On Negation and Necessities about Contingent Existents

This is a long, wide-ranging, difficult to understand post. It will be easier to understand in connection with other posts, some of which already exist and are linked to, some of which have yet to come.

There is an irritating issue with some typical examples of the necessary a posteriori. This issue has to do with existence, negation and presupposition. The purpose of this post is to investigate what our options are with respect to this issue.

Kripke raises the issue and says some things about it in Lecture III of Naming and Necessity:

One qualification: when I say 'Hesperus is Phosphorus' is necessarily true, I of course do not deny that situations might have obtained in which there was no such planet as Venus at all, and therefore no Hesperus and no Phosphorus. In that case, there is a question whether the identity statement 'Hesperus is Phosphorus' would be true, false, or neither true nor false. And if we take the last option, is 'Hesperus = Phosphorus' necessary because it is never false, or should we require that a necessary truth be true in all possible worlds? I am leaving such problems outside my considerations altogether. If we wish to be somewhat more careful, we could replace the statement 'Hesperus is Phosphorus' by the conditional, 'If Hesperus exists then Hesperus is Phosphorus', cautiously taking only the latter to be necessary. Unfortunately this conditional involves us in the problem of singular attributions of existence, one I cannot discuss here.

So, the issue lies, in the first instance, with canonical Kripkean examples of the necessary a posteriori involving names referring to objects which exist contingently. Besides the example of empirical identity statements, this also affects, for example, subject-predicate statements ascribing what we might think of as necessary properties to contingently existing individuals, e.g. 'N is spatiotemporal' where 'N' names some physical object.

Kripke's Gappy Option

Since Kripke is talking of necessity in terms of truth-values at possible worlds, the issue for him involves the question of truth-value gaps. If the cases in question fall into truth-value gaps when their putative referents don't exist, then we can take the option of using 'necessary' to mean 'not false in any possible world', and these cases come out necessary without having to replace them with conditionals whose antecedents assert the existence of the putative referents. Call this 'Kripke's gappy option'.

Kripke's gappy option is quite open to controversy, since the idea that such sentences fall into truth-value gaps when their putative referents don't exist may be, and has been, resisted. If there is no King of France, is 'The King of France is bald' false (as Russell held), or neither true nor false (as Strawson held)? If there is no Santa, is 'Santa Claus came to my house and spoke to me yesterday' false, or neither true nor false? I am not sure that this is a sensible or important question – isn't there perhaps room for quite harmless stipulation either way here?, and isn't there room to quite harmlessly just not care (compare “don't care cases” in computer science)? – but for what it's worth, I am inclined to think it's more plausible that the definite description case is neither true nor false, than it is that the Santa case (a proper name case) is neither true nor false. It seems very natural to me to call the Santa sentence false. And since it is proper name cases that we have to do with, Kripke's gappy option seems predicated on something quite open to doubt and controversy.

The Conditionalizing Option

The other option Kripke identifies is to give up the idea that 'Hesperus is Phosphorus' and the like are strictly speaking necessary, and to replace these examples with conditionals like 'If Hesperus exists, then Hesperus is Phosphorus'. Call this 'the conditionalizing option'. Kripke shies away from this, saying that it 'unfortunately' involves us in the problems of singular existence attributions. To contemporary philosophers, this probably doesn't seem like much of an obstacle; OK, so there might be difficult philosophical problems pertaining to singular attributions of existence, but surely we can understand and legitimately use them, and so this strategy doesn't really involve us in the problems of singular existence attributions in any urgent way. It seems that things looked different when Kripke was lecturing, however – in large part because those very lectures had not yet had their great effect. The above-quoted passage continues thus, shedding light on this:

In particular, philosophers sympathetic to the description theory of naming often argue that one cannot ever say of an object that it exists. A supposed statement about the existence of an object really is, so it's argued, a statement about whether a certain description or property is satisfied. As I have already said, I disagree. Anyway, I can't really go into the problems of existence here.

The conditionalizing option is more open to us, however, since singular existence statements are now widely conceded to be legitimate and understandable, and furthermore because I do go into the problems of existence, and try to deal with them (for a start, in this post) – and so, to any extent that the conditionalizing option does involve us in these problems, that isn't a bad thing at all, since they are being dealt with.

So the conditionalization option is viable. But is it the only option? We may feel that there is another – that, by construing or modifying our account of necessity in some simple way (which may take some finding), we can have 'Hesperus is Phosphorus' itself come out necessary.

This is somewhat desirable, since a lot of people call that proposition necessary, without any implicit intention to really be talking about the conditionalized version, and it seems more plausible that they have their concepts arranged in a way that makes what they say true, than to suppose that they are simply mistaken. But even apart from that, and apart from any considerations in favour of preferring an account on which 'Hesperus is Phosphorus' comes out necessary, it is instructive to see whether and how we might have one. If we get one, we've just enriched our conceptual resources – we then have (at least) two notions available, closely related, which we can use 'subjunctively necessary' for – or we can use two different terms. (We may ask which one best fits most extant uses of the relevant terminology in the philosophical literature, but I can't see that this is an especially important question.) We will now pursue this.

Options Arising on My Account of Necessity

I say that a proposition is necessary iff it is, or is implied by, a proposition which is both true and inherently counterfactually invariant.

In the context of my account of necessity, the issue we have been discussing does not come so inevitably upon the issue of truth-value gaps, since I do not give my account of necessity in terms of truth-values at possible worlds. Truth-values are involved, of course: to be necessary, a proposition must be true – it must be, or be implied by, a true and inherently counterfactually invariant proposition. But the bit which does all the special work, the notion of inherent counterfactual invariance, is not spelt out in terms of truth-values, but rather in terms of negation and the contents of counterfactual scenario descriptions: a proposition is inherently counterfactually invariant if it is such that, if one comes to believe it, its negation does not appear in any counterfactual scenario description permitted by one's system.

Thus, with the account given here, the issue becomes about negation and the space of counterfactual scenario descriptions permitted by a system. We will first consider the option of approaching the issue via negation, in particular, by disambiguating the reference to 'its negation' – the negation of a proposition – in the characterization of inherent counterfactual invariance rehearsed above. On one reading, the cases at issue will come out necessary, on another, contingent.

Following that, we will consider the option of approaching the issue via the space of counterfactual scenario descriptions held to be relevant to inherent counterfactual invariance, and considering a restriction on this space such that, if it is in place, the cases at issue come out necessary, and contingent otherwise. One reason for considering this as well as the (more natural, I think) negation approach is that the idea that 'not' is ambiguous, or even the idea that a proposition can be negated in more than one way, is controversial. I will defend this idea, but even given its correctness, it may be dialectically useful to have another approach available, and it is interesting in any case to see that there is such an alternative.

Internal and External Negation

What is the negation of 'Hesperus is Phosphorus'? Two sentences we might give are 'Hesperus is not Phosphorus' and 'It is not the case that Hesperus is Phosphorus'. Is there more than one natural reading for these sentences? That is, can we give them different natural readings, or give one of them more than one natural reading?

It seems we can. It seems particularly clear that we can give one natural reading to one of them, and another to the other: we can read the first as implying that Hesperus (and perhaps a distinct object Phosphorus) exists, and read the second as having no such implication. Consideration of cases involving empty names (i.e., where the relevant existence-implications will be false) gives this intuitive support; if someone says 'Santa's not happy', we might take this as saying that Santa's state of mind is not a happy one – that is, we might understand it to be saying something which, in order to be true, requires that Santa exists. If someone says 'It is not the case that Santa is happy', we might understand this as having no such requirement – roughly, as merely denying that it is true to say that Santa is happy. It may also be that one of these sentences can naturally be understood both ways, or that both can (whether or not we should say this will not concern us here).

So, it seems natural to suppose that we have here a difference between two kinds of negation of a proposition, and should therefore regard the phrase 'the negation' as ambiguous, or as failing to refer due to a failure of uniqueness. Let us for now use, in a rough way, the terms 'internal negation' and 'external negation' to talk about these two kinds, in advance of any particular account of how negation works in the different cases, or how the difference gets made, and without assuming that there aren't further distinctions to be made, for example between different sorts of internal negation. We will now consider some different strategies for accounting for this distinction.

Internal Negation as Dispredication

Looking at 'Santa is not happy' and 'It is not the case that Santa is happy', we might have the idea that the difference between them consists in this: that the first purports to name Santa and deny the predicate 'is happy' of him, much as 'Santa is happy' affirms it, whereas in the second case the negation applies to the whole proposition about Santa. Thus the first cannot be true without Santa existing, while the second can.

We may think of these two propositions as being of different logical forms. The first is of the subject-predicate form (or, we might say, a closely related dispredicational form) whereas the second is a truth-functional compound with one argument. On this way of construing the latter, the phrase 'It is not the case that' is taken as forming a symbol, so that the proposition may be rewritten 'Not-(Santa is happy)' or '~(Santa is happy)'.

Alternatively, we may think of external negation in natural language propositions like 'It is not the case that Santa is happy' as a special case of internal negation, namely where the subject is a proposition (or proposition-meaning, or whatever is apt to be the case) which is specified with a that-clause. And so while 'It is not the case that p' propositions are equivalent, and effectively the same as, those of the more artificial operatorial forms, we really only need internal negation, and the difference between 'Santa is not happy' and 'It is not the case that Santa is happy' is that the subject of the first is Santa, and the subject of the second is this 'It' – that Santa is happy. The first dispredicates happiness of Santa. The second dispredicates being the case of a certain proposition, or proposition-meaning (including both internal meaning and external projective relations), or whatever being-the-case-apt thing it is, and could be rendered without the 'It' as 'That Santa is happy is not the case'. (Note that not all similar-looking forms can easily be regarded as having the same kind of subject. For example, 'It is good that …'. It seems that for 'It is not good that p' to be true, 'p' needs to be true. We might say that the subject here is a state of affairs or a fact.)

Irrespective of this alternative regarding external negation, we have a problem with internal negation. The trouble with the dispredicational approach to internal negation lies in generalizing it. Recall that our aim is to spell out a distinction between two kinds of negation of a proposition, internal and external, so that cases like 'Hesperus is Phosphorus' come out necessary on our account if we use one sort of negation in our characterisation of inherent counterfactual invariance, and contingent if we use the other. It will be internal negation which yields the former verdict, external which yields the latter; since there are no counterfactual scenarios where Hesperus exists and fails to be Phosphorus, the internal negation of 'Hesperus is Phosphorus' should not be part of any counterfactual scenario description. But since there are counterfactual scenarios where Hesperus doesn't exist, the external negation can appear in descriptions of those.

For this strategy, as outlined, to work, it will have to be the case that every proposition has an internal and an external negation. For the notions of necessity and contingency, and therefore our special notion of inherent counterfactual invariance in terms of which we define them, are meant to apply (positively or negatively, and borderline cases aside) generally to all kinds of propositions.

The only other options I can see, short of denying modal status to some propositions of a non-borderline sort which seem to have modal statuses, are the following variations on the strategy as outlined: (i) simply stipulate some modal status for all truths and for all falsities for which there is no internal negation (or for which 'internal negation' is undefined), (ii) take a disjunctive approach to counterfactual invariance (and in turn necessity) by making the internal negation of the proposition in question the relevant thing when there is one, the external negation otherwise. Neither of these seems very appealing, and below we will encounter reasons for thinking that, in addition to seeming ad hoc and messy, they do not behave in any satisfying way: (i) will wind up giving intuitively wrong answers about a great many propositions to which the dispredicational account of internal negation does not apply (such as quantifications, to anticipate), and (ii) will yield an account which behaves in a way which intuitively seems unsatisfyingly non-uniform, even if every verdict it gives is correct on some natural, intuitive understanding of the modal notions.

So, how might we extend the dispredicational account of the distinction between internal and external negation so that it gives all propositions both an internal and external negation?

The case of external negation presents no difficulties on this score, whether or not we regard it as a special case of internal. Every proposition can have the truth-functional negation operator applied to it, and every proposition (or proposition-meaning, or whatever) can be made the subject of an 'It is not the case that p' proposition.

But what are the internal negations of propositions which do not seem to be of the subject-predicate form supposed to be? We will consider three classes of such propositions: relational propositions, truth-functions, and quantifications.

Relations. We began by considering 'Hesperus is Phosphorus', which is often construed relationally, as an ascription of the identity relation. We are perhaps able to stave off, as far as identity statements go, the problem of extending our dispredicational account of internal negation; in other posts we considered issues which arise when identity is construed as a relation, and offered an alternative construal of identity statements as subject-predicate propositions of a special kind. But that is small comfort, since we must face the issue anyway – we will certainly need to be able to use our account of internal negation to rule some relational statements about contingently existing individuals to come out necessary, others contingent, others impossible, others contingently false. (Consider: 'Hesperus is in the same location as Phosphorus' - necessary if colocation is reflexive, 'Hesperus is near Mars' - contingently true on a suitable construal of 'near', 'Hesperus is in a different place from Phosphorus' – impossible, 'Comet A fell to Hesperus' – could be contingently false.)

So, what might the internal negations of 'John loves Mary' and 'John told Paul about Mary' be?

One simple enough option is to treat these as subject-predicate propositions, with 'John' as the subject, and 'loves Mary' and 'told Paul about Mary' as the predicates, and then treat their internal negations as simple dispredications. But the results will not require Mary (and in the second case, Paul as well) to exist, as they might if construed differently. On different ways of construing them (according to whether we have just one subject, or two or more, or equivalently how we construe the predicate (how many places, and where)) their counterfactual invariance status may vary. But if the positive statements being negated all imply each other under these various construals, even if their negations don't, then these differences will not “percolate up” to necessity and other modal statuses. Nevertheless, we have got ourselves into a messy situation, and the dispredicational approach is not looking very appealing, even if it can be argued to happily deliver the desired results. I will try to unpack this a bit, before moving on to consider how the dispredicational approach to internal negation might be extended to truth-functions.

Suppose 'If a and b exist, then aRb' is necessary on an external negation construal of counterfactual invariance, and suppose a and b are distinct, contingently existing and independently existing things. Now, we want a dispredicational construal of the internal negation of 'aRb' such that this comes out necessary when we make that the relevant negation for counterfactual invariance. Since a and b exist contingently and independently of each other, we can divide counterfactual scenarios into four kinds. Using 'a' in this sentence to mean 'a exists' and '~a' to mean 'a does not exist', we have scenarios in which (i) a & b, (ii) ~a & b, (iii) a & ~b, and (iv) ~a & ~b.

Now, let us consider what happens when we take 'a' in 'aRb' as subject and 'Rb' as predicate, i.e. when we construe 'aRb' as 'a is (R b)'. The internal negation of this, which in effect says that a lacks the (non-contingently existing) property of bearing R to b, can appear in descriptions of scenarios of type (iii); when b doesn't exist but a does, a must lack that property, since possessing it would require b to exist. And so on this constual, 'aRb' is not counterfactually invariant – its internal negation does appear in counterfactual scenarios permitted by the system to which it belongs.

On the other hand, when we take a and b both as subjects, or for clarity and neatness's sake the ordered pair <a, b>, i.e., when we construe 'aRb' along the lines of '<a, b> (falls under R)', we get a different result. Instead of saying that a lacks the property of being R-related to b, this says in effect that the pair of things a and b lack the property of falling under R. So this cannot appear in scenarios of type (iii), since that pair doesn't exist in those scenarios. On this construal, therefore, 'aRb' is counterfactually invariant.

(Note that this divergence doesn't show up with true identity statements, even when we do construe them relationally, since in their case the categories (ii) and (iii) will be empty.)

It may seem that, since 'aRb' differs on these construals with respect to counterfactual invariance status, it must differ with respect to necessity vs. contingency as well. After all, this is not one of the disjunctive sorts of cases which was seen to motivate the closure of necessity under implication in my account of necessity, and so it may seem like propositions like 'aRb' are necessary only if they themselves are counterfactually invariant, and since 'aRb' isn't counterfactually invariant on one of its construals (namely 'a is (Rb)'), it isn't necessary on that construal either.

This can, and arguably should, be resisted, on the grounds that the '<a, b> (falls under R)' construal implies the 'a is (Rb)' construal. Indeed, they very arguably imply each other, even though their dispredicational internal negations do not. And so, while 'aRb' on the 'a is (Rb)' construal is not counterfactually invariant, it is implied by a proposition ('aRb' on the '<a, b> (falls under R)' construal) which is both counterfactually invariant and true, so it comes out necessary.

Nevertheless, all this complication is unpleasant – these multiple ways of grouping relational statements into subject, or subjects, and predicates don't seem very instructive, and what they are capturing might plausibly be expected to be capturable in a more illuminating way. (To anticipate, I will end up suggesting that we do this by considering the way a proposition may be construed as having more or less presuppositions.)

Truth-functions. This could perhaps be dealt with by laying it down that, to obtain the internal negation of a truth-functional proposition, one must translate it into disjunctive normal form (a disjunction of conjunctions of non-truth-funcional propositions or negated non-truth-functional propositions) or conjunctive normal form (a conjunction of disjunctions of atomic non-truth-functional propositions or negated non-truth-functional propositions), and then give each negated non-truth-functional proposition an internal reading.

For example, the internal negation of 'p or q', then, will be, taking the option of disjunctive normal form, '(p and q) or (<p's internal negation> and q) or (p and <q's internal negation>)'.

Presupposing that internal negation is defined for all non-truth-functional propositions, we can extend the definition to truth-functions in this artificial way. The artificiality of this procedure compared with the account of internal negation we will eventually settle on is a reason for favouring the latter, but this will be overshadowed by problems arising in the consideration of quantifications below.

Quantifications. So far, we have been able to wangle, using fairly elaborate means, extensions of the dispredicational account of internal negation to relations and truth-functions. But the account founders more seriously on quantifications.

Consider, for example, the proposition 'There are feathers in Robin Hood's cap'. It seems we can say 'There are no feathers in Robin Hood's cap' and read this as requiring Robin Hood's existence – effectively, asserting the featherlessness of Robin Hood's cap. And on the other hand it seems we can say 'It is not the case that there are feathers in Robin Hood's cap', and read it as making no such requirement.

(It may be protested that 'There are no feathers in Robin Hood's cap' can be dealt with by regarding 'no' not as a negation sign, but as like a number-term, making the proposition effectively the same as 'There are zero feathers in Robin Hood's cap'. That is plausible enough for this particular case, but consider 'There aren't feathers in Robin Hood's cap', or this exchange: A: 'There are feathers in Robin Hood's cap', B: 'There are not!'.)

I do not think we can extend the dispredicational approach to internal negation to cover this case, but let us consider two routes which may present themselves: (i) treating quantifications in a way inspired by Frege and Russell as second-order predications, and (ii) treating quantifications as infinite conjunctions or disjunctions.

Both Frege and Russell had a version of what Scott Soames calls a 'properties of properties' analysis of quantification. The basic idea, in formal logical terms, is first to abstract a property from the open sentence bound by the quantifier and then to see the quantifier itself as predicating of that property, in the case of universal quantification, the property of being possessed by everything, or in the case of existential quantification, the property of being possessed by something.

In simpler terms, 'Everything is F' (i.e. 'Everything has the property F') is analyzed along the lines of 'being F is possessed by everything', i.e. a subject-predicate proposition. Or to take a slightly more complicated example, 'All men are mortal' is analyzed along the lines of 's is P', where 's' denotes the set of things which are such that, if they are men, then they are mortal, and 'P' is a predicate true of properties which are possessed by everything.

Brilliant and insightful as this is, it doesn't meet our present needs, since the subject here will always be a property. What we needed was an ability to make contingently existing things whose existence is presupposed by the proposition in question the subjects. So a proposition like 'There are no feathers in Robin Hood's cap', if this is taken to presuppose the existence of Robin Hood, cannot be rendered in this way, at least not without regarding the property of being a feather in Robin Hood's cap to be a contingently existing thing, which would be controversial and difficult. (We will come back to this example below and discuss it a bit further, when we consider and reject the option of a disjunctive approach to inherent counterfactual invariance, whereon the internal negation is what is relevant when there is one, the external negation otherwise, such as in quantificational cases.)

What about the second option of analyzing quantifications as infinite truth-functional propositions – universal quantifications as infinite conjunctions, and existential quantifications as infinite disjunctions? It is a commonplace in contemporary analytic philosophy that this doesn't work – the treatment of them as such by the Tractatus, for example, is widely regarded as one of the fundamental flaws in the Tractarian system.

Why exactly it doesn't work – or, how best to argue that it doesn't work – is actually not nearly so clear and well-established as that it doesn't work. I will not try to give a full treatment of this matter here, but I will briefly look at one quite natural but bad argument that it doesn't work, before discussing why it doesn't work from my point of view.

The natural but bad argument I want to consider is instanced by Richard Holton and Huw Price in 'Ramsey on Saying and Whistling: A Discordant Note', where they consider a reconstruction of one of Ramsey's arguments for treating quantificational sentences as not being (or expressing) genuine propositions. The argument proceeds via an argument that they cannot be treated as infinite conjunctions or disjunctions. Here is the reconstruction, with a good criticism following it:

If we treat universally quantified sentences as expressing propositions we will be forced to see them as equivalent to conjunctions which, since they are infinite, ‘we cannot express for lack of symbolic power’. But that is no good: ‘what we can’t say we can’t say, and we can’t whistle it either’.

Is this argument convincing? At first sight, apparently not, for consider an analogy. What do you get if you divide one by three? If you try saying the result as a decimal expansion you will never stop: 0.33333... However, that doesn’t mean that you can’t say it, only that you need to express it in a different way: as the fraction 1/3.

Holton and Price go on to consider another interpretation of Ramsey, which brings the argument closer to other arguments he gave for the same conclusion. We won't bother with that; our motive was largely to bring into focus that appealing to our finitude, our limitations, in resisting the infinite truth-function analysis of quantifications isn't the point.

What is the point? From my point of view, we can distinguish eight angles from which to see that the truth-functional analysis of quantification fails. For four of these (namely (1), (2), (6) and (8)), I will draw on Wittgenstein's discussion of 'Generality' in Philosophical Grammar (part II, section II, pages 257 – 279), which includes a subsection called 'Criticism of my former view of generality', this former view being the truth-functional analysis of quantification as given in the Tractatus.

(1) Phenomenology of sense: It just doesn't seem like a simple existential quantification (for example), such as 'There are horses', means the same as some long, or infinite, disjunction. We feel like the particular disjuncts in such a disjunction deal with cases, and that these cases to not enter into the sense of the quantification. The quantificational proposition is simple, whereas the long disjunction is complicated.

Here are three passages from Philosophical Grammar which express this point:

Let us take the particular case of the general state of affairs of the cross being between the end-lines.

|--X------| |-----X---| |-------X-| |---X-----|

Each of these cases, for instance, has its own individuality. Is there any way in which this individuality enters into the sense of the general sentence? Obviously not. (p. 257)

There is one calculus containing our general characterization and another containing the disjunction. If we say that the cross is between the lines we don't have any disjunction ready to take the place of the general proposition. (p. 258)

Suppose I stated a disjunction of so many positions that it was impossible for me to see a single position as distinct from all those given; would that disjunction be the general proposition (Ex).Fx? Wouldn't it be a kind of pedantry to continue to refuse to recognize the disjunction as the general proposition? Or is there an essential distinction, and is the disjunction totally unlike the general proposition?

What strikes us is that the one proposition is so complicated and the other so simple. … (p. 262)

(2) Favourable cases are clearly special: Cases which do seem to have justice done to them by the truth-functional analysis of quantification are clearly special – they clearly fulfil conditions which aren't generally fulfilled by quantifications. So, by applying the Wittgensteinian method (made explicit later, in the Investigations) of considering 'a language-game for which this account is really valid' (PI, 48), we will be able to see that the account is not generally valid, since its validity in the favourable cases turns on features not shared by all quantificational propositions.

Of course it is correct that (Ex)Fx behaves in some ways like a logical sum and (x)Fx like a product; indeed for one use of the words “all” and “some” my old explanation is correct, - for instance for “all the primary colours occur in this picture” or “all the notes of the C major scale occur in this theme”. But for cases like “all men die before they are 200 years old” my explanation is not correct. (p. 268)

These amenable cases are clearly special; what makes them amenable is the fact that the concept terms 'primary colour' and 'note of the C major scale' determine, by their very meaning (or 'grammar'), their extensions. Any term used in such a way that something other than {C, D, E, F, G, A, B} (where these letters are taken as names of notes) is its extension simply wouldn't express the concept of a note of the C major scale. Clearly, not all quantifications involve such concepts – for example, the one about men all dying before the age of 200 doesn't. From this we can see that the truth-functional analysis isn't generally valid.

(3) The “that's all” problem: Suppose there are just three objects, a, b and c. (This is just for simplicity's sake – the problem I which will now emerge could be stated for the case of reality, too.) Now, on the basic truth-functional analysis of quantification, 'Everything is F' means 'a is F and b is F and c is F'. But suppose someone believes falsely in a fourth object, which they call 'd'. Surely believing in an object which doesn't exist doesn't stop them correctly understanding 'Everything is F', and using it with its actual meaning. But if they thought d wasn't F, they would deny that everything is F, but they may consistently accept that a is F and b is F and c is F.

This actually enables us to see two problems: the present “that's all” problem, and the next problem on our agenda, the problem of meaning varying with the domain.

The “that's all” problem is this: the conjunction above doesn't capture the meaning of 'Everything is F' because it leaves it open whether there are other objects not mentioned by it – it lacks a “that's all” implication.

This shows the basic truth-functional analysis of quantification, exemplified above, to be wrong. But what if the “that's all” problem could somehow be solved?

There are three sorts of attempts at solutions to consider:

(i) Attempts which deny the need for quantification propositions, upon analysis, to say anything to the effect that “that's all”.

(ii) Attempts to modify the analysis so that quantificational propositions somehow have a “that's all” implication, while preserving non-circularity.

(iii) Attempts to modify the analysis by adding a “that's all” clause, without trying to avoid circularity.

Wittgenstein in the Tractatus is the representative of (i) which I have in mind. The dilemma I propose for this approach is: either there is something deeply right about Wittgenstein's contention that it is nonsensical to name a bunch of objects and say that they are all the objects that there are, or there isn't.

If there is (as I suspect), then this just shows the whole truth-functional approach to quantification to be wrong: many quantificational propositions clearly do say everything they try to say. While it may be important to see that certain “that's all” type propositions are really pseudo-propositions, this appearance of an unsayable element which we may try, and inevitably fail, to express in a proposition, obviously doesn't percolate up to quantificational propositions, as it would if the analysis were correct. It is simply beyond the pale to say that all quantificational propositions involve this sort of “nonsense”, or this attempt to say what can only be shown, or whatever it is, and equally beyond the pale to say that they show something along “that's all” lines which we may try and fail to say – to bite the bullet on this would be to embrace a cripplingly narrow view of the typology and functioning of propositions.

If there isn't, then this sort of attempt is misguided through-and-through.

The second sort of attempt, modifying the analysis to get a “that's all” implication, but without circularity, seems to me to be a non-starter. It may perhaps be argued that this might just be due to a failure of imagination on my part, but I'm inclined to think this isn't so, and that considerations along the lines of the Paradox of Analysis would apply to any materially adequate attempt: any linguistic device able to pull off the trick of giving you a “that's all” implication would thereby qualify as a quantificational device, and so no analysis pulling off this trick could possibly be non-circular.

The third sort of attempt is interesting; perhaps an analysis of quantification in terms of a truth-function with a “that's all” clause tacked on may be true, even if the “that's all” clause involves quantification, making the analysis circular; it may be a non-vicious circularity. Furthermore, the analysis may answer to the task of explaining how quantificational propositions may have dispredicational internal negations: the negation would now apply to a truth-function involving (what may be unnegated) quantification, rather than directly to a quantification, and so the artificial strategy suggested above for getting dispredicational internal negations of truth-functions may apply.

This third attempt is in many ways the most promising – the others seem quite hopeless. At least we get an analysis here, something which we can work with and assess, and one which clearly doesn't suffer from the “that's all” problem. But this attempt brings out the fact that our example above of someone who believes in something which doesn't exist, but still gives quantifications the right meaning, shows up two further problems: the problem of excess content, and the problem of meaning varying with the domain. These show the truth-functional approach to be wrong, with or without a circularity-making “that's all” clause.

(4) The problem of excess content: with the “that's all” problem above, we were entertaining considerations which suggest that the truth-functional approach to quantification yields analyses which aren't logically strong enough in a certain respect: they fail to imply that there is nothing else in the world not covered by the proposition. But there is an opposite problem as well: the analyses yielded are too strong in certain respects, having implications which the propositions they are meant to be analyses of do not have.

For example, 'All men are mortal', if analyzed as an conjunction with a conjunct for every thing, saying that that thing is either mortal, or not a man, then it would involve, for example, a conjunct saying: Venus is either mortal, or not a man. And this implies that Venus exists. But 'All men are mortal' doesn't imply that Venus exists, so the analysis is wrong.

This holds whether or not we have a “that's all” clause. If we do, the problem just gets worse: not only is 'All men are mortal' falsely predicted to imply, of any existing thing you care to mention, that that thing exists, it is also falsely predicted to say that there is nothing besides what it names – in other words, it is predicted to specify which things exist, which it surely does not do (no matter how deeply you analyze it!).

(5) The problem of meaning varying with the domain:

This problem is, in the abstract, that a truth-functional analysis of quantification falsely predicts that the meaning of quantificational propositions varies with the domain.

Suppose (again, for simplicity's sake) that there are just three things, a, b and c. On the truth-functional approach, 'Everything is F' can now be analyzed as 'a is F and b is F and c is F' (or perhaps this with a “that's all” clause added).

The problem can now be seen from an epistemic (or doxastic) angle: suppose someone in this three-object world believed falsely in a fourth object, d, then they wouldn't accept the analysis, and yet intuitively they might understand 'Everything is F' just as well as we do, be just as good at analysis, and mean the exact same thing by it.

If we take a temporally dynamic view of existence, it can also be seen from the angle of the domain changing over time: 'Everything is F' doesn't mean 'a is F and b is F and c is F', since, if something new, d were to come into existence and come to our attention, and if we didn't know whether it was F yet, we wouldn't say 'Everything is still F, according to what we used to mean by that'. Rather, we would bring the question of whether d is F to bear on our proposition 'Everything is F' without it having changed its meaning.

What about if we had a “that's all” clause? In that case, if the analysis were right, and we carried on using 'Everything is F' with the same old meaning (not changing it to cover d), the mere existence of d ought to make us judge it false (since you can no longer say “that's all” of a, b and c taken together). But in fact, we don't change its meaning, but we don't just it false just in view of d's existence either.

Finally, we can see the problem from a counterfactual or “other worlds” angle: our proposition 'All men die before the age of 200', with the meaning it has, may be true of circumstances in which some things which actually exist don't exist (i.e. where the domain varies). But if the truth-functional analysis were right, it would come out false: if some actually existing thing a failed to exist in these circumstances, then in the analysis of 'All men die before the age of 200', the conjunct which covers a, and says of it that it either dies before 200 or isn't a man, would come out false, falsifying the whole. But that is clearly not how the proposition analyzed works: we can speak of counterfactual scenarios in which all men die before the age of 200, but in which you or I don't exist.

(6) The problem of the relevant propositions not existing: On this approach, the infinite conjunctions and disjunctions we would need just don't exist, or even cannot exist – and not just because they are infinite. Rather, also because they would have to contain propositions which name objects, where there just aren't any such propositions, and perhaps couldn't be.

Here is Wittgenstein's expression of this point in Philosophical Grammar:

Criticism of my former view of generality

My view about general propositions was that (Ex)Fx is a logical sum and that though its terms aren't enumerated here, they are capable of being enumerated (from the dictionary and the grammar of language).

For if they can't be enumerated we don't have a logical sum.

…

Of course, the explanation of (Ex)Fx as a logical sum and of (x)Fx as a logical product is indefensible. It went with an incorrect notion of logical analysis in that I thought that some day the logical product for a particular (x)Fx would be found. (p. 268)

This point – which is quite well known in a slightly different guise, namely as a problem for substitutional quantification (which I will discuss in a future post) – is likely to be controversial, since it relies on taking a certain kind view of the nature and existence of propositions. This kind of view may be called a “down to earth” view: propositions are the sort of things which come out of our mouths and get written down, or types thereof.

Many philosophers think about propositions and the like in ways which are not down-to-earth in this sense. If, for example, we conceive of a name with a referent in abstract mathematical terms, as for example an ordered pair consisting of an abstract typographical object and an object (the referent), then this problem doesn't really come through - although another, Benacerraf-esque problem arises: what's to say which “names” are chosen for the RHS of the analysis, since if the pair <the letter A, n> exists, where n is some highly obscure object which no one will ever really name, and perhaps couldn't, then so too does the pair <the letter B, n>.

(7) Cardinality problems: If there exists a non-denumerable infinity of real numbers, for example, then it may be argued that there are not enough names and propositions to go around – and this isn't due to our finitude: even infinitely many names and infinitely many propositions about infinitely many objects may not be enough, even on an abstract, non-down-to-earth view (in the sense of point (6) above), if there are only denumerably many names.

(8) The problem of domain indeterminacy: The “universal domain”, as well as various subdomains of discourse, cannot (without significant idealization) be regarded as constituting a determinate totality. Many domains are such that you can talk literally, without making any idealization, about a determinate, non-fuzzy set which collects all and only the elements of the domain together. This may lead us to fail to see that there really is no such thing in the universal case, and in many subdomains as well. Despite the idea of a determinate totality being ultimately chimerical here, that doesn't mean we can't quantify over all things, or use quantificational propositions in connection with subdomains not constituted by determinate totalities.

From Philosophical Grammar:

[W]hat matters, I believe, isn't really the infinity of the possibilities, but a kind of indeterminacy. Indeed, if I were asked how many possibilities a circle in the visual field has of being within a particular square, I could neither name a finite number, nor say that there were infinitely many (as in a Euclidean plane). Here, although we don't ever come to an end, the series isn't endless in the way in which | 1, ξ, ξ + 1 | [Wittgenstein's sign for the series of positive integers] is.

Rather, no end to which we come is really the end; that is, I could always say: I don't understand why these should be all the possibilities. – And doesn't that just mean that it is senseless to speak of “all the possibilities”? (p. 276)

This seems closely related to the Tractarian idea, which came in in connection with point (3)(i) above, that you can't actually list a bunch of objects and sensically say “that's all” of them.

We have now seen eight reasons why quantificational propositions cannot be analyzed truth-functionally.

We now conclude, then, that there is no satisfactory way to extend the dispredicational approach to characterising internal negation to the case of quantificational propositions – it cannot be done with them as they are, and they cannot be analyzed into a more amenable form either.

Here is a good place to say something about what is wrong with a disjunctive approach to selecting the negation relevant to inherent counterfactual invariance status, i.e.: take the internal negation when there is a suitable one, otherwise (such as in the case of quantifications) take the external negation. Then one reading of 'inherently counterfactually invariant' will just involve external negation, and the other will be this disjunctive one. This may be coherent, but it is complicated, and doesn't behave in any uniform, satisfying way: the point of the latter sort of reading was to get 'Hesperus is Phosphorus' to come out necessary without having to posit truth-value gaps or conditionalize it. But there will be true quantificational propositions which, by parity, we ought to count as necessary along with 'Hesperus is Phosphorus', but which will come out contingent. For example, 'All things identical to Hesperus are identical to Phosphorus'. (This could arguably be construed so as not to require the existence of Hesperus or Phosphorus, but still, it seems clear that it can be construed as presupposing their existence, so that its internal negation ought to presuppose their existence as well.)

The Presuppositional Account

I propose we account for the distinction between internal and external negation is in terms of the notion of presuppositions; 'Hesperus is Phosphorus', intuitively, doesn't actually say that Hesperus/Phosphorus exists, but presupposes it. We can see the internal negation as preserving this presupposition, and the external negation as cancelling it. Rather than thinking of this as some kind of effect of the two sorts of negation, which we would have to do anyway if we adopted another account of the internal/external negation distinction, we should use this to characterise the two sorts, and the difference between them.

We can be more nuanced too and differentiate between, not just cases where all presuppositions are cancelled and cases where none are, but cases where some presuppositions are in force and others aren't. In this way, we can capture what was awkwardly captured on the dispredicational approach by means of different groupings of subject(s) and predicate.

This approach to making a meaning-distinction between two sorts of negation based on the notion of presuppositions is taken up, in a particular version, by Pieter Seuren in his papers 'Presupposition and Negation' and 'Presupposition, Negation and Trivalence'.

Seuren's approach involves a third truth-value, since Seuren wants both internal and external negations to be expressible as unary truth-functional propositions. Propositions whose presuppositions fail are 'radically false', while propositions whose presuppositions are met but which still aren't true are 'minimally false'. This then leads to several options and complications which Seuren tackles undaunted.

I want to abstract away from this way of going. Why, for example, do we need to regard internal negation as a truth-functional propositional operator? If we don't, then do we really need three truth-values, or even gaps for that matter? For example, can't we just distinguish between cases of what Seuren calls the 'radically false' and the 'minimally false' by taking directly about presuppositions, or the truth-values of their internal negations, while calling them all just 'false'?

Our conclusion, then, is simply that we should distinguish two kinds of negation, internal and external, and that we should characterise this distinction by means of the notion of presupposition: internal negation preserves (some or all) presuppositions, and external negation cancels them.

Negation and Ambiguity

Some linguistic constructions may quite unambiguously be internal or external negations, other constructions may tend to signal one or the other, and still other constructions may not carry anything in themselves to suggest one or the other, this being left to context.

Since I do not want to get deep into the linguistic details of how things are in these respects, it will be difficult to discuss questions of potentially controversial ambiguities implied by this approach with any concreteness. For example, should we say 'not' is ambiguous on this approach? Or can we avoid this by locating the difference, not in the meanings of expressions, but their form or mode of composition? (Or even their subject matter: recall in this connection the idea adumbrated above of construing external negation expressed with 'It is not the case that' as dispredicating being the case of a proposition (or proposition-meaning) referred to by 'It' and the 'that' clause.)

Without going into the details of negation-expressions themselves (like 'not'), we may still consider questions of ambiguity in two ways: (i) by considering, in the abstract, the question of how any ambiguities which may arise on this approach may be made sense of and legitimized, and (ii) by considering the “meta” but concrete case of the expression 'The negation of …' where the dots are filled in with an expression referring to a proposition.

Here is what Seuren has to say about negation and ambiguity:

The ambiguity of negation in natural language is different from the ordinary type of ambiguity found in the lexicon. Normally, lexical ambiguities are idiosyncratic, highly contingent, and unpredictable from language to language. In the case of negation, however, the two meanings are closely related, both truth-conditionally and incrementally. Moreover, the mechanism of discourse incrementation automatically selects the right meaning. These properties are taken to provide a sufficient basis for discarding the, otherwise valid, objection that negation is unlikely to be ambiguous because no known language makes a lexical distinction between the two readings.

While Seuren and I are in agreement that we can profitably speak of an ambiguity between internal and external negation, I want to say that the case of negation is not so special as these remarks of Seuren's may make it seem.

Indeed, the topic of negation just provides another case, or bunch of cases, of ambiguities which fail the Kripke test (roughly: when faced with the question of whether an expression is ambiguous, look at whether other languages use two different expressions instead of one – if not, that casts doubt on the ambiguity).

I will dicuss ambiguities failing the Kripke test, in connection with semantic granularity, in a future post, and from that discussion it will be clear that there is nothing which isn't 'ordinary' or 'normal' here (Seuren's words for what the case of negation isn't), and that the Kripke-test-based objection he considers is not 'otherwise valid': there is a large class of cases for which the Kripke test is valid, and a large class for which it is not, and granularity considerations can help us see what characterizes these classes.

Restricting the Space of Scenarios

There is an alternative to the approach of getting 'Hesperus is Phosphorus' and the like to come out necessary on my account by defining inherent counterfactual invariance in terms of internal negation. It is perhaps a bit less natural, but it may have an ecumenical advantage, in not requiring there to be an internal/external negation distinction.

Rather than defining inherent counterfactual invariance in terms of all counterfactual scenarios, as in:

A proposition is inherently counterfactually invariant iff it is inherently such that, if you come to believe it, its negation doesn't appear in any counterfactual scenario description.

We can say:

A proposition is inherently counterfactually invariant iff it is inherently such that, if you come to believe it, its negation doesn't appear in any counterfactual scenario description according to which the presuppositions of that proposition are met.

An Analogous Issue with the Contingent A Priori, Treated Differently

Recall that the issue we have been looking at here affect necessary truths about contingently existing things. There is an analogous issue with cases of the contingent a priori. When a name, say 'N', has been stipulated to refer to the concrete particular, if there is one, which fulfils certain conditions F, it is often said that the proposition 'N is F' will then be contingent a priori. But this statement entails that N, some concrete particular, exists (and is F), and surely this cannot be a priori. In this case, I think there is no very natural other way, and so I simply deny that such propositions are a priori, speaking strictly. (There is of course room for a slightly artificial use of 'a priori' defined in terms of a more natural use with the express purpose of making such cases come out a priori – e.g. something is a priori in this special sense iff its conditionalization (in the sense of the conditionalizing option discussed above) is a priori.) 'If N exists, N is F', on the other hand, will be genuinely contingent a priori.

Conclusions

Our main conclusions are as follows. We can distinguish two construals of inherent counterfactual invariance, so that on one 'Hesperus is Phosphorus' is not strictly speaking necessary (while 'If Hesperus exists, then Hesperus is Phosphorus' is), and on the other it is. There are at least two viable ways to get the second construal (and so, you might say, we can really distinguish three construals, but two of them line up): (i) by means of a distinction between internal and external negation, where the former preserves presuppositions and the later cancels them, or (ii) by defining inherent counterfactual invariance, not in terms of all counterfactual scenario descriptions permitted by the system of language to which the proposition in question belongs, but rather in terms of the subset of permitted counterfactual scenario descriptions which describe scenarios in which the presuppositions of the proposition in question hold.

Along the way, we reached some other conclusions, which may be of independent interest. One of these was that the dispredicational approach to the internal/external negation distinction cannot be satisfactorily extended to all propositions. In the course of seeing this, we also saw that quantificational propositions cannot be analyzed as truth-functions, and furthermore, we got a good look at why not, distinguishing eight angles from which it can be seen.