Sunday, 12 July 2015

An Account of the Analytic/Synthetic Distinction

Some of the intuitive characterisations given, in the last post, of the notion of internality of truth-value – such as 'internal meaning determines truth-value' - sound a lot like a common post-Kantian way of characterising or defining analyticity, namely as 'truth in virtue of meaning'. This raises the question of whether the class of a priori truths is the class of analytic truths, and the question of whether there are, or should be, distinct notions here at all. My answers to these questions will be No and Yes respectively.

The aim here will be to try to clarify an interesting notion of analyticity which is conceptually and extensionally distinct from all the notions of truth a priori identified in the last post (internality of truth, non-Twin-Earthability of truth, Chalmers' epistemic two-dimensionalist account, and traditional conceptions). It is distinct from, but builds on, our internality conception of the a priori.

The account I will give of this notion is inspired by Kant's account of the analytic-synthetic distinction in the Critique of Pure Reason, as well as Wittgenstein's remarks on the synthetic a priori and concept-formation in the Remarks on the Foundations of Mathematics.

It is well known that Kant's definition, or principal explication, of 'analytic' and 'synthetic' is given in terms of subject and predicate:

In all judgments wherein the relation of a subject to the predicate is cogitated (I mention affirmative judgments only here; the application to negative will be very easy), this relation is possible in two different ways. Either the predicate B belongs to the subject A, as somewhat which is contained (though covertly) in the conception A; or the predicate B lies completely out of the conception A, although it stands in connection with it. In the first instance, I term the judgment analytical, in the second, synthetical.

Since modern logic and philosophy of language has taught us not to regard every proposition as being composed of a subject and a predicate, this definition can't be adequate for us. But it is suggestive, and even moreso are some of the other things Kant says about the analytic-synthetic distinction. He says of analytic and synthetic propositions respectively that 'the former may be called explicative, the latter augmentative'. And consider this elaborated version he gives of his main question, that of how synthetic a priori knowledge is possible: 'If I go out of and beyond the conception A, in order to recognize another B as connected with it, what foundation have I to rest on, whereby to render the synthesis possible?'.

The idea that synthetical judgments are 'augmentative', that they 'go out and beyond' 'conceptions', can, I think, be generalized or abstracted from Kant's discussion in such a way that it does not depend on construing all propositions as being of the subject-predicate form. And we get a hint of how to do this from the following passage about the syntheticity of the proposition '7 + 5 = 12':

We might, indeed, at first suppose that the proposition 7 + 5 = 12 is a merely analytical proposition, following (according to the principle of contradiction) from the conception of a sum of seven and five. But if we regard it more narrowly [my emphasis], we find that our conception of the sum of seven and five contains nothing more than the uniting of both sums into one, whereby it cannot at all be cogitated what this single number is which embraces both. The conception of twelve is by no means obtained by merely cogitating the union of seven and five; and we may analyse our conception of such a possible sum as long as we will, still we shall never discover in it the notion of twelve. We must go beyond these conceptions, […]

This regarding-more-narrowly will be the key for us. We said above that a proposition is a priori iff it contains its truth value, i.e. iff its internal meaning determines its truth-value. Our idea now is that a proposition is analytic iff its internal meaning regarded more narrowly in a certain way – or iff a certain sort of part or fragment of its internal meaning – determines its truth-value. And so the next task is to try to clarify what characterises the aspects of internal meaning we are restricting our attention to here.

To do this, we will use the notion of concept- or conceptual-structure-possession, and the notion of understanding. We will not need to involve considerations of knowledge, judgement, being-in-a-position-to-see-that, or anything like that. (Later, we will consider how what we say may shed light on accounts which do involve such considerations.)

As a first approximation, we will say that a proposition is analytic iff the bits of conceptual structure – the part of its internal meaning - one must possess in order to understand it, determines its truth-value. (This involves a terminological departure from the possibly more common procedure of regarding analyticity as implying truth – we say that an analytic proposition can be true or false, just as we say an a priori proposition can be true or false. This has the nice feature of giving us a simple division among propositions in general, not just truths, so that we can say that for propositions in general, being analytic is just not being synthetic, and vice versa.)

We can make this definition easier to handle and more memorable by giving it in two parts:

The meaning-radical of a meaningful expression consists in the bits of conceptual structure, i.e. the part of its internal meaning, one must possess in order to understand it.

A proposition is analytic iff its meaning-radical determines its truth-value.

(As an added bonus, we now have the general concept of a meaning-radical, which we can apply to sub-propositional expressions as well as propositions, and perhaps also to super-propositional expressions such as arguments.)

Consider the fact that we can come to believe false arithmetical propositions - for example on the basis of miscalculation, or misremembering, or false testimony - and that we can apply them.

Contrast the case of a paradigm analytic proposition, such as 'All bachelors are unmarried'. (To get around the irrelevant problem that in English 'bachelor' very arguably doesn't mean 'unmarried man', let us just suppose that it does mean exactly that.) To be sure, someone can assent to the sentence 'Not all bachelors are unmarried', and dissent from 'All bachelors are unmarried', but in such a case we would say that they don't understand this latter as we do – they don't understand our proposition 'All bachelors are unmarried'. So they don't believe – and here we are using words with our meanings kept intact – that not all bachelors are unmarried.

Kant says that we can become 'more clearly convinced' of the syntheticity of arithmetical propositions 'by trying large numbers'. Let us now, therefore, try to illustrate the notion of a meaning-radical, and in turn that of analyticity, by considering an example of a false arithmetical proposition involving numbers larger than 7, 5 and 12. Say '25 x 25 = 600'.

Despite being false a priori, the proposition '25 x 25 = 600' is something we can mistakenly believe and apply while still understanding it correctly (in some suitably minimal, and natural, sense of 'understand'). We have – wrongly – made a connection between our conception of the product of 25 and 25, and our concept of 600.

Why do we say that we understand the proposition in its ordinary sense and are wrong, rather than saying that we are operating in a different system, in which the sentence '25 x 25 = 600' is true, and that we (therefore) don't understand the proposition in its ordinary sense? It is not hard to see what sorts of things make it that way. If we worked it out on a calculator, or calculated it again ourselves, we would unmake the connection. Such developments would show that we did understand the proposition correctly (i.e. in its ordinary sense).

(Suppose an illegal move is made in chess, say that someone moves their king into check (so that it need not be immediately obvious that it is an illegal move). If the maker of this move can easily be brought to accept that their move was illegal, we can maintain that they understand how to play chess and were playing it according to the ordinary rules, but playing wrongly. If they cannot, then either they simply do not understand chess, or are insisting on playing according to deviant rules.)

The fact that in the case of the false belief that '25 x 25 = 600', there is this other option here, if I may put it that way, of saying that we are operating in a different system – an option which we will have to reject because of many things about how things are, so in that sense not an option, but still something which makes sense – shows that there is a possible system, compatible with the meaning-radical of '25 x 25 = 600', in which that sentence holds. That is, the meaning-radical of '25 x 25 = 600' – that bit of conceptual structure – can be incorporated into a larger structure wherein the concept of the product of 25 and 25 (although we might not want to call it that anymore) is connected to that of 600 in such a way that the sentence '25 x 25 = 600' is true. It will have a different internal meaning from our proposition '25 x 25 = 600', despite the system it belongs to incorporating the meaning-radical of our proposition. This is what makes '25 x 25 = 600' synthetic.

On this picture, the full internal meaning of a concept or proposition-meaning may involve connections which do not have to be made in order to understand it.

So, a proposition is analytic iff it has its truth-value in virtue of the bits of conceptual structure someone has to possess in order to understand it. That is, iff the bits of conceptual structure one must have in order to understand it cannot be embedded in a context such that the proposition-radical of that proposition gets a completion such that the resulting proposition has a different truth-value from the proposition in question.

More briefly, a proposition is analytic iff its meaning-radical determines its truth-value.

All a priori propositions, then, on the account I am giving here, will be such that their internal meanings determine their truth-values. But analytic propositions have the further property that their radicals determine their truth-values, whereas the radicals of synthetic a priori propositions can be incorporated into both true propositions and false ones.

A Complication

One complication: perhaps there is a mistaken assumption of uniqueness built into my talk above of the meaning-radical of a proposition. The bit of conceptual structure one must possess in order to understand it. Perhaps one and the same proposition can be understood from more than one angle, as it were, in which case it may be better to talk about multiple meaning-radicals – distinct bits of conceptual structure all of which individually and minimally suffice for understanding.

This gives rise to a choice: if that's how things are, should we call the analytic propositions those which are such that all their meaning-radicals determine their truth-values? Or those which have at least one meaning-radical which determines their truth-value?

I do not want to try to settle the issue of whether we should recognize a possibility of multiple radicals. Furthermore, I have no opinion about which use of terminology is best, in case we should – perhaps it just doesn't matter. If we use 'analytic' for the first, we may say 'weakly analytic' for the second. Or, if we use 'analytic' for the second, we may say 'strongly analytic' for the first. Or we might make 'analytic' mean 'either strongly or weakly analytic'. Or drop it entirely, and always specify 'strong' or 'weak'.

Sunday, 14 June 2015

On a Semantic Account of the A Priori

This post is an attempt at stating and evaluating an approach to analyzing the concept of apriority as it applies to propositions. 

Follow-up: An Account of the Analytic/Synthetic Distinction

My conception of propositions, which sees them as having an internal nature constituted by their place in the system of language and thought to which they belong, but sometimes also external projective relations to the world, is highly suggestive of an approach to analysing the a priori-a posteriori distinction.

Using 'a priori' in such a way as not to imply truth, so that a proposition can be a priori true or a priori false, we might give expression to the basic idea by saying:

A proposition is a priori iff either truth or falsity is an internal property of it.

Or, using 'a priori' like 'necessary', in such a way as to imply truth (that is, meaning what 'true a priori' means on the above usage):

A proposition is a priori iff truth is an internal property of it.

(We will stick to the first usage.)

An account of this sort is attractive from my point of view for two related reasons. Firstly, it explains the feeling that definitions or accounts of the a priori which involve considerations of a knowing subject, and what such a subject can (in some sense) do without need (in some sense) of experience (in some sense), fail to get at the heart of the matter. That is, the feeling that a proposition's being a priori or a posteriori is a matter of the nature of the proposition itself, and that the stuff about being able to know an a priori proposition independent of experience (in some elusive sense) holds as a consequence of that proposition's being a priori, rather than constituting that property.

Secondly, many philosophers are skeptical of the notion of the a priori, or of the idea that anything genuinely falls under it. But topics at the centre of the present work, such as the existence of the necessary a posteriori and the contingent a priori, involve a notion of the a priori, and furthermore, one which is held not to be vacuous. If I can provide a new account of the notion I intend here, perhaps these skeptics will be able to enter into my discussion further than they would be able or willing to otherwise.

On this account, the concepts 'a priori' and 'a posteriori' are broadly logical. They can also be called epistemological if one wishes, but there is a danger in that, since by itself it leaves one free to overlook the distinction between properties like that of being a priori, which have to do with the nature of the propositions which possess them, and blatantly epistemological properties like that of being known, that of being hard to understand, that of being easy to verify, etc., the very constitution of which involves relationships to knowers.

That said, it is of course open to anyone to stipulate that 'a priori' is to have a meaning given in terms of a knowing subject and what they can do. But I would think of the word in that usage as expressing a property the possession of which is explained by possession of the property expressed by the word as I use it - a broadly logical property. I use 'a priori' that way because I find this latter to be of more fundamental interest, but I'm not concerned to insist on or argue for such a usage. The point about a broadly logical property explaining, or having as a consequence, stuff about what a knowing subject can do is supposed to help motivate the notion or notions I want to propose, but even that is secondary. My primary concern is just to propose them and try to make them clear

What do I mean by saying that a priori propositions' truth-values are internal to them? I do not mean that a priori propositions – individuated the way we are individuating propositions here, such that external projective relations are held fixed – necessarily have the truth-values they have. (In that case, it would be difficult or impossible to allow the class of a priori truths to differ from the class of necessary truths.)

The thought is, rather: with many propositions, their internal meanings – that is, their positions in the language-system to which they belong – do not by themselves determine a truth-value; rather, this depends on their external connections to reality, and what lies on the reality end of the connections. But with the a priori propositions, there is no such dependence; internal meaning determines truth-value. Or we might use the pair of locutions 'in virtue of' and 'irrespective of': a posteriori propositions have the truth-values they have partly in virtue of what (if anything) lies on the reality end of external projective relations borne by them, whereas a priori propositions have the truth-values they have irrespective of that. We might even simply say that their internal meanings, in contrast to a posteriori propositions, have truth-values already, all by themselves.

Is Our Notion of A Priority Explicable in Terms of Twin-Earthability?

I think this idea, as explained in various ways above, and guided by our pre-existing, traditional conception of a priority, can have considerable philosophical value. It will turn out to be worthwhile, however, to see what happens if we try to explicate this notion of internality of truth-value by means of the concept of Twin Earthability. We may say:

A proposition is Twin Earthable iff in some possible situation, a proposition with the same internal meaning has a different truth-value.

And then:

A proposition is a priori iff it has its truth-value internally iff it is not Twin Earthable.

This seems a natural strategy, since if a proposition has its truth-value irrespective of its external connections to reality, then Twin Earthing it shouldn't be able to change that. And that is correct, but for the strategy to succeed, we also need it to be the case that Twin Earthing is unable to change a proposition's truth-value only if it has this truth-value irrespective of its external connections to reality. And certain kinds of propositions might seem like counterexamples to this, such as 'I exist', 'Language exists' (where 'language' means concrete linguistic phenomena) and 'I am uttering a sentence now' (where 'uttering' is taken to mean a spatiotemporal process).

I am indebted to discussions of Chalmers ('The Foundations of Two-Dimensional Semantics', 'Epistemic Two-Dimensional Semantics') for these examples, and for the notion of Twin Earthability (although he defines it over sentence-tokens), which Chalmers derived in turn from Putnam's famous Twin Earth thought experiment.

In Chalmers' discussion, the examples are given as counterexamples to a different sort of account of a priority, and for different reasons. Chalmers' target is a particular class of interpretations of what he calls 'The Core Thesis': 'For any sentence S, S is a priori iff S has a necessary 1-intension'. Namely, those interpretations on which intensions are regarded are 'any sort of linguistic or semantic contextual intensions'. (Chalmers goes on to argue for what he calls 'epistemic intensions'.) Chalmers is testing this account against a notion of a priority understood along traditional lines, in terms of knowledge and experience. We, on the other hand, are, at least in the first instance, testing non-Twin-Earthability against our notion of a priority as internality of truth-value. (We will eventually come back and consider how these two notions - non-Twin-Earthability and internality – might line up with a more traditional notion involving knowledge and experience.)

First, it might seem that 'Language exists' and 'I am uttering now', construed the way they are supposed to be construed, namely as implying certain spatiotemporal goings-on, do not carry their truth-values inside themselves. It might seem like they must reach out and get their truth from those spatiotemporal goings-on.

They have non-Twin-Earthability, not because they are true irrespective of what lies on the reality end of their external projective relations to reality (that isn't the case), but rather because any Twin-Earthing of them, any proposition with the same internal meaning, is sure to bear external projective relations to reality such that what lies on the reality end of these relations makes the proposition true. (They are non-Twin-Earthable for what we might call transcendental reasons: their instantiation guarantees their truth; their truth conditions are subsumed under their instantiation-conditions.)

Regarding 'I exist', we might think: this cannot be true in virtue of internal meaning, since what makes it true is that I actually exist – this proposition reaches out to me.

There are numerous unclear and discomforting things about all this, however, especially the 'I exist' case. (If we imagine 'I exist' reaching out to us, aren't we thinking of our bodies? And isn't there a way of construing 'I'-propositions such that they don't imply the existence of bodies?). We will try to address them below.

(To anticipate, since this may ease comprehension: I will end up going along with the train of thought above regarding 'Language exists' and 'I am uttering now', but rejecting it for 'I exist', provided 'I' is construed so as not to imply the existence of a body. These are very confusing topics, however, and I don't wish to be dogmatic.)

On the Peculiarity of these Examples

First, a word about the peculiarity – the weird basicness, so to speak – of these examples.

Throughout my investigation of the difficulties arising with these examples, I have been heartened but also troubled by how peculiar they are – all of them, but especially 'I exist'.

Heartened, because since this proposition ('I exist') is so peculiar, the fact that it raises confusing problems in connection with our analysis of a priority should worry us less than if a less peculiar, more worldly sort of example raised these problems. The confusing problems presumably have a lot to do with the peculiarity of 'I exist', and we know that “the first person” raises confusing problems anyway.

It might even look like this is a peculiarly philosophical proposition, what Wittgenstein might have called a pseudo-proposition, especially when we reflect that it may not even concern a body.

But – and this is why it is troubling – that wouldn't absolve us in any clear way of the need to square our account of a priority with it – it is one thing to be dismissive of certain propositions when you aren't putting forward an account of a general notion in propositional typology, but I am doing that: I'm saying that a proposition (any proposition) is a priori if it has its truth-value internally.

That leaves the option of arguing that 'I exist' is no proposition at all, but that's not an inviting prospect. It seems dogmatic and ad hoc.

Secondly, and connected with this last point: it isn't clear that 'I exist' really is a peculiarly philosophical proposition with no practical use. Couldn't people (indeed, don't they sometimes?) assert 'I exist' in order to make someone more mindful and just toward them? And couldn't someone whose existence is in doubt, but whose supposed appearance is well-known, appear to the doubters and say 'I exist'? (Furthermore, aren't these instances of the very same (internal) proposition-meaning as Descartes tried to establish with the cogito? And that's not to say that weird, peculiarly philosophical things weren't happening there. Clearly they were.)

Accordingly, I will face up to these difficulties as best I can. I will now discuss 'I exist' and argue that it can be maintained to have its truth-value internally, when 'I' is construed so as not to imply the existence of a body. Following that, I will consider the other examples, 'Language exists' and 'I am uttering now'. (These also seem peculiar, and this seems to have to do with their being good candidate instances of Wittgenstein's remark in On Certainty (#83): The truth of certain empirical propositions belongs to our frame of reference. We will discuss this in turn.)

'I Exist'

Here is one consideration which throws doubt on the idea that 'I exist' doesn't have its truth-value internally, since it must reach out to me – i.e. a consideration which suggests that perhaps it does have its truth-value internally after all.

Suppose a computer system used by an administrator allows them to enter queries such as 'user23 space?', and the computer will tell them how much disk space user23 has left. The administrator's username is 'admin', so they can find out how much disk space they have left with 'admin space?'. But to save inputting time, a first-personal pronoun is introduced, so that the administrator and other users can enter 'I' place of their username in such queries. Let us suppose that a command involving 'I' is, under the hood, first transformed so that its occurrences of 'I' are replaced with the username of whoever is logged in (this username, we may suppose, is sitting in a particular memory location, ready for this purpose), and then passed on to be executed.

Now suppose there is a query 'loggedin?', so that the administrator can find out if user23 is logged in with 'user23 loggedin?'. When the administrator enters 'I loggedin?', the computer first looks up the username of the current user where it waits in memory (in this case 'admin'), plugs that in in place of 'I', and sends that to be executed. Suppose the system now searches a constantly updated list for the name 'admin', and returns 'Yes' if it reaches that name, 'No' if it reaches the end without finding it.

There is an obvious inefficiency here. Pointless as the 'I loggedin?' query might be in practise, if for some reason it had to be executed a trillion trillion times (say, at the whim of a rich eccentric, or for an art project), it might be worth modifying the software so that queries containing 'I' are first checked for 'loggedin?', in which case 'Yes' is immediately returned. And note that we don't need to first study the results of 'I loggedin?' working in this inefficient way – don't need to study what's on the dynamic list of users – in order to see this. We look into the system and see that it can only go one way.

It seems to me that if we think of an instance of 'I exist' as functioning like 'I loggedin?' in the unmodified computer system, we may be inclined to think that its truth-value is not internal to it, but if we think of it as functioning more like 'I loggedin?' in the modified system, we will judge that its truth-value is internal to it. On that way of thinking of it, we might say there is a tight conceptual connection, an empirically indefeasible connection, between 'I' and 'exist'. (More on conceptual connections in a future post.)

But what about the first option: thinking of 'I exist' as functioning like 'I loggedin?' in the first system? What might this come to? Well, the truly analogous procedure would be something like: when I ask myself whether I exist, I take 'I' as an abbreviation for my name, and then accordingly ask myself 'Does TH exist?', and then I use the same method I use when I consider whether someone else exists – perhaps trying to observe my body, or traces of my activity. That is plainly not how it works. But perhaps there is a half-way reasonable sort of procedure we can imagine, which contrasts with the immediate verification of 'I exist' based on a tight conceptual connection. Descartes' procedure comes to mind: I must exist, since I would have to exist in order to be thinking about whether I exist. (Perhaps there is a tight connection between 'I' and 'think.)

But this is not an empirical procedure, and it doesn't involve looking out into the world. This suggests that 'I exist', despite what we might have been inclined to say (perhaps on the basis of thinking of 'I' as requiring a body), can be maintained to have its truth-value internally, and so be a priori according to the account proposed.

Here I am using notions like 'empirical procedure', and so making appealing to elements of the more traditional, pre-existing conception of a priority. This would be fishy if I were trying to fully explain those existing conceptions by means of other, independently understandable conceptions. But that's not how I think of it: traditional conceptions of a priority help explain internality of truth-value as well as vice versa. To anticipate: we will end up with two mutually supporting conceptions, giving us two angles on a single important division in propositional typology.

'Language Exists' and 'I am Uttering Now'

What about 'Language exists' and 'I am uttering now'?

Remember, it is stipulated that 'language' and 'uttering' here mean physical, spatiotemporal phenomena.

It seems to me that, so interpreted, these cases clearly do not have their truth-values internally. They do reach out and get their truth, not just their meaning, from outside language – that train of thought is one I want to go along with. (Its not applying, despite possible appearances, to 'I exist', turns on the fact that 'I exist' as construed does not imply the existence of a body.)

There are, however, superficial appearances to the contrary. When we ask ourselves 'Does language exist?' and 'Am I uttering now?', we don't actually do any “looking out into the world”, as I have, speaking figuratively, said these propositions themselves do. It seems that we just immediately, or after a moment's reflection, see that the answer is 'Yes' in both cases. Just like with 'Do I exist?'.

And yet I am insisting that there is a fundamental difference of category here. I think we can get clearer here by considering further the category that 'Language exists' and 'I am uttering now' belong to – considering, that is, what is characteristic of them as opposed to the propositions I want to call a priori?

I think this was one of the main topics, if not the main topic, of Wittgenstein's last work On Certainty. Wittgenstein was responding to Moore's work in epistemology – his 'Defence of Common Sense', wherein he claims to know he has a hand. Apparently Normal Malcolm had some role turning Wittgenstein's attention to this late in the latter's life.

The following remarks seem to characterise the peculiar class to which 'Language exists' etc. belong (I will them the 'Moorean' propositions):

83. The truth of certain empirical propositions belongs to our frame of reference.

136. When Moore says he knows such and such, he is really enumerating a lot of empirical propositions which we affirm without special testing; propositions, that is, which have a peculiar logical role in the system of our empirical propositions.

137. ... Moore's assurance that he knows... does not interest us. The propositions, however, which Moore retails as examples of such known truths are indeed interesting. Not because anyone knows their truth, or believes he knows them, but because they all have a similar role in the system of our empirical judgments.

138. We don't, for example, arrive at any of them as a result of investigation.

Note that these Moorean propositions won't generally be non-Twin-Earthable, like 'Language exists' and 'I am uttering now' (which are clearly special in that their truth is guaranteed transcendentally, i.e. by the preconditions of formulating them). It is the combination of non-Twin-Earthability and Mooreanness which makes these cases, and their nature, require special consideration from the point of view of our account of a priority as internality of truth-value.

Wittgenstein calls the peculiar role of Moorean propositions a 'logical' role, and it is not hard to see why: that we have empirical propositions playing this role, i.e. being part of the framework which guides thought and enquiry, is part of how our practise of thought and inquiry works.

But a proposition's playing this role is not an immutable or intrinsic fact about that proposition, and that is the very thing – I want to say – which makes Moorean propositions empirical rather than a priori.

Here is a metaphor I find helpful: imagine a cluster of metal frames, some of which have two feet (I will call them 'biframes'), others of which have three ('triframes'). The biframes need support in order to stand – they are build to be able to stand or fall. The triframes can stand by themselves. At the periphery of the cluster there are both biframes and triframes, and some of the biframes are held up by bits of wood. No biframe could be supported by wood alone; but some are held up by frames and wood, others by frames alone (which may themselves be held up by wood, or by further frames held up by wood, etc.). Any biframe, if you rearranged the cluster sufficiently and removed certain bits of wood, would fall, but some may be very hard or impossible to get at. (On Certainty #255: 'What I hold fast to is not one proposition but a nest of propositions'.)

Frames correspond to propositions. Their standing corresponds to truth or belief. Wood corresponds to experience, empirical confirmation. Triframes correspond to a priori propositions, biframes to empirical propositions.

The biframes which stand without being in contact with wood, i.e. which are supported by other frames, correspond roughly to Moorean propositions: the special category of empirical propositions with which we are concerned, and with which Moore and Wittgenstein were concerned. Or perhaps better: out of these standing biframes not in direct contact with wood, some are more robust than others with respect to the removal of bits of wood. Some might fall right away if you remove a single bit of wood near them, and perhaps we should not count them in this category (depending on how exactly we project the metaphor). Likewise perhaps those which would fall if a small number of easily identifiable bits of wood were removed. Then, the biframes which correspond to propositions of this special category – 'Language exists', 'I have a hand' etc. – are those which don't clearly depend on any particular bits of wood.

These special biframes are no mere epiphenomena, however: they belong to our 'frame of reference'. So in a sense they are foundations:

401. I want to say: propositions of the form of empirical propositions, and not only propositions of logic, form the foundation of all operating with thoughts (with language). ...

But this is then squared with, for example, the metaphor of biframes, with this brilliant remark:

248. I have arrived at the rock bottom of my convictions. And one might almost say that these foundation-walls are carried by the whole house.

In other words: don't be mislead by a “bottom-up” metaphor into thinking that Moorean propositions are not themselves supported, in a diffuse way, by non-Moorean empirical propositions. The image of the cluster of metal frames (or Wittgenstein's 'nest') helps avoid this: we do not imagine the cluster extending high into the air, but rather getting denser and covering a wider area.

And what's the difference between these and what we call a priori propositions? In our frame metaphor, the correlates of a priori propositions – triframes – stand by construction. Here is what Wittgenstein says about the difference:

655. The mathematical proposition has, as it were officially, been given the stamp of incontestability. I. e.: "Dispute about other things; this is immovable - it is a hinge on which your dispute can turn."

656. And one can not say that of the proposition that I am called L. W. Nor of the proposition that such-and-such people have calculated such-and-such a problem correctly.

657. The propositions of mathematics might be said to be fossilized. - The proposition "I am called...." is not. …

Putting this 'fossilization' talk from On Certainty together with the bits of wood from the metal frame metaphor above, we might say: bits of wood supporting biframes can petrify and become part of the frames themselves. (But that is not a plausible story about the actual origin of each a priori proposition taken individually. Perhaps it is plausible, however, that when we enter new regions of the a priori, so to speak, we often do so in this way. Or that, when humankind apprehended a priori propositions initially, we got there in this way. But this is all by the by.)

But in this case, fresh wood may still grow in the same area. And this sheds light on a distinct metaphor from Wittgenstein's Remarks on the Foundations of Mathematics which helped inspire the one about metal frames above: A mathematical proposition stands on four feet; it is over-determined.

This isn't the same metaphor, but I wager that these four feet do not all correspond to the same sort of thing: one of them can be distinguished as the 'fresh wood' alluded to above: for example, nothing can refute '7 + 5 = 13', it is not held hostage to experience (as we can see from the standard, convincing critiques of Mill's view that arithmetical propositions are empirical generalizations), and yet!: experience bears it out in some sense. We put seven and five things together, and we usually find we have thirteen. Or better: our experience is such that this proposition is useful, is possible even (i.e. is something we can grasp, use, instantiate).

Another Angle on the Difference Between Moorean and A Priori Propositions

Another angle from which to see the difference between Moorean propositions and a priori propositions, in particular 'I exist' as we construed it in the previous section, is with a thought experiment involving experiences radically different from those most of us have had. Suppose everything went black and all bodily (kinaesthetic) sensation ceased, and a voice, claiming to be a demon, or some kind of scientist, but not belonging to the world of our experience, announced that space, or everything in it, has been destroyed, so that there is no language (construed as spatiotemporal occurrences) and no uttering (construed likewise).

If this happened to me, I would be unable to refute this. Or at least, no immediate knock-down objection would come to mind; the most near-to-hand strategy for refuting it would probably involve appeal to a belief in psycho-physical parallelism, which the demon would gainsay. I wouldn't know what to think.

If, on the other hand, the voice told me that I no longer exist, it would be totally different: I would steadfastly deny that, and nothing the demon could say to me would shake my belief that I exist. We will come back to this in a moment, when we consider how the conceptions of internality of truth-value and non-Twin-Earthability line up with the traditional conception of a priority.

To summarize our conclusions so far:

'I exist' is a priori (i.e. has its truth-value internally) and non-Twin-Earthable.

'Language exists' and 'I am uttering' are a posteriori and non-Twin-Earthable.

This is when we construe 'I' as not implying the existence of a body, and 'language' and 'uttering' as meaning spatiotemporal phenomena. If we construe 'I' bodily, 'I exist' falls in line with the other two. Conversely, if we construe 'language' and 'uttering' as not necessarily being spatiotemporal, 'Language exists' and 'I am uttering now' may be argued to fall in line with 'I exist'.

So, we appear to have two different notions here, with different extensions, both of which can be legitimate parts of our typology of propositions: internality of truth-value, and non-Twin-Earthability. We shall continue to reserve the term 'a priori' for the notion of internality of truth value. All a priori propositions are non-Twin-Earthable, but not the other way around.

Are All Non-Twin-Earthable A Posteriori Propositions Moorean?

The above discussion of propositions which are non-Twin-Earthable yet a posteriori focused on Moorean propositions – propositions it is hard or impossible to doubt, and which play a peculiar logical role in our thought and language which Wittgenstein tried to describe in On Certainty – such as 'Language exists' and 'I am uttering now'. Their being like that makes it important to distinguish them from a priori propositions, since our traffic with them resembles our traffic with certain a priori propositions in striking ways: we don't have to look out into the world of experience to verify them.

The question now arises: are all the propositions which are non-Twin-Earthable yet a posteriori like that? That is, are they all Moorean?

The answer, and this seems quite definite, is no: take, for example, the conjunction 'Language exists and first-order logic is undecidable'. Adding the second conjunct leaves non-Twin-Earthability unaffected, since it is a priori, but destroys Mooreanness. (Or, if you think it's Moorean that first-order logic is undecidable, pick some less basic a priori fact from the formal sciences.)

Comparison with the Traditional Conception

How do these two notions – non-Twin-Earthability, and determination of truth-value by internal meaning alone (which latter is what we have been using 'apriority'/'a priori' for), line up with the traditional conception of a priori truths as those which can be known without recourse to experience?

Our choice of terminology above has probably given the game away: the a priori propositions (those whose truth-values are internal to them) just are those whose truth-values can be known a priori in the traditional sense, i.e. without recourse to experience. And, since the non-Twin-Earthable propositions outrun those whose truth-values are internal to them (the a priori propositions, on our usage), they also outrun the propositions which are traditionally a priori.

I will not embark on an extensive discussion of the traditional idea, but will be quite rough and ready with it.

To repeat: non-Twin-Earthable propositions are not all traditionally-apriori, but the a priori (on our usage) propositions just are the traditionally-a priori ones. I regard the first part of that suggestion as more certain, and more robust with respect to different precisifications of traditional-apriority, than the second; .

The non-Twin-Earthable propositions which are not traditionally-apriori are the 'transcendent' ones: those whose concrete instantiation ensures their truth: as we just saw, these include both Moorean propositions such as 'Language exists' and 'I am uttering now', but also more complicated, non-obvious transcendental propositions. They give information concerning what goes on in space and time, and no propositions traditionally regarded a priori do that (perhaps an exception must be made for the most extreme Liebnizian rationalism, but even he distinguished between truths grounded by the principle of non-contradiction and those grounded by the principle of sufficient reason).

The case of 'I am uttering now' seems particularly clear: when we know that, we are clearly relying on experience: in the most straightforward case, the very experience of uttering. It could conceivably fail to come off.

A more complicated case could be imagined where some antecedent condition is empirically and reliably connected with me uttering 'I am uttering now', I might observe the condition and be said to know the proposition already, i.e. not by means of my experience of uttering it.

But hold on: is that right? It might be thought that there is a mistake here, similar to the mistake that would be made by saying that the way we know our own intentions is by observation and empirical correlation (cf. Wittgenstein's Remarks on the Philosophy of Psychology). We may just know that we intend to utter, not by observing ourselves, and know that our intention will be carried out, and then utter 'I am uttering now' knowingly. But in that case we may say: very well, but you are still drawing on past experience in knowing that your intention will be carried out. Furthermore, we can maintain high standards of knowledge and say: you don't really know you're uttering now until you experience it (hear it, see it, feel it etc.), because something could go wrong with your attempt at uttering – your larynx could disappear, for example.

So, not all non-Twin-Earthable propositions are traditionally a priori: 'I am uttering now' is the former but not the latter.

Does the converse hold? Are all traditionally a priori propositions non-Twin-Earthable? I say Yes, but I will not bother arguing this directly, since it falls out from the other things I am arguing: that the a priori propositions are just the traditionally a priori propositions, and all a priori propositions are non-Twin-Earthable.

One challenge to the idea that the a priori propositions – i.e. those whose truth-values are determined by their internal meanings alone – just are the traditionally a priori propositions is that already troublesome proposition, 'I exist'. Above, we made a case for saying that – provided the 'I' is construed in such as way as not to require the existence of a body – this proposition is a priori: its internal meaning determines its truth-value. And our intuitive considerations in favour of that were not unrelated to the traditional conception of apriority: we spoke of 'looking out into the world', 'looking into the language-system', and reasoning.

I think we should follow through with this, and say that 'I exist' (on our “bodiless” construal of it) is traditionally a priori: you can know it without recourse to experience.

However, there is an opposing line of thought here. We see it, for example, in Chalmers. Chalmers' idea about 'I exist' is that we know it via introspection, which is a kind of experience. But we may want a narrower concept of experience which this doesn't fall under.

It is admitted by Chalmers that it is 'somewhat controversial' that 'I exists' can only be known on the basis of experience. 'I am uttering now', on the other hand, Chalmers regards as a clear case of something that can only be known the basis of experience.

I propose to go along with this, and say that 'Language exists' and 'I am uttering now', construed so that they give information about spatiotemporal goings-on, is neither traditionally a priori nor a priori in our internality sense.

It is clear that 'Language exists' and 'I am uttering now', construed the way they are supposed to be construed, give information about particular contents of space and time – and that makes us feel that in some sense, surely, these can only be known on the basis of experience. But it seems there is a conceptual gap between 'I exist' and 'I occupy space and time' (or anything else which says something about the contents of space and time) – otherwise Descartes' recovery of his old beliefs would have been considerably easier.

(Here we come upon an interesting difference between space and time. Kant's calling time 'the form of the internal sense' and space 'the form of the external sense' is suggestive of it, so perhaps he comments on it somewhere. Roughly: while perhaps I could be taken out of the “time-space” I was in and put into another one, so that there is no fact about whether I am presently earlier or later than my birth in external time (in the sense of Lewis's distinction between personal and external time), but I cannot imagine not being in any time-space at all. If everything went dark and all bodily sensation ceased, and I heard a voice telling me I have been taken out of space, I would be unable to refute this. Or at least, no immediate knock-down objection would come to mind; the most near-to-hand strategy for refuting it would probably involve appeal to a belief in psycho-physical parallelism, which could only be founded on experience. If the voice, on the other hand, told me I had been deprived of all temporal locatedness and relations, I would immediately be able to see that this was in some important sense wrong. We might say that spatial locatedness is a posteriori, temporal locatedness (in some sense, at least – i.e. Lewis's personal time) a priori.)

Another major source of potential difficulty for my thesis that the traditionally a priori propositions just are the a priori ones in our internality sense, perhaps the most fundamental source of difficulty, is the synthetic a priori; substantial a priori truths which don't seem to be “true in virtue of meaning” in the sense that 'A priori propositions are those whose truth-values are internal to them', for example, is in this discussion (since we stipulated that we would use 'a priori' for internality of truth-value).

To begin to meet this difficulty, we must give an account of the analytic-synthetic distinction. I will make a start on this in the next post.

Monday, 1 June 2015

Forthcoming in Philosophia


My paper 'A Problem for Hofweber's Ontological Project' is forthcoming in Philosophia. It grew out of this blog post. The final draft is available at my homepage and at PhilPapers.

My preoccupation with arguing against Hofweber goes back further than that blog post, and he has always been very gracious and encouraging about it.

Monday, 25 May 2015

Illusory Explanatory Benefits in Philosophy

This post was prompted by a recent blog post of Wolfgang Schwarz's. I will follow it up with another post, prompted by a recent post of Alexander Pruss's.

“The difficult thing here is not, to dig down to the ground; no, it is to recognize that the ground that lies before us is the ground” - Wittgenstein (Remarks on the Foundations of Mathematics, VI 31, p.333).

I think we are often dogged, when doing philosophy, by a tendency to give credence to false theories on the grounds that they provide an explanation of something, when really the explanation is a pseudo-explanation, and where nothing of the kind is required if we see things aright.

In such situations, the false theory gives us something to say about some fact which resembles a real explanation, but gets us nowhere, and charmed by the idea that here we have an explanation where before there was none, we think better of the false theory. But having an explanation where before there was none is only a virtue if the explanation is a real one - if it actually helps us understand something, and does not merely have the form of an explanation.

A recent blog post by Wolfgang Schwarz called 'Magic, worlds, numbers and sets' contains an interesting example of this. It begins as follows:

'In On the Plurality or Worlds, Lewis argues that any account of what possible worlds are should explain why possible worlds represent what they represent. I am never quite sure what to make of this point. On the one hand, I have sympathy for the response that possible worlds are ways things might be; they are not things that somehow need to encode or represent how things might be. On the other hand, I can (dimly) see Lewis's point: if we have in our ontology an entity called 'the possibility that there are talking donkeys', surely the entity must have certain features that make it deserve that name. In other words, there should be an answer to the question why this particular entity X, rather than that other entity Y, is the possibility that there are talking donkeys.

It might be useful to consider parallel questions about mathematical entities.'

The example I want to concentrate on here is the one about mathematical entities, coming right after this passage. The post goes on to explore all kinds of weird stuff about Lewis, and I am not responding to it as a whole - I am just helping myself to something which occurs early on in the post, and using that as a vivid illustration of the particular failure mode in philosophy that I am trying to isolate and warn against.

(Before that, some sidenotes on the possible worlds case. The case is difficult, in part because there are various different ways of understanding 'possible worlds' in philosophy. We have some on which they really exist, some - perhaps for this reason closer to ordinary language - on which, apart from the actual world, they do not. We have some on which they are all the same sort of thing as the real, actual world, and some on which they are not. But on a lot of these, I too have sympathy for the idea that there is nothing here to explain. However, I think putting the point in terms of an emphatic identification of possible worlds with ways things might be missing the mark - for there are reasons due to Stalnaker in 'Possible Worlds' and elaborated on by Yablo in 'How in the World' for thinking that possible worlds are not to be identified with ways at all.

Secondly, regarding the point about an entity called 'the possibility that there are talking donkeys' (which of course need not be thought of as maximal or world-like) having to have certain features in virtue of which is deserves that name: perhaps that isn't so wrongheaded, but why can't the answer be along the following lines?: yes, one such feature is that, in this possilibity, there are donkeys. Another is that, in this possibility, they talk - or at least some of them do.)

To continue quoting:

'Mars has two moons, Phobos and Deimos. So here is a fact about the number 2: it is the number of moons of Mars. Following Lewis, one might argue that any account of numbers should explain in virtue of what the number 2 has this property. If we have numbers in our ontology, surely it can't be a brute fact that precisely this one is the number of moons of Mars.

The von Neumann construction of numbers gives a plausible answer to the Lewisian challenge. Here the number 2 is identified with the set { {}, { {} } }. This set has two members. The set of moons of Mars also has two members. And that is why 2, i.e. { {}, { {} } }, is the number of moons of Mars. In general, a von Neumann cardinal n is the number of Xs iff there is a one-one map between the members of n and the Xs.

By contrast, consider a primitive platonism about numbers on which the numbers are irreducible extra entities, distinct from sets, sticks, Roman emperors, and everything else. I do think the Lewisian objection has some bite here. One of the Platonic entities, call it X, is supposed to be the number 2. But what makes it the case that X, rather than Y, is the number 2, and thereby the successor of 1, and the number of moons of Mars? How come our label '2' picks out X rather than Y?

There seems to be an argument here for reducing numbers to sets.'

I want to criticize the line of thought indicated here. Firstly, regarding the idea that an account of numbers must explain in virtue of what the number 2 has the property of being the number of moons of Mars: aren't we being misled here by a phrasing which puts the focus on 2 instead of Mars? Intuitively, it is not an intrinsic feature of 2 that it is the number of moons of Mars, but an extrinsic one.

It is indeed plausible that it can't be a brute fact that the number 2 is the number of moons of Mars, but it doesn't follow from this that an account of numbers is the place to look for the explanation. Rather, the explanation we feel the lack of is that of, as we would more naturally say, the fact that Mars has two moons. And so I suggest, the apparent non-bruteness of the fact in question lies in its being explicable in astronomical terms. It seems like there is, whether or not we are able to figure it out, a story to tell about the formation of the planets and their moons which explains why Mars has two of them. I don't think there are any good reasons to believe that, with such an explanation on board, we would have further explaining to do as to why the number 2 is the number of moons on Mars. (Indeed, from a practical standpoint the idea seems ridiculous. But perhaps a practical standpoint isn't everything.)

I say 'I don't think there are any good reasons' above - but is that really the point? What force does my argument have? I am doing two things: firstly, I am suggesting that there is potentially a kind of bait and switch going on in our getting to the point of feeling that we need an account of numbers that explains why the number two is the number of moons of Mars: the fact calls for astronomical explanation, but if we consider the matter abstractly, we may just feel that it needs some explanation, and then the weird explanation involving the von Neumann construction is wheeled in.

Secondly, I am proposing that there is no explanatory gap between 'There are 2 Fs' and '2 is the number of Fs'. But at this point my arguments give out. Indeed, I think the best approach at this point is to stop arguing for the correct viewpoint, and switch to trying to trace the origin of the incorrect viewpoint. And I think in this case it lies in our misunderstanding expressions like '2 is the number of Fs', due to their superficial resemblance to expressions which work in a different way. Am I saying there is no such thing as the number two, or that it doesn't really have such properties as being the number of moons of Mars? Of course not. Of course, this is just echoing Wittgenstein. Unsympathetic readers will probably find this very annoying. Still, I stand by all of this and think it's important.

Sunday, 17 May 2015

Philosophical Percolations

There's a new group blog called Philosophical Percolations and I'm one of the authors. Sprachlogik will continue as usual.

Saturday, 25 April 2015

An Interpretation of Quantified Modal Logic Formulae

What does it even mean to ask whether the formulae of quantified modal logic can be given a coherent interpretation? Obviously we can in a sense interpret all of them to mean, say, 'Snow is white', but that doesn't count.

It is important to see that our guiding idea cannot be completely precise – for here it can be said that if we knew exactly what we were looking for, we would have found it (or, perhaps, seen that there could be no such thing). We can say, however, that it is crucial that the interpretation of the quantifiers should give them a meaning which is at least related to that of quantifiers elsewhere – and the closer the match, the better. Also, it is crucial that the interpretation of the modal operators should give them a meaning is closely related to that of 'necessary' as a predicate of propositions.

In the last post in this series, I offered an account of de re ascriptions of subjunctive necessity – an account of what it means to say that some object necessarily has a certain property.

Here I want to propose an interpretation of QML formulae which does not rely on this account of de re modal ascription, but which employs a similar strategy at one point (but in a different place – namely, in the interpretation of quantification). The strategy involves adopting a special sort of de dicto or substitutional interpretation of the existential quantifier, yielding what I call 'strengthened substitutional quantification'. The method of interpretation will be translation into technical natural language, rather than any kind of formal semantics.

I will now rehearse the syntax of QML, and give the general plan of interpretation (identifying the parts of the syntax requiring special treatment). I will then give the interpretation in the following section, and then go on to discuss briefly its significance.

I take the existential quantifier and the necessity operator as basic (so any formulae involving universal quantification or possibility operators must first be translated in the customary way). The formulae of QML may be built, in the familiar way, out of the following components:

- The existential quantifier (Ǝ).
- The necessity operator ([]).
- The truth-functional connectives.
- A stock of n-place predicates for n >= 1 (F, G).
- The two-place predicate of identity (=).
- A stock of individual variables (x, y, z)
- A stock of constants (a, b, c)

That is, first-order logic with identity plus the box. The formation rules are just those for first-order logic with identity, plus the clause: if A is a well-formed formula, then [](A) is a well-formed formula.

Extra things such as universal quantification and a possibility operator can be added by means of definitions. I will not consider function expressions, but I do not expect that they would create any special problems here.

The plan of translation is as follows. Truth-functional connectives are given no translation, and may be regarded as part of the technical natural language we translate into. We will likewise leave predicates, constants and atomic formulae involving them untranslated, but translation of these could be added as a final step. We now proceed to interpret the quantifiers and variables (i.e. quantified formulae), and the necessity operator (i.e. box-formulae).

4.5.1. Strengthened Substitutional Quantification

The translation of quantified formulae is key to the task at hand. The aim is to get them into a form such that open formulae no longer appear in the scope of box-formulae, i.e. so that quantification into modal contexts no longer occurs, and then apply the interpretation of box-formulae, which can proceed in terms of necessity conceived of as an attribute of propositions.

I call the interpretation, or the translation strategy, 'strengthened substitutional quantification'. It is designed to behave as much like ordinary objectual quantification as possible while fulfilling the above desiderata. I will argue that, in terms of material adequacy (i.e. something like truth-value match across cases), it is a perfect match.

It is worth emphasizing that ordinary objectual quantification is used in the translation, so strengthened substitutional quantification must not be regarded as a free-standing alternative to ordinary objectual quantification. The point of it, if not already clear, should become clearer as we proceed with the interpretation, and when we discuss its philosophical significance below.

I will introduce the translation in two stages, first giving a simplified version, showing how quantifiers iterate, and then adding a final feature which removes the simplification and deals with a remaining difficulty.

A quantified formula '(Ǝx) … x …' is to be translated as:

'There is an object o such that, if you were to substitute for 'x' in '… x …' a term rigidly designating o, the result would be true.'

For example, '(Ǝx) Mx' becomes:

There is an object o such that, if you were to substitute for 'x' in 'Mx' a term rigidly designating o, the result would be true.

'(Ǝx)(Ǝy) Lxy':

There is an object o such that, if you were to substitute for 'x' in 'There is an object o such that, if you were to substitute for “y” in “Lxy” a term rigidly designating o, the result would be true' a term rigidly designating o, the result would be true.

(We could have, instead of the two inner occurrences of 'o', used a different letter to help avoid confusion, but this is not strictly necessary.)

I hope these translations can be seen, when looked at carefully, to be intelligible. To begin to make the motivation for their particular features more intelligible, we will now consider some problems concerning unnamed and unnameable objects which have been raised in connection with traditional versions of substitutional quantification.

We will distinguish the primary problems of unnamed and unnameable objects respectively from a secondary problem (which can be set up with either unnameds or unnameables). We will then show how the translation scheme given above deals with the primary problems. That being done, we will add a final feature to our translation scheme, showing how it deals with the secondary problem.

It should be emphasized that these are not problems pertaining to the intelligibility of substitutional interpretations of the quantifiers. Rather, they pertain to potential failures of match with objectual quantification.

The Primary Problems of Unnamed and Unnameable Objects

Many writers, notably Quine, have pointed out that substitutional quantification, as ordinarily understood, does not allow quantification over unnamed and unnameable objects. For example, if the only object which has the property of F'ness is unnamed, then, while '(Ǝx)Fx' will be true on a standard objectual reading of the quantifier, it will not come out true if '(Ǝx)Fx' is interpreted to mean something like 'Some substitution of a name for 'x' in 'Fx' yields a truth'. This failure of match, in the case of merely unnamed objects, may be called 'the primary problem of unnamed objects'. The failure of match in the case of unnameable objects may be called 'the primary problem of unnameable objects'.

We avoid these failures of match on our understanding of quantification, since our translation of '(Ǝx)Fx', namely:

There is an object o such that, if you were to substitute for 'x' in 'Fx' a term rigidly designating o, the result would be true.

is given in terms of objectual quantification ('There is an object o such that') and a counterfactual conditional. This clearly deals with the problem of unnamed objects: if something has the property of being F, then clearly I would say something true if I were to name that thing and ascribe F'ness to it, even if it isn't in fact named. And vice versa: if there is something such that, if I were to name it and ascribe F'ness to it, I would say something true, then there is something which is F, although it may not actually be named.

What about the primary problem of unnameable objects? I distinguish this as a separate problem, because it involves considerable further considerations to see that strengthened substitutional quantification very arguably succeeds in avoid this problem, i.e. matching objectual quantification here too.

What the response to this problem requires depends on what is meant by 'unnameable'. 'Being unnameable' might mean, among other things:

  • Being such that no one who ever has existed or will exist (i.e. no actual object) can manage (in some practical sense) to name you.
  • Being such that no one who ever has existed or will exist (i.e. no actual object) could subjunctively possibly have named you.
  • Being such that you could not subjunctively possibly have been named. That is, being such that there is no subjunctively possible scenario in which you are named. Or again: having the property of unnamedness necessarily, rather than contingently.

The unnameability in the first two senses, and other like them, poses no further problems for our translation scheme: it should be uncontroversial that we can frame substantial, non-vacuous counterfactuals about what would be the case if certain things were named, provided that there are possible worlds in which they are named.

The third, and other similarly strong construals of unnameability, is a little more delicate. But: even if an object is unnameable, we can maintain, there are still non-vacuous counterfactuals about what truth-values certain sentences would have if they contained (per impossible) a name for that object.

This seems to contradict the view held by some philosophers (for example David Lewis in Counterfactuals) that counterfactuals with impossible antecedents are all vacuous – i.e. that they are all trivially true, or all trivially false, or all lack truth-values. That view is controversial and doesn't seem very plausible to me, so I am not too bothered by this. Since others might be, however, it is worth pointing out that there is no problem here in securing substitutional quantification over objects which are merely unnameable in some modally restricted sense – for example, unnameable because no would-be namers have the technology to reach them, or the knowledge to pick them out descriptively, or unnameable because they are outside the light cones of any would-be namers. It is only objects which are unnameable in a very strong sense, i.e. where it is metaphysically or a priori or 'logically' impossible that they should be named, which require a contradiction of the vacuity view of counterpossible counterfactuals. And it is not even clear (to me, at any rate) that there are, or could be, such objects.

We will come back to this issue, that the counterfactuals used in strengthened substitutional quantification may have to non-vacuously involve scenarios which are impossible in some strong sense, below, where it will also be considered in connection with our account of de re modal attributions.

From what we have said already, it should be clear that strengthened substitutional quantification avoids the primary problem of unnamed objects, and also that of unnameables when their unnameability isn't construed in a very strong sense. Furthermore, it should seem quite plausible to many that the problem is also solved for strongly unnameables as well. For example, to those who have learned elsewhere to be accommodating of counterfactuals with (strongly) impossible antecedents, or those who have reasons to think that there are and could be no strongly unnameable objects.

Now to the secondary problem, which pertains to both unnamed and unnameable objects. To solve this we will add a feature to our account.

The Secondary Problem of Unnamed and Unnameable Objects

What about '(Ǝx)Fx' in the case where F'ness is the very property of unnameability, or that of unnamedness? Our present translation of that, with the atomic sentence rendered in English, is:

There is an object o such that, if you were to substitute for 'x' in 'x is unnamed' a term rigidly designating o, the result would be true.

(There is a minor wrinkle here, since the translation uses the notion of a rigid designator rather than the narrower category of a name. That gets more at the heart of the matter, but a translation using the notion of a name instead would be equivalent, provided everything counted as a name is a rigid designator. For this discussion let us just stick to the present translation but interpret 'unnameable' specially to mean 'not rigidly designatable' and 'unnamed' to mean 'not rigidly designated'.)

This translation seems to get things wrong. For there are surely unnamed objects (and in fact all we require is that there could have been), so '(Ǝx)Fx' is true (or at least could have been). But it's not the case (nor could it have been the case) that there are any unnamed objects (i.e. objects which aren't rigidly designated) such that, if you rigidly designated them, it would be true to say that they are unnamed. You would have made that false by your act of designation. This is the secondary problem.

The solution I propose is that '(Ǝx) … x …' is to be translated as:

'There is an object o such that, if you were to substitute for 'x' in '… x …' a term rigidly designating o, the result would be true of things as they actually are.'

(This is just our original proposal with 'of things as they actually are' tacked on the end.)

An alternative, employing, instead of this notion of 'true of', the notion of the state of affairs asserted to obtain by a proposition, is:

'There is an object o such that, if you were to substitute for 'x' in '… x …' a term rigidly designating o, the state of affairs asserted to obtain by the result would be one which actually obtains.'

How This Enables Us to Apply My Account of Necessity

With quantified formulae being translated in this way, any free variables in the operands of box-formulae will be gotten rid of, and so we may now translate '[] …', as ' “…” is necessary', and then apply our analysis of necessity as an attribute of propositions.

So '(
Ǝx)[] Px' is first translated into:

There is an object o such that, if you were to substitute for 'x' in '[] Px' a term rigidly designating x, the result would be true of things as they actually are.

And then translating the box:

There is object o such that, if you were to substitute for 'x' in '“Px” is necessary' a term rigidly designating x, the result would be true of things as they actually are.

As a final step, we may now apply our account of necessity:

There is object o such that, if you were to substitute for 'x' in '“Px” is implied by a proposition which is both inherently counterfactually invariant and true' a term rigidly designating x, the result would be true of things as they actually are.

That concludes the exposition of the translation scheme for QML formulae.

We will now consider a bit further the issue of the counterfactuals in the strengthened substitutional translation of the quantifiers, and the counterfactuals in the analysis of de re modal ascriptions, having to deal with impossibilities. We will then finish off by considering briefly what, if anything, we have achieved with these two proposals.

What are the Relevant A-Scenarios?

The account of de re modal acriptions we have proposed may be stated, with the expository simplifications which were in force in the last post removed, as follows:

An object x possesses a property y necessarily iff: if you were to rigidly x that it possesses the property y, you would say something necessary.

And our account of how to translate quantified formulae so that quantification into modal contexts can be made sense of using my account of necessity as an attribute of propositions is:

'(Ǝx) … x …' is to be translated as:

'There is an object o such that, if you were to substitute for 'x' in '… x …' a term rigidly designating o, the result would be true of things as they actually are.'

This raises the question of how these counterfactuals are to be interpreted, i.e. which A-scenarios are to be regarded as required to be C-scenarios?

The first thing to say is that there is an important difference between the two cases, in connection with the vexing issues of strongly unnameable objects and strongly unstateable facts. The problem of strongly unnameable objects we have argued can be dealt with in the case of strengthened substitutional quantification by means of the '… of things as they actually are' clause, and by allowing the set of relevant A-scenarios to involve ones in which, per impossibile, strongly unnameable objects get named.

A similar move for our account of de re modal ascriptions, on the other hand, is not so attractive. For there, the consequents of the counterfactuals are about necessity – necessary truth – rather than just truth. The analogous move would yield:

An object x possesses a property y necessarily iff: if you were to rigidly x that it possesses the property y, you would say something necessarily true of things as they actually are.

And this notion of being 'necessarily true of' is not clear. Furthermore, I am repelled by the idea of trying to make it so – I am inclined to think, although I have no argument for this, that the prospects (at least from the point of view of our desiderata) are bad.

As a result, I conclude that the account of de re attributions must be prepared to countenance more extreme, further out, possibilities than the strengthened substitutional translation of quantification, insofar as we are to recognize strongly unnameable objects and strongly unstateable facts. I will try to explain this.

In the de re attribution case, if we countenance strongly unnameable objects, we cannot get away from the problem that, in scenarios where you designate them rigidly, they are no longer unnamed, by adding an actualizing clause. We must instead stretch the bounds of possibility further than we would have to for the translation of quantification, and talk as it were about scenarios in which – per impossible – you designate something and predicate something of it without it thereby being unnamed. That is, we hold fixed the unnameability/unnamedness, even though it isn't consistent with the supposition that the object in question is being designated.

This, while very worrying for an analytic philosopher, is, I think, not completely absurd. It seems we can make some sense, for example of the following:

If you were to say of an unnamed object that it was unnamed, you would say something true.

There is certainly an available reading on which this is false, but I think we can also give it a reading on which it is true. This may be brought out by the following somewhat strange but I think not absurd expansion:

If you could somehow say of an unnamed object that it was unnamed, without thereby changing its namedness status, you would say something true.

So far, I have been talking about the impossible scenarios we may need in our sets of relevant A-scenarios. But we must also consider the question of which ones we may need to leave out. I will not go into this this time.

What Has Been Achieved

A final word about the motive of the above accounts. There is no denying the fact that they are not very simple or straightforward. In mitigation of this, it may be said that some of the main difficulties only came up on certain conditions, where these conditions and whether they are met may yet be clarified to our advantage – the two main ones are: the question of whether acriptionally identical propositions can differ in ICI/modal status, and the question of whether we need to worry about objects which are unnameable, and facts which are unstateable, in some modally very strong sense.

I think if we take a step back, these proposals do indeed shed light on the meaning of de re modal attributions and quantifying into modal contexts. Rather than saying that they 'give the meaning' of such talk (although I think we could say this at a suitable granularity), we can say that they are equivalent, and that seeing their connection to such talk can shed light on the latter. They can also be regarded as partly determinative of the meaning of such talk, guiding its intuitive meaning more definitely along certain channels.

Furthermore, the fact that such connections can be made may help ward of scepticism, born of misunderstanding, about my whole approach to subjunctive modality, even on the approach's home ground of necessity as an attribute of propositions. This is connected with the general confusion surrounding subjunctive modality and Kripke's achievement in isolating it so clearly.

If we can make some intuitive sense of both de re subjunctive modal talk and quantification into subjunctive modal contexts, as well as of talk of necessity as an attribute of propositions, and if we feel that we are in some sense dealing with the same notions here, it would be a bad sign for my account of necessity as an attribute of propositions if connections such as we have proposed couldn't be made. So showing that they can helps, not just to make the account more powerful, but to make it more attractive on its home turf.

In case this is not already clear from the above, it remains to be emphasized that the accounts given above are not supposed to be the only way of approaching the issues of de re modal ascription and quantification into modal contexts. Just way. So, there is plenty of room for such talk to already make sense and work as it were “standalone”, as well as for there to be other accounts of a quite different nature, giving quite other connections and explanations.

Finally, it is worth remarking that the device of strengthened substitutional quantification may also be of use in accounting for quantification into other contexts besides that of a subjunctive modal operator – for example, epistemic modal contexts, or the context of an a priority operator, and propositional attitude contexts.