Comments on: A Connection between Bayesian and Mainstream Epistemology http://choiceandinference.com/?p=90 Mon, 06 Sep 2010 15:34:57 +0000 hourly 1 http://wordpress.org/?v=3.0.1 By: Tomoji Shogenji http://choiceandinference.com/?p=90&cpage=1#comment-161 Tomoji Shogenji Mon, 08 Jun 2009 18:11:43 +0000 http://choiceandinference.com/?p=90#comment-161 This is a belated reply to Isaac’s May 8 post that refers to my paper (I was away when his post appeared). In <a href="http://www.ric.edu/faculty/tshogenji/Justification_Leuven.pdf" rel="nofollow">the paper</a> that Isaac refers to, I proposed the following measure J(h, e) of epistemic justification. $$! J(h,e)=\dfrac{log P(h|e)-log P(h)}{-log P(h)}$$ In the process of formulating this measure, I reasoned that P(h|e) determines the risk of h being false, while P(h) determines the potential gain in truth, which I took to be the amount of information h carries. Isaac reminds us in his post that he argued against this approach in his article “Inference and Information” in Synthese (1967) and other places. The central thesis of Isaac's 1967 Synthese article is that “information does not decrease with an increase in probability in the sense in which probability determines fair betting quotients” (370). I have my reason for holding onto my own view (I do not accept Isaac’s principle A-2 on page 374), but I am not going into this issue here. I want to point out instead that the main part of my paper is not tied to the fate of the particular view about information that I adopted in the paper. For example, Isaac distinguishes the probability function $$M_{K}(h)$$ that determines the informational value of h, from the credal probability function $$Q_{K}(h)$$. Since $$Q_{K}(h)$$ determines the degree of risk and $$M_{K}(h)$$ determines the amount of potential gain in information, their roles are comparable, respectively, to those of P(h|e) and P(h) in my paper. So, those who like this part of Isaac’s view can systemically replace P(h|e) and P(h) in my paper with $$Q_{K}(h)$$ and $$M_{K}(h)$$, respectively. The resulting measure of epistemic justification would be J*(h, e) instead of J(h, e) above. $$! J*(h,e)=\dfrac{log Q_K(h)-log M_K(h)}{-log M_K(h)}$$ Of course, Isaac uses $$Q_{K}(h)$$ and $$M_{K}(h)$$ to determine the expected epistemic utility of h in his decision-theoretic approach to belief change, while I make no use of decision theory in my reasoning for the J measure. (My proposal is guided, instead, by the idea that if h1 and h2 are mutually irrelevant, e.g. h1 is about the demise of the Roman Empire while h2 is about the salmon’s immune system, then the acceptance of h1 should not affect the epistemic evaluation of h2.) Given this fundamental difference in the approach, I do not expect Isaac to find J*(h, e) appealing, but my point here is that the main part of my paper would be of interest to those who are open to a non-decision-theoretic approach to increasing true beliefs and avoiding false beliefs, even if they doubt that P(h) determines the potential gain in truth (information). This is a belated reply to Isaac’s May 8 post that refers to my paper (I was away when his post appeared). In the paper that Isaac refers to, I proposed the following measure J(h, e) of epistemic justification.
J(h,e)=\dfrac{log P(h|e)-log P(h)}{-log P(h)}

In the process of formulating this measure, I reasoned that P(h|e) determines the risk of h being false, while P(h) determines the potential gain in truth, which I took to be the amount of information h carries. Isaac reminds us in his post that he argued against this approach in his article “Inference and Information” in Synthese (1967) and other places. The central thesis of Isaac’s 1967 Synthese article is that “information does not decrease with an increase in probability in the sense in which probability determines fair betting quotients” (370). I have my reason for holding onto my own view (I do not accept Isaac’s principle A-2 on page 374), but I am not going into this issue here. I want to point out instead that the main part of my paper is not tied to the fate of the particular view about information that I adopted in the paper.

For example, Isaac distinguishes the probability function M_{K}(h) that determines the informational value of h, from the credal probability function Q_{K}(h). Since Q_{K}(h) determines the degree of risk and M_{K}(h) determines the amount of potential gain in information, their roles are comparable, respectively, to those of P(h|e) and P(h) in my paper. So, those who like this part of Isaac’s view can systemically replace P(h|e) and P(h) in my paper with Q_{K}(h) and M_{K}(h), respectively. The resulting measure of epistemic justification would be J*(h, e) instead of J(h, e) above.

J*(h,e)=\dfrac{log Q_K(h)-log M_K(h)}{-log M_K(h)}

Of course, Isaac uses Q_{K}(h) and M_{K}(h) to determine the expected epistemic utility of h in his decision-theoretic approach to belief change, while I make no use of decision theory in my reasoning for the J measure. (My proposal is guided, instead, by the idea that if h1 and h2 are mutually irrelevant, e.g. h1 is about the demise of the Roman Empire while h2 is about the salmon’s immune system, then the acceptance of h1 should not affect the epistemic evaluation of h2.) Given this fundamental difference in the approach, I do not expect Isaac to find J*(h, e) appealing, but my point here is that the main part of my paper would be of interest to those who are open to a non-decision-theoretic approach to increasing true beliefs and avoiding false beliefs, even if they doubt that P(h) determines the potential gain in truth (information).

]]> By: Gregory Wheeler http://choiceandinference.com/?p=90&cpage=1#comment-150 Gregory Wheeler Sun, 24 May 2009 13:01:01 +0000 http://choiceandinference.com/?p=90#comment-150 Dear Isaac & Horacio, Thanks much for this excellent exchange. Horacio, I think you are right about Risky Knowledge not quite capturing acceptance in EP. Here is another data point: Both description logics and EMN are complete, but one of the last conversations that I had with Henry was about a completeness result for EP itself. He waved this away as a category mistake to even ask. (!). (Kyburg & Teng have a soundness result in <i>Uncertain Inference</i>). If either of you recall discussions with Henry on this point, I'd be keen to hear about it. Dear Isaac & Horacio,

Thanks much for this excellent exchange. Horacio, I think you are right about Risky Knowledge not quite capturing acceptance in EP. Here is another data point: Both description logics and EMN are complete, but one of the last conversations that I had with Henry was about a completeness result for EP itself. He waved this away as a category mistake to even ask. (!). (Kyburg & Teng have a soundness result in Uncertain Inference). If either of you recall discussions with Henry on this point, I’d be keen to hear about it.

]]> By: Isaac Levi http://choiceandinference.com/?p=90&cpage=1#comment-149 Isaac Levi Sun, 24 May 2009 06:54:00 +0000 http://choiceandinference.com/?p=90#comment-149 Dear Greg and Horacio I agree with Horacio that if NA&NB -> N[A&B] is abandoned where ‘NA’ is interpreted as “it is highly probable that”, there is nothing objectionable philosophically as far as I can see. Indeed, rejection of the principle is the right thing to do. But if the box operator represents ‘it is fully believed that’ or ‘it is certain that’ or ‘it is known that’ where the set of certainties or corpus of knowledge is the standard of serious possibility, there is plenty to object to even if the ‘underlying logic’ is classical. We would have systems of modal judgment like it is not seriously possible that ~A and it is not seriously possible that ~B yet it is seriously possible that ~Aor~B. Of course, as Horacio, if I remember rightly points out, one can abandon the duality of the possibility operator and the necessity operator. But that does not help. It only reinforces the point that the full beliefs construed using the box cannot be the standard for serious possibility. This is not a merely verbal point. P(X/~Aor~B) would be well defined but neither P(X/~A) nor P(X/~B) would be. Horacio seems to think that Henry’s position is better off if one abandons ‘Kripkean Semantics’ for a version of neighborhood semantics. Not if the necessity operator is to serve as a standard for serious possibility. There may be other useful applications of the approach Horacio favors. We should always keep an open mind about that. But Henry’s romance with high probability (and I would include the romance with high probability as infinitesimally less than 1) cannot satisfy our requirements for a standard for serious possibility Of course, as LJ Cohen rightly noted (see also Gambling with Truth) one might reinterpret probability so that high probability rules for adding new information to the belief state yield a deductively closed set. This is easy to do using Shackle measures. The point of emphasizing this is that the yearning for high probability rules that so many authors display might be assuaged by the use of Shackle measures. I want to insist that high probability rules are neither necessary nor sufficient for developing the contributions that make Henry's work so important. But that is another story. Isaac Dear Greg and Horacio

I agree with Horacio that if NA&NB -> N[A&B] is abandoned where ‘NA’ is interpreted as “it is highly probable that”, there is nothing objectionable philosophically as far as I can see. Indeed, rejection of the principle is the right thing to do.

But if the box operator represents ‘it is fully believed that’ or ‘it is certain that’ or ‘it is known that’ where the set of certainties or corpus of knowledge is the standard of serious possibility, there is plenty to object to even if the ‘underlying logic’ is classical.

We would have systems of modal judgment like it is not seriously possible that ~A and it is not seriously possible that ~B yet it is seriously possible that ~Aor~B.

Of course, as Horacio, if I remember rightly points out, one can abandon the duality of the possibility operator and the necessity operator. But that does not help. It only reinforces the point that the full beliefs construed using the box cannot be the standard for serious possibility.

This is not a merely verbal point. P(X/~Aor~B) would be well defined but neither P(X/~A) nor P(X/~B) would be.

Horacio seems to think that Henry’s position is better off if one abandons ‘Kripkean Semantics’ for a version of neighborhood semantics. Not if the necessity operator is to serve as a standard for serious possibility.

There may be other useful applications of the approach Horacio favors. We should always keep an open mind about that.

But Henry’s romance with high probability (and I would include the romance with high probability as infinitesimally less than 1) cannot satisfy our requirements for a standard for serious possibility

Of course, as LJ Cohen rightly noted (see also Gambling with Truth) one might reinterpret probability so that high probability rules for adding new information to the belief state yield a deductively closed set. This is easy to do using Shackle measures.

The point of emphasizing this is that the yearning for high probability rules that so many authors display might be assuaged by the use of Shackle measures.

I want to insist that high probability rules are neither necessary nor sufficient for developing the contributions that make Henry’s work so important. But that is another story.

Isaac

]]> By: Horacio Arló-Costa http://choiceandinference.com/?p=90&cpage=1#comment-148 Horacio Arló-Costa Sat, 23 May 2009 23:33:55 +0000 http://choiceandinference.com/?p=90#comment-148 Dear Greg: I basically agree with you. But I have a doubt that is based on recent work done by Makinson and Hawthorne and Paris (independently). Notice that neither Kyburg and Teng or our paper with Eric (Pacuit) prove a representation result. The idea is to propose a logical system that seems adequate given that it satisfies properties that it needs to satisfy and fails to satisfy other problematic properties like C. Now independently Makinson and Hawthorne and more recently Paris have looked at the non-monotonic notions of logical consequence generated by the following equation (for all measures w, and threshold t in [0,1]: $$\theta \,|\!\!\!\sim\,_{w,t} \psi \Longleftrightarrow w(\theta \wedge \psi) \geq tw(\theta)$$ Paris shows in an impressive paper (published in the Review of Symbolic Logic) that the notion of consequence that thus arises cannot be axiomatized. This continues the interesting work initiated by Makinson and Hawthorne. Now consider: $$\top \,|\!\!\!\sim\,_{w,t} \psi \Longleftrightarrow w(\psi) \geq t$$ The notion $$\top \,|\!\!\!\sim\,_{w,t} \psi $$ can be seen as a notion of belief. Is this notion completely axiomatizable? Astonishingly I do not think that this has been investigated, although this is a natural question to ask. Logically, perhaps it is the most natural question to ask here. The notion of belief should obey the axioms of EMN among others. But do these axioms suffice to characterize the operator of high probability (now a doxastic operator, rather than a conditional)? I posed the question in a way that it can be checked by looking at the recent proofs of non-axiomatizability. Unfortunately Paris's paper is not easy to read (it presumes a fair amount of high powered mathematics). Makinson and Hawthorne's paper is more clear and perhaps it suffices. In any case, it seems to me that we still do not understand clearly the logical commitments of a high probability test. In conditional logic the question has been avoided for years by changing theme and embracing tests based on infinitesimal probability (which are well behaved logically). Paris and Makinson and Hawthorne focused their attention on the high probability test that Kyburg and others proposed. Perhaps the results established by Paris can be presented in a more direct way, but prima facie the result indicates that the notion of consequence induced by the test cannot be characterized logically. It would be interesting to see whether the notion of belief induced by a natural modification of the test is characterizable. If the answer is not, this would indicate that Kyburg and Teng's notion of `risky knowledge' is not graspable via the use of standard logical methods. As I said above it is incredible that this is still an open question at this point. Best, H. Dear Greg: I basically agree with you. But I have a doubt that is based on recent work done by Makinson and Hawthorne and Paris (independently). Notice that neither Kyburg and Teng or our paper with Eric (Pacuit) prove a representation result. The idea is to propose a logical system that seems adequate given that it satisfies properties that it needs to satisfy and fails to satisfy other problematic properties like C.

Now independently Makinson and Hawthorne and more recently Paris have looked at the non-monotonic notions of logical consequence generated by the following equation (for all measures w, and threshold t in [0,1]:

\theta \,|\!\!\!\sim\,_{w,t} \psi \Longleftrightarrow w(\theta \wedge \psi) \geq tw(\theta)

Paris shows in an impressive paper (published in the Review of Symbolic Logic) that the notion of consequence that thus arises cannot be axiomatized. This continues the interesting work initiated by Makinson and Hawthorne. Now consider:

\top \,|\!\!\!\sim\,_{w,t} \psi \Longleftrightarrow w(\psi) \geq t

The notion \top \,|\!\!\!\sim\,_{w,t} \psi can be seen as a notion of belief. Is this notion completely axiomatizable? Astonishingly I do not think that this has been investigated, although this is a natural question to ask. Logically, perhaps it is the most natural question to ask here. The notion of belief should obey the axioms of EMN among others. But do these axioms suffice to characterize the operator of high probability (now a doxastic operator, rather than a conditional)? I posed the question in a way that it can be checked by looking at the recent proofs of non-axiomatizability. Unfortunately Paris’s paper is not easy to read (it presumes a fair amount of high powered mathematics). Makinson and Hawthorne’s paper is more clear and perhaps it suffices. In any case, it seems to me that we still do not understand clearly the logical commitments of a high probability test. In conditional logic the question has been avoided for years by changing theme and embracing tests based on infinitesimal probability (which are well behaved logically). Paris and Makinson and Hawthorne focused their attention on the high probability test that Kyburg and others proposed. Perhaps the results established by Paris can be presented in a more direct way, but prima facie the result indicates that the notion of consequence induced by the test cannot be characterized logically. It would be interesting to see whether the notion of belief induced by a natural modification of the test is characterizable. If the answer is not, this would indicate that Kyburg and Teng’s notion of `risky knowledge’ is not graspable via the use of standard logical methods. As I said above it is incredible that this is still an open question at this point.

Best, H.

]]> By: Gregory Wheeler http://choiceandinference.com/?p=90&cpage=1#comment-147 Gregory Wheeler Sat, 23 May 2009 21:28:24 +0000 http://choiceandinference.com/?p=90#comment-147 Dear Horacio & Isaac, I agree with Horacio's take on this, but would put the emphasis a bit differently: technically, closure conditions and revision operations are possible to build on a qualitative version of Kyburgian evidential probability. That's not to say that the mere possibility of doing so is an argument for adopting the Kyburgian view; but, the relative ease of getting these two features does remove these two technical objections from use in this philosophical argument. As an aside, I have taken to viewing Kyburg's EP as essentially a description logic. It is a system for reasoning about (reference) classes, basically, where the probability intervals enter in as terms simply for determining which classes to close under the rules of richness and specificity (i.e., those whose intervals 'conflict') and which to ignore. The actual numeric values of the interval do not show up until the end, when you apply the strength rule to the surviving classes. What seems promising about this view is that it yields tremendous technical advantages over a (convex) Bayes probability logic. (Good luck computing with credal sets!) Again, I agree that this doesn't settle the philosophical dispute between the views. But, in so far as there have been complaints lodged against EP for not being a proper logic or for EP being technically unwieldy, it appears to me that this type of objection in the end may very well be stood on its head: the technical features of the system, stemming from the (apparent) ability to translate the core of the theory into various types of guarded fragments of first-order logic with extremely nice properties, may well be EP's strongest advantage. Dear Horacio & Isaac,

I agree with Horacio’s take on this, but would put the emphasis a bit differently: technically, closure conditions and revision operations are possible to build on a qualitative version of Kyburgian evidential probability. That’s not to say that the mere possibility of doing so is an argument for adopting the Kyburgian view; but, the relative ease of getting these two features does remove these two technical objections from use in this philosophical argument.

As an aside, I have taken to viewing Kyburg’s EP as essentially a description logic. It is a system for reasoning about (reference) classes, basically, where the probability intervals enter in as terms simply for determining which classes to close under the rules of richness and specificity (i.e., those whose intervals ‘conflict’) and which to ignore. The actual numeric values of the interval do not show up until the end, when you apply the strength rule to the surviving classes. What seems promising about this view is that it yields tremendous technical advantages over a (convex) Bayes probability logic. (Good luck computing with credal sets!)

Again, I agree that this doesn’t settle the philosophical dispute between the views. But, in so far as there have been complaints lodged against EP for not being a proper logic or for EP being technically unwieldy, it appears to me that this type of objection in the end may very well be stood on its head: the technical features of the system, stemming from the (apparent) ability to translate the core of the theory into various types of guarded fragments of first-order logic with extremely nice properties, may well be EP’s strongest advantage.

]]> By: Horacio Arló-Costa http://choiceandinference.com/?p=90&cpage=1#comment-145 Horacio Arló-Costa Sat, 23 May 2009 17:32:02 +0000 http://choiceandinference.com/?p=90#comment-145 Dear Isaac and Greg: I think that Isaac is right in pointing out that the philosophical problem related to the normative adequacy of acceptance rules in terms of high probability is not solved by appealing to a neighborhood representation. But I also think that the use of classical modalities to represent high probability operators is a big improvement with respect to previous attempts to determine the logical commitments of philosophers like Kyburg who are convinced about the philosophical adequacy of these rules of acceptance. Previously to this development Kyburg suggested that the classical rule of Adjunction has to be abandoned if you embrace these rules. The representation via classical modalities shows that what needs to be abandoned is the Kripkean semantics for the modalities (and there are independent reasons that indicate that the Scott-Montague approach offers an interesting and more flexible alternative to Kripke's semantics). Kyburg does not need to touch the underlying logic which remains classical. What is abandoned is the axiom C: $$\Box(A) \wedge \Box(B) \rightarrow \Box(A \wedge B)$$. Of course one can use a non-adjunctive logic rather than deploying a modal operator, but the representation in terms of neighborhoods shows that one does not need to do that. One can embrace high probability acceptance rules without giving up classical logic. This does not offer a philosophical argument supporting the conceptual adequacy of high probability acceptance rules. But it shows that the logical commitments of philosophers like Kyburg are less radical than previously thought. Moreover once one sees the connection between high probability operators and classical modalities one can formulate these bridges at the first order level. Then an epistemic interpretation of first order classical logic is apparent. For example, there is an interesting connection between the Barcan schema and the lottery paradox as I indicated in various recent articles. This connection has not been yet explored in detail, I think. All this seems to indicate that the neighborhood representation is the right logical tool for philosophers who appeal to high probability acceptance rules. But still the follower of Kyburg needs to establish that these rules can be applied successfully. Of course Kyburg thought that this is indeed the case, but this has been disputed by other philosophers, like Isaac. The development of a neat logical account for the acceptance rules eliminates some obstacles for its application and makes possible a better understanding of the way the rules work, but per se it cannot replace a philosophical debate about the normative or descriptive adequacy of the rules. Best, H. Dear Isaac and Greg:

I think that Isaac is right in pointing out that the philosophical problem related to the normative adequacy of acceptance rules in terms of high probability is not solved by appealing to a neighborhood representation. But I also think that the use of classical modalities to represent high probability operators is a big improvement with respect to previous attempts to determine the logical commitments of philosophers like Kyburg who are convinced about the philosophical adequacy of these rules of acceptance. Previously to this development Kyburg suggested that the classical rule of Adjunction has to be abandoned if you embrace these rules. The representation via classical modalities shows that what needs to be abandoned is the Kripkean semantics for the modalities (and there are independent reasons that indicate that the Scott-Montague approach offers an interesting and more flexible alternative to Kripke’s semantics). Kyburg does not need to touch the underlying logic which remains classical. What is abandoned is the axiom C: \Box(A) \wedge \Box(B) \rightarrow \Box(A \wedge B).

Of course one can use a non-adjunctive logic rather than deploying a modal operator, but the representation in terms of neighborhoods shows that one does not need to do that. One can embrace high probability acceptance rules without giving up classical logic. This does not offer a philosophical argument supporting the conceptual adequacy of high probability acceptance rules. But it shows that the logical commitments of philosophers like Kyburg are less radical than previously thought. Moreover once one sees the connection between high probability operators and classical modalities one can formulate these bridges at the first order level. Then an epistemic interpretation of first order classical logic is apparent. For example, there is an interesting connection between the Barcan schema and the lottery paradox as I indicated in various recent articles. This connection has not been yet explored in detail, I think. All this seems to indicate that the neighborhood representation is the right logical tool for philosophers who appeal to high probability acceptance rules. But still the follower of Kyburg needs to establish that these rules can be applied successfully. Of course Kyburg thought that this is indeed the case, but this has been disputed by other philosophers, like Isaac.

The development of a neat logical account for the acceptance rules eliminates some obstacles for its application and makes possible a better understanding of the way the rules work, but per se it cannot replace a philosophical debate about the normative or descriptive adequacy of the rules.

Best, H.

]]> By: Isaac Levi http://choiceandinference.com/?p=90&cpage=1#comment-144 Isaac Levi Fri, 22 May 2009 12:30:51 +0000 http://choiceandinference.com/?p=90#comment-144 Dear Greg, I am not concerned with whether you can define a revision operator. My interest is in justifying changes in states of full belief where agent X is committed to such a state when X is committed to distinguish between what is seriously possible and what is not in terms of consistency with that state and where the set of standard real valued credal probability functions Q(x/y) that are permissible according to X are restricted to those where y is a serious possibility and for fixed y, Q(x/y) is a probability defined over the powerset over the set of serious possibilities. Best, Isaac Dear Greg,

I am not concerned with whether you can define a revision operator. My interest is in justifying changes in states of full belief where agent X is committed to such a state when X is committed to distinguish between what is seriously possible and what is not in terms of consistency with that state and where the set of standard real valued credal probability functions Q(x/y) that are permissible according to X are restricted to those where y is a serious possibility and for fixed y, Q(x/y) is a probability defined over the powerset over the set of serious possibilities.

Best,
Isaac

]]> By: Gregory Wheeler http://choiceandinference.com/?p=90&cpage=1#comment-143 Gregory Wheeler Fri, 22 May 2009 09:49:11 +0000 http://choiceandinference.com/?p=90#comment-143 Dear Isaac, It is fairly straight-forward to define an AGM revision operator for the class of modal logics that contains EMN, and I anticipate that there will be available options other than AGM to choose from. So, I don't see technical barriers to doing revision on non-adjunctive logics, either. Now, the devil is in the details -- of course. But, these ideas seem new and perhaps worth mentioning. Best, Greg Dear Isaac,

It is fairly straight-forward to define an AGM revision operator for the class of modal logics that contains EMN, and I anticipate that there will be available options other than AGM to choose from. So, I don’t see technical barriers to doing revision on non-adjunctive logics, either.

Now, the devil is in the details — of course. But, these ideas seem new and perhaps worth mentioning.

Best, Greg

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