Therefore, by the rule we call conservation of expected evidence, reasoning through a belief system and deriving a conclusion consistent with the premise you started with should increase your credence.
This is correct but (slightly) less relevant to the point you are trying to make in this particular section that it might come across as at first. Valid reasoning increases your credence, but circular reasoning, by itself, does not.
“Circular reasoning is valid” only means that circularity does not create any additional logical invalidity issues; it does not mean that a line of logical reasoning is made to be valid just because it contains circularity. Indeed, just add a bit of circularity in between (or as part of) a few other steps of otherwise invalid reasoning: the validity does not change. The titles of the sections in your post, as well as the explanation you gave in the part I quoted, give off a slight impression that you are conflating circularity and validity,[1] while in reality the latter screens off the effects of the former.
Of course, you can say that observing a single instance of circularity and determining that the argument carries through locally gives a modicum of evidence towards the overall line of reasoning being correct.[2] But the magnitude of this update should depend a fair bit on how the argument is presented to you, given that my impression in real life is that circular steps are not generally the true load-bearing part in most complex chains of reasoning.
Infinitism: Infinite chains of justification are allowed. This position seems rare.
It might be rare to a non-Bayesian, but if you take the position that you must update on observations seriously, it sems rather straightforward to me to say that there is an infinite sigma-algebra of events that you assign probabilities to over time (if you are an idealized, as opposed to bounded, reasoner), and an infinite set of possible observations that you can update on. Everything affects everything else, in the reinterpretation of “justification” as conditioning. And your priors relative to a certain Bayesian updating are really your posteriors from the Bayesian update that came immediately before it, which were modified by the update right before that, and on, and on, and on. (And there is nothing theoretically wrong with updating continuously over time, either, such that when you start with a prior at time t=0, at t=1 you have already gone through an “infinite” chain of justification-updates)
Of course, you can say the idea of “justification” doesn’t really carve reality at the joints once you adopt a Bayesian epistemology. You just select a large-enough probability space to contain events you care about, start with your prior, make sure you are capable of updating, and then you just update. Who needs any “justification”? Just run the math, bro.[3]
Alternatively, you can also simply abandon Bayesianism entirely in favor of another theory that builds on top of it (which I get the impression is the approach you endorse?)
I think the temptation to outlaw circular arguments as a form of justification comes mainly from trying to construct an objective third-person perspective to judge disagreements. In other words, the concept “justification” is doing double duty. We cannot consistently use it for both honest philosophical examination and constructing arguments to persuade others.
Ah, but now that we bring real life into this,[4] a couple of additional problems (which were not relevant back in nice-old math-land) start coming into play:
Firstly, the fact that somebody used a circular argument in favor of a conclusion they support should, in most circumstances, be treated as (usually really weak) evidence against their conclusion, because a conclusion that is correct is more likely than one that is false to have much better justifications than circularity.
In a conversation where another person explains why they believe something, or they are trying to convince you of it, they are more likely than not to try to find the best arguments they can.[5] So, conditional on you observing that they have nonetheless selected an argument that uses circularity, it becomes much more likely that they do not have a better argument to give than that, which lowers your credence that they have “good reason” to believe it, and also lowers your expectation that the conclusion is correct.[6]
This somewhat ties into how clever arguers can be used successfully as part of a truthseeking scheme regardless of whether they themselves are not truthseeking and are only trying to persuade you for personal reasons. Just do what the legal system does (at least in an adversarial system): put clever arguers on both sides, make sure they are incentivized to fight for their positions as hard as possible (while not violating ethical norms of behavior), and then read their arguments: an arguer advocating for the “false” side might nonetheless seem convincing in an absolute sense, but they will (on average) be less convincing than you would expect them to, conditional on them being clever arguers. Bertrand Russell has said that “In a man whose reasoning powers are good, fallacious arguments are evidence of bias”; I would modify that to “weak evidence that his conclusion is false.”
Secondly, and perhaps more straightforwardly, people want to ban circular arguments because they dislike circular arguments. And they dislike circular arguments because they are used very often by people who are so inexperienced at reasoning that they don’t understand what the problem with circularity even is. By which I mean, people who don’t understand that, for example and as you said, double counting is an issue.
It’s the same reason people sometimes try to ban other formally fallacious lines of reasoning in organized debates (or try to get moderators to publicly call them out): sure, you could go with a full-on libertarian approach and say that banning (or even mentioning) fallacies is unnecessary because all participants in the discussion will readily identify the reasoning as fallacious all by themselves and thus immediately reverse their updates towards the conclusion, but in practice we know that this obviously doesn’t work and all human beings, regardless of how rational, are susceptible to changing their views in an epistemically unjustifiable manner because of strongly-presented evidence that’s not “logical.”
So if someone makes an argument that’s circular, the thinking goes, someone needs to point this out immediately so that people don’t get fooled into assigning higher probative value to it than it deserves. Or better yet, to disallow such argumentation entirely.[7]
To be clear, I think this is just an impression I get from your specific writing choices here; I strongly suspect you are not actually making such an error in your thinking.
Because if you have 10 logical steps, in order for the argument to be valid you need all 10 of them to be valid, so seeing that one is valid makes it more likely that the entirety of it is.
Of course, they often expect short inferential distances and don’t tailor the argument to be as persuasive to the other person as possible, etc., so this is not guaranteed, but in expectation, the sign of the update you should make in such a spot seems clear to me.
As already mentioned, this should be a small update because you should have already expected that most arguments would be bad and most arguers would be incompetent, regardless of the truth of the position they are arguing for. But, ceteris paribus, “If you want to convince people of something, it’s much easier if it’s true”, as Paul Graham has said.
Of course, you can say the idea of “justification” doesn’t really carve reality at the joints once you adopt a Bayesian epistemology. You just select a large-enough probability space to contain events you care about, start with your prior, make sure you are capable of updating, and then you just update. Who needs any “justification”? Just run the math, bro.[3]
Alternatively, you can also simply abandon Bayesianism entirely in favor of another theory that builds on top of it (which I get the impression is the approach you endorse?)
I admit I am a bit inconsistent about how I use the term “Bayesian”; sometimes I use it to point to the “classic Bayesian picture” (like Dogmatic Probabilism here and Reductive Utility here). However, I think the core idea of Bayesian philosophy is subjective probability, which I still think is quite important (even within, say, infrabayesianism). Granted, I also think frequentist notions of probability can be useful and have an important role to play (EG, in the theory of logical induction).
Anyway, I think there’s something right about “justification doesn’t carve reality at its joints”, but I think stopping at that would be a mistake. Yes, I think justification works different ways in different contexts, and the seeming paradox of the regress argument comes mostly from conflating the “trying to convince someone else” context with the “examining your own beliefs” context. However, I do think there’s something interesting going on with both of those notions of justification, and it seems potentially fruitful to examine them rather than throw them out. “Just run the math” only works if you’re not in the business of examining your beliefs (eg, the faith you have in the math) or justifying said math to others.
It might be rare to a non-Bayesian, but if you take the position that you must update on observations seriously, it sems rather straightforward to me to say that there is an infinite sigma-algebra of events that you assign probabilities to over time (if you are an idealized, as opposed to bounded, reasoner), and an infinite set of possible observations that you can update on. Everything affects everything else, in the reinterpretation of “justification” as conditioning. And your priors relative to a certain Bayesian updating are really your posteriors from the Bayesian update that came immediately before it, which were modified by the update right before that, and on, and on, and on. (And there is nothing theoretically wrong with updating continuously over time, either, such that when you start with a prior at time t=0, at t=1 you have already gone through an “infinite” chain of justification-updates)
I’m not really sure what infinite chain of justification you are imagining a Bayesian position to suggest. A posterior seems justified by a prior combined with evidence. We can view this as a single justificatory step if we like, or as a sequence of updates. So long as the amount of information we are updating on is finite, this seems like a finite justification chain. I don’t think it matters if the sigma-algebra is infinite or if the space of possible observations is infinite, although as you seem to recognize, neither of these assumptions seems particularly plausible for bounded beings. The idea that everything effects everything else doesn’t actually need to make the justificatory chain infinite.
I think you might be conflating the idea that Bayesianism in some sense points at idealized rationality, with the idea that everything about a Bayesian needs to be infinite. It is possible to specify perfect Bayesian beliefs which can be calculated finitely. It is possible to have finite sigma algebras, finite amounts of information in an update, etc.
The idea of continuous updating is interesting, but not very much a part of the most common Bayesian picture. Also, its relationship to infinite chains seems more complex than you suggest. If I am observing, say, a temperature in real time, then you can model me as having a prior which gets updated on an observation of [the function specifying how the temperature has changed over time]. This would still be a finite chain. If you insist that the proper justification chain includes all my intermediate updates rather than just one big update, then it ceases to be a “chain” at all (because ALL pairs of distinct times have a third time between them, so there are no “direct” justification links between times whatsoever—any link between two times must summarize what happened at infinitely many times between).
The titles of the sections in your post, as well as the explanation you gave in the part I quoted, give off a slight impression that you are conflating circularity and validity, while in reality the latter screens off the effects of the former.
I agree that that’s a bad misunderstanding that I want to avoid, so although I’m a bit incredulous at people reading it this way, I’ll try to edit to make this misunderstanding less plausible.
This is correct but (slightly) less relevant to the point you are trying to make in this particular section that it might come across as at first. Valid reasoning increases your credence, but circular reasoning, by itself, does not.
“Circular reasoning is valid” only means that circularity does not create any additional logical invalidity issues; it does not mean that a line of logical reasoning is made to be valid just because it contains circularity. Indeed, just add a bit of circularity in between (or as part of) a few other steps of otherwise invalid reasoning: the validity does not change. The titles of the sections in your post, as well as the explanation you gave in the part I quoted, give off a slight impression that you are conflating circularity and validity,[1] while in reality the latter screens off the effects of the former.
Of course, you can say that observing a single instance of circularity and determining that the argument carries through locally gives a modicum of evidence towards the overall line of reasoning being correct.[2] But the magnitude of this update should depend a fair bit on how the argument is presented to you, given that my impression in real life is that circular steps are not generally the true load-bearing part in most complex chains of reasoning.
It might be rare to a non-Bayesian, but if you take the position that you must update on observations seriously, it sems rather straightforward to me to say that there is an infinite sigma-algebra of events that you assign probabilities to over time (if you are an idealized, as opposed to bounded, reasoner), and an infinite set of possible observations that you can update on. Everything affects everything else, in the reinterpretation of “justification” as conditioning. And your priors relative to a certain Bayesian updating are really your posteriors from the Bayesian update that came immediately before it, which were modified by the update right before that, and on, and on, and on. (And there is nothing theoretically wrong with updating continuously over time, either, such that when you start with a prior at time t=0, at t=1 you have already gone through an “infinite” chain of justification-updates)
Of course, you can say the idea of “justification” doesn’t really carve reality at the joints once you adopt a Bayesian epistemology. You just select a large-enough probability space to contain events you care about, start with your prior, make sure you are capable of updating, and then you just update. Who needs any “justification”? Just run the math, bro.[3]
Alternatively, you can also simply abandon Bayesianism entirely in favor of another theory that builds on top of it (which I get the impression is the approach you endorse?)
Ah, but now that we bring real life into this,[4] a couple of additional problems (which were not relevant back in nice-old math-land) start coming into play:
Firstly, the fact that somebody used a circular argument in favor of a conclusion they support should, in most circumstances, be treated as (usually really weak) evidence against their conclusion, because a conclusion that is correct is more likely than one that is false to have much better justifications than circularity.
In a conversation where another person explains why they believe something, or they are trying to convince you of it, they are more likely than not to try to find the best arguments they can.[5] So, conditional on you observing that they have nonetheless selected an argument that uses circularity, it becomes much more likely that they do not have a better argument to give than that, which lowers your credence that they have “good reason” to believe it, and also lowers your expectation that the conclusion is correct.[6]
This somewhat ties into how clever arguers can be used successfully as part of a truthseeking scheme regardless of whether they themselves are not truthseeking and are only trying to persuade you for personal reasons. Just do what the legal system does (at least in an adversarial system): put clever arguers on both sides, make sure they are incentivized to fight for their positions as hard as possible (while not violating ethical norms of behavior), and then read their arguments: an arguer advocating for the “false” side might nonetheless seem convincing in an absolute sense, but they will (on average) be less convincing than you would expect them to, conditional on them being clever arguers. Bertrand Russell has said that “In a man whose reasoning powers are good, fallacious arguments are evidence of bias”; I would modify that to “weak evidence that his conclusion is false.”
Secondly, and perhaps more straightforwardly, people want to ban circular arguments because they dislike circular arguments. And they dislike circular arguments because they are used very often by people who are so inexperienced at reasoning that they don’t understand what the problem with circularity even is. By which I mean, people who don’t understand that, for example and as you said, double counting is an issue.
It’s the same reason people sometimes try to ban other formally fallacious lines of reasoning in organized debates (or try to get moderators to publicly call them out): sure, you could go with a full-on libertarian approach and say that banning (or even mentioning) fallacies is unnecessary because all participants in the discussion will readily identify the reasoning as fallacious all by themselves and thus immediately reverse their updates towards the conclusion, but in practice we know that this obviously doesn’t work and all human beings, regardless of how rational, are susceptible to changing their views in an epistemically unjustifiable manner because of strongly-presented evidence that’s not “logical.”
So if someone makes an argument that’s circular, the thinking goes, someone needs to point this out immediately so that people don’t get fooled into assigning higher probative value to it than it deserves. Or better yet, to disallow such argumentation entirely.[7]
To be clear, I think this is just an impression I get from your specific writing choices here; I strongly suspect you are not actually making such an error in your thinking.
Because if you have 10 logical steps, in order for the argument to be valid you need all 10 of them to be valid, so seeing that one is valid makes it more likely that the entirety of it is.
Unless you’re dealing with problems of embeddedness or reflective reasoning or indexicality or anthropics… Hmm, kind of a lot of stuff, actually.
By talking about the “temptation” and thinking of actual people on this matter.
Of course, they often expect short inferential distances and don’t tailor the argument to be as persuasive to the other person as possible, etc., so this is not guaranteed, but in expectation, the sign of the update you should make in such a spot seems clear to me.
As already mentioned, this should be a small update because you should have already expected that most arguments would be bad and most arguers would be incompetent, regardless of the truth of the position they are arguing for. But, ceteris paribus, “If you want to convince people of something, it’s much easier if it’s true”, as Paul Graham has said.
I am not saying I necessarily endorse this perspective, of course.
I admit I am a bit inconsistent about how I use the term “Bayesian”; sometimes I use it to point to the “classic Bayesian picture” (like Dogmatic Probabilism here and Reductive Utility here). However, I think the core idea of Bayesian philosophy is subjective probability, which I still think is quite important (even within, say, infrabayesianism). Granted, I also think frequentist notions of probability can be useful and have an important role to play (EG, in the theory of logical induction).
Anyway, I think there’s something right about “justification doesn’t carve reality at its joints”, but I think stopping at that would be a mistake. Yes, I think justification works different ways in different contexts, and the seeming paradox of the regress argument comes mostly from conflating the “trying to convince someone else” context with the “examining your own beliefs” context. However, I do think there’s something interesting going on with both of those notions of justification, and it seems potentially fruitful to examine them rather than throw them out. “Just run the math” only works if you’re not in the business of examining your beliefs (eg, the faith you have in the math) or justifying said math to others.
I’m not really sure what infinite chain of justification you are imagining a Bayesian position to suggest. A posterior seems justified by a prior combined with evidence. We can view this as a single justificatory step if we like, or as a sequence of updates. So long as the amount of information we are updating on is finite, this seems like a finite justification chain. I don’t think it matters if the sigma-algebra is infinite or if the space of possible observations is infinite, although as you seem to recognize, neither of these assumptions seems particularly plausible for bounded beings. The idea that everything effects everything else doesn’t actually need to make the justificatory chain infinite.
I think you might be conflating the idea that Bayesianism in some sense points at idealized rationality, with the idea that everything about a Bayesian needs to be infinite. It is possible to specify perfect Bayesian beliefs which can be calculated finitely. It is possible to have finite sigma algebras, finite amounts of information in an update, etc.
The idea of continuous updating is interesting, but not very much a part of the most common Bayesian picture. Also, its relationship to infinite chains seems more complex than you suggest. If I am observing, say, a temperature in real time, then you can model me as having a prior which gets updated on an observation of [the function specifying how the temperature has changed over time]. This would still be a finite chain. If you insist that the proper justification chain includes all my intermediate updates rather than just one big update, then it ceases to be a “chain” at all (because ALL pairs of distinct times have a third time between them, so there are no “direct” justification links between times whatsoever—any link between two times must summarize what happened at infinitely many times between).
I agree that that’s a bad misunderstanding that I want to avoid, so although I’m a bit incredulous at people reading it this way, I’ll try to edit to make this misunderstanding less plausible.