Saturday, 25 April 2015

An Interpretation of Quantified Modal Logic Formulae

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

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

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

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

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

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

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

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

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

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

4.5.1. Strengthened Substitutional Quantification

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

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

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

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

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

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

For example, '(Ǝx) Mx' becomes:

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

'(Ǝx)(Ǝy) Lxy':

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

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

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

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

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

The Primary Problems of Unnamed and Unnameable Objects

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

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

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

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

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

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

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

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

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

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

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

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

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

The Secondary Problem of Unnamed and Unnameable Objects

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

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

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

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

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

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

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

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

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

How This Enables Us to Apply My Account of Necessity

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

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

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

And then translating the box:

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

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

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