How can the strength of evidence (both positive and negative) for a proposition P be assessed and aggregated objectively?

Question

Questions such as Is there more evidence for God than for Russell’s teapot? and Is evidence for the "big bang" superior to evidence for "god"? are challenging to answer objectively without first establishing a common standard for evaluating the strength of evidence for a proposition P—whether P represents "God exists", "the Big Bang happened", "there is a teapot orbiting the Sun between Mars and Jupiter", or even "my parents exist (or existed)". Moreover, in the comments section of the the first question, @Conifold highlights the importance of the concept of negative evidence:

A common view is that Russel's teapot analogy fails at its intended purpose, due to ignoring negative evidence, among many other things. "Clearly we have a great deal of evidence against teapotism. For example, as far as we know, the only way a teapot could have gotten into orbit around the sun would be if some country with sufficiently developed space-shot capabilities had shot this pot into orbit. No country with such capabilities is sufficiently frivolous to waste its resources by trying to send a teapot into orbit", Plantinga.

This means that when assessing a proposition P, we should be careful to ponder not only the evidence in favor of it (positive) but also the evidence against it (negative).

Is there an agreed-upon standard among epistemologists that could help us assess the strength of evidence for a proposition P objectively?

Such a standard would presumably need to address the following considerations:

• Determine what constitutes positive evidence for P (i.e., what qualifies as evidence?).
• Determine what constitutes negative evidence against P.
• Evaluate the strength of both positive and negative evidence.
• Perform some form of aggregate calculus to determine the overall strength of the proposition P.

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UPDATE

Originally the question referenced God explicitly in the title, but this prompted one of the mods to tag it as a duplicate of Can we assign probabilities to God and is the argument from improbability from Dawkins valid?. To address this, God has been removed from the title (and the tags) so that the question is understood to be as generic as possible. Think of propositions involving God, such as P = "God exists", as just particular challenging propositions to take into account for whatever general-purpose evidence framework that is proposed.

Note also that the linked question is explicitly tied to the concept of probability. I acknowledge that probabilities are extremely useful in general but in principle this question is not tied to probabilities a priori.

Note also that there is no question (at least that I'm aware of) that explicitly addresses the issue of pondering and balancing both positive and negative evidence for/against a proposition.

More on this on the meta question: Charge of duplicate seems unjustified. Shouldn't my question be reopened?

Answer 1

From a Bayesian perspective (and possibly more generally), "no", at least in the case of the existence of God. This is because the strength of the evidence depends on your model of what God is. There are hypotheses though where the strength of the evidence might be evaluated objectively for all reasonable intents and purposes. Nothing is objective under unreasonable radical skepticism and a degree of pragmatism is required to make progress (see comment by @Dcleve in chat linked below).

In Bayes rule we have:

P(G|E) = P(E|G)P(G)/P(E)

where E is the evidence, and G is the proposition that God exists.

We might use the Bayes factor as a measure of the strength of the evidence for G, my taking the ratio of P(E|G) and P(E|~G)

P(G|E)   P(E|G)    P(G)
------ = ------- x -----
P(~G|E)  P(E|~G)   P(~G)

The first factor is the Bayes factor and it shows us how much the ratio of our prior beliefs are modified by the evidence in forming our posterior belief.

The key point is that it depends on P(E|G), the probability of observing the evidence if G is true (God actually does exist). So first you have to reach agreement on what God is, i.e. a model of God. That is not going to be a subject on which theists and atheists are likely to agree (and really why they don't agree about the value of the evidence).

IFF P(E|G) can be objectively justified, then it may be possible to objectively evaluate the strength of the evidence.

As an example, for coin flipping we may be trying to decide whether the coin has heads on both sides (H). In this case, we can probably all agree on P(E|H) and P(E|~H), ignoring negligible complications like the coin coming to rest on it's edge or being taken mid-flight by a falcon, or hallucinations or conjuring tricks. In that case, we probably could meaningfully talk about objective evidence.

Note that the strength of the evidence, represented by the Bayes factor doesn't depend on the prior belief P(G). So any analysis where people argue about the strength of the evidence on the basis of their prior beliefs are likely making a mistake and will be double-counting their prior in arriving at their posterior belief.

As to the sub-questions in the bullet points:

• If the Bayes factor is greater than one, that means E is positive evidence for G, because observing E means the posterior odds ratio is higher than the prior odds ratio. An increase in belief in G is justified by E.
• If the Bayes factor is less than one, that means E is negative evidence against G. Observing E will make the posterior odds ratio lower than the prior odds ratio.
• The further away from 1, the stronger the evidence for or against G (note it will probably make more sense on a logarithmic axis as the Bayes factor operates multiplicatively).

• For an aggregate calculus, if we have multiple pieces of evidence that are independent of eachother, we can just multiply their Bayes factors:

    
  P(G|E1,E2)    P(E1|G)    P(E2|G)    P(G)
  ----------- = -------- x -------- x -----
  P(~G|E1,E2)   P(E1|~G)   P(E2|~G)   P(~G)
  

  If they are not independent, it is a bit more complicated as we have to use conditional probabilities

  P(G|E1,E2)    P(E1|G)    P(E2|G,E1)    P(G)
  ----------- = -------- x ----------- x -----
  P(~G|E1,E2)   P(E1|~G)   P(E2|~G,E1)   P(~G)
  

  In such cases, Bayesian networks (https://en.wikipedia.org/wiki/Bayesian_network) can be useful in identifying the dependence structure between variables, to minimise the use of conditional probabilities.

It would be nice to see how this is dealt with in other epistemic frameworks, but sadly the question is closed. I'm off now.