logicandlanguage.net

So at FEW there were a few people (I think the commentator) who suggested you only proved a weakened version of these barrier theses. In particular someone argued that since you didn't address intermediate sentences which involve elements on both sides of the barrier this didn't really show what you wanted.

In particular they argued (in Hume's law) that one might know that Napolean existed (N) and thus deduce ~N->O[A] for any act A. The argument is that "If Napolean hadn't existed one ought to do blah" sounds like a moral statement so you somehow fall short of Hume's law because you only rule out deriving entierly moral claims. Furthermore, I think they argued that you further fall short as you don't say anything about whether agents who believe these mixed sentences can derive moral claims.

First of all I suggest that our intuition that suggests sentences like the above are properly moral rest on a non-material reading of the conditional above. That is we think the above sentence is moral because we imagine what we would be ought to do in worlds with Napolean even though the material conditional doesn't authorize this reading at all. If we simply restate the sentence as "It is actually the case that Napolean existed or it is actually the case that one ought to blah," and banish the Gricean conventional implicature that one never asserts an or if one knows one disjnt is true the intuition that this is moral disappears.

Moreover, I think any requirement that you treat mixed sentences is a misreading of these laws. The intuition behind Hume's law is that no amount of observation of the world is ever going to entail any moral facts. Now any mixed statements you somehow manage to deduce from observational facts will be totally empty. ANY assignment of moral atoms (ought blah) will be compatible with these observations, i.e., the semantic options for morality are not decreased by these observational facts. If there were semantic restrictions and your system is complete this would allow one to deduce some completely moral statement

I think your proof shows that no agent who only makes observations will never put any more semantic restrictions on morality than pure logic does alone. Moreover, I think this exhausts the content of Hume's law so it is unfair to ask any more. It might be a little unclear when formulated syntacticly as one has those empty moral disjunctions hanging around and being confusing. However, I think it is quite clear semantically if I'm not making some mistake. In other words can't you just redefine a statement with moral implications as one which puts a restriction on the semantic assignment of ought?

Also while I would very much like to see a version with an alethic modal operator I don't see why ought implies can should be entaield by our logic. Sure we have an intuition that this is true but I'm skeptical that ought implies can is analytic much less logical.

Well, if you can't get "can" from "is" either, then there's no problem, right?

Can, like ought, requires an understanding of possibilities. Maybe there's no problem?

Cheers,
-MP

Related to logicnazi's point - because you've proven a barrier from sentences of type A to sentences of type B, there's not too much of a problem for the sentences in between (which seem at least in part to be disjunctions of a sentence of type A and one of type B). These intermediate sentences can themselves further be partitioned into the ones that can be proven from type A sentences one knows, and the ones that prove type B sentences, given what one knows. Because of the overall inference barrier, these two classes will be disjoint (unless one's "knowledge" is inconsistent). I suppose they probably won't fully partition the set though, and might be relative to background knowledge.

Also, I'll look up a good summary of those "preservation theorems" I mentioned before, which seem to be duals of the definitions you use of particular and universal sentences. I know they're definitely discussed in Chang & Keisler's "Model Theory" and Hodge's "Model Theory", but I'm not certain if they're available in any thinner book.

I don't see why we want to partition the entire set ay priory like into moral and descriptive claims. In fact it seems like this is far too much to hope for as the disjunction cases would simply make Hume's law false in normal logic. Although something stronger might be true in relevance logics (i.e. can never get a non-descriptive sentence from descriptive ones). I don't remember if you said you were doing anything about this in your talk.

Kenny, in this

These intermediate sentences can themselves further be partitioned into the ones that can be proven from type A sentences one knows, and the ones that prove type B sentences, given what one knows

I think we need to restrict the second "what one knows," to forbid the type B sentences one knows. Should it be again "given the type A sentences one knows"?

Maybe we could go for:

type A sentences
type C sentences, which can be proved from the type A sentences one knows
type D sentences, which can prove type B sentences given the type A and C sentences one knows
type B sentences

but again no reason to think this would be exhaustive.

I'm going to have a look at Entailment to see if relevance logic gets us anything interesting here--one of the things I realized at FEW (I think I mentioned this to logicnazi) is that I'd forgotten whether DeMorgan's even applies in R. (Turns out it does.)

That division works if you require complete knowledge of type A sentences (that is you know every sentence or its negation). However, if you don't require this an exhaustive division can't be done.

Let p be a type A sentence one neither knows nor knows it's negation. Let q be a type B sentence.

Now pvq can prove a type B sentence if you learn ~p but it can be proved from your type A sentences if you learn p. This is why I don't think you can ask for any more in terms of Hume's law then what Gillian has already shown.

LN, we can save that if we go to relevance logic. In relevance logic you can't infer q from ~p and pvq. (I have always found this a little hard to wrap my head around.)



Thinking it over, given a group of A sentences the following is definitely an exhaustive partition:



C sentences: Sentences that can be proved from some set of A sentences in R

D sentences: Sentences that can't



The question is whether it's a trivial partition, and whether all B sentences turn out to be D sentences. (OK, that's two questions.) The set of C sentences had better not be closed under negation, or we'll run into the problem Gillian and Greg brought up in slide 4; but the set of A sentences actually can be closed under negation, because R doesn't let you infer everything from a contradiction.



I think just using G & G's descriptive sentences as the set of A sentences will actually work, in that their normative sentences will all turn out to be D sentences, and even sentences of the form P -> O(Q) will turn out to be D sentences (as Vranas desired). ~P v O(Q) will be a C sentence, but we don't want it to be normative, because taken together with P it doesn't imply O(Q).



But I haven't proved that. Since the semantics of R are pretty intractable (and found in the volume of Entailment I don't have), the only way I can think of to prove that would be to take your favorite axioms for deontic logic and do independence matrices. If someone wants to point me to their favorite axioms for deontic logic, I might try to see if some of the existing matrices can be exploited in some obvious way.



None of this is to say that we should really desire an exhaustive partition. Just in the spirit of "let's see if we can get one."

Gillian,

Thanks for the reply to my post. I am quite out of the loop when it comes to modal logic; could you explain what "bi-modal" logic is, and why, exactly, you getting around the point I made depends on having a good understanding of it.
If your post actually is just about bi-modal logic, and I am ignorant, that would be helpfull too, since I understood the post.

Matt, I think your claim about ~p not allowing you to derive q from pvq depends on how you choose to translate pvq into the basic connectives in relevance logic. That is the treatments of relevance logic I have seen simply define statements involving ors in terms of statements involving -> and &. Sure they usually define pvq as ~(~p&~q) however the 'real' analog of the proposition I am interested in is actually (~p)->O(q) (where this is the relevant conditional). However, now I believe this is simply not going to be in C.

Now there are two ways to read what Vranas wanted. First is that he wanted to divide all the sentences up into descriptive and normative sentences. This is clearly false as Putnam pointed out we have plenty of mixed statements. More reasonably he is asking for a set containing all the descriptive sentences and no normative sentences which is deductively closed. In other words he wants to make sure there are no sentences like the pvq example I gave which can be proved from one set of descriptive facts and can prove a normative claim from another set of descriptive facts.

Now one would thus also need to verify your set C is deductively closed. This isn't obvious as the corresponding set in standard logic is not deductively closed (the consequence of one set of descriptive facts can be used to prove a normative fact with another set). I suspect you might be able to just take the deductive closure of your set C and be fine but I just don't know enough relevance logic to say for sure.

Still I think you are essentially right that relevance logic gives us what we want. In fact the relevance logic WR (slightly weaker than R) gives us exactly what we want ( read about it here ). In this logic no collection of premises can prove a result which has a propositional letter not in the premises. So if we let L(D) represent the language containing the descriptive propositions and L(N) the normative ones it is a trivial result that L(D) is closed under deductive consequence (i.e. if you start out mentioning only descriptive facts you end only mentioning descriptive facts).

Of course though if we move to this kind of relevance logic the result becomes pretty trivial. In any case I still think the better way to go is to prove it in full first order logic and show that no amount of descriptive facts can constrain the semantic assignment of truth to normative facts.

C as I've defined it is definitely deductively closed. It's defined as the set of sentences that are relevantly implied by some collection of A-sentences. Suppose P is relevantly implied by C1...Cn in C. Each of the Ci is relevantly implied by some set of A-sentences Si; the union of the Si will relevantly imply P. (Actually I retract the definitely; it's possible that relevance leads to a snarl in there. But I'm almost certain it doesn't.)

Thanks for the link to W(R).

I agree that ~p->O(q) won't be in C; it doesn't follow relevantly from p.

(Brendan: I'm pretty sure that bi-modal means that 'ought' is one kind of modality and necessity is another kind of modality, so a logic that contains them both is bi-modal. So Gillian needs to understand it because your point involves both 'ought' and 'can'.)

Interesting convo. I have a question.

Is a descriptive focus an appropriate one for a legal theorist to adopt?