Cumulative Cases and the Conjunction fallacy

[Athée]

Hi all,

Friendly atheist question. If you are a believer who finds the notion of the cumulative case for the truth claims of your religion compelling, how do you avoid falling prey to the conjunction fallacy?

This link is a short read (5 minutes or so) that describes the fallacy and gives a few examples.

Burdensome Details - LessWrong 2.0

The basic idea is that the probability of A + B is going to be less than A on its own or B on its own.

The intuition I am pushing is that when we hear, for example, that the case for God's existence or for the resurrection of Jesus, is a cumulative case, our immediate reaction should be to worry about this conjunction fallacy.

For example if the hypothesis we are testing is that God raised Jesus from the dead and we are told we know this because it is the best explanation for the combination of facts 1,2,3,4,5,6,7,8...12 (based on Mike Winger's recent case on. YouTube - seems like a good guy (Christian) so go follow), our reactions should not be "ah indeed, all these extra details add credibility", instead we should think, "wow all those details multiply the unlikelyness of this hypothesis".

This will not be a debate thread on my end, I am genuinely interested in how those of you who find the cumulative case compelling, relate to the problem of the conjunction fallacy. Maybe this will cause you to lose faith and convert ... Or more likely (and my hope) it will cause you to think carefully about this one particular aspect of your belief structure. With a. It of extra luck someone here (my bet is 2PhilloVoid) will have a reasonable account of this and can help me understand it better from the Christian perspective.

Peace all,
Athee

 

[zippy2006]

This is simple: cumulative reasoning does not commit the conjunctive fallacy.

Suppose you come to the conclusion that it is lightly raining outside. There are three reasons:

1. Your view of the city in the distance is slightly blurry
2. There are droplets of water on your window
3. The puddle on your sidewalk is receiving a ripple effect from small particles hitting the surface

Cumulative reasoning says that the presence of (1), (2), and (3) makes it more likely that it is raining than any subset of those reasons alone. That is true. For example, (1) without (2) and (3) would present a less likely conclusion than (1 and 2) or (1, 2, and 3).

The conjunctive fallacy says that there is a lesser probability that (1, 2, and 3) are true, than that (1) alone is true, or that (1 and 2) are true. This is just basic logic: the truth of a conjunction presupposes the truth of each of its conjuncts.

Cumulative reasoning says:

(1 ^ 2 ^ 3) -> "It is raining outside"
​

The conjunctive fallacy just says that (1 ^ 2 ^ 3) is less likely than, say, (1) alone. Cumulative reasoning is not at all committed to the contradictory claim, namely that (1 ^ 2 ^ 3) is more likely than (1) alone.

(Note that cumulative reasoning is not committed to each cumulative reason being a necessary condition. It's not as though when someone overdetermines a conclusion with additional reasons they are walking a precarious line where, if any one of those reasons is falsified, the conclusion itself is also falsified. Just because the view of the city is not blurry does not mean it isn't raining.)

 

[Athée]

This is simple: cumulative reasoning does not commit the conjunctive fallacy.

Suppose you come to the conclusion that it is lightly raining outside. There are three reasons:

1. Your view of the city in the distance is slightly blurry
2. There are droplets of water on your window
3. The puddle on your sidewalk is receiving a ripple effect from small particles hitting the surface

Cumulative reasoning says that the presence of (1), (2), and (3) makes it more likely that it is raining than any subset of those reasons alone. That is true. For example, (1) without (2) and (3) would present a less likely conclusion than (1 and 2) or (1, 2, and 3).

The conjunctive fallacy says that there is a lesser probability that (1, 2, and 3) are true, than that (1) alone is true, or that (1 and 2) are true. This is just basic logic: the truth of a conjunction presupposes the truth of each of its conjuncts.

Cumulative reasoning says:

(1 ^ 2 ^ 3) -> "It is raining outside"
​

The conjunctive fallacy just says that (1 ^ 2 ^ 3) is less likely than, say, (1) alone. Cumulative reasoning is not at all committed to the contradictory claim, namely that (1 ^ 2 ^ 3) is more likely than (1) alone.

​

So to apply your thinking to the cumulative case for the resurrection (as an example) it would need to be the case that each of the items in that cumulative case be as certain as your observation of the blurriness, the water droplets and the ripples, otherwise (I think, and correct me if I am wrong) the uncertainty introduced by each successive non-certain item, would decrease the probability of the initial hypothesis, quite possibly overwhelming it.

So if, to pick one at random, the evidence for the empty tomb is less than 100% certain, it might be better to leave it out of the cumulative case, rather than allow it to introduce extra uncertainty into the equation. As I wrote that I see a weakness in my thinking. I guess if an element of the cumulative case is so essential to the initial hypothesis, that to remove it is instantly fatal, then it would be better to include it and bite the bullet on the uncertainty it introduces. In our example of the resurrection, for example, if you take out the piece of evidence about anyone believing they had any type of experience of the risen Jesus, the whole case falls apart And to to keep things on the rails, that premise gets included no matter what probabilistic cost it incurs.

Am I on the right track?

 

[zippy2006]

So to apply your thinking to the cumulative case for the resurrection (as an example) it would need to be the case that each of the items in that cumulative case be as certain as your observation of the blurriness, the water droplets and the ripples, otherwise (I think, and correct me if I am wrong) the uncertainty introduced by each successive non-certain item, would decrease the probability of the initial hypothesis, quite possibly overwhelming it.

I added something to my post that may be helpful here:

(Note that cumulative reasoning is not committed to each cumulative reason being a necessary condition. It's not as though when someone overdetermines a conclusion with additional reasons they are walking a precarious line where, if any one of those reasons is falsified, the conclusion itself is also falsified. Just because the view of the city is not blurry does not mean it isn't raining.)​

So if, to pick one at random, the evidence for the empty tomb is less than 100% certain, it might be better to leave it out of the cumulative case, rather than allow it to introduce extra uncertainty into the equation. As I wrote that I see a weakness in my thinking. I guess if an element of the cumulative case is so essential to the initial hypothesis, that to remove it is instantly fatal, then it would be better to include it and bite the bullet on the uncertainty it introduces. In our example of the resurrection, for example, if you take out the piece of evidence about anyone believing they had any type of experience of the risen Jesus, the whole case falls apart And to to keep things on the rails, that premise gets included no matter what probabilistic cost it incurs.

Am I on the right track?

No... Cumulative reasoning does not place strong inferential weight on any one conjunct. That's sort of the whole point. It is each piece of the puzzle working together that leads to the cumulative conclusion. No one piece bears excessive weight, and certain pieces may well be incorrect. Cardinal John Henry Newman wrote a book on this called An Essay in Aid of a Grammar of Assent. He calls our cumulative reasoning "The Illative Sense."

I think you are basically misunderstanding the conjunction fallacy. The essence of that fallacy is a simple relation between the probability of a conjunction and its conjuncts. IMO it is just objecting to our psychological inclination to believe an elaborate story over a simple story due to the fact that elaborate stories are harder to make up. It is objecting to this psychological move, not cumulative reasoning. As noted above, cumulative reasoning does not commit such a fallacy.

One of the conclusions you seem to have drawn from the conjunction fallacy is that one ought to minimize conjuncts in order to maximize probability. Unfortunately that is wildly false. It would only hold in a sort of random-mathematical world absent any methods of verification. Just because the probability of a conjunction is generally less than the probability of one conjunct does not mean cumulative reasoning is weaker than single-variable reasoning. If multiple reasons all point to the same conclusion the conclusion is more likely to be true.

Heck, try out your approach on my example above:

• Ben: I think it is raining. My view of the city is blurry, there are droplets on my window, and something is creating ripple effects on a puddle in the middle of my sidewalk.
• Joe: I think it is raining. My view of the city is blurry.

Who do you have more reason to believe? Who has made a stronger case?

Ben has asserted more conjuncts, therefore according to the conjunction fallacy it is less likely that all three of his conjuncts are true than that just one is true. But this is just a very small piece of the puzzle. It's like saying that a person who speaks more is more likely to speak a falsehood. Okay...sure. But there are all sorts of other variables. Some people are smarter. Some are more careful in their speech. Some people who speak infrequently may do so due to stupidity. And what sort of percentage change are we even talking about? If the verificational process is generally sound the percentage change will be miniscule.

When extra reasons overdetermine a conclusion the inference itself is strengthened. At the same time the probability that each and every conjunct is true technically decreases by a small amount, but there is nothing overly problematic with a small percentage of conjuncts being false. All of our premises and inferences have the possibility of being false. It's not even clear to me what solution your objection leads to. Cumulative reasoning is far superior to single-variable analysis when we are talking about abductive reasoning. I mean, abductive reasoning is inferior to demonstrative reasoning, but I think we're in agreement that historical sciences rarely have access to demonstrative reasoning. ...So what's the alternative to cumulative reasoning?

 

[Athée]

I added something to my post that may be helpful here:

(Note that cumulative reasoning is not committed to each cumulative reason being a necessary condition. It's not as though when someone overdetermines a conclusion with additional reasons they are walking a precarious line where, if any one of those reasons is falsified, the conclusion itself is also falsified. Just because the view of the city is not blurry does not mean it isn't raining.)​

No... Cumulative reasoning does not place strong inferential weight on any one conjunct. That's sort of the whole point. It is each piece of the puzzle working together that leads to the cumulative conclusion. No one piece bears excessive weight, and certain pieces may well be incorrect. Cardinal John Henry Newman wrote a book on this called An Essay in Aid of a Grammar of Assent. He calls our cumulative reasoning "The Illative Sense."

I think you are basically misunderstanding the conjunction fallacy. The essence of that fallacy is a simple relation between the probability of a conjunction and its conjuncts. IMO it is just objecting to our psychological inclination to believe an elaborate story over a simple story due to the fact that elaborate stories are harder to make up. It is objecting to this psychological move, not cumulative reasoning. As noted above, cumulative reasoning does not commit such a fallacy.

One of the conclusions you seem to have drawn from the conjunction fallacy is that one ought to minimize conjuncts in order to maximize probability. Unfortunately that is wildly false. It would only hold in a sort of random-mathematical world absent any methods of verification. Just because the probability of a conjunction is generally less than the probability of one conjunct does not mean cumulative reasoning is weaker than single-variable reasoning. If multiple reasons all point to the same conclusion the conclusion is more likely to be true.

Heck, try out your approach on my example above:

• Ben: I think it is raining. My view of the city is blurry, there are droplets on my window, and something is creating ripple effects on a puddle in the middle of my sidewalk.
• Joe: I think it is raining. My view of the city is blurry.

Who do you have more reason to believe? Who has made a stronger case?

Ben has asserted more conjuncts, therefore according to the conjunction fallacy it is less likely that all three of his conjuncts are true than that just one is true. But this is just a very small piece of the puzzle. It's like saying that a person who speaks more is more likely to speak a falsehood. Okay...sure. But there are all sorts of other variables. Some people are smarter. Some are more careful in their speech. Some people who speak infrequently may do so due to stupidity. And what sort of percentage change are we even talking about? If the verificational process is generally sound the percentage change will be miniscule.

When extra reasons overdetermine a conclusion the inference itself is strengthened. At the same time the probability that each and every conjunct is true technically decreases by a small amount, but there is nothing overly problematic with a small percentage of conjuncts being false. All of our premises and inferences have the possibility of being false. It's not even clear to me what solution your objection leads to. Cumulative reasoning is far superior to single-variable analysis when we are talking about abductive reasoning. I mean, abductive reasoning is inferior to demonstrative reasoning, but I think we're in agreement that historical sciences rarely have access to demonstrative reasoning. ...So what's the alternative to cumulative reasoning?

You have done an excellent job of showing me some of the errors in my thinking! Thanks, now I don't have to be wrong in the same way anymore

It seems to me that we would still worry about increasing conjuncts of dubious certainty
Eg
Hypothesis: it is raining
1. I see droplets and ripples in the puddles outside.
2. I think maybe there are droplets on my widows ..?
3. My view outside might be blurry... But maybe not...

It seems like in this case you should still worry about the added uncertainty but you are correct that the cumulative reasoning remains valid. I guess it just adds some problem to something like a Bayesian analysis of the problem.

It would still be the case that it is more probable that God exists than that God exists and Jesus was resurrected. But that would not mean that you can't use the truth of the resurrection to lend credibility to the hypothesis that God exists.

Right?

 

[zippy2006]

You have done an excellent job of showing me some of the errors in my thinking! Thanks, now I don't have to be wrong in the same way anymore

Sure, it's an interesting question.

It seems to me that we would still worry about increasing conjuncts of dubious certainty
Eg
Hypothesis: it is raining
1. I see droplets and ripples in the puddles outside.
2. I think maybe there are droplets on my widows ..?
3. My view outside might be blurry... But maybe not...

It seems like in this case you should still worry about the added uncertainty but you are correct that the cumulative reasoning remains valid. I guess it just adds some problem to something like a Bayesian analysis of the problem.

Yeah, I guess you have to ask yourself how much you care whether an additional, dubious conjunct turns out to be false. There are cases where the only thing we have access to are weak conjuncts and one is forced to make an inference to the best possibility on weak data. In that case you would just arrive at a weaker conclusion.

If someone is just piling on superfluous conjuncts to artificially inflate the perceived probability this would be a problematic form of deception, though I'm not sure that the deception results from the conjunction fallacy so much as misidentifying weak conjuncts for strong conjuncts. Either way a good analysis will look at each conjunct in isolation.

It would still be the case that it is more probable that God exists than that God exists and Jesus was resurrected. But that would not mean that you can't use the truth of the resurrection to lend credibility to the hypothesis that God exists.

Right?

Right. In these cases the conjuncts should never be mutually exclusive.

I guess it's a little bit hard for me to imagine a scenario where the conjunction fallacy becomes crucial in evaluating a cumulative argument. Doing things like analyzing the individual conjuncts and assessing the coherence of the conjunction would seem to deserve more weight than the general rule that "more claims are less likely than one claim."

Hopefully someone else will have more light to shed on that, because that premise seems too abstractly mathematical to me, almost like the way a slot machine operates. It is of course unlikely for all of the icons on a slot machine to line up perfectly, but that is precisely because of the random chance involved. Reasoning and verification processes are actually designed to minimize randomness and produce reliable, true results. And the better the scholar, the less randomness; the better the scholar, the less the conjunction fallacy even applies. Beyond that, cumulative reasoning strikes me as too important and pervasive in everyday life to strongly challenge.

 

[2PhiloVoid]

Hi all,

Friendly atheist question. If you are a believer who finds the notion of the cumulative case for the truth claims of your religion compelling, how do you avoid falling prey to the conjunction fallacy?

This link is a short read (5 minutes or so) that describes the fallacy and gives a few examples.

Burdensome Details - LessWrong 2.0

The basic idea is that the probability of A + B is going to be less than A on its own or B on its own.

The intuition I am pushing is that when we hear, for example, that the case for God's existence or for the resurrection of Jesus, is a cumulative case, our immediate reaction should be to worry about this conjunction fallacy.

For example if the hypothesis we are testing is that God raised Jesus from the dead and we are told we know this because it is the best explanation for the combination of facts 1,2,3,4,5,6,7,8...12 (based on Mike Winger's recent case on. YouTube - seems like a good guy (Christian) so go follow), our reactions should not be "ah indeed, all these extra details add credibility", instead we should think, "wow all those details multiply the unlikelyness of this hypothesis".

This will not be a debate thread on my end, I am genuinely interested in how those of you who find the cumulative case compelling, relate to the problem of the conjunction fallacy. Maybe this will cause you to lose faith and convert ... Or more likely (and my hope) it will cause you to think carefully about this one particular aspect of your belief structure. With a. It of extra luck someone here (my bet is 2PhilloVoid) will have a reasonable account of this and can help me understand it better from the Christian perspective.

Peace all,
Athee

Ah, gee! I'm humbled... but to tell you the truth, I take this as a 'shot' across the bow of Pascal's Wager, a cheap shot, really. I guess it's a good thing that in my obsession with spiritual gambling, I don't have to be a fan of "cumulative case faith." If I was a fan of it, then I'd have something to worry about! ... then again, I'm not a fan of Bayesian probability type faith, either, but that may be beside the point.

 

[Athée]

Ah, gee! I'm humbled... but to tell you the truth, I take this as a 'shot' across the bow of Pascal's Wager, a cheap shot, really. I guess it's a good thing that in my obsession with spiritual gambling, I don't have to be a fan of "cumulative case faith." If I was a fan of it, then I'd have something to worry about!

Wait that's not following the rules! You have to be a cumulative case Christian and I have to be an "I just lack a belief in God and have no positive claim" atheist

 

[2PhiloVoid]

Wait that's not following the rules! You have to be a cumulative case Christian and I have to be an "I just lack a belief in God and have no positive claim" atheist

So, what part of your OP would you like me to address being that I think it's rather moot in nature?

 

[Athée]

So, what part of your OP would you like me to address being that I think it's rather moot in nature?

Nothing if you don't find it interesting. Do you see the conjunctive fallacy as being something to worry about when doing cumulative case type reasoning? Why are you not a fan of the cumatove approach? How badly have I messed up the philosophy ( I ways seem to, sigh).

 

[2PhiloVoid]

Nothing if you don't find it interesting. Do you see the conjunctive fallacy as being something to worry about when doing cumulative case type reasoning?

It could be something to be concerned about, but the extent to which this is the case will depend on exactly who is attempting to use a cumulative case type of reasoning, what the specific praxis is being used, and if the person reasoning in this way is someone like Jim Baker versus someone like Jim Wallace. We might want to consider what someone like Jim Wallace could say in response, even if we don't agree with the finality of his overall conclusions ...

Why are you not a fan of the cumatove approach?

On one hand, I will admit that on a very, very practical level, I have a more or less Coherence style of thought when it comes to the Christian faith, but unlike my fellow Christians who hold onto cumulative case belief structures in their epistemology, I hold as a part of the Hermeneutical Whole the fact that the following fields present some mitigation to any kind of sheer reliance upon Coherence in the Christian faith:

1) The field of Biblical Epistemology
2) The field of Philosophical Hermeneutics
3) Jewish "Thought Culture" studies
4) The field of Christian Existentialism and Fideism​

When cumulative case reasoning is used to supposedly arrive at some final state of deliberative affairs where one can claim "faith," I remain somewhat skeptical of that person's state of epistemic assurance. This kind of goes in line with the fact that I also have a similar allergy to Bayesian thinking when it's applied to the Christian faith as well, whether it's for it, or (like Carrier) against it. I think we talked about this a few years ago, but that remains essentially another matter ...

How badly have I messed up the philosophy ( I ways seem to, sigh).

I couldn't tell you how you've messed it up since I'm no expert on the topic; my insight here is cursory even though some of it indirectly ties in to the social sciences and educational research stat methods I had to study years ago.

In scratching the surface of your OP, one of the articles I came across seems to back up your concern for the ways in which the Conjunctive Fallacy can be committed by those persons who are prone to believing conspiracy theories or entertain other paranormal beliefs; I suppose that in some ways this article can be indirectly relevant to Christians, and I thought I'd share it since it is pertinent---and I like the little piece of research regarding the fictional person of "Linda" :

Brotherton, Robert, and Christopher C. French. "Belief in conspiracy theories and susceptibility to the conjunction fallacy." Applied Cognitive Psychology 28, no. 2 (2014): 238-248.​

Despite this interesting academic appeal and the fact that it contributes to the skeptic's inclination to apply the Conjunction Fallacy to the typical, modern day Christian, I think we need to keep in mind that there are a few caveats in the "ease" by which any of us should cite and apply this fallacy to all Christian folks across the board. One of the additional caveats in addition to those I've mentioned above is that the Conjunctive Fallacy also doesn't clearly apply to Pascal's Wager ... even though this fallacy is often a part of and relevant to the psychological study of Gambling Problems or Gambling Addictions.