Relations and Nonexistents

 Consider the following two sentences: 

a) Lions are smaller than dragons.
b) Mice are smaller than elephants.

From this datanic base a puzzle emerges. 

1) The data sentences are both true.
2) 'Smaller than' has the same sense in both (a) and (b).
3) In both (a) and (b), 'smaller than' has the same reference: it refers to a dyadic relation.
4) No relation holds or obtains unless all its relata exist.

What we have here is an aporetic tetrad. The four propositions just listed are individually plausible but collectively inconsistent: they cannot all be true. What we have, then, is a philosophical problem in what I call canonical form. Any three  of the above four, taken in conjunction, entails the negation of the remaining one.  Which limb of the tetrad should we reject?

One might reject (4) while upholding (1), (2), and (3).  Accordingly, some relations connect existents to non-existents.  It is true that lions are smaller than dragons despite it being the case that dragons do not exist.  The sense of 'smaller than' is the same in both (a) and (b).  And 'smaller than' picks out one and the same dyadic relation in both (a) and (b).

The idea here is that there is nothing in the nature of a relation to require that its obtaining entails the existence of all its relata.  Contrast thinking about the Trevi Fountain in Rome and thinking about the Fountain of Youth. Some will say that in both cases the intentional nexus is a genuine relation since there is nothing in the nature of a relation (to be precise: a specific relatedness) to require that all of its relata exist.  It is the same relation, the intentional relation, whether I think of an existing item or think of a non-existent item.

If you don't like this solution you might try rejecting (2) while upholding the remaining limbs: 'smaller than' does not have the same sense in our data sentences. Accordingly, 'are smaller than' in (b) picks out a relation that actually connects mice and elephants.  But in (a), 'are smaller than' does not pick out that relation.  In (a), 'is smaller than' has the sense  'would be smaller than.'  We are thus to understand (a) as having the sense of 'Lions would be smaller than dragons if there were any.'

(2)-rejection arguably falls afoul of Grice's Razor, to wit: one ought not multiply senses beyond necessity. Here is what Grice himself says:

[O]ne should not suppose what a speaker would mean when he used a word in a certain range of cases to count as a special sense of the word, if it should be predictable, independently of any supposition that there is such a sense, that he would use the word (or the sentence containing it) with just that meaning. (Grice, 1989, pp. 47-48, Quoted from Andrea Marchesi, "A radical relationist solution to intentional inexistence," Synthese, 2021.)

Pick your poison.

 

Comments

[Oz]

I am working on this subject as we speak, for reasons you will appreciate.

Your proposition (4) is unclear.

(4) No relation holds or obtains unless all its relata exist.

What does ‘its relata’ mean? I assume it means the relata of the relation, the relata that it has. I.e. there is a relation expressed by the verb ‘has,’ the relation by which the relation stands to the relata. So can we replace your (4) by

(4*) No relation holds or obtains unless all it has relata

I would say that is enough. Does the relation ‘are smaller than’ hold between mice and dragons. Certainly, if the relation has mice and dragons as relata. Why would the fact that the relata which it has are non-existing things? The ‘having’ relation is enough.

If by contrast the relation does not have such relata, your expression ‘its relata’ refers to nothing. I am assuming that your genitive ‘its’ signifies the relation of having or possession. If you mean something else then say so.

There is a separate exegetical question about what Brentano means by ‘eine Beziehung zu einem immanenten Objekt’. Does ‘Beziehung’ mean a relation, or what? Brentano is unclear on the question. More later.

[BV]

(4) is perfectly clear. Why do you stumble over it? It is a substantive metaphysical claim the truth of which is in dispute.

(4*) is nonsense as it stands. Deleting 'all' it makes sense, but is utterly trivial. It is true by definition.

Therefore, (4) cannot be replaced by (4*). That, to my mind, is perfectly clear. Why are you quibbling over this?

Brentano confuses Inhalt und Gegenstand. His view of intentionality is indefensible. But intentionality theory can no more be identified with Brentano's views, early or late, than with some radically externalist view -- which is the exegetical mistake that you make.

[Cyrus]

Possibly too quick, but how about a truthmaker reply? Consider the following:

(1a) Mice are smaller than dragons.
(1b) Mice are smaller than chimeras.
(1c) Mice are smaller than the golden mountain.

For (1a) where “dragons” stands in for a kind of fictional entity, perhaps the statement is true in virtue of essential conjunctive or kind properties of mice and the sum of everything written about dragons.

For (1b), where “chimeras” stands in for a kind of impossible entity (conceptually or essentially incoherent, i.e. incoherent in terms of what would be its essence), then I might just have to bite the bullet and say that either (1b) collapses into a form of (1a) or (1b) is just well-disguised nonsense. (I'm not sure I actually believe that it's well-disguised nonsense, but I'm channeling my favourite Australian right now. I might be able to defend it.)

For (1c), where “the golden mountain” stands in for a merely possible entity, conjunctive or kind properties of mice, gold atoms, and relational properties of mice.

Do we still need a referent for “smaller than”? Can we deflate the problem out of existence with Charlie Martin and dear old Armstrong's theory?

[Cyrus]

In my most previous, "relational properties of mice" s/b "relational properties [constituting mountains]". Sorry.

[Oz]

There was a typo, as you note.

Is my statement trivial? No, I am making a fundamental point here. Does “A has B” imply “B exists”? I think you will say not, because A could ‘have’ some non-existing thing. Then it follows that a relation R can possibly ‘have’ relata that do not exist. But then R stands in the ‘having’ relation to its relata, which is a paradox, for on your account the ‘having’ relation cannot hold or obtain unless the relata exist.

On the other hand, if “A has B” implies that B exists, then as you say it is trivial that every relation holds between its relata, i.e. the relations that it has, and which by definition exist. I hope that makes the paradox clearer.

[oz]

On the Brentano exegesis point, I think he says different and contradictory things at different points. E.g.

"Everything that we experience so evidently inwardly shows a relationship to an immanent object. I grasp a representation in myself; this is not without something that is presented, that is: without an immanent object there, be it present outside or not."

I can’t understand that passage unless I take ‘immanent object‘ to mean Jupiter, either the planet (which is present outside) or the god (which is not present outside). He clearly doesn’t mean a representation or idea of Jupiter. I can find many passages like this.

[BV]

Oz @9:08 >>Does “A has B” imply “B exists”?<<

That's a weird way of talking. I suspect you are assuming an Aristotelean-scholastic theory of relations according to which there are no irreducible relations in ultimate reality. There are substances and their accidents, and relations reduce to monadic foundations in substances which are relational accidents. Call this view foundationism.

Given that Socrates is shorter than Simmias, I guess you will say that the sentence 'Socrates is shorter than Simmias' is to be analyzed as a conjunction neither conjunct of which contains a relational expression: 'Socrates has shortness-with-respect-to-Simmias AND Simmias has tallness-with-respect-to-Socrates.' Thus there is no relation hovering between the two men.

Is that the way you think of relations?

[oz]

> I suspect you are assuming an Aristotelean-scholastic …

No on the contrary I am thinking purely of logical implication.

1. Jake owns something
2. Something is owned by Jake
3. Something owned by Jake exists

You probably accept the move from 1 to 2 where we turn a relational statement into a categorical one. The move from 2 to 3 depends on the Brentano-Venn thesis, which ironically was an innovation of Brentano himself. See my page here http://www.logicmuseum.com/wiki/Brentano-Venn_thesis

Likewise, the relational statement “R has relata” implies “some things are relata of R”. By Brentano-Venn we can turn this into “some relata of R exist”, in which case your (4) above is trivially true – if R has relata, the relata exist.

Now if you choose to deny the Brentano-Venn thesis, AND you hold (4) above, you get the paradox I identified above. R could not stand in the ‘having’ relation to its non-existing relata, if by your account the ‘having’ relation cannot hold or obtain unless the relata exist.

This is all a matter of logic and nothing to do with metaphysics.

What I suspect you are having trouble with is understanding that 'R has relata' is itself a relational expression.

[oz]

Oddly, you refer to the Brentano Venn thesis in your paper "Brentano on Existence", History of Philosophy Quarterly Vol. 18, No. 3, 311-327, see page 320 on 'some man is sick' etc.

You argue against the thesis, but that is irrelevant to my argument, which is that if you accept the thesis, your (4) is trivial, but if you don't accept it, you run into problems.

[David Brightly]

Morning Bill,

I am wondering if there is an aporia here at all. Let X be the extension of the relation referred to by 'smaller than'. Let L be the set of lions and D the set of dragons. D is empty. (4) says that there is no pair (x,d) in X with d in D. This is consistent with D being empty. But it's also consistent with (a) being true. Read (a) as 'every lion is smaller than every dragon':

∀x. x ∈ L → (∀y. y ∈ D → (x,y) ∈ X)

The expression in parenthesis is 'vacuously' true because there are no such y (since D is empty), and hence the whole expression is true. Alternatively, there is no counterexample y ∈ D such that (x,y) ∉ X, that would make (a) false, again because D is empty.

[BV]

Good morning, David. I apologize for not responding to your earlier comment. No job, no kids, no social entanglements, NO TIME!

Your comment is much better than that of your fellow Londoner, OZtrich.

Since you are a maths and IT man, your comment is not surprising. It is also very good, and I am having trouble thinking of a good response.

Would you say that a relation just is a set of ordered n-tuples? Then the 'relation' connecting a thinker and a nonexistent item is the null set, right? Or would it be the set consisting of the null set: {{ }}?

[oz]

Brightly: "(4) says that there is no pair (x,d) in X with d in D."

Does (4) say that?

"there is no pair (x,d) in X with d in D" is broadly equivalent to "there is no mouse that is smaller than any dragon". But this is (4):

(4) No relation holds or obtains unless all its relata exist.

>Your [Brightly's] comment is much better than that of your fellow Londoner, OZtrich.

You still haven't addressed the beautiful paradox that results from the non-trivial reading of (4).

The paradox is that if a relation has non-existent relata, then the relation signified by 'has' must hold between the relation, and its non-existent relata. But (4) refers to 'its relata', i.e. the relata of the relation, the relata which the relation has.

[Cyrus]

Ostrich:

Now if you choose to deny the Brentano-Venn thesis, AND you hold (4) above, you get the paradox I identified above. R could not stand in the ‘having’ relation to its non-existing relata, if by your account the ‘having’ relation cannot hold or obtain unless the relata exist.

By 'the "having" relation', do you mean what philosophers variously call the 'instantiation' or 'exemplification' relation? This is, if I understand right (I may not!), very strange talk for an ostrich.

[Cyrus]

I ask in part because I'm assuming you (OZ) know your interlocutor's corpus (BV's) fairly well, and (as part of following your back-and-forth) I'm trying to figure out why you would think this:

What I suspect you are having trouble with is understanding that 'R has relata' is itself a relational expression.

When he has written ad nauseam about problems of instantiation relations, and seems almost tiresomely aware that 'R has relata' is a relational expression.

[oz]

Cyrus
>By 'the "having" relation', do you mean what philosophers variously call the 'instantiation' or 'exemplification' relation? This is, if I understand right (I may not!), very strange talk for an ostrich.

Ostriches are OK with instantiation of linguistic items (as opposed to 'universals').

>When he has written ad nauseam about problems of instantiation relations, and seems almost tiresomely aware that 'R has relata' is a relational expression.

Possibly, but he insists that (4*) below is trivially true whereas (4) is not trivially true but rather a ‘substantive Metaphysical Truth’.

(4) No relation holds or obtains unless all its relata exist.
(4*) No relation holds or obtains unless it has relata

But if (4) is not trivially true, there must be an example of a relation R such that (a) R has relata, but (b) R does not hold because the relata do not exist. But (a) and (b) cannot both be true, so (4) must be trivially true as well.

[Cyrus]

OZ:

>By 'the "having" relation', do you mean what philosophers variously call the 'instantiation' or 'exemplification' relation? This is, if I understand right (I may not!), very strange talk for an ostrich.

Ostriches are OK with instantiation of linguistic items (as opposed to 'universals').

I'm currently sitting at a dinner table in Thamesmead looking out at two apartment buildings. One of the buildings is smaller than the other. How would you assay this? (What does it mean here to say the two apartments are instantiating linguistic items? Further, what is instantiating the linguistic items? other linguistic items? If not other linguistic items, what is holding between the linguistic and non-linguistic items? If other linguistic item, what is holding between the linguistic items?) I realize that I'm asking a lot of questions, but I'm genuinely a bit baffled and trying to pin down what your views are.

Possibly, but he [BV] insists that (4*) below is trivially true whereas (4) is not trivially true but rather a ‘substantive Metaphysical Truth’.

(4) No relation holds or obtains unless all its relata exist.
(4*) No relation holds or obtains unless it has relata.

But if (4) is not trivially true, there must be an example of a relation R such that (a) R has relata, but (b) R does not hold because the relata do not exist. But (a) and (b) cannot both be true, so (4) must be trivially true as well.

I suspect that all he means is that (4*) is true purely in virtue of the meanings of its terms (i.e. having relata is just what means for a relation to 'hold' or 'obtain'), whereas (4) claims something contentious. To put it another way, (4*) is compatible with every ontological theory (in other words, it doesn't 'take sides' in the ontological debate), whereas (4) 'takes sides' by ruling out Meinong's theory.

[David Brightly]

Hello Bill,

>> Would you say that a relation just is a set of ordered n-tuples?

No. In the monadic case that would reduce a property to the set of its instances. The extension of a relation that had no instances would indeed be the empty set.