Contradictions and Conjunctions

Epistemic Humility

            Perhaps the costliest of all mistakes in thinking is to affirm a contradiction.  Most people recognize this intuitively.  What can a juror think when the person on the witness stand says he saw the accused commit the crime and that he did not see the accused commit the crime?  Common sense is flummoxed by such blatant self-contradiction.  So we almost never hear it.  Instead, the witness may say he saw the accused commit the crime and later, under close questioning, admit that he was in no position to see the crime.  The attorney examining the witness would then draw the contradiction to the attention of the jurors, confident that they would disregard testimony that contradicts itself.

            Formal logic demonstrates exactly how bad self-contradiction really is.  Suppose someone affirms a contradiction; say, for example, “Orange juice contains vitamin C and orange juice does not contain vitamin C.”  We would symbolize this as follows.

1.     O + ~O

Proposition 1 is a self-contradiction.  It is necessarily false.  But what happens if someone affirms it?  How about this: “Phil is President of the United States”?  We would symbolize this as P.  Given the self-contradiction in 1, it is easy to prove P:

1.     O + ~O     /therefore P

2.     O               from 1, by “simplification” (a basic rule of logic: if p+q, then p)

3.     ~O                        from 1, by simplification again

4.     O v P         from 2, by “addition” (another basic rule: if p, then p or q)

5.     P                from 4 and 3, by “disjunctive syllogism” (p or q, but not p, so q)

Given an explicit contradiction it follows that I am President.  Of course, this isn’t the only bizarre conclusion that follows from a self-contradiction.  Given an explicit self-contradiction, any proposition whatever follows with deductive necessity.  People who affirm self-contradictions imply the truth of every possible proposition.

Someone might say: “Okay, so what?  Common sense and logic both condemn self-contradictions.  Let’s all avoid contradicting ourselves and move on to something more interesting.”

But there is a problem: all sane people believe at least one implicit contradiction.  “Really?  How so?” you might ask.  The reason is that our beliefs do not obey the rules of logic.  Remember that some weeks ago in Story and Meaning I published this little argument.

I.          One of the easiest and most obvious inference rules in logic is the rule of conjunction.  If A is true and B is true, then A and B is true.  We make students create the truth table for this inference mostly so they can practice building truth tables.  No one doubts that propositions are conjunctive.

1.     A

2.     B                /therefore A+B

3.     A+B            1, 2, conjunction

II.         It may seem counterintuitive, but the rule of conjunction does not apply to beliefs.  It is axiomatic that if a person believes something, she believes that it is true.  At first guess, we might think that if two beliefs are held to be true, the conjunction of those beliefs must also be believed.  But this is not so. 

III.        The best way to prove this is to assume the opposite and see what happens.  We will assume that beliefs are conjunctive.  On this assumption, if a person considers proposition D and finds that she believes it, and then considers proposition E and finds that she believes it, then she should believe a new proposition: D + E.

            Let us imagine that our subject considers all her beliefs sequentially.  We will let B₁ stand for her first belief, B₂ for her second, and so on.  After considering B₁ and B₂, she should find that she believes B₁ + B₂.  Soon after, she should also discover that she also believes B₁ + B₂ + B₃.

            Now, human beings hold an indefinitely large number of beliefs, but the number of beliefs a person holds may be finite.  Let’s assume that it is.  (If people hold an infinite number of beliefs, my overall argument still holds.)  Given enough time, our subject should find that she believes the following proposition, which we will call proposition omega:

            B_ω = {B₁, B₂, B₃, B₄, …B_ω} where B_ω is the last of her beliefs. 

            It will interest some people that B_ω is one of the beliefs that our subject believes; that is, the subject’s class of beliefs is self-referential.  The argument does not depend on this fact.

            The problem is this.  No reflective person believes B_ω.  To the contrary: it is safe to say that all reflective people believe the opposite, i.e. they believe  ~B_ω.  To see that this is so, consider the meaning of ~ B_ω in ordinary English: "It is not true that all of my beliefs are true," which is the same as saying, “At least one of my beliefs is false.”

IV.  All of us know that among our indefinitely large class of beliefs, there are almost certainly some that are wrong.  If it were possible to consider individually each one of our beliefs (neuropsychologically speaking, this may be impossible) we would find that we think each one is true.  After all, that’s what it means to believe something; you think that it is true.  And yet, we do not believe that all our beliefs are true.  We have very great confidence that at least one of our beliefs is false.  But if conjunction rules over beliefs, we ought to believe B_ω.  The fact that we do not believe B_ω shows that conjunction does not rule over our beliefs.

            The fact that conjunction does not rule over our beliefs should cause us some pause.  Conjunction is a basic inference rule in formal logic.  Common sense supports the conjunction rule as firmly as it condemns self-contradiction.  (Imagine serving as a juror when the witness testifies that “roses are red” is true, and “violets are blue” is true, but “roses are red and violets are blue” is not true.) 

Here’s the problem.  If our beliefs obeyed the rule of conjunction we would all believe B_ω.  But we don’t.  We believe ~B_ω.  There is a contradiction implicit between what we would believe if we believed in accord with logic (B_ω) and what we actually believe (~B_ω).  If we actually affirmed this contradiction, we would thereby imply every proposition.

The contradiction is only implicit.  I can only consciously think about a tiny fraction of my beliefs at any time.  If the beliefs I am presently aware of seem coherent and true, I am satisfied with them.  Let us call the conjunction of all my present beliefs proposition CB (for conjuncted beliefs).  I may well think CB is true while still affirming ~B_ω, because I have indefinitely many other beliefs, and at least one of them is false.

What should we do?  When someone points out to me that two of my beliefs contradict each other (something I wrote a year ago contradicts what I say today), I face the awful danger of affirming a contradiction.  Common sense says I should reexamine my contradictory beliefs and modify or abandon one of them.  Much experience shows that we ought to be fallibilists.  Very often we find that our belief about this or that turned out to be wrong.

            The upshot of this meditation is this.  We need to be appropriately humble about our epistemic efforts.  Our beliefs do not in fact obey the rules of logic.  That doesn’t mean we can abandon logic.  But the logical implication of my current beliefs may show that one or more of my beliefs stand in need of improvement.