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).

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 de dicto:

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 a 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.

Toward an Account of De Re Modal Ascriptions

This is the third post in a series on de re modality and quantification into modal contexts. This one is quite exploratory and anything but final. The first two posts are here and here.

Let us begin by considering a simple proposal and two problems with it:

a is necessarily F iff 'a is F' is necessary.

The first problem may best be called a potential problem. It affects this approach if the question 'Can propositions which ascribe the same property (or relation) to the same object (or n-tuple of objects) differ in modal status?' is correctly answered in the affirmative.

If propositions can be ascriptionally identical and yet differ in modal status, then while my proposition 'a is F' may be necessary, someone else may have a proposition in another system, let us say using the sign 'b is G' (but of course it may also be the same sign as I use), which is ascriptionally identical but contingent. In such a case, we might not want to say that a, that object, is necessarily F, since some propositions which ascribe the property F'ness to the object a are not necessary – we might want to say, generally that an object fails to be necessarily F if there is any proposition which ascribes F to it and isn't necessary.

This gives rise to various terminological opportunities and options – e.g. we might want to distinguish 'weak' and 'strong' necessity, and may go different ways on questions like 'If something is F, but is not strongly necessarily F, is it contingently F?'. We will come back to this. For now, we will go along with saying that a thing fails to be necessarily F if some proposition says of it that it is F, and that proposition fails to be necessary.

Furthermore, we will go along with the idea, while actually remaining agnostic, that ascriptionally identical propositions can differ in (ICI, and in turn) modal status – that is, we will try to solve this potential problem, without actually deciding for sure that it is a problem.

The second problem, unlike the first (potential) problem, does not threaten the truth or validity of the account, but rather its power. Recall the simple proposal we began with:

a is necessarily F iff 'a is F' is necessary.

The second problem lies with generalizing this: the above, if it is not read as being about some specific proposition, is a schema. And getting a general statement, a universal quantification, about when an object x, say, is necessarily F (or necessarily has some property y), is still a non-trivial task given the above, since the schematic letters occur on the right hand side in a quotational context.

I will now pursue the first problem for a long and tortuous stretch (this will hopefully be instructive). In the end, it will emerge that by solving the second problem in a certain way, we can modify the result so that it solves the first problem (using, for this modification, what we will have learned by that point about the first problem). This solution is, in essentials, the solution we will offer in the next post to the problem of quantifying into modal contexts, although the success conditions there may be a bit different. Therefore in this section, at the end of the long and tortuous stretch, I will just briefly state the solution, and explain how it solves the second problem as well.

Again: the first problem is, roughly, that instances of the simple schema might come out false if, while my proposition 'a is F' is necessary, there are other propositions ascribing F'ness to a which are not.

This naturally suggests the following:

a is necessarily F iff all propositions ascribing F'ness to a are necessary.

One problem with this which is, I think, not hard to surmount, lies with the possibility of non-rigidly designating an object and ascribing a property to it, in the sense of: ascribing that property to whatever falls under the description. I mean, for example, propositions such as:

The number I have written on this piece of paper is odd.

This can certainly be read as a contingently true proposition. Suppose I have '3' written on a piece of paper. Now, we will want to say that the number three, that very object, is necessarily odd. But since I might have written a different number, the above proposition, on the reading I have in mind, is contingent. And yet we might say that this proposition, so construed, satisfies the condition 'is a proposition ascribing F'ness to a'. The solution is to add the condition that the proposition rigidly designates a.

So really, what we want to consider is:

a is necessarily F iff all propositions rigidly designating a and ascribing F'ness to it are necessary.

Another problem is that a proposition may rigidly designate a and ascribe F'ness to it, but also do a bunch of other things, such as designating b and ascribing G'ness to it. And this extra stuff may make them contingent. For example: '3 is odd and this piece of paper has a 3 on it'.

The solution to that problem is to add a “that's all” clause – e.g. to talk about propositions which just rigidly designate a and ascribe F'ness to it, and do nothing else.

These problems, then, are easily solved. In the discussion of more serious difficulties which follows, I will not incorporate these solutions in order to keep things simple.

So, what (besides the two problems we saw how to fix) is wrong with:

a is necessarily F iff all propositions ascribing F'ness to a are necessary?

The problem is: what if there just aren't enough relevant propositions around in the actual world? (Whether this is a problem depends on the view of propositions one takes.)

And that leads to the thought:

a is necessarily F iff all possible propositions ascribing F'ness to a are necessary.

Disambiguation of 'Possible Propositions'

There is an unfortunate ambiguity here in talking about 'possible propositions'. I will not try to fix the terminology, but only explain the ambiguity: this means 'a proposition which can exist'. By contrast, when I speak of a proposition being necessary, I mean being subjunctively necessary, necessarily true in the Kripkean sense. I don't mean a proposition which must exist. A subjunctively possible proposition, then, is one which is true and not necessary – but the talk here of 'possible propositions' does not mean this. Fortunately this ambiguity is, for me, largely confined to these modes of construction, rather than particular constructions, since I hardly ever speak of the property of being subjunctively possible, and I never – except in this note – speak of propositions which must exist: so 'possible proposition' always means 'proposition which can exist', and 'necessary proposition' means 'proposition which is necessarily true'.

The 'All Possible Propositions' Strategy

We were considering the thought: a is necessarily F iff all possible propositions ascribing F'ness to a are necessary

This raises two worries: (i) is there a circularity problem here?, and (ii) what about impossible propositions, or perhaps better: what about objects and properties such that no possible proposition can say of the object that it has the property?

Regarding the first worry, it is not obvious that there is a circularity. Recall that we are not trying to analyze all modal notions in terms of other notions (indeed, the very idea of doing that may, for all that is said in this book, be chimerical) – inherent counterfactual invariance, for instance, is characterized in terms of all counterfactual scenarios a system can produce. Furthermore, the use of 'possible' here doesn't on the face of it seem to be the sort of de re modal attribution we are concerned to analyze. It's not about properties or relations possibly holding of actual things, but about possible things (in this case propositions), things which might exist, and that is very different. Secondly, the modal space in question may best be regarded as broader and more inclusive in certain respects than subjunctive modal space.

Furthermore, even if there is a circularity here (which may be quite indirect and subtle – i.e. may be present even if the 'possible' here is not itself to be regarded as directly invoking subjunctive modality), perhaps it's not a vicious circularity – for instance, we could say that we have still reduced the mysteries of necessary property possession (de re modality) to the mysteries of logical space.

Regarding the second worry, about the possibility of things and states of affairs which no possible proposition can refer to or represent: perhaps this can be overcome by taking 'possible' in a very wide sense.

Accordingly, I think this analysis may not be without value, but these worries create difficulty enough that a somewhat different approach seems desirable.

I think something like the following: intuitively, part of what the truth of a proposition of the form 'a is necessarily F' reflects is an internal connection between a proposition's ascribing F'ness to a and its modal status. One strategy we might try for capturing this is two-pronged: semantically ascend and invoke a priority. As a first pass:

a is necessarily F iff 'All propositions ascribing F'ness to a are necessary' is a priori.

Or equivalently:

a is necessarily F iff 'If a proposition ascribes F'ness to a, it is necessary' is a priori.

But this cannot be quite right, for necessity implies truth, and some necessary propositions are a posteriori. If 'a is F', for example, is just such a necessary a posteriori proposition, then it can't be a priori that if a proposition ascribes F'ness to a, it is necessary. Just like with our main analysis of necessity, i.e. as a category of propositions, we have to separate truthmaking from necessity-making.

This suggests employing, as we did in the main analysis of necessity, the notion of inherent counterfactual invariance:

a is necessarily F iff (a is F and 'All propositions ascribing F'ness to a are inherently counterfactually invariant' is a priori).

This is a definite improvement, but now out analysis falls victim to the same type problem which motivated our holding that necessity is closed under implication. Recall that we can't say:

A proposition is necessary iff it is inherently counterfactually invariant and true.

Since a disjunction of a necessary a posteriori proposition and a contingent proposition, where the necessary disjunct makes it true, is not inherently counterfactually invariant (since if it is held true on the basis of the second disjunct only, it will be allowed to vary across counterfactual scenario descriptions), but this disjunction will be necessary in the case that its necessary disjunct makes it true, so that the above analysis undergenerates: it says that, e.g., 'All cats are animals or I had lunch today' is not necessary, when it is. And recall that this problem is avoided by the account advocated:

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

We get a similar problem with the above analysis of de re modal attribution, but involving disjunctive properties rather than truth-functional, propositional-level disjunction. Consider for example:

'Hesperus is either identical to Phosphorus or a common object of philosophical examples'

Or, to remove any possibility of a truth-functional construal:

'Hesperus has the property of either being identical to Phosphorus or being a common object of philosophical examples'.

(Instead of 'being identical to' I will just say 'being'. I will also abbreviate 'being a common object of philosophical examples' as 'being a comex'.)

Now, according to the rough, dimly seen intuitive meaning of de re modal attributions which we are trying to analyse, it would seem we should say, since Hesperus is Phosphorus and in view of Kripkean considerations:

'Hesperus necessarily has the property of either being Phosphorus or being a comex'.

But this doesn't come out true on the analysis we are now considering. Plugging it in, we get:

Hesperus necessarily has the property of either being Phosphorus or being a comex iff:

- Hesperus has the property of either being Phosphorus of being a comex, and

- 'All propositions ascribing being either Phosphorus or being a comex [or, more strictly uniformly, having the property of being either etc.] to Hesperus are inherently counterfactually invariant' is a priori.

And the second clause fails to be true – far from being true a priori, the proposition mentioned is not true at all, since it is possible to hold it true while disbelieving that Hesperus is Phosphorus but believing that Hesperus is a comex, in which case it would be allowed to vary across counterfactual scenario descriptions (since things could have been such that quite other objects were comexes). Indeed, the mentioned proposition is false a priori.

But if we close under implication, as in our main analysis of necessity:

- 'For all propositions ascribing either being Phosphorus or being a comex to Hesperus, there is some inherently counterfactually invariant proposition which implies that proposition' is a priori.

we get something true, as required. We are making progress, but while both clauses come out true in this case, the analysis will still not give intuitively right results. Now it will overgenerate in some cases. Consider, for example:

Hesperus necessarily has the property of either being Saturn or being a comex.

This is intuitively false, since Hesperus is, intuitively, necessarily not Saturn, and only contingently a comex.

But the following both hold:

- Hesperus has the property of either being Saturn or being a comex, and

- 'For all propositions ascribing the property etc., there is some inherently counterfactually invariant proposition which implies that proposition' is a priori.

The second clause comes out true, because 'Hesperus is Saturn', while false, is inherently counterfactually invariant and does imply 'Hesperus has the property of either being Saturn or a comex'. And presumably, for any other proposition which might also ascribe the property in question to Hesperus, there would be some proposition identifying it with Saturn which implies it.

This would be solved by somehow requiring the (possibly hypothetical) implying propositions to be true as well as ICI, without jeapoardizing a priority. But it is not clear to me how this could be done.

For if we just tack on 'and true' to 'some inherently counterfactually invariant' above, yielding this as a second clause:

- 'For all propositions ascribing the property etc., there is some inherently counterfactually invariant and true proposition which implies that proposition' is a priori.

We are back to our problem of the second clause failing to be true as required for the case of Hesperus necessarily either being Phosphorus or a comex: its not a priori that the implying proposition, 'Hesperus is Phosphorus', is true, even though it is true.

We want our second clause, in general, to say something like: for all propositions P ascribing F'ness to a, there is some true proposition Q such that it is a priori that Q implies P.

But if we say that we have forgone the semantic ascent part of our two-pronged strategy, taking us back to our problems of non-existent and impossible propositions (or things for which there are no possible propositions of the relevant kind).

I find it surprising that it is apparently impossible to solve all these problems at once. I am far from sure that I haven't overlooked a possibility (i.e. an analysis quite close to the last few above, involving the strategy of semantic ascent together with the invocation of a priority, or a similar strategy, but which doesn't face such blatant material adequacy problems).

Be that as it may, there is still a further issue with any account along these lines. And it happens that, by considering this further issue which would still arise and describing that issue in a natural way, a quite different strategy comes into view.

Would Semantically Ascending Achieve Anything, or Just Mask Something?

It may seem that our move from talking about, say, 'all possible propositions' (with all its attendant difficulties) to talking about whether a priority is possessed by a proposition which says that all propositions of a certain kind are a certain way (namely a priori) – even if it could surmount the difficulties found above – would be a silly move, merely complicating things and getting us nowhere.

It might seem that this is so, because the right hand side of the analysis involves mentioning a proposition about the object in question, and so we can only state it when dealing with an object which we can talk about. But note that this doesn't stop it from being the case that all instances of:

a is necessarily F iff <one of our last analyses' RHSs>

are true (although other things were seen to). What this does get us is a way of dealing with any 'a is necessarily F'-form proposition, as it comes a long. We cannot apply it to an object which we cannot speak about – or rather, 'applying it to an object which we cannot speak about' makes no sense here. But given that we have such a proposition, what the analyses were designed to do was avoid any problems pertaining to nonexistent (and perhaps in a sense impossible) propositions: if some of those ascribe (or would ascribe if they existed) F'ness to a and aren't (wouldn't be) necessary, we do not want to say that a is necessarily F.

This shows that the strategy was not totally idiotic. Better than that, the last sentence (especially the clauses in brackets) suggests another approach to the first problem: solve it by first solving the second problem along conditional lines, where the antecedent condition (which very arguably doesn't have to be possible) covers all the cases which might have caused the first problem. (This should become clearer along with the proposal.)

So, we will go back to our original simple schema, and propose a conditional approach to our second problem – the problem of generalizing it. The simple schema was:

a is F iff 'a is F' is necessary.

Now as we saw, if we go along with the first problem, not all instances of this will be true. Let us ignore this for the moment, and consider how we might generalize it along conditional lines. We might say the underlying point of this (faulty) schema is something like:

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

Now, if we interpret this conditional on the right hand side in one way, we get the first problem again, just as we did with the simple, faulty schema. (And that is fitting, for a generalization of the schema.) But if we interpret it in another way (in broadly Lewisian terms, by widening the class of relevant A-worlds), it is no longer a generalization of the schema, and is no longer vulnerable to the first (potential) problem.

I will briefly explain this here, but leave discussion of certain further difficulties of interpretation to the discussion of quantification into modal contexts in the next post, where the approach taken to quantification is very much along the lines of this approach to de re modal attributions. I will first state two basic assumptions about counterfactual conditionals. (Not that they're absolutely required – see below.)

Two Basic Assumptions About Counterfactual Conditionals

I will take as a basic assumption about how counterfactuals work that they can be understood as requiring a set of A-scenarios (scenarios at which the antecedent is true) to all be C-scenarios (scenarios at which the consequent is true). This is not to commit to any particular story (for example, David Lewis's) about how the relevant A-scenarios are determined. It also doesn't commit us to only dealing with possible scenarios, as for example Lewis does. It also doesn't commit us to any particular story about the nature of scenarios.

The second basic assumption is that (and here I agree with Lewis) the relevant set of A-scenarios will not always be the same. And it isn't just that different forms of words induce different relevant sets – the counterfactual conditional in the analysis above, for example, can be intended and interpreted different ways, making different A-scenarios relevant. And in the present philosophical context, we need to explicitly specify and discuss different interpretations (that is, I know of no other way of inducing contexts in which they get the readings I am interested in, and have no reason to think there should be a way).

These assumptions can in principle be jettisoned, by trading in the counterfactual conditional form in the proposed analysis above (and in our special treatment of quantification below) for explicit talk about what 'all relevant scenarios' are like, and then specifying which they are (but this time not as a way of fixing a reading of some conditional). But making the assumptions serves a heuristic purpose, since they are very plausible and the counterfactual conditional form is highly familiar to us.

Two Interpretations of the Words of the Account

Now, recall that the account I propose, in its simplest (but in a sense ambiguous) form, runs:

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

Now, we must ask, about the counterfactual on the right hand side: which A-scenarios (scenarios in which you say of x that it is y) are required to be C-scenarios (scenarios in which you say something necessary) here? Which facts about the actual world are to be held fixed, and which allowed to vary? Or briefly, what is the relevant set of A-scenarios?

The thing to see is that, if we take a “closest worlds” approach, or at least if we take such an approach in a natural and simple way, we will run into the (potential) problem which would stem from ascriptionally identical propositions being able to differ in modal status. If, on the other hand, we take an approach on which a wider set of A-scenarios must be C-scenarios, we may avoid it. (The propriety of doing this without departing from the natural meaning of counterfactuals can I think be defended, but again this is not absolutely essential, since regarding it as a technically modified sort of counterfactual, or going straight for talk about a relevant set of A-scenarios and abandoning the counterfactual form, are both options.)

Suppose, for example, that your proposition 'a is F' is necessary, but that some other proposition, in some other system, which ascribes the same property to the same object, is contingent. In that case, we will (according to the kind of usage I want to go with here) not want to say that a is necessarily F.

Nevertheless, the relevant counterfactual comes out true, if the relevant A-scenarios are to be kept close to the actual world: on this approach, we can say that, indeed, if you were to say of a that it possesses the property F, you would say something necessary – because if you were to do that, you would do it using your proposition 'a is F'!

This is a perfectly legitimate interpretation of that counterfactual, but it is not the one we want. We want to hold less things fixed, and allow more things to vary (which sounds like it amounts to the same thing, but it may not, since we may have to explicitly widen the overall space of scenarios in question, i.e. explicitly allowing impossible scenarios). In this way, for any (actual, possible, or maybe even impossible) proposition which might make trouble for a being necessarily F, by ascribing F to a and not being necessary, will be covered – it will be what you said in some relevant A-scenario – so that the conditional will be falsified as required.

We will return to the question of how to get a better grip on what our relevant sets of A-scenarios for instances of our proposed analysis must be like in the next post in this series, once we have our special interpretation of quantification on the table, since that will raise a similar question.

For now, our account may be regarded as partially but not wholly specified.