## Monday, January 30, 2012

### Modus Ponens versus Affirming the Consequent

Consider these two rules of doxastic practice:

• Modus Ponens (MP): If you believe that p and you believe that if p, then q, then infer q.
• Affirming the Consequent (AC): If you believe that q and you believe that if p, then q, then infer p.
MP is a good rule of inference. AC is a fallacy. But why is MP better? An obvious pair of relevant modal facts is:
1. Necessarily, if it is true that p and it is true that if p, then q, then it is true that q.
2. Possibly, it is true that q, it is true that if p, then q, but it is not true that p.
These facts suggest that
1. MP is a more effective way of getting to truth than AC.

But (3) does not necessarily follow from (1) and (2). For instance, from (1) and (2), we get these claims:

1. Necessarily, if all your beliefs are true, and you apply MP to generate a new belief, your beliefs will still be all true.
2. Possibly, all your beliefs are true, and you apply AC to generate a new belief, and your resulting beliefs are not all true.
But obviously (4) and (5) tell us nothing about whether MP is better than AC for us, since the antecedent in (4) is not satisfied in our situation: it is false that all our beliefs are true.

Imagine Sam. Most of Sam's beliefs are true. But in cases in which he believes that if p, then q, it is more often true that the converse conditional is true than that this conditional is true. It could very well be the case for Sam that following AC is a more effective way of getting to truth than following MP is.

Or imagine Dory. While most of her beliefs are true, and it is more often the case when she believes that if p, then q, that this conditional is true than that the converse conditional is true, nonetheless due to some cause she happens to tend to apply AC or MP almost only in cases where only the converse conditional is true. Again, for her following AC is a more effective way of getting to truth than following MP is.

Of course, I expect that for most if not all of us MP is a more effective way of getting to truth than AC. But there is no necessity in this. In particular, that MP is a more effective way of getting to truth than AC is not a thesis of logic (but of what? psychology? natural theology?). Nothing surprising about that, of course.