Here, I explain how probability theory doesn’t refute the Coherence Theory of Justification.* [ *Based on: “Does Probability Theory Refute Coherentism?”, Journal of Philosophy 108 (2011): 35-54. ] Caveat: This is a bit technical. Aside: This paper holds my record for the longest review time: four and a half years (including one revision and resubmission).
I dunno, Mike, whenever some parties in a legal dispute agree to settle the case that A and thereby agree that none assert A, such settlements often seem to support that A. (just joking around)
Many years ago I outlined a slightly crazy theory for replacing epistemology in toto with the probability calculus read in a very literally subjective Bayesian way. The key idea was that it is a category error to talk about justifying beliefs at all- beliefs are never justified or unjustified (although they can be incoherent I suppose). Rather, only updates to beliefs can be evaluated vis a vis their justification- let me see if I can rustle it up:
**Through-going subjective Bayesianism as a solution to the problem of scepticism**
A common argument for the existence of God is that there is something rather than nothing. There are many good replies to this argument, but one of the more sophisticated challenges our sense that ‘nothing’ is the ordinary state of things and ‘something’ is an exception that needs explanation. There is an enormous, uncountably infinite plurality of ways the world could be. The ‘nothing’ world is simply one of these ways, and no more inherently ‘natural’ than the others. Why must the initial state of the universe be nothing unless we can give a reason otherwise?
We might talk then of the ‘nothingness is natural’ fallacy in relation to the initial conditions of universes. Scepticism seems to me to commit a variant of this fallacy in relation to epistemology. The set of permissible beliefs is assumed to start empty or near-empty, and then we have to fill it. A very low probability judgment for all statements is assumed to be the real correct default. Any exception is in need of justification.
There is an alternative view of epistemology that does away with scepticism. I see it as, broadly speaking, a development of G.E. Moore’s argument against scepticism to a whole epistemology by way of subjective Bayesianism, though this might simply be reading my own ideas into Moore’s work. It is a development of subjective Bayesianism into a full epistemological standpoint, a kind of alternative to coherentism and foundationalism, or a dissolution of the questions they attempt to dissolve, depending on how you look at it.
The view goes like this. There are no rationality requirements beyond consistency on the initial assignment of priors. Rationality only governs the updating or changing of our beliefs and does not impose any constraints on our initial beliefs beyond consistency requirements of the kind given by the Dutch book argument.
The real province of normative epistemology is purely the assessment of changes in probability assignments. The demand for some ultimate justification of the initial judgements is an illusory over-generalisation from the practice of justifying changes in levels of belief. Our priors start out non-sceptical, and so our beliefs remain throughout a lifetime of updating—there is no higher ledge to stand on from which to critique this. On the subjective understanding of probability, we do, after all, have to have some priors, and there is no known principled way of assigning them. Thus we arrive at some rather simple anti-sceptical proofs. For example, mirroring Moore’s proof of an external world, we simply observe that our prior belief in an external world is rather high, and nothing has decreased it.
So why does the problem of scepticism appear to be a real problem? Over generalization. If I tell you that Susie is a lecturer in mathematics, you might ask how I came to know that, because you judge that it is very unlikely that it’s simply built in to my priors that Susie is a lecturer in mathematics. This is true of basically all statements worth talking about outside a philosophical context, thus we get into the habit of thinking in terms of justifying our beliefs. When I then tell you that I believe there is an
external world, and you ask how I know this, and I cannot give an adequate reply, it appears something is wrong. Really though this is a sort of illusion, all we are ever doing is justifying why we have decreased or increased our credence. Since your a priori belief that Susie is a mathematician is presumably low you need to explain why it has risen but the same is not true of belief in an external world, or causation- both of which start with high priors.
Edit: How do I respond to the objection that Bayesianism is probably not a complete explanation of belief formation- e.g. the problem of new theories? I’ve given an account in which beliefs themselves never need be justified- only changes to beliefs, and I’ve argued this arises from subjectivist Bayesianism. Now, it is likely that Bayesianism needs to be modified in order to work- and we can’t know what a final Bayesian theory would like, but until shown otherwise, there’s no reason to think it would involve justifying priors.
It seems to me that the statements of foundationalism and coherentism are extreme. Is it possible to regard coherentism as a technique for casting a wide net for evidence by investing tentative belief in lots of claims just because they're consistent with things you already believe? There may be *some* foundational beliefs your just know are true, but not enoughb of them to do much work. With all large coherent system of beliefs, a lot more evidence is relevant, when it comes. And then deliberative thought leading to reflective equilibrium will cause the updating based on evidence to affect a large domain of belief. What do you think?
It is generally agreed that if you have at least some justification for P, and some justification for Q, then if P and Q fit together well, that enhances the justification of both P and Q. Both foundationalists and coherentists agree with that.
I want to focus on where they disagree. Which is about whether, even if you have *no* justification for P or Q to begin with, the fact that they fit together well can provide justification for both.
I am having trouble interpreting the probability distribution diagram. Does the entire rectangle on the left represent P(A)? Does the square on the top left corner represent P(A|W1)? I would do a web search, but I’m not sure what terms to search for. When I search for probability distribution diagram it shows me probability density or frequency diagrams, none of which resemble this diagram.
Areas represent probabilities. The whole square has an area of 1 and represents the whole range of possiblities. The rectangle on the left with dashed lines around it represents the possible situations in which A is true. (You can see that it has an area of 1/4, so P(A) is 1/4.)
P(A|W1) is represented by the *portion* of the "W1" area that is also part of the "A" area. (You can see that it is 1/2.) I hope that helps.
I was thrown off by my intuition that there should also be an area representing the case where neither witness testifies about A, or denies A. I guess we could just consider that to have the probability of zero.
I was thinking that white region represented the fact that, at least, in this instance, it is never the case that A is false and both of the witnesses testify that it is true. That is different from a situation where either neither of the witnesses testify at all, or both deny A, when it is true. This case does not seem to be represented in the diagram, unless we think of it as having probability zero, and therefore can be represented by an area with a volume of zero located anywhere on the diagram.
I dunno, Mike, whenever some parties in a legal dispute agree to settle the case that A and thereby agree that none assert A, such settlements often seem to support that A. (just joking around)
Many years ago I outlined a slightly crazy theory for replacing epistemology in toto with the probability calculus read in a very literally subjective Bayesian way. The key idea was that it is a category error to talk about justifying beliefs at all- beliefs are never justified or unjustified (although they can be incoherent I suppose). Rather, only updates to beliefs can be evaluated vis a vis their justification- let me see if I can rustle it up:
**Through-going subjective Bayesianism as a solution to the problem of scepticism**
A common argument for the existence of God is that there is something rather than nothing. There are many good replies to this argument, but one of the more sophisticated challenges our sense that ‘nothing’ is the ordinary state of things and ‘something’ is an exception that needs explanation. There is an enormous, uncountably infinite plurality of ways the world could be. The ‘nothing’ world is simply one of these ways, and no more inherently ‘natural’ than the others. Why must the initial state of the universe be nothing unless we can give a reason otherwise?
We might talk then of the ‘nothingness is natural’ fallacy in relation to the initial conditions of universes. Scepticism seems to me to commit a variant of this fallacy in relation to epistemology. The set of permissible beliefs is assumed to start empty or near-empty, and then we have to fill it. A very low probability judgment for all statements is assumed to be the real correct default. Any exception is in need of justification.
There is an alternative view of epistemology that does away with scepticism. I see it as, broadly speaking, a development of G.E. Moore’s argument against scepticism to a whole epistemology by way of subjective Bayesianism, though this might simply be reading my own ideas into Moore’s work. It is a development of subjective Bayesianism into a full epistemological standpoint, a kind of alternative to coherentism and foundationalism, or a dissolution of the questions they attempt to dissolve, depending on how you look at it.
The view goes like this. There are no rationality requirements beyond consistency on the initial assignment of priors. Rationality only governs the updating or changing of our beliefs and does not impose any constraints on our initial beliefs beyond consistency requirements of the kind given by the Dutch book argument.
The real province of normative epistemology is purely the assessment of changes in probability assignments. The demand for some ultimate justification of the initial judgements is an illusory over-generalisation from the practice of justifying changes in levels of belief. Our priors start out non-sceptical, and so our beliefs remain throughout a lifetime of updating—there is no higher ledge to stand on from which to critique this. On the subjective understanding of probability, we do, after all, have to have some priors, and there is no known principled way of assigning them. Thus we arrive at some rather simple anti-sceptical proofs. For example, mirroring Moore’s proof of an external world, we simply observe that our prior belief in an external world is rather high, and nothing has decreased it.
So why does the problem of scepticism appear to be a real problem? Over generalization. If I tell you that Susie is a lecturer in mathematics, you might ask how I came to know that, because you judge that it is very unlikely that it’s simply built in to my priors that Susie is a lecturer in mathematics. This is true of basically all statements worth talking about outside a philosophical context, thus we get into the habit of thinking in terms of justifying our beliefs. When I then tell you that I believe there is an
external world, and you ask how I know this, and I cannot give an adequate reply, it appears something is wrong. Really though this is a sort of illusion, all we are ever doing is justifying why we have decreased or increased our credence. Since your a priori belief that Susie is a mathematician is presumably low you need to explain why it has risen but the same is not true of belief in an external world, or causation- both of which start with high priors.
Edit: How do I respond to the objection that Bayesianism is probably not a complete explanation of belief formation- e.g. the problem of new theories? I’ve given an account in which beliefs themselves never need be justified- only changes to beliefs, and I’ve argued this arises from subjectivist Bayesianism. Now, it is likely that Bayesianism needs to be modified in order to work- and we can’t know what a final Bayesian theory would like, but until shown otherwise, there’s no reason to think it would involve justifying priors.
It seems to me that the statements of foundationalism and coherentism are extreme. Is it possible to regard coherentism as a technique for casting a wide net for evidence by investing tentative belief in lots of claims just because they're consistent with things you already believe? There may be *some* foundational beliefs your just know are true, but not enoughb of them to do much work. With all large coherent system of beliefs, a lot more evidence is relevant, when it comes. And then deliberative thought leading to reflective equilibrium will cause the updating based on evidence to affect a large domain of belief. What do you think?
It is generally agreed that if you have at least some justification for P, and some justification for Q, then if P and Q fit together well, that enhances the justification of both P and Q. Both foundationalists and coherentists agree with that.
I want to focus on where they disagree. Which is about whether, even if you have *no* justification for P or Q to begin with, the fact that they fit together well can provide justification for both.
I am having trouble interpreting the probability distribution diagram. Does the entire rectangle on the left represent P(A)? Does the square on the top left corner represent P(A|W1)? I would do a web search, but I’m not sure what terms to search for. When I search for probability distribution diagram it shows me probability density or frequency diagrams, none of which resemble this diagram.
Areas represent probabilities. The whole square has an area of 1 and represents the whole range of possiblities. The rectangle on the left with dashed lines around it represents the possible situations in which A is true. (You can see that it has an area of 1/4, so P(A) is 1/4.)
P(A|W1) is represented by the *portion* of the "W1" area that is also part of the "A" area. (You can see that it is 1/2.) I hope that helps.
I was thrown off by my intuition that there should also be an area representing the case where neither witness testifies about A, or denies A. I guess we could just consider that to have the probability of zero.
So that would be the region outside W1 and outside W2. Which is the completely white region in the square. (P(~W1 & ~W2) = 6/16.)
So, is P(A|~W1 & ~W2) = 0, or equal to the area of the white rectangle?
P(A|~w1,~w2) = 0; there's no overlap between the A region and the (~w1,~w2) region.
P(~A|w1,w2) = 0; all of the (w1,w2) region overlaps with A.
I was thinking that white region represented the fact that, at least, in this instance, it is never the case that A is false and both of the witnesses testify that it is true. That is different from a situation where either neither of the witnesses testify at all, or both deny A, when it is true. This case does not seem to be represented in the diagram, unless we think of it as having probability zero, and therefore can be represented by an area with a volume of zero located anywhere on the diagram.
Yes, the diagram represents that if both witnesses assert A, A is guaranteed to be true. This is a strong form of coherence justification.
Also, if both witnesses fail to assert A, then A is guaranteed to be false.