Wednesday, June 2, 2010

Property entailment as the ground of modality

Consider a view on which all modality is grounded in property entailment: the relation between properties F and G expressed by "having F entails having G". Jubien has defended a view that might sound like this (though see comments at the end). One way to make this precise is to say that the theory of necessity is generated by the axiom schema

  1. (x1)...(xn)(A(x1,...,xn)→B(x1,...,xn)),
where A expresses a relation that entails the relation expressed by B (I take relations and properties to be the same thing here, just that we use different words depending on the adicity), and where → is material implication, together with the rule of inference that
  1. from a subproof of p that reiterates no assumptions other than instances of (1), we can derive Necessarily(p).

Here is one quick problem. This fits best with a Platonic metaphysics of properties (Jubien certainly does that). On a Platonic metaphysics of properties:

  1. Necessarily(a is a circle → circularity exists).
Actually, standard Platonists will say that the consequent holds necessarily, independently of the antecedent, but we don't need this for the argument. But (3) cannot be proved via (1) and (2), unless the system is inconsistent. Why? Here is a simple way to see this. The axioms make no reference to circularity. There are instances of (1) that use the predicate "is a circle", but that's not good enough.[note 1] But then take any proof of (3) and replace "circularity" with a non-referring singular term. We will then get a proof that the referrent of the non-referring singular term exists, and given that the axioms are all true (this is uncontroversial), the only way we can get a proof of a falsehood is if the system is incoherent.

One might think that something could be done if existence is a property and there are entailment relations like that being a property entails existence. But that won't help unless we add to the axioms that circularity is a property. But the axioms are automatically necessary by (2), so now we are no longer just giving a property entailment account. We are adding axioms that directly force cerain essentialist claims, like that any property is essentially a property.

Alright, maybe that's unfair. Maybe any Platonist who has a property entailment view of modality will also have among the axioms the schema:

  1. exists(P)
for every property P. But what about other very plausible necessities, like:
  1. Necessarily(circle(a) → circularity is a property).
We had better not make existence entail propertyhood, as then Obama becomes a property.

Of course, we can add things like (5) to our axiom scheme. But the theory is now really swelling, and it is no longer true that it grounds modality in property entailment. It grounds modality in provability from a whole bunch of axiom schemata, one of which is the property entailment one.

My fairly quick glance at Jubien doesn't show him discussing this. But it does show him discussing a related issue, Kripkean arguments that a certain particular table must be made of wood. Jubien says that the table has a "table-essence" being this table and being this table entails being made of wood. So he could handle (5) by saying that circularity has an "object-essence" (I think it had better be an obejct essence, in his terminology), being circularity, and that being circularity entails being a property. Fine so far, but what about the following necessities:

  1. Necessarily(exists(circularity) → circularity has being circularity).
So what this shows is that on the proposed account we need additional necessity-generating axioms governing object essences. When Jubien introduces object essences, he explicitly says that they have modal properties, such as that an object necessarily has its object essence. So perhaps Jubien is not someone we should describe as giving an entailment view. For maybe he has an axiom schema that generates (6), as well as the Platonist schema that generates (5). Now as long as the theory of necessity was generated by (1) and (2), it was a pretty cool partial reduction. But once we had to add the schemata (4) and something generating (6), we start to wonder: what guarantees that no further schemata need to be added? All the axioms are automatically necessary, after all, and once we have a plurality of schemata, we have failed to explain what they all have in common—what makes them all be necessary. The account becomes in effect disjunctive.

Here is a different kind of problem. Consider this claim:

  1. Necessarily(wrongs(x,y) → agent(x)).
Only agents can wrong. But (7) isn't an instance of (1). Now, maybe, "wrongs(x,y)" is an abbreviation for something one of whose conjuncts ascribes to x a monadic property that entails being an agent. But what if it's not like that? And is it really plausible that all relations that entail a monadic property in one of the relata are abbreviations for stuff that includes a monadic property attribution? (Here is an example every Platonist should accept: Instantiates(a,circularity) entails circle(a).)

So, it seems, the system needs to be extended to include entailments between relations of different adicities. Moreover, it needs to be extended to include entailments between relations of the same adicity but with the relata reorganized:

  1. Necessarily(wrongs(x,y) → iswrongedby(y,x)).
We no longer have just entailment between relations, then, at the base of the system. We have what one might call "twisted entailment". Specifically, if f is a function from {1,...,m} to {1,...,n}, we can say that the n-adic property P f-twistedly-entails the m-adic property Q provided that:
  1. Necessarily(A(x1,...,xn) → B(xf(1),...,xf(m))),
where A and B express P and Q respectively.

The view is still non-trivial. But it is messy, and it is difficult to see what motive one has for believing it rather than just giving up on the grounding project altogether—or going for my view. :-) After all, f-twisted-entailment is not such a natural property as entailment.

[Fixed definition of twisted entailment.]


Alexander R Pruss said...

I suppose we also want the rule that from necessity we get actuality. This doesn't affect what I say.

sophia said...

I just couldnt leave your website before saying that I really enjoyed the quality information you offer to your visitors... Will be back often to check up on new stuff you post!