Friday, January 18, 2013

Why Evidentialism Sucks (p.3)

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Why Evidentialism Sucks
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How Can the Believer Be Justified?

How could a theist be justified in believing in God without evidence or proof? Here’s one way: God designed humans (whether via evolution or otherwise) in such a way that when our cognitive faculties are functioning properly, we intuitively apprehend God. True enough, a pretty sizeable portion of humanity has believed in God or at least something like God. I thus think this hypothesis is probable, at least if God exists. An atheist will find the hypothesis highly improbable of course, but it seems that about the only way to disprove the hypothesis is to show that God doesn’t exist, not to argue that people are irrational for believing in God even if God exists, because if God does exist then our intuitions of his existence are veridical and likely the direct or indirect product of God himself.

What about the Christian embracing Christianity without evidence? The same sort of situation for theism applies here. If Christianity is true, it’s plausible that the Holy Spirit encourages the believer when he or she hears the message and receives it. On this view, the intuition that the Christian faith is true is thus at least in part the result of the inner testimony of the Holy Spirit. The atheist may find this view implausible, but again the best way to refute it seems to be refuting Christianity itself, and skip arguing for the idea that Christians are irrational even if Christianity is true. It’s relatively trivial to come up with a flavor of Christianity that is rational to believe if that flavor of Christianity is true (e.g. the inner testimony of the Holy Spirit thing).

Conclusion

Evidentialism sucks. Even if it’s well-intentioned, it sucks. The infinite regress problem (evidence for one’s evidence for one’s evidence...) is enough to make it unworkable, but evidentialism also runs into trouble from mathematics. Consider for example the hypothesis you are recently created (say, within the past few years) brain in a vat of chemicals hooked up to a supercomputer that gives you all the memories, sense experiences, and intuitions you now have. We are rational to believe that crazy hypothesis is false, and yet mathematics shows us that we can’t really have any evidence against that hypothesis (since any memory, sense experience, or intuition you have is predicted by said crazy hypothesis).

The upshot is that it is rational to believe some things without evidence or proof, e.g. that the crazy brain-in-the-vat hypothesis is false. There is also room for rationally believing in God and Christianity without evidence (at least if God and Christianity are true).

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Why Evidentialism Sucks (p.2)

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Why Evidentialism Sucks
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The Mathematical Problem for Evidentialism

Ironically enough, mathematics and probability theory pose serious problems for evidentialism. For those whose math is a bit rusty, I recommend reading this quick introduction to Bayes’ theorem before moving forward.

Ready? Good. Here’s some other math stuff you’ll need to know here. E is evidence for hypothesis H if and only if the following is true:
Pr(H|E) > Pr(H)
Where Pr(H) is the prior probability.[1] E is evidence for hypothesis H if it makes the hypothesis more likely than it would have been without E. To oversimplify it a bit, something being evidence for a hypothesis means it makes the hypothesis more likely to be true.

One form of Bayes’ theorem, when comparing hypotheses, goes like this:
Pr(H1|E)
Pr(H2|E)
 = 
Pr(E|H1)
Pr(E|H2)
 × 
Pr(H1)
Pr(H2)
For example, one can use hypothesis H and it’s negation ¬H as follows:
Pr(H|E)
Pr(¬H|E)
 = 
Pr(E|H)
Pr(E|¬H)
 × 
Pr(H)
Pr(¬H)
Notice the importance Pr(E|H) and Pr(E|¬H) have in changing the likelihood ratio of Pr(H|E) to Pr(¬H|E). As one might suspect from the equation above, for this to get bigger via evidence E:
Pr(H|E)
Pr(¬H|E)
Pr(E|H) must be greater than Pr(E|¬H), and a bit of math shows that for E to be evidence for hypothesis H, this has to be true:
Pr(E|H) > Pr(E|¬H)[2]
Similarly, for something to be evidence against a hypothesis, the following has to be true:
Pr(E|¬H) > Pr(E|H)[3]
These math facts are part of a recipe of doom for evidentialism. Let ES be the set of all memories, sense experiences, and intuitions you now have. Let HB be the hypothesis you are a recently created (say, within the past few years) brain in a vat of chemicals hooked up to a supercomputer that gives you all the memories, sense experiences, and intuitions you now have. Thus, Pr(ES|HB) = 1. You are rational to believe that hypothesis HB is false, but here’s the problem: you cannot cite any evidence against it. Thanks to mathematics, for some would-be piece of evidence E to be evidence against HB, the following has to be true:
Pr(E|¬HB) > Pr(E|HB)
Hypothesis HB however predicts with 100% certainty any memory, sense experience, and intuition you now have, thus Pr(E|HB) = 1, and it is impossible for any evidence E to be such that Pr(E|¬HB) > Pr(E|HB).

This actually teaches us an important lesson: some things are rational to believe without proof or evidence, e.g. the falsity of hypothesis HB. It also teaches us that evidentialism sucks; if evidentialism were true, we could never reject that crazy brain-in-the-vat hypothesis.

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[1] The Pr(H|E) > Pr(H) inequality is oversimplifying it a bit; I haven’t mentioned that e.g. the background knowledge for Pr(H) doesn’t already have E (though I later say E is evidence if “it makes the hypothesis more likely than it would have been without E”), because if our background knowledge for Pr(H) included E then Pr(H|E) = Pr(E) even when E is evidence for H. Hopefully though the general raising-the-probability-of-the-hypothesis idea is apparent.

[2] This holds for when Pr(E) ≠ 0, which is the situation that interests us (presumably, any evidence E we are interested in is the sort that Pr(E) ≠ 0). For math nerds who are interested in the mathematical proof for this, I will first prove a few lemmas.

Lemma (1): (H ∩ E) and (¬H ∩ E) are disjoint (mutually exclusive). We can show that no element in the universe can be a member of both (H ∩ E) and (¬H ∩ E). Let x be an arbitrary element and let’s suppose x is a member of both (H ∩ E) and (¬H ∩ E). With a bit of math logic, we show that there can’t be any x such that x ∈ (H ∩ E) and x ∈ (¬H ∩ E) by assuming there is such an x and deriving an impossibility, like so:
  1. x ∈ (H ∩ E) and x ∈ (¬H ∩ E)
  2. (x ∈ H and x ∈ E) and (x ∈ ¬H and x ∈ E), from (1) and definition of ∩
  3. x ∈ H and x ∈ E and x ∈ ¬H and x ∈ E, from (2)
  4. x ∈ H and x ∈ ¬H, from (3)
Of course, it’s impossible for there to be an element that is a member of a set and its complement, since (¬H ∩ H) = ∅. Thus (H ∩ E) and (¬H ∩ E) are disjoint, i.e. (H ∩ E) and (¬H ∩ E) = ∅.

With this in mind, let ξ be the universal set.
E ∩ ξ = E
⇔ E ∩ (H ∪ ¬H) = E
⇔ (E ∩ H) ∪ (E ∩ ¬H) = E
Lemma (2): Pr(H|E) = 1 − Pr(¬H|E) for Pr(E) ≠ 0.

Proof: since (E ∩ H) ∪ (E ∩ ¬H) and are mutually exclusive as proven in Lemma (1), by the rules of probability:
Pr(E ∩ H) + Pr(E ∩ ¬H) = Pr(E)
⇔ Pr(E) × Pr(H|E) + Pr(E) × Pr(¬H|E) = Pr(E)
⇔ Pr(E) × [Pr(H|E) + Pr(¬H|E)] = Pr(E)
⇔ 1 × [Pr(H|E) + Pr(¬H|E)] = 1
⇔ Pr(H|E) + Pr(¬H|E) = 1
⇔ Pr(H|E) = 1 − Pr(¬H|E)
This holds for all Pr(E) ≠ 0. Thus, Pr(H|E) = 1 − Pr(¬H|E) for Pr(E) ≠ 0.

Lemma (3): If Pr(H|E) > Pr(H), then Pr(¬H|E) < Pr(¬H)

Proof: assume for sake of argument that Pr(H|E) > Pr(H) is true. We can then see how this leads to Pr(¬H|E) < Pr(¬H) using Lemma(2), where Lemma (2) implies that Pr(H|E) = 1 − Pr(¬H|E). If you’ve understood what’s going on thus far, I’ll assume you’re already aware of the well-known probability fact that Pr(¬A) = 1 − Pr(A) and that Pr(A) = 1 − Pr(¬A).
Pr(H|E) > Pr(H)
⇔ 1 − Pr(¬H|E) > 1 − Pr(¬H)
⇔ −Pr(¬H|E) > −Pr(¬H)
⇔ Pr(¬H|E) < Pr(¬H)
The components of the odds form of Bayes’ theorem are these:

posterior
odds
  likelihood
ratio
  prior
odds
            
Pr(H|E)
Pr(¬H|E)
 = 
Pr(E|H)
Pr(E|¬H)
 × 
Pr(H)
Pr(¬H)


Pr(H|E) > Pr(H) entails Pr(¬H|E) < Pr(¬H). If Pr(H|E) > Pr(H) and Pr(¬H|E) < Pr(¬H), then the posterior odds are greater than the prior odds, and the only way for this to happen is if the likelihood ratio is greater than one, which requires Pr(E|H) > Pr(E|¬H). Thus, if Pr(H|E) > Pr(H) then Pr(E|H) > Pr(E|¬H) for all E such that Pr(E) ≠ 0.

[3] To prove this, the task is to show that if Pr(H|E) < Pr(H), then Pr(E|¬H) > Pr(E|H) for all E such that Pr(E) ≠ 0.

One proof for this is quite similar to the one above, and so we can borrow from the same lemmas. Note that Lemma (2) from above entails that Pr(H|E) = 1 − Pr(¬H|E), for Pr(E) ≠ 0. From this we can construct a new lemma.

Lemma (4): Pr(H|E) < Pr(H) entails Pr(¬H|E) > Pr(¬H) for Pr(E) ≠ 0.
Pr(H|E) < Pr(H)
⇔ 1 − Pr(¬H|E) < 1 − Pr(¬H)
⇔ −Pr(¬H|E) < −Pr(¬H)
⇔ Pr(¬H|E) > Pr(¬H)
The components of the odds form of Bayes’ theorem are these:

posterior
odds
  likelihood
ratio
  prior
odds
            
Pr(H|E)
Pr(¬H|E)
 = 
Pr(E|H)
Pr(E|¬H)
 × 
Pr(H)
Pr(¬H)


Pr(H|E) < Pr(H) entails Pr(¬H|E) > Pr(¬H). If Pr(H|E) < Pr(H) and Pr(¬H|E) > Pr(¬H), then the posterior odds are less than the prior odds, and the only way for this to happen is if the likelihood ratio is less than one, which requires Pr(E|¬H) > Pr(E|H). Thus, if Pr(H|E) < Pr(H) then Pr(E|¬H) > Pr(E|H) for all E such that Pr(E) ≠ 0.

Why Evidentialism Sucks

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Why Evidentialism Sucks
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Intro

While the term evidentialism can mean somewhat different things in different contexts (as is all too often the case with philosophical terms), here I’ll use the term to refer to the idea that a belief is justified if and only if it is backed by sufficient evidence, such that a belief that has zero evidence for it is not justified.

An atheist might criticize religious people for believing things without sufficient evidence on evidentialist grounds. While there is evidence for theism and Christianity, if we’re honest many if not most Christians really couldn’t cite much evidence for their faith. Some don’t have the time or the resources to do so. Are these Christians irrational for being Christians?

In one sense evidentialism is understandable; there are many cases where it’s important to have evidence, and people often undervalue evidence when making decisions. Doesn’t it seem rational then that all beliefs should have sufficient evidence?

The Regress Problem for Evidentialism

Suppose I believe something on the basis of evidence E1, say, the testimony of my friend Joe. For this to work I have to believe that that my evidence is reliable, e.g. I have to believe that Joe is reliable source of information. But evidentialism says I shouldn’t believe anything without sufficient evidence, so I need to have sufficient evidence E2 to believe that evidence E1 is reliable; I need evidence for my evidence. But if I believe evidence E2 is reliable, I’ll need evidence E3 for my belief that evidence E2 is reliable; I need evidence for my evidence for my evidence. Evidentialism thus gives us a problem of a regress. There are three possibilities here:
  1. A vicious infinite regress (E1 is evidence for E2 which is evidence for E3...).
  2. Circular reasoning (e.g. E1 is evidence for E2 and E2 is evidence for E1).
  3. Some stopping point (something that is not believed on the basis of “sufficient” evidence).
Option 1 is unworkable, and option 2 employs a notorious fallacy. An example of circular reasoning is this:
Alice: What reason do you have for believing the book is a reliable source of information?

Bob: The author of the book says it’s a reliable source of information.

Alice: What reason do you have for believing the author is a reliable source of information?

Bob: The book says the author is a reliable source of information.
Circular reasoning doesn’t really give us any genuine evidence or reason at all.

We believe our evidence is reliable to at least some degree, but because of this, if we accept evidentialism any piece of evidence we have is going to suffer from some form of regress or circular reasoning (though one could conceive of a very complex web of circular reasoning). It seems to be the case that eventually we need to reach one or more stopping points.

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