Friday, February 4, 2011

Probability in Deductive Arguments

Many agnostics and atheists are putting forth what I call the “probability objection” against deductive arguments. This objection is not lodged against any premise of any theistic argument, but rather against our warrant for saying the argument’s conclusion is true.

Probability is related to background knowledge. It is assigned a value from 0 (being impossible) and 1 (being certain). For instance, the flip of a coin will gain a probability of .5 for both sides. When dealing with multiple values, or propositions, to understand the probability of each proposition’s being true together, one must multiply the individual probabilities. So, for example, if we want to know the probability of both “Jodi picking up soda” and “Jodi getting into a car accident,” we should multiply both numbers together (arbitrarily, we’ll say each is .6). When we do this, we get a probability of both statements occurring of just .36 (.6 x .6=.36). This means it is less likely to happen than not, and thus we shouldn’t believe they will both occur.

Thus, in an argument such as the kalam cosmological argument, which looks like this

1. Whatever begins to exist has a cause.
2. The universe began to exist.
3. Therefore, the universe has a cause.

we must understand the probability of each premise’s being true, and then multiply that together (since we need both premises to establish the conclusion). But if we assign a .7 to both, this yields just a .49 probability, which means the conclusion is less likely to be true than not, and thus we should reject it.

However, there are major problems with this approach. First, in a logically-valid deductive argument, the conclusion follows necessarily from the premises. So long as the premises are true, the conclusion is guaranteed to be true! Probability just doesn’t enter the picture. Second, one should not attempt to find out if an individual premise is true by knowing if the other premise is true! Aside from being circular,[1] do you really need to know that “I am at the mall” in order to know that “if I am at the mall, then I am having fun”? It seems that we may establish some degree of plausibility for a statement wholly independent of the other statement’s truth. If this is the case, however, then if we have reason to think that each statement in a deductive syllogism is true, then we have reason to think the conclusion is true! Just keep this in mind in case any agnostic or atheist brings this up.
                [1] If each premise’s truth cannot be evaluated apart from the other, then no premise’s truth could ever be evaluated probabilistically, since every premise would rely on another!


  1. This doesn't seem to be interacting with the force of the objection.

    Sure, if the premises are true then the conclusion of a valid deductive argument follows necessarily. The point is, in the real world, we often find ourselves in doubt about just those premises. So what happens when you are perhaps fairly sure but not all that confident of the premises of a deductive argument?

    The Atheist/Agnostic objection here is surely on the money (although, Tim McGrew suggested similar things before). We should multiply our credences together, and this gives us the lower bound for the probability we should assign to the conclusion.

    Consequently a) Craig's mantra that more plausible than negation is enough is simply wrong*, b) doubts multiply with dubious premises - thus arguments reliant on plausible but not decisively convincing premises tend to be insufficient to motivate belief in their conclusions on their own.

    * Reductio.

    1) Dice 1 came up 1-4
    2) Dice 1 came up 1-4
    N) Dice N came up 1-4

    C) All N dice came up 1-4

    Each premise is more probable than it's negation (2/3), and the conclusion of this argument follows by conjunction introduction (form: A / B // A^B). Yet the conclusion is probably false with 2 dice, and it's probability trends asymptotically to zero as N increases!

    Craig's condition is necessary for a good deductive argument (something that persuades someone of its conclusion): if the probability of any premise is less than one half, the lower probability bound of the conclusion will be less than one half. Yet it isn't sufficient for a good argument - furthermore, deductive arguments that don't have a lower bound greater than 0.5 might still be useful, as the lower bound by the argument might be higher than the current assignment of that person.

  2. Hi polemical, thanks for the comment! You readily admit that if the premises are true, the conclusion follows.

    I suppose perhaps you meant (2') Dice 2 came up 1-4. In that case, however, note that the proposition is non-specific. It is really a disjunction which says:

    "Either Dice 1 comes up 1, or 2, or 3, or 4." In order for a disjunct to be true, only one part needs to be true.

    It seems your objection boils down to your last paragraph--that a premise's being more plausible than its negation is necessary but not sufficient; yet no one (not even Craig) is saying that it is! This article simply points out that one does not need to know the probability of all premises in order to know the probability of each individual premise. The only reason the proffered counterexample is such is because (probably) one of them will eventually not roll a 1-4--but this violates the criteria of soundness! Finally, if this argument is turned into a deductive argument, it seems we can actually reject it out of hand!

    1. Every die rolled will come up 1-4.
    2. N is a die.
    3. Therefore, N will come up 1-4.

    We can reject (1) precisely because of our background knowledge of (1) alone; not because we took the probability of (1) and multiplied it by the probability of (2).

  3. Return to the counter example.

    1) Dice 1 comes up 1-4
    2) Dice 2 comes up 1-4
    C) Both Dice 1 and Dice 2 come up 1-4.

    Yet, although this argument is valid, and even though each premise is more likely true than its negation, the conclusion is less likely true than its negation (moreso if you add more dice). It therefore counterexamples the idea that an argument is persuasive if we satisfy the argument is valid and the premises more true than their negations.

    If Craig only meant to say it was a necessary condition, then this seems a bit irrelevant: we are surely interested in whether an argument is in fact good or persuasive or whatever, not simply that we can come up with a set of criterion which do not rule it out as good or persuasive.

    To find out whether a deductive argument is indeed a good argument, one should do what I said previously - multiply together your credences of each premise. This gives you the lower bound of what your assignment for the conclusion should be. If *this* is greater than one half, then you have a good argument. "An deductive argument is a good one if the the conjunction of all premises used is more plausible than its negation" specifies out the sufficiency condition we should be using. If Craig's arguments fail this test but pass his own less rigorous necessity condition, that should be little comfort to him.

  4. Hi polemic, glad to hear from you! However, there is some confusion here. You have already agreed, as far as I can tell, that probability doesn't apply to the conclusion itself. That is to say, if the premises are true, the probability of the conclusion's being true is 1.

    What you ought to do is to say that one of the premises in your counterexample is false. That you don't know it a priori is just irrelevant, as you may choose to accept a premise on any evidence you wish. We can resolve this problem by simple experimentation. We can designate one Die 1 and the other Die 2. If they are rolled and both true, the conclusion follows. If one is false, the conclusion is false. The criteria for a good argument is that the argument is logically valid, that we believe the premises to be more plausibly true than false, and that the argument is indeed sound (as we could be mistaken in the second criterion ultimately). Your counterexample does not succeed in all three if it functions counter at all! So long as the criteria met is necessary and sufficient, it doesn't matter that your proffered criteria demands greater probabilistic conjunction (especially since you don't really need to know "Jim is at the mall" is true in order to know "if Jim is at the mall, then he is having fun" is true! I've yet to see anyone explain why we would need it!); it just means you have a higher but unnecessary standard.

  5. "If one is false, the conclusion is false." Rather, I meant the conclusion does not follow. :)


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