Thursday, February 26, 2015

Counting up arguments

Some time in the fall, Ted Poston asked me how I thought one should model the force of multiple arguments for the existence of God in a Bayesian setting. There are difficulties. For instance, when we discover a valid argument, what we are discovering is the necessary truth that if the premises are true, the consequent is as well. But necessary truths should have probability one. And it's hard to model learning things that have probability one. Moreover, the premises of the arguments are typically not something we are sure of. At the time, I suggested that we conditionalize on the contingent propositions reporting that the premises seem true. Poston ended up going with an urn model instead.

I want to try out another model for counting up the force of multiple arguments, one where we not worry about what is and what isn't necessary. I will develop the story with a toy model that has prior probabilities that make calculation easy, leaving it for future investigation to weaken my assumptions of prior independence and equiprobability.

So, suppose we're looking at decent (say: valid, non-question-begging) arguments for and against a conclusion q, and we find that there are m arguments for and n arguments against. How likely is q given this? Start the model by identifying in each argument the controversial premise. (If there is more than one, conjoin them.) Thus, we now have m+n controversial premises. Let's say that premises p1,...,pm support arguments for q and pm+1,...,pm+n support arguments for ~q.

Prior to the discovery of the arguments, in my model I will take the propositions p1,...,pm,pm+1,...,pm+n,q to be all independent, and, further, to each have probability 1/2.

I now model the discovery of the arguments as a discovery of material conditionals. Thus, we discover the m material conditionals p1q,...,pmq that favor q and the n material conditionals pm+1→~q,...,pm+n→~q that favor ~q. How do we model this discovery? We simply ignore all the messy details that the discoveries were at least in part a matter of discovering logical connections (though perhaps only in part; some of the premises beside the controversial premise might have been empirical). We simply conditionalize on the m+n discovered material conditionals.

What's the result? Well, we could use Bayes' Theorem, but that's just a tool for computing conditional probabilities, and sometimes other methods work better. We have m+n+1 "propositions of interest" (i.e., q and the pi). Our prior probabilities assign equal chances to each of the 2m+n+1 possible ways of assigning True or False to the propositions of interest. When we conditionalize on the material conditionals we rule out some combinations. For instance, if we assign True to p1, we had better assign True to q as well, and we had better assign False to pm+1,...,pm+n, all on pain of contradiction.

We can say something about how many truth assignments remain after the conditionalizations:

  1. Assign False to all the pi and False to q: one combination
  2. Assign False to all the pi and True to q: one combination
  3. Assign False to p1,...,pm and True to at least one of pm+1,...,pm+n and False to q: 2n−1 combinations
  4. Assign True to at least one of p1,...,pm and False to all of pm+1,...,pm+n and True to q: 2m−1 combinations.
All the combinations remain equally likely after conditionalizations. The final probability of q then is just the proportion of the combinations where q comes out true amongst all four types of combinations. Now, q comes out true in combinations of types (2) and (4). Thus, the final probability of q given the discovery D of the arguments is:
  • P(q|D)=2m/(2n+2m).
Try some numbers. Say that we have 3 arguments in favor and 1 against. P(q|D)=23/(23+21)=8/10=0.8. With 4 arguments in favor and 1 against, P(q|D)=16/18=0.89 (for this, think of the theism-atheism debate as pitting the cosmological, design, religious experience and moral arguments against the argument from evil). If we have 10 arguments in favor and 2 against, then P(q|D)=1024/(1024+4)=0.996.

For a more realistic model, we will need to change our priors for the controversial premises so that they aren't all 1/2. Some of the controversial premises of the arguments will be fairly plausible and they will have priors higher than 1/2. Some may not be all that plausible and will have priors lower than 1/2. And maybe the conclusion q will have a prior other than 1/2. Furthermore, there may be mutual dependencies among the controversial premises over and beyond the dependencies induced by the fact that some of them imply q and others imply ~q (the latter dependencies are handled by our conditioning). All of this would require fiddling with the priors, and the simple "counting combinations" method of calculating the posterior P(q|D) will need to be replaced by a more careful calculation. Nonetheless, the principle will be the same.


Heath White said...

Another way the model needs to be refined is that we might discover undercutting defeaters for the original set of arguments. For example, the discovery of natural selection does not at all tend to show that God does not exist, but it does tend to show that the traditional biological form of the teleological argument is not persuasive. And skeptical theism is essentially a view on which premises about evil cannot generate an argument against the existence of God.

Alexander R Pruss said...

Maybe those defeaters can simply be seen as removing the corresponding material conditionals from the things we took ourselves to know. This isn't quite the Bayesian way to handle it, but it's a quick and dirty solution.

Alexander R Pruss said...


Suppose the controversial premises of the arguments in favor of q have prior probability a, and those against q have prior probability b, but we still keep the prior for q at 1/2. Then:
P(q|D) = (1-b)^m / [(1-b)^m + (1-a)^n].

This leads to some fun numbers. For instance, suppose we have one argument against q, with controversial premise at prior probability 0.8, and the arguments in favor of q are much weaker--their controversial premise is only 0.2. We can ask how many of these weaker arguments for q do we need to overcome the force of the stronger argument against q. Answer: eight. Eight weaker arguments in favor of q overcome the one stronger argument against q and yield a posterior greater than 1/2. How many of the weaker arguments would we then need in order to have 0.95 confidence in q? Answer: 21.

Mark Rogers said...

So if I bring 21 arguments to Heath as to why all life did not evolve from the descent of all living species from a common ancestor he will be open to the suggestion that he may be wrong about natural selection?

Mark Rogers said...

(1) Considering the vastness of the universe it seems possible that life could have started independently elsewhere and considering the known durability of some microbes we are aware of it seems possible that some form of life could have arrived on earth from elsewhere and thrived here.

Mark Rogers said...

(2) If one considers it possible that a necessary being exists, then that necessary being would be quite distinct from the life forms we are familiar with. If it is possible that there are two distinct forms of life then it seems possible that there could be three distinct forms of life.

Mark Rogers said...

(3) if a necessary being did exist it would seem possible that it could create or destroy species at will.

Mark Rogers said...

(4) There are many examples in the fossil record of mass extinctions followed by explosions of new species. Not many examples of the gradual evolution of species that one might expect from natural selection.

Mark Rogers said...

(5) If a necessary being existed who desired to remain hidden existed, it would not be surprising that there could be a number of theories that explain the origin of species.

Alexander R Pruss said...

The reliance of the arguments on a necessary being damages the independence condition, unless we're assuming there is such a being.

Mark Rogers said...

(5) If a necessary being desires to become even more hidden, what better way than becoming obsolescent?

Mark Rogers said...

(6) It is difficult to explain the extent and complexity of eyes which appeared in the Cambrian.

Alexander R Pruss said...

And you'd have to enumerate the arguments on the other side. It's harder to do this when it's empirical stuff based on lots of observations.

Mark Rogers said...

(7) Natural Selection is a nice theory, but possibly scientists will come up with a better theory. Consider what Thomas Aquinas said:

"Reason may be employed in two ways to establish a point: firstly, for the purpose of furnishing sufficient proof of some principle [...]. Reason is employed in another way, not as furnishing a sufficient proof of a principle, but as confirming an already established principle, by showing the congruity of its results, as in astronomy the theory of eccentrics and epicycles is considered as established, because thereby the sensible appearances of the heavenly movements can be explained (possunt salvari apparentia sensibilia); not, however, as if this proof were sufficient, forasmuch as some other theory might explain them."


“The suppositions that these men [Ptolemaic astronomers] have invented need not necessarily be true: for perhaps, while they save the appearances under these suppositions, they might not be true . For maybe the phenomena of the stars can be explained by some other schema not yet discovered by men” (Book II, lecture 17).