Topics Logic

Question I studied philosophy in university and I recall that one of my tutors for symbolic logic was trying to walk me through a problem by saying that if you have a large enough set of premises, two of them will inevitably contradict one another. I've always had trouble understanding (and consequently, accepting) this proposition because: if one conceives of reality as a set of claims (e.g., I am right-handed, electron X is in position Y, 2 + 2 = 4, etc.) there are an infinite number of "premises" to the "argument" that is reality and consequently reality is self-contradictory. Am I missing something here? Can you explain which of us is right about this and in which sense? I should mention that I don't necessarily have a problem with reality being self-contradictory, but that really throws symbolic logic out the window (and doesn't throw it out the window at the same time)! Thanks to all respondents for their time. -JAK

Accepted:
May 17, 2008

Comments

I'm not sure what your tutor was getting at either. If your tutor meant that one can always enlarge a set of premises to make it an inconsistent set, that's obviously true: simply add the negation of one of the premises already in the set. If he meant that any axiom system with infinitely many premises (say, one that employs an axiom schema) is inconsistent, then there's no reason to believe that.

Better: There's a fairly simple proof that this is false. Just consider the theory consisting of the sentence letters p1, p2, .... This theory is clearly consistent. It'd be an amusing question just how easy the proof is, i.e., exactly what sorts of theories are needed for the proof.

Maybe the tutor was thinking something like this (I seem to recall it from Popper). Let's consider only atomic (simple) contingent propositions (A, B, C, ... , Z) that are logically independent of each other. The probability that an atomic contingent proposition is true is less than 1 and greater than 0 (an atomic contingent proposition is true in at least one possible world and false in at least one possible world). Suppose P(A) is X (where 0 < X < 1). And suppose P(B) is Y (where 0 < Y < 1). Sure, the set {A, B} is consistent (or satisfiable), and the conjunction A&B will be true in at least one possible world. But there's a hitch: P(A&B) is the product of X and Y, which means that P(A&B) is less than P(A) and less than P(B). Let's increase the number of atomic propositions in our set to get {A1, A2, A3, ... , An}. Again, the set will be consistent, but the probability that the set (or long conjunction) is true is the product of the individual probabilities of each proposition; the more propositions we put into the set, the smaller this probability becomes, approaching (but never reaching) zero. Hence (I think it follows), the more premises you have in an argument, the less likely it is that the argument is sound. It does not follow that the premises will "inevitably" issue in a contradiction. Moral of the story: the more you say, the less likely it is that the totality of what you say is true. Silence is golden. (Panelists: feel free to correct my mistakes. Use as few propositions as possible.)