September 21, 2008

Is Deduction Justification-Preserving?

Believe it or not, I have never had a course in epistemology (I am taking a class on skepticism this quarter, though). Of course, I have dealt with it in passing in a number of other courses, and, since I do a lot of early modern work, I am very familiar with the dispute between the rationalists and empiricists of that period. Anyway, the reason I am starting this post off this way is simply to note that there might very well be a huge literature debating the subject I am about to discuss, but if there is I haven't read any of it.

There is a popular class of theories of epistemology called "Justified True Belief" (JTB) theories. According to these theories (which have their ancestor in Plato) a belief counts as knowledge just in case the content of the belief is true and the belief itself is justified. (For simplicity, I will ignore the differences between the uses of the terms 'justification' and 'warrant.') The reason I call this a class of theories, rather than a single theory, is that no one can agree on what 'justification' is other than to say that it is that property which, when added to truth, makes a belief knowledge.

The Gettier cases are a famous pair of objections to JTB accounts. Many more cases have been added since the original publication in 1963. Now, both of the original Gettier cases and, as far as I know, all of the more recent ones, involve a type of deduction called a vacuous introduction. The following are vacuous introduction rules for propositional logic (P and Q are meta-variables representing any proposition, and ⊨ is the entailment operator):

  • P ⊨ P ⋁ Q

  • ~P ⊨ ~(P & Q)

  • ~P ⊨ P -> Q


The important thing to note here is that in each case the value of Q is irrelevant to the deduction. So, for instance, today is Sunday, and from that fact I can deduce that today is Sunday or pigs can fly, but I can also deduce that today is Sunday or 1+1=2. What happens in Gettier cases is that we have a justified belief P and perform a vacuous introduction involving a proposition Q, but, unbeknownst to us, P is actually false, and our conclusion is true only, as it were, 'accidentally,' on account of the Q we selected at random. (See the original paper for concrete examples.)

I have thought for some time that we could probably solve the Gettier problem by simply 'biting the bullet' and acknowledging that vacuous introductions are not justification-preserving - that is, that if you are justified in believing P, it doesn't follow that you are justified in believing a proposition, such as P ⋁ Q derived by a vacuous introduction from P. Now, the reason this is 'biting the bullet' - the reason it is troubling - is that deduction is supposed to model sound reasoning and deduction is certainly truth-preserving: if P, then necessarily P ⋁ Q. Still, I don't suppose it's so bad to place a few rules of deduction off-limits to avoid these cases.

However, in a recent post on his blog The Maverick Philosopher, Bill Vallicella raises a point called "the preface paradox" which threatens to create broader problems for deduction and justification. Suppose I write a preface to a book of my writings, and suppose I haven't changed my mind about anything, but I know that, like other human beings, I am fallible. I might, in my modesty, make a statement in my preface to the effect that the book probably contains at least some false claims. Now I am effectively claiming that:

  • Proposition 1 is true

  • Proposition 2 is true

  • ...
  • Proposition n is true

  • At least one of the preceding n propositions is false.

This, of course, is a contradiction. Yet, one would suppose, I am perfectly rational in believing it (unless there is some independent irrationality in my believing propositions 1 through n).

Bill considers the following argument:

  1. It is rational for the author to believe that each statement in his book is true.

  2. It is rational for the author to believe that some statement in his book is not true.

  3. Therefore
  4. There are cases in which it is rational to believe statements of the form (p & ~p).

He attempts to defuse the objection by arguing that justification, unlike truth, is relative to what factors are considered. However, even if he is correct, I'm not certain that this will defuse all problems of this kind. Suppose a person has the following three beliefs:
  1. The theory of general relativity is true

  2. Quantum field theory is true

  3. General relativity and quantum field theory are logically inconsistent, and so cannot both be true.

Now, I think most people who study this stuff believe that one or both contain certain errors, or are mere approximations, or something, and so they don't really believe both of them to be true in the strict sense, but suppose that, for some reason, neither of them could be an approximation or have minor errors. Suppose that we knew they were either perfectly true or utterly false. Wouldn't this still be a case in which it was rational to believe P and Q while denying (P & Q)? And isn't this kind of like saying that I believe every one of my beliefs, and yet believe that I have some false beliefs, just as in the previous case?

Bill might try to use the same response in this case: he might say that relative to the evidence for each one, we are justified in believing that that one is true, but relative to our knowledge of the contradiction between them we are justified in believing that at least one of them is false. This does make good sense of how the things we claim to 'know' and the things we regard as 'certain' shift given different discursive contexts. To me, though, it seems irrational to completely ignore certain relevant pieces of knowledge, especially knowledge of logical contradictions, and remove them from what counts as 'background knowledge'. There are, of course, such things as 'useful fictions' which we may want, for practical reasons, to act as if we believe, but to actually believe things that, based on all of our relevant knowledge we are not justified in believing seems irrational, and certainly ought not to constitute a form of 'knowledge'.

The problem is that, if we go this route, it seems that we are forced to a very troublesome conclusion: perhaps deduction in general is not justification-preserving! After all, if the rule {P, Q} ⊨ P&Q falls, there's not much left. That one's pretty central, as is its inverse. But if that's the case, then it is irrational to use deduction in belief formation, and then what's to become of us?

Posted by Kenny at September 21, 2008 9:14 PM
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Comments

Degrees of belief seem like a nice solution to the latter problems.

Incidentally, the 'vacuous introduction' deduction you describe is inessential to the Gettier problem. I discuss a couple of alternative examples here.

Posted by: Richard at September 21, 2008 11:02 PM

Richard - I do think the degrees of belief idea can help, especially if we use actual Bayesian analysis. When we make deductions from uncertain premises, the uncertainty of the conclusion will generally be greater than the uncertainty in any of the premises, taken individually. However, I'm not sure this will solve my physics case in quite the way my intuitions do.

As for your other cases, are those called Gettier cases in the literature? It is unclear to me whether they are actually manifestations of the same problem as Gettier's original paper or not. Certainly, however, they are a problem for JTB, and disallowing vacuous introductions will not solve them.

Posted by: Kenny at September 21, 2008 11:29 PM

Most people are not anywhere near fully rational or fully informed so they believe contradictory things like your quantum mechanics example. An idealized thinker on the other hand has a very complex set of confidence profiles (Or in a world where processing time and information storage matters - a list of heuristics).

So I would have thought that normal deduction was not justification preserving (because we are just humans) and idealized deduction was justification persevering by definition.

Posted by: GNZ at September 22, 2008 4:04 PM

GNZ - right. So is it your claim that formal logic does not, in fact, embody 'idealized deduction'?

Posted by: Kenny at September 22, 2008 4:08 PM

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