**Debbie:** We argue a lot, yet what do we need to do to be convinced by it? What makes some arguments good, and others bad? How should we assess an argument, and whether or not to believe it?

**Adam:** Take a recent example. William Lane Craig offers something called a Kalam Cosmological argument, which runs like this:

- Everything that begins to exist has a cause.
- The universe began to exist
- The universe had a cause.

The argument is valid – that is, the conclusions follow from the premises. The question is whether the premises are true. Craig’s standard conception of what makes a good argument is whether or not the premises are more plausible than their negations.

Is that sufficient? Suppose that you do find each premise more plausible than it’s negation, but not by much: so you assign something like P=0.6 for each premise. Yet, given this the probability of the conclusion is only around 0.36, so you shouldn’t accept it.

**Debbie:** Hmm. It obviously isn’t right that P(conclusion) is the multiplication of each P(premise). Otherwise one could disprove god by producing an awful (but valid) argument that concludes he exists. The right answer should be something like the multiplication of the premises give a *lower bound* on the probability of the conclusion.

Yet this example is right insofar as it goes. It isn’t enough to say that the premises are more likely than not means you should accept the conclusion. Here’s an easy counter-example: suppose I tell you I’ve rolled two dice, and give you this argument.

- Dice A came up 1-4
- Dice B came up 1-4
- Both die came up 1-4.

Now that’s also a valid argument (A. B. A&B). Both premises are more likely true than their negations. Yet it would be silly to actually believe the conclusion – because it is probably false! In this case, the lower bound of the probability (2/3 * 2/3 = 4/9) is in fact it’s actual value, yet this is not more plausible than it’s negation. To make the point even stronger, just add more dice.

**Adam:** There are parallel examples to the dice case. For all of my beliefs (at least those about questions with only two answers), I believe they are more likely true than their negations. Yet I am fairly confident that at least some of these are wrong. Similarly authors often offer a pre-emptive apology for any errors made in the foreword.

**Debbie:** Right. I think for these sorts of deductive arguments to be persuasive, you need to believe that the *conjunction* of premises is more likely true than not, not that all of them meet some threshold. Because to believe that the argument is sound (that is, a valid argument with true premises) it needs to be that all the premises are true and the argumentative form is valid. So, popping back under our epistemic veil, we need to be confident of just these things. We can usually quickly work out whether an argument is valid or not, so its the premises that are the issue. Yet, in cases where we’re in doubt about multiple premises, these doubts, well, multiply. So – using the credences you offered earlier – even though we’d accept P1 or P2 of Craig’s argument, we wouldn’t accept that P1 and P2 are probably both true together. So we shouldn’t accept the conclusion.

Further, the P=0.5 threshold doesn’t matter the other way either – not only are arguments with P(premise) > 0.5 not necessarily persuasive, also arguments with P(premise) < 0.5 aren’t necessarily unpersuasive. Suppose I was arguing with someone who took P(the universe had a cause) to be very very low – 0.01, say. Yet, if they hold to P=0.6 for the premises of the Kalam, they should raise their estimate to at least 0.36 (or, if they’re feeling Moorean, drive down their assignments for the premises). So I’m not even sure it’s a very good heuristic – any better than “more plausible premises are better than less plausible ones for a persuasive argument” for example.

**Adam:** In that case, then, all these valid arguments really do is point to links between certain beliefs: that certain combinations of beliefs entail others. That we can’t hold conjunctions of premises with a probability higher than their entailments.

That leads to some interesting asides. For example, argumentative force leaks away with the addition of controversial premises. If you need only two premises to make your argument, then having each at P=0.9 is good enough. Yet, with several premises, even if you are very confident of each one, you shouldn’t necessarily be confident of the truth of their conjunction – even if you’re sure of each, there are plenty of chances to ‘get unlucky’. Perhaps this is why we are wary of arguments that rely on things a far way removed from the topic at hand.

**Claire:** It also means that these sorts of arguments don’t add power in concert – they are only as good as their strongest link, because an argument that gives a lower bound less than your current assignment isn’t interesting. So, when Craig offers half a dozen arguments for the existence of God, it isn’t the case that each makes the case more likely. He might think each argument is sufficient to raise a reasonable persons estimate of God’s existence above 0.5, but they can’t incrementally convince one of the conclusion.

**Adam:** That isn’t quite right. Suppose I have two arguments, such that A/B //C and D/E //C. Suppose again the probability for he premises (A,B,D,E) is 0.6. Yet the probability of C is not 0.36, but 0.59. This is because now you are being asked the probability of (A&B)V(D&E), which amounts to the probability of tossing at least one head in two shots when P(head)=0.36. And that is more likely than not. ^{1}

In effect, multiple arguments offer a chance at entrapment. The multiple arguments demand you commit to a large number of auxiliary denials to avoid these argument (“So you believe C, well, you need to deny at least one of A or B, and at least one of D or E, etc.”) As these mount up, the tension in these denials might become unbearable. So if all of Craig’s arguments purchased significant plausibility, then Atheist would be in trouble. However, it wouldn’t be true to claim that only one of Craig’s arguments need be successful to carry the day, unless it satisfies the condition above.

# Abductive and inductive concerns

**Adam:** How does this work with a cumulative case? Because it seems wrong to use what we said above to defeat a cumulative case in detail by showing that each element doesn’t force the conclusion in mind – for that is the purpose of the cumulative case in the first place.

**Claire:** Yes. We want there to be these sorts of arguments. They are doing different things.

Our deductive case is something simple like.

- A → B
- A
- B

And so our relevant probability calculus is simply something like P(A)*P(B|A), which gives us our probability of B. Yet, for an abductive argument, things are a bit different. Suppose something like a fine tuning argument.

- Fine tuning exists
- Fine tuning is better explained under Theism than under Naturalism
- Theism is confirmed over Naturalism.

The probabilistic calculus here is something more complicated like:

P(F|T) > P (F|N).

[P(T|F)/P(N|F)]= [P(F|T)/P(F|N)] * [P(T)/P(N)]

// P(T|F) > P(T)

So here we’re looking at relative probabilities, or the balance of probabilities between two competitors. So we can say that some evidence favours our hypotheses, even if it needn’t necessarily drive it above P=0.5. The probability of the evidence in question isn’t fundamental – the lower it is, the more the confirmatory push is ‘screened off’, but even likely-to-be-false evidences can still push one towards one hypothesis over another.

**Adam:** Perhaps we can put it another way: deductive arguments concern themselves with a single ‘confirmatory route’. Any doubt in the premises saps away from the conclusion – on the plus side, you end up with an absolute assignment. Adding multiple arguments can help persuade by plotting other confirmatory routes, and perhaps making it harder to deny that all different sets of argument have premises more likely to be true than not. The sort of probabilistic argument above doesn’t worry about any absolute assignment – instead of ‘linking’ premises to conclusions, it shows how one belief ‘pulls’ upon another – that, on considering certain data, we should adjust our confidence in other beliefs we hold. Other arguments to the same confusion add directly, like tributaries onto a river.

**Claire:** Yes, although the cost is the conclusions are vague: instead of “so X is probably true”, we end up with “you should now believe X is more likely to be true (or maybe ‘much more likely to be true’). But you could still accept the argument and yet still believe not-X: you just evaluate the confirmatory ‘shove’ as insufficient to get from your prior to over 0.5.

Also, they have varying degrees of robustness. Multiple deductive arguments for the same conclusion don’t necessarily sum well. Unless one of them reaches the ‘silver bullet’ territory of the set of premises being probably true, then it, alone, is unpersuasive. It needs its peers to begin to do the sort of entrapment you have in mind. Yet an abductive or hypothetico-deductive idea can still ‘work’ even if it is both implausible and unsupported. It can still offer some confirmatory shove, even if the effect is meagre.

Notes:

- I owe this to Andrew Johnson ↩