# Greg Wheeler’s paper on Lottery and Preface

Greg’s forthcoming paper is here. It’s about the notion of rational acceptance and discusses the Hawthorne-Bovens’ paper on the paradoxes. Check it out!

#### Greg Wheeler’s paper on Lottery and Preface — 6 Comments

1. Hi Greg,

I like your paper very much! (It’s about a paper written my me and Luc! What’s not to like!!!) But I want to better understand what you have in mind by a “structured system” for a consequence relation, or for a logic of rational acceptance. For example, does the well known system for the preferential consequence relations, P, count as “structured”? More generally, do any ,or all, of the systems O, P, Q, and R described in the previous posting (about defeasible conditionals that behave like conditional probabilities )count? If not, can any of these logics be turned into structured ones? — and if so, what would one need to do?

2. Hi Jim, thanks! Yours was a fun and stimulating paper to think about.

The short answer to your question with respect to what counts as a structured system for rational acceptance is: System P and stronger, No; the systems you describe below system P, likely (!).

(1) The reason P isn’t a structured system for rational acceptance is because, I would argue, such systems gut the notion of probabilistic acceptance: the acceptance threshold is 1 and must be for the strong and-axiom and the cautious monotonicity property to hold, both of which are necessary properties for P. The argument then is that since structural constraints are additional constraints to what I take to be the standard view to the paradoxes of rational acceptance which calls for a substantive resolution, then adopting System P is effectively not structured since it resolves the issues surrounding closure by trivializing the notion of acceptance.

It should be stressed that this last argument does not appear in the present paper. It needs to be developed with an example of a proposal to resolve the paradoxes from “P and above”, which is a project that is in the queue for me to do. It would be nice to have both arguments in hand so that one could view the present paper as exercising the constraints on one end (strong notion of probabilistic acceptance; syntactically weaker langauge) and exercising the constraints on the other (weak notion of probabilistic acceptance; syntactically strong language), to put the matter very roughly. A point that falls out of this is that I view my reply to your paper as a first attempt to articulate these constraints, rather than view the constraints as a single purpose construction for the reply.

(2) What motivates what I’m calling the structural constraints is my belief that the action on rational acceptance is below system P, which is where O and Q are. The prescriptive methodological idea to require accounts to offer some syntax is motivated by a belief that (much of) philosophical logic is actually (or rapidly becoming) a subfield of applied logic. There is equal concern in applied logic for understanding the formal properties of one’s framework on the one hand but also concern for the ‘fit’ between your framework and the problem domain, just as there is (or should be) in philosophical logic (as opposed to philosophy of logic). (There is also concern for implementation, which I am also sensitive too.)

This is another way to say that I take one of the comments here (in which thread I’ve forgotten, hence by whom is lost to me as well–forgive me) to heart: namely, this formal stuff we’re working on needs to hook up to the epistemology.

(3) Now, to my fudged answer regarding systems O and Q. They certainly are in the spirit of what I had in mind when thinking about these constraints, since I explicitly had in mind system P as being too strong. So, in one sense I think of the constraints as trying to spell out why the slogan ‘the logic for rational acceptance is the logic below system P’ is reasonable. For these reasons, I’m inclined to say, ‘Sure, it’s structured’.

But O and Q raise (deep, I think) questions about the conditional at the heart of each system, on the one hand, and raise another question about even the plausibility (not probabilistic soundness, mind you) of axiom 4, which is at the heart of both systems. I was probing this topic a bit in my reply to your note posted here. Is this a case of a structured view with questionable axioms or a case of a very peculiar conditional that stretches the typical meaning of a logical connective? My own view is that I think these are the right questions to grapple with, whatever the status as a structured theory.

(4) The last question raises a meaty philosophical question, namely the merit of viewing acceptance in terms of conditional probabilities. The other main motivating idea behind the structural constraints and the pitch for syntax and the applied logic paradigm is to try to put our fingers a bit more precisely on these questions so that we can make more reasonable choices about which type of approach is appropriate for what. The payoff would be of both philosophical and practical significance.

This is bit too long. However, hopefully saying a bit about my motives will help direct critical discussion of the structural view, if one should develop.

3. What worries me about the Hawthorne-Bovens paper (as well as about Foley’s) is the direction of the main argument: Isn’t the Lockean Thesis that belief corresponds to a sufficiently high degree of confidence in deep trouble because belief and degrees of confidence behave very differently with respect to conjunction? For simplicity, let us say the Lockean Thesis is just BEL(p) = CON(p) > 1-x, for an appropriate small x. Now, if there are any plausible rational constraints on qualitative or full belief, then this is a pretty safe candidate: If BEL(p1) & BEL(p2), then BEL(p1&p2). On the other hand, degrees of confidence obey the usual probablity axioms (see Hawthorne-Bovens Fn6), such that CON(p1&p2) = CON(p1) * CON(p2), if p1 and p2 are probabilistically independent. If the Lockean Thesis were true, the following must be true: If CON(p1) > 1-x and CON(p2) > 1-x (such that the believer believes p1 and believes p2), then the product CON(p1) * CON(p2) > 1-x. But, as is clear, in many cases this product will be even smaller than 1-x. So, why should we take the Lockean Thesis as our starting point?

4. Greg,

I’m a little unclear on what you see as the problem with P. Is it that although it counts as “structured”, it is wrong for an account of “rational acceptance”? Or does it not even count as “structured”. I guess I’m not clear on what it means for a logic to be “structured”. Perhaps it would help me if you can tell me whether a standard deduction system for predicate logic counts as structured — e.g. does a Gentzen system count as structured?

About the problem with axiom 4, here is axiom 4:
4. if (CÂ·B)–>A , (CÂ·~B)–>A then C–>A.
Notice that this rule appears to be sound for lots of kinds of conditionals (e.g. the material conditional, counterfactual conditionals, indicitive conditionals). Indeed, if we replace ‘–>’ by the logical entailment relation, it is sound for that as well.
Can you tell me more about what you find troubling about axiom 4. Is the problem with its “intuitiveness” as a rule about acceptance, or with its syntax (with how the rule is written), or something else?

Sometimes it looks to me (from things you say in your paper) like you want the rules to be expressed in a formal syntax. Would doing something like this (putting the rules in a formal clausal form) help:
4. -[(CÂ·B)–>A] V -[(CÂ·~B)–>A] V [C–>A]
where ‘-‘ and ‘V’ are not part of the syntax for sentences that go on the left and right sides of ‘–>’, but are “up a level” from there? Or, would it be better if the whole rule were stated in the lowest level object language (in terms of ‘Â·’ and ‘~’)?
Perhaps my question here about syntax is way off base. I’m just trying to get a handle on what you have in mind with your concern about the syntax needed for a structured system.

5. Michael (isn’t it?),

You say, “Now, if there are any plausible rational constraints on qualitative or full belief, then this is a pretty safe candidate: If BEL(p1) & BEL(p2), then BEL(p1&p2).” But is that really such a safe candidate? Consider the “preface paradox”. Can the author consistently BEL(no error on pg 1), BEL(no error on pg 2), …, BEL(no error on pg n), and BEL(error on pg 1 or error on pg 2 or … or error on pg n) (i.e. that there is an error somewhere in the book)? Many of us think that such beliefs are perfectly consistent. But if your “safe candidate” is accepted, then the author must also BEL(~A1 & …& ~An & (A1 v … v An)) — i.e. must Believe a logical contradiction. Giving up the “safe candidate” avoids this problem, and that is just what the Lockean thesis recommends. What would you recommend instead?

6. Hi Jim,

(1) It is assumed that a resolution to the paradoxes of rational acceptance calls for a substantive solution, which is understood to be a proposal that reconciles the general conflict between adopting a probabilistic notion of acceptance and closing sets of so-accepted sentences under consequence. In other words, we wouldn’t find appealing a proposal to remove the antinomy by simply giving up one of these features. (Also, by probabilistic notion of acceptance I mean one that allows for acceptance thresholds that are less than unity; denying this is what I mean by ‘gutting’ rational acceptance.)

The structural constraints are proposed as additional constraints on what we should expect from a proposal, which I’ve spelled out better in the paper than I’m liable to reproduce here in shorter form. True enough, the restrictions are motivated from thinking about some devices within proof theory, and the sequent calculus of course is a sterling example. However, to the best of my knowledge, a Gentzen style sequent calculus would not satisfy the structural constraints proposed here since it would not be a candidate for a substantive solution: it is a deductive system, hence incorporates all axioms of system P. (more on this next).

(2) The conditions we’re discussing are in section 2. It is important to stress that they are necessary conditions. They are designed to prod one to consider syntax which, I’m betting, will allow us to get a better handle on some of the issues that get smoothed over when we just work within model theory. But, since the first two structural conditions are discussing the way in which to represent rational acceptance it is reasonable to count systems satisfying P as not being structural, if you accept the argument that the notion of probabilistic acceptance is gutted. Also, notice that the third condition discusses systems that make the relationship between rational acceptance and logical consequence transparent—which corresponds to a feature that is used to evaluate the syntax of other formal languages. I also note that this is imprecise, but its imprecision is general one: we’ve jettisoned Frege’s notation; polish notation is nice for machines to read but not for people (except maybe Poles and Russians); intro logic students on the continent learn the sequent calculus, North America students typically learn Fitch style natural deduction; and so on.

The upshot is that the structural conditions targeted for systems that are proposed for modeling rational acceptance are systems that offer substantive proposals, which are systems weaker than system P. (Pity I didn’t make this argument about P in the paper, but it is fairly long as it stands.)

(3) Axiom 4…can I hold off on commenting before seeing the logic? I have worries about stacked applications involving this conditional, which a syntactic connective should allow; but this is a vague worry. I’d be more comfortable commenting after I see the completed paper rather than how I imagine your results will turn out.

(3′) In case line (3) sounds like an unfair dodge, let me offer this: under all of this, I am suspicious of modeling the notion of probabilistic acceptance in terms of conditional probabilities. The reasons for my suspicions are rooted partly in practice, partly in my training, neither of which are entirely satisfying reasons to me. The structural view is a proposal to pin down what my suspicions amount to and whether they are well-founded or not. That’s what is driving this proposal; hence, I suspect that if I could give a more precise answer (now) to your inquiry about the structural constraints, I wouldn’t be in the position of needing to propose them. Moreover, judging from the variety of proposals for resolving the lottery that exist in the literature, it seems that making this proposal to narrow the field public may have some value to the community.

(4) Syntax: I would like a language within which to formally express multiple premise arguments composed of rationally accepted propositions and from which to evaluate, based upon the formal features of the language, whether a conclusion is rational to accept of the premises are rational to accept. I had in mind doing this in the object language, but I don’t wish to rule out other approaches.

For instance, I’m toying with an approach that uses two inference rules that takes as premises collections of accepted sentences separated by a comma and returns either a boolean disjunction that is bounded below by a lower probability, or a boolean conjunction that is bounded below by a lower probability. To model multiple probability measures, I do this in terms of inner measures; one could do this in terms of sets of probabilities as well. These rules are sound. The attractive point is that you can have conjunction and disjunction of arbitrary accepted propositions without the particular overhead that comes with viewing the matter in terms of belief states; it is not, however, complete (nothing is for free!). But, observe that the weaker systems below system P are not complete, either.

Oh, by the way: Halpern et. al. gives a probabilistic semantics for system P that is complete, if I’m not mistaken.