude that there are no cats in w. However, if we consider w as actual, the situation is rather different. Upon seriously entertaining the hypothesis that w obtains, we should rationally be led to the conclusion that cats are demons. What’s more, if we consider our world @ as counterfactual, under the presupposition that w is actual, then we should deny that the cat-like animals in @ are real cats after all – for real cats, we suppose, are demons. The upshot of all this is that the essence of cathood is left an open indicative possibility, but once fixed, it holds of subjunctive necessity. (This is because the essential animality of @-cats, along with the essentially demonic nature of w-cats, is fixed in either sense. What’s left open is simply which of these is found in our actual cats.)

We may generalize and precisify this account with the apparatus of two-dimensional possible-worlds semantics. We begin by associating expressions with an extension, or what they pick out in the world. So here ‘GWB’ and ‘The U.S. President’ both extend to Bush. But recall our earlier discussion (§1.1) of how these terms come apart in other possible worlds. This is reflected in their respective intensions, which are functions from possible worlds to extensions. Intuitively, we may think of intensions as determining what an expression picks out in each possible world. A rigid designator has an invariant intension. Hence, the intension of ‘GWB’ returns Bush for every possible world where he exists, but the intension of ‘The U.S. President’ is more variable, sometimes returning Kerry, for instance.

Two-dimensionalism comes in when we remember that there are two ways to consider a world. While philosophers standardly employ subjunctive or “secondary” intensions, as introduced in the previous paragraph, we might also construct a new semantic value of “primary intensions”, to mirror indicative possibility. Here the function from worlds to extensions reflects our judgment of what the given expression picks out on the hypothesis that the world in question is actual. It is related to ‘verification’ rather than ‘satisfaction’ (as defined above). So, for instance, the primary intension of ‘cat’ picks out the cat-like creatures in each possible world. This is so even though the secondary intension of ‘cat’ is restricted to the kinds of creatures that actual-world cats are.

We may now apply this apparatus to the Kripkean necessary a posteriori: there we have claims that are subjunctively, but not indicatively, necessary. A Kripkean sentence S is satisfied at all worlds although its negation, ~S, is verified at some worlds. S thus has a necessary secondary intension, but a contingent primary intension. All this derives from the simple fact that S is true at all worlds considered counterfactually, but not at all worlds considered as actual. This explains why it is subjunctively (or “metaphysically”) necessary without being a priori. There are ways the world could be such that, if the world is that way, then S is false. It takes empirical investigation to be sure that the antecedent here fails. But given that this is actually the case, the truth of S then holds of subjunctive necessity.

Recall the intuitive argument for CT presented in the introduction. There it was suggested that a claim should be a priori just in case it will be true no matter how the world turns out, i.e. necessarily true. This ran into problems because our actuality-relative semantics entails that whether a claim is true of a counterfactual world may depend on the a posteriori fact of which world is actual. But we are now in a position to see that the original argument can be saved, it simply needs to be interpreted in the indicative mood rather than the subjunctive. There may be a posteriori claims true of all worlds considered counterfactually; but apriority should instead be linked to truth in all worlds considered as actual – that is, verification by all worlds, rather than satisfaction by all. We may thus revise the apriority-necessity coincidence thesis as follows:

(CTI): For any statement S: S is a priori iff S is indicatively necessary.

CTI is still far from trivial, and will be explicitly challenged in later chapters. But for now it suffices to note that the standard Kripkean cases offer no counterexample to it. Further, CTI paves the way for modal rationalism. Indicative possibility reflects the space of possible worlds, albeit under a certain “mode of presentation”, i.e. their being considered as actual. So a priori access to indicative modal truths entails a kind of a priori access to modal space itself. The nature of this connection can be further clarified by the notion of semantic neutrality. Let us say, following Chalmers, that an expression is semantically neutral just in case its primary and secondary intensions coincide. Intuitively: the expression is unaffected by whether we consider worlds as actual or as counterfactual. Hence, a neutral statement will be indicatively necessary just in case it is subjunctively necessary. So even the original CT will at least hold true of this restricted class of statements. (Examples include descriptive terms like ‘cat-like creature’ or ‘watery stuff’, as opposed to the rigidified ‘cat’ or ‘water’.) We may thus restate it:

(CT*): For any semantically neutral statement S: S is a priori iff S is necessary.

Finally, it is plausible that any possibility can be (re-)described in semantically neutral vocabulary, thereby providing us a priori access to it. Modal rationalism is thus restored.


  1. Arr. I have a usage question about one of the examples you use in 1.1 --

    "To clarify this point, consider the statement:
    (A) “S is as long as S actually is”
    While (A) is presumably a priori, it may yet be false in other possible worlds, so long as the “actually” operator is understood as fixedly referring to our world. There are worlds where S is a different length from what it actually is, after all. So statements like (A) are contingent despite being a priori.

    To make this more precise, let us understand the modal operator ‘actually’ as signifying ‘in @’, where ‘@’ names the actual world. Consider some contingent proposition P that is actually true. Because P is contingent, there will be some possible world w at which it is false. At such a world, P is false, despite it being the case that P is true at @, i.e. ‘actually P’ is true. Because ‘P’ and ‘actually P’ differ in truth-value at world w, we find that ‘P iff actually P’ is false at w, and thus not metaphysically necessary."

    Is it common to have 'actual' rigidly designate the actual world? I realise you're jumping from Kripke here, but -- at least in On the Plurality of Worlds -- David Lewis appears to want it a bit more flexible than that, where 'the actual world' is non-rigid and is used to denote the 'own world' of whichever counterfactual citizen was under discussion.

    I've never seen it used to rigidly designate our world. That said, it doesn't seem to interfere with the logic of the example, and it may be that what you're doing is legitimate. [I haven't read much of this stuff.] Is it common to use 'actual' in the way that you have?

  2. Hmm, I think it's pretty standard to treat 'actually' as a rigidifying operator. We can talk about Twin Earth, and how it lacks any of "the actual watery stuff". Our use of "the actual president" will pick out Bush, even when we apply it to Kerry-worlds, etc.

    But the standard use might be as a rigidified indexical ("this world"). Then folk in Twin Earth would speak truly when they say "the actual watery stuff - i.e. water - is XYZ". But that's just because their term "actual" (like "water") has a different semantic value from ours. I might need to be clearer about that. But the key logical point is clarified in one of my missing footnotes: for any possible world w, [our sentence] ‘actually P’ is true in w iff P is true in @. See, e.g., Davies and Humberstone (1981) ‘Two notions of necessity’.

    See also this old post for a related (but slightly messy) discussion.


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