Quantification and 'Extra Constant' Semantics

(The following is a companion piece to this offsite post.)

In a fascinating new paper entitled 'Truth via Satisfaction?', N.J.J. Smith argues that the Tarskian style of semantics for first-order logic (hereafter 'FOL'), which employs the special notion of satisfaction by numbered sequences of objects, does not provide an explication of the classical notion of truth - the notion of saying it like it is - but that the second-most prominent style of semantics for FOL, which works by considering what you get if you introduce a new constant, does. I agree with him about the first claim but disagree with him about the second.

My main point in this post, however, is not to argue that Smith's preferred style of semantics for FOL fails to explicate the classical notion of truth. I will do that a bit at the end - although not in a very fundamental way - but the main point will be to draw out a moral about how we should think about the 'extra constant' semantics for FOL, and more generally, about how we need to be careful in certain philosophical contexts to distinguish mathematical relations (such as 'appearing in an ordered-pair with') from genuinely semantic ones (such as 'refers to'). The failure to do this, in fact, is what made Tarski introduce his convoluted satisfaction apparatus which others have muddle-headedly praised as some sort of great insight. (I blogged about this debacle, to this day largely unrecognized as such by the logical community, offsite in 2015.)

By way of intuitive explanation of the universal quantifier clause of his preferred semantics for FOL, Smith writes: 

Consider a name that nothing currently has—say (for the sake of example) ‘Rumpelstiltskin’. Then for ‘Everyone in the room was born in Tasmania’ to say it how it is is for ‘Rumpelstiltskin was born in Tasmania’ to say it how it is—no matter who in the room we name ‘Rumpelstiltskin’. (p. 8 in author-archived version).

But this kind of explanation is not generally correct. Get a bunch of things with no names and stick them in a room. Now, doesn’t this purported explication of what it is for quantified claims to be true run, in the case of the claim ‘Everything in this room is unnamed’, as follows: for ‘Everything in this room is unnamed’ to say it how it is is for ‘Rumpelstiltskin is unnamed’ to say it how it is--no matter what in the room we name ‘Rumpelstiltskin’? And this, I think, is very clearly false; by hypothesis, everything in the room in question is unnamed, so surely ‘Everything in this room is unnamed’ says it how it is. But if we name one of the things in the room‘Rumpelstiltskin’, then ‘Rumpelstiltskin is unnamed’ will certainly not say it how it is.

Now, as Smith pointed out to me in correspondence, this problem with unnamedness can be avoided by considering another method of singling out objects, such as attaching a red dot to them. (The worry arises that some objects are abstract and so it makes no sense to talk about attaching a red dot to them, but I won't pursue that here.) Then you can use a slightly different form of explanation, and say that for 'Everything in the room is unnamed' to say it how it is is for 'The thing with the red dot on it is unnamed' to say it how it is no matter which thing in the room has the red dot on it. Now we will of course get a counterexample involving 'red-dotlessness' but we can then just consider a different singling-out device.

But this slightly different style of explanation is also not generally viable, as becomes clear when we consider, not unnamedness, but unreferred-to-ness. Things which haven't been named but have been referred to, say by a definite description, count as unnamed but not as unreferred-to. And let's stipulate that we are talking only about singular reference - so that even if 'All the unreferred-to things' in some sense refers to the unreferred-to things, it doesn't singularly refer to them, so this wouldn't stop them from being unreferred-to in the relevant sense.

Now, applying the new style of explanation involving an arbitrary singling-out method to the case of 'Everything in this room is unreferred-to', we get:

For 'Everything in this room is unreferred-to' to say how it is is for 'The thing with the red dot on it is unreferred-to' to say how it is, no matter which thing we put the red dot on.

And this is wrong, not because the thing has a red dot on it, but because 'The thing with the red dot on it is unreferred-to' can't be true, whereas the quantified claim can be.

No analogous problem arises in the formal setting. If we specify that 'G' is to be mapped to the set of things in some room and 'F' is to be mapped to the set of unreferred-to things, and consider '(∀x)(Gx ⊃ Fx)', then neither Smith's preferred style of semantics for FOL nor the silly Tarskian style cause any sort of problem, since for there to exist a function which maps some constant c to an object o is compatible with o being unreferred-to. Thus we get the desired truth-value for '(∀x)(Gx ⊃ Fx)'.

(You might now think: OK, but what if we replace unreferred-to-ness with not-being-mapped-to-by-any-function, or whatever? Don't we then get the wrong truth-value? Well, no, because - at least on a classical conception of functions - nothing is unmapped-to-by-any-function.)

So, quantified propositions are not correctly explicated by talking about the truth-values of propositions you get by naming things. Nor are they correctly explicated by adopting a non-semantic singling-out device and then considering propositions which talk about 'The thing' singled out. This in itself shouldn't really be news, but also noteworthy is that, despite such explications being incorrect, the style of semantics for FOL which works via consideration of an extra constant gives no undesired results, and is arguably better than the Tarski-style semantics, which is needlessly complicated and is born of philosophical confusion. (Still, it does create a danger that students of it will wrongly think that you can explain quantified propositions in the way shown here to be incorrect.)

What does this mean for Smith's claim that 'extra constant' style semantics for FOL explicates the classical conception of truth, the conception of saying it like it is? Well, I think that's an interestingly wrong idea anyway, and probably deeper things should be said about it, but: Smith's incorrect informal gloss of the formal quantification clause - which gloss, as we have seen, cannot be corrected by moving to an arbitrary singling-out device and talking about 'The thing' singled out - certainly seems to be doing important argumentative work in his paper. His main claim, bereft of the spurious support of the informal gloss, is as far as I can see completely without support.

Many thanks to N.J.J. Smith for discussion.

Forthcoming in Philosophia

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

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

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.

Facts: The Object-Property-Relation Model

While the notion of a fact is not itself central to the main accounts I have been developing (e.g. of names, of necessity, analyticity and a priority, and of propositions), it is intimately connected with notions which are central, e.g. that of a proposition. In this series of posts, I want to consider what might be said about facts in certain connections in light of those ideas.

The Object-Property-Relation Model

An idea that has been very influential in analytic philosophy is that facts, or a certain class of basic facts at least, are complexes of objects, properties and relations – or objects, properties and relations arranged in a certain way.

There are many familiar issues with this sort of conception. Some of these are:

- How are the elements of a fact glued together? (The problem of the unity of the proposition, but for facts.)

- Are there negative facts? What are they like?

- Are there general (quantificational) facts? What are they like? (Generality can not be reduced to truth-functions.)

- What other forms of facts are there, and how are they related to basic facts? (E.g. propositional attitude facts, conditional facts.)

The first item on this rough list, and related worries, may be avoided in large part without thereby avoiding the rest, and the further ones I will raise. I will not attempt a comprehensive discussion of this problem, but I will reproduce a brief selection of remarks from a Wittgenstein typescript, 'Complex and Fact', which seem to me to contain considerations which break the grip of it, and try to explain how they do so. These will not be the only considerations relevant to the problem, since the other problems remain and will require separate treatment which independently tells against the basic model of facts at work in the first problem. Here are the quotations:

Complex is not like fact. For I can, e.g., say of a complex that it moves from one place to another, but not of a fact.

But that this complex is now situated here is a fact.
…

A complex is composed of its parts, the things of a kind which go to make it up. (This is of course a grammatical proposition concerning the words 'complex', 'part' and 'composite'.)

To say that a red circle is composed of redness and circularity, or is a complex with these component parts, is a misuse of these words and is misleading. (Frege was aware of this and told me.)

It is just as misleading to say that the fact that this circle is red (that I am tired) is a complex whose component parts are a circle and redness (myself and tiredness).
…

'To describe a fact', or 'the description of a fact', is also a misleading expression for the assertion stating that the fact obtains, since it sounds like: 'describing the animal that I saw'.

Of course we also say: 'to point out a fact', but that always means; 'to point out the fact that …'.
…

A chain, too, is composed of its links, not of these and their spatial relations.

The fact that these links are so concatenated, isn't 'composed' of anything at all.

What do these considerations achieve? By drawing attention to the actual grammar of 'fact', and showing how it differs from that of 'complex' (and 'animal', and 'chain') they can break the grip of the worry about how the components of facts are glued together; this rests on a false grammatical assimilation.

The same grammatical points may also help rid us of 'shadowiness' worries: worries about what queer sort of things facts could be. (I will discuss worries of this sort at greater length in future posts.)

However, it seems to me that a certain core of the object-property-relation view can survive considerations such as those above – or at least, such considerations do not effectively neutralize this core. Also, once the core is finally neutralized, shadowiness worries will arise again, and will have to be dealt with.

The plan for this series of posts is as follows. Firstly, to identify the core of the object-property-relation view. Secondly, to neutralize it – to show that it doesn't work, and give due weight and care to the observations which tell against it. I will do this firstly by considering the remaining worries on the list above, and secondly by means of the examples of facts about identity and singular existence. These are the tasks of the present post.

In one subsequent post, I will discuss facts in connection with granularity. In others, I will discuss the worries which arise in the wake of the demolition of the object-property relation model. Broadly, these could be called shadowiness worries – but I will distinguish and separately discredit a special form of worry - roughly, the worry that facts are fictions arising from an illicit projection of features of language and thought onto the world.

This series of posts should not be seen as being all about the object-property-relation model of facts. Although it begins with that, it is just as much a discussion of skepticism about facts – since it deals with the problem of what facts could be, if the object-property-relation model doesn't work (which it doesn't). The skeptic can be thought of as someone who sees that that model doesn't work, but can't see any acceptable way of thinking about facts either.

The Remaining Core

I will now try to characterize the remaining core of the object-property-relation model of facts, explicitly abstracting away from differences between different views which share it.

There are basic or atomic facts, and these are what make the most basic true propositions true. This is part of the core. Views incorporating this core may differ on the following points (1) whether these basic facts include negative facts, and (2) whether there are complex facts, built out of basic facts, or whether complex true propositions are just made true by the atomic facts somehow (in which case 'basic' or 'atomic' is actually superfluous, or even misleading, in relation to facts; they're all basic).

While views incorporating the core may differ on the point of (2), it is essential that they are committed to one of those two alternatives (or at least something very like one of them). Views on which there are quite other facts, which are not in any way built out of these basic facts (whose characterization we are still at work on), do not count as incorporating the core. Incorporating the core involves, loosely speaking, seeing all facts as being ultimately a matter of objects, properties and relations – or objects possessing or standing in properties or relations, or not possessing or standing in them. (The view I advocate is that this is a good way of thinking of some facts, but that it doesn't work at all for others.)

An important part of the core, regarding its conception of the basic facts themselves, is the way they are individuated. This can be brought out quite precisely in the following terms. According to the core of the object-property-relation model, each basic fact maps onto one of a set of n-tuples whose elements are objects, properties and relations arranged according to a certain scheme, in such a way that two distinct facts never get mapped to the same n-tuple.

For example, we may use the set of n-tuples whose first element is an property or relation, and whose remaining elements are objects. (There need be no assumption that properties and relations cannot be construed as objects.)

Then, the fact that Socrates is mortal (assuming Socrates and the property of mortality to be basic enough) is a basic fact, and gets mapped to <mortality, Socrates>. The fact that John loves Mary gets mapped to <love, John, Mary>, etc. Negative facts, if they are posited as basic facts, pose no problem here: the fact that Mary does not love John gets mapped to <love, Mary, John>; if that's a fact, then there's no fact that Mary loves John around requiring mapping to the same tuple, so uniqueness of mapping remains. (These n-tuples, it must be remembered, are not supposed to be representations of basic facts, but are being used to make a point about individuation. If negative facts are not posited, they could serve as such representations. Otherwise, they could be conceived of as partial representations. But that is not necessary for the present point.)

The Remaining Familiar Worries

These are:

- Are there negative facts? What are they like?

- Are there general (quantificational) facts? What are they like? (Generality can not be reduced to truth-functions.)

- What other forms of facts are there, and how are they related to basic facts? (E.g. propositional attitude facts, conditionals.)

Of these remaining issues, the consideration of generality seems particularly telling against the object-property-relation model. Negative facts seem to require less of a departure from the ingredients of basic atomic facts. We might imagine that the arrangement of these ingredients in the fact is a “negative arrangement”, or some such thing.

As for truth-functions, there are prospects (which, in a later post, we will see Russell appealing to) for saying that they are made true by atomic facts. Alternatively, molecular, truth-functional facts may be posited. (It may be argued that the core-incorporating theorist who makes this latter moves owes us an account of how these facts are composed, or constituted, but it is clear that they can be regarded as in some sense being made up of non-truth-functional facts, so we will not attack the core of the object-property-relational model on the point of truth-functions.)

As for conditionals, there are prospects for analyzing these in terms of truth-functions and quantification over worlds or scenarios.

However, there seems no getting away from general, quantificational propositions which cannot be analyzed as truth-functions. It seems overwhelmingly plausible that these are genuine, fact-stating propositions. (Contra Ramsey.) But the facts they state just can't be represented with the object-property-relation model.

The Last Straw: Identity and Existence

There is another class of facts, however, which are very telling against the object-property-relation model, and in a particularly worrying way not shared with general facts: facts about identity.

It is a fact (assuming, as I am whenever I use examples such as the following, that the story of Superman is true) that Clark Kent is Superman. We can also say that it's a fact that Clark Kent is Clark Kent, but this is not the same fact. At least, that's a very natural thing to say. It is very natural to say that, if Lois Lane learned that Clark Kent is Superman, she would have learned a new fact which she didn't know before, namely the fact that Clark Kent is Superman.

What makes this 'particularly worrying' from the point of view of the object-property-relation model is, I believe, this: that here, the individuation of facts in some sense involves concepts, senses, meanings, modes of presentation or something of that sort. And not just in the sense that we need concepts or etc. to grasp objects, properties and relations which we then individuate by. Rather, we bring concepts or etc. to bear on the individuation itself; we count the fact that Clark Kent is Clark Kent as distinct from the fact that Clark Kent is Superman. (No such thing is suggested by the problem of general facts.)

As clear as it is that the fact that Clark Kent is Clark Kent is not the same as the fact that Clark Kent is Superman, this peculiar involvement of concepts (or whatever) in the individuation of facts which aren't about concepts, but other things like people, planets or numbers, appears outrageous from a certain confused point of view. Since (as I hope to make clear) it is a confused point of view, there is probably no such thing as a genuinely clear statement of what the problem is supposed to be with this involvement, but only more or less apt and characteristic evocations of the worry. The worry, then, is something like: facts are supposed to be objective, “out there”, and mind- and language-independent (at least when they are not facts about mental or linguistic things, or things which depend on them), but if their individuation can involve concepts or modes of presentation or whatever, how could they be? What are these strange, shadowy things, which seem to multiply before us when we bring new concepts on board? Are we not guilty of some confused reading of features of our language and thought into reality?

These are worries we will be dealing with in later posts. But for now, we will observe that there is a further class of facts upon which the core breaks down. Namely, the class of singular negative existential facts, such as the fact that Santa Claus doesn't exist, and the fact that Peter Pan doesn't exist. These are different facts, but they can't be sensibly mapped to any n-tuple which accords to the scheme considered above, let alone a distinct one each, because there are no (real, existing) objects which they are about.

So, facts about identity and existence destroy the core – the only way out, short of an unnatural and counterintuitive account of facts, would be to try to analyze propositions about identity and singular existence so that the problem disappears. I plan to argue in future that any such attempt will be futile. (Furthermore, any half-way plausible analysis of singular existence propositions would surely involve quantification, and quantification conflicts with the core, albeit perhaps in a less worrying way than identity and existence do.)

Propositions involving other constructions, such as propositional attitudes, or adverbs, may also be argued to conflict with the core, but this will not have the simple and dramatic effect that considerations of generality, and, especially, identity and existence, have. But the damage is already done; the object-property-relation model, at its core, just doesn't work as a general model of facts (or even just of facts not built out of further facts).

In the next post, we will consider the extent to which granularity considerations can come into the individuation of facts. In later posts, we will try to deal with some of the philosophical worries which arise in connection with facts once they are seen not to conform in general to the object-property-relation model.

Breckenridge and Magidor on Arbitrary Reference: An apparent counterexample

[This is an early draft of a paper which, since being posted, has grown and changed title. Email me if you would like a copy. - TH 9/4/15]

In an interesting paper forthcoming in Phil. Studies, Breckenridge and Magidor argue for this thesis:

Arbitrary Reference (AR): It is possible to fix the reference of an expression arbitrarily. When we do so, the expression receives its ordinary kind of semantic-value, though we do not and cannot know which value in particular it receives.

Their primary argument in favour of AR is that it can be used to give an attractive account of 'instantial reasoning' such as this (their 'Argument 1'):

(1) There is someone x such that for every person y, x loves y [Premise]
(2) Let John be such a person
(3) For every person y, John loves y [Existential Instantiation on 1]
(4) Let Jane be an arbitrary person
(5) John loves Jane [Universal Instantiation on 3]
(6) There is some person x such that x loves Jane [Existential Generalisation on 5]
(7) But since Jane was an arbitrary person, for every person y there is some person x such that x loves y [Universal Generalisation on 6] 

I will not attempt to rehearse, or even summarize, their arguments, since they state them well and their paper is freely available on Magidor's website. My purpose here is to give an apparent counterexample to the claim that AR can be used to give an attractive account of instantial reasoning.

The following appears to be a logical truth: 

(Unref) If (all unreferred-to objects are white and there is an unreferred-to object), then there is a white object.

(By 'unreferred-to object', I mean an object which is never referred to by anyone or anything.) Here is a quasi-formal argument for (Unref):

(1) All unreferred-to objects are white and there is some unreferred-to object. [Assumption]
(2) All unreferred-to objects are white. [Conjunction Elimination on 1]
(3) There is some unreferred-to object. [Conjunction Elimination on 1]
(4) Let O be such an object.
(5) O is white. [Universal Instantiation on 2]
(6) There is some white object [Existential Generalization on 5]

(Unref) now follows from (1) - (6) by conditional proof.

This seems to be a valid argument. But the theory of instantial reasoning advanced by Breckenridge and Magidor seems to imply that the expression 'O' above refers to an unreferred-to object, which is absurd.

Tristan Haze
The University of Sydney

Reference
Breckenridge, Wylie & Magidor, Ofra (forthcoming). 'Arbitrary reference'. Philosophical Studies. 

There is a post about this paper on Ross Cameron's blog here.

A Note on Hofweber's Distinction between Internal and External Quantification

ABSTRACT: Thomas Hofweber's distinction between internal and external quantification is crucial to the solution he offers to his now well-known puzzle about ontology. Here I argue that this distinction is not well motivated by the considerations he employs.

In a series of interesting papers (2000, 2005b, 2007), Thomas Hofweber has identified a puzzle about ontology and developed a novel solution. Briefly, the puzzle is that questions such as 'Do numbers exist?' seem trivial from one point of view, but highly contentious from another. On the one hand, it is obvious that, e.g., there are even numbers smaller than 6. It follows trivially from this obvious statement that there are numbers. On the other hand, it is hotly disputed among philosophers whether or not there are numbers. Hofweber himself denies their existence. Nonetheless, he agrees that, e.g., there are even numbers smaller than 6.

Hofweber's solution to this puzzle crucially involves a distinction between two kinds of quantification which he calls 'internal' and 'external'. External quantification is familiar; externally quantified statements impose conditions on a domain of objects. Hofweber plausibly argues (2000, 2005b) that we must also recognize a kind of quantification which does not impose such conditions. His strategy is to highlight a certain 'inferential role' which quantifiers play in natural language, which enables them to function as place-holders for incomplete information; suppose we knew that Fred admires Thomas Edison, but then forgot this, remembering only that there is someone Fred admires. Hofweber argues that 'this situation is completely general', and that 'the only instances of the quantifier might be things that don't exist'.¹

With this distinction between internal and external quantification on board, Hofweber's solution to the puzzle about ontology is that the "trivial arguments" to the existence of contentious entities are indeed trivially valid, on the proviso that the quantifiers in their conclusions are given an internal reading. Questions about what there is, where 'what there is' is construed externally, thus remain as a non-trivial subject matter for ontology.

My object here is to show that the distinction Hofweber intends to make is not what it may appear to be at first glance, and furthermore that it cannot in fact be motivated solely by means of the considerations (indicated above) which he employs.

Let us begin with the question: why can't external quantification play the role of facilitating the expression of incomplete information? From the considerations offered, it seems that the only reason is that, as Hofweber says, the only instances of the quantifier might be things that don't exist. Thus we might think of internal quantifiers as characterised by the fact of ranging over both merely intentional objects and not-merely-intentional objects, in contrast to external quantifiers, which range over not-merely-intentional objects only. (I will call this 'the simple intentional-permissive understanding' of internal quantification.) This, however, is not how Hofweber conceives the distinction.

This becomes clear once we look at his views about arithmetical discourse with the distinction between the merely intentional and the not-merely-intentional in mind.² You can look for a prime between 24 and 28, and thus be looking for something. However, you will not find one: in this sense, there is no such thing. Hofweber fully recognizes this distinction, while nonetheless believing all quantification in arithmetic to be internal. Thus Hofweber's distinction between internal and external quantification cannot be understood in terms of the pre-existing distinction between the intentional and the not-merely-intentional. And yet this pre-existing distinction seems a natural and sufficient basis for a notion of quantification fit to play the inferential role Hofweber identifies. Therefore his consideration of this role is not by itself a good motivation for his internal-external distinction.

Note carefully that this argument does not require that the simple intentional-permissive understanding of internal quantification be a suitable basis for a solution to the puzzle about ontology. Furthermore, it does not rule out Hofweber's using the puzzle itself as a motivation for a special reading of quantification. The point is that he has not succeeded in establishing an independent motivation for such a reading.

It might be objected that I have not made an adequate case for the possibility of the simple intentional-permissive understanding of internal quantification. But I am not seeking to establish this conclusively; only, given that Hofweber has identified an inferential role which calls for a non-external reading of the quantifiers, the simple intentional-permissive conception is prima facie a better candidate than one based on Hofweber's internal-external distinction (considered apart from any puzzle about ontology). It may seem as though I'm not being quite fair, since I haven't really made his distinction clear in its own right. But I have no idea how to do this. Hofweber wants a reading of quantification such that the following comes out true:

There is an x such that x is not a merely intentional object, and x does not exist (in the external sense).

It has not been made sufficiently clear that such a reading is available.

Finally, one might wonder how Hofweber's internalism about arithmetical discourse avoids trivializing arithmetic. For on this conception, so-called "existence statements" about merely intentional objects (e.g. the largest prime) can easily come out true. Hofweber handles this with a supplementary doctrine to the effect that quantification in arithmetic is generally restricted to statements which have instances containing number words or numerals ('one', '46', etc.). However, and as Hofweber himself acknowledges, this sort of account cannot be extended to the reals, since we do not have number words for all of them.³ This gives rise to the worry that whatever the truth is about our quantification over the reals, it may also account for our quantification over natural numbers in arithmetic, rendering internalism about arithmetic theoretically superfluous.

Tristan Haze

The University of Sydney

References

Hofweber, T. 2000. 'Quantification and Non-Existent Objects', in Empty Names, Fiction and the Puzzles of Non-Existence, eds. Everett, A. and Hofweber, T. CSLI Publications.

Hofweber, T. 2005a. 'Number Determiners, Numbers, and Arithmetic', The Philosophical Review 114:2.

Hofweber, T. 2005b. 'A Puzzle about Ontology', Nôus 39:2.

Hofweber, T. 2007. 'Innocent Statements and their Metaphysically Loaded Counterparts', Philosophers' Imprint 7:1, <www.philosophersimprint.org/007001/>.

These papers are available on Hofweber's homepage: http://web.mac.com/hofweber/Thomas_Hofwebers_homepage/Papers.html

1 (2000), p 16.

2 These views are indicated in his (2005a).

3 Thanks to Thomas Hofweber for helpful correspondence on this and related points.