Posts Tagged ‘Koopman’

Missed Article

Saturday, 21 November 2020

I found an article that, had I known of it, I would have noted in my probability paper, A Logic of Comparative Support: Qualitative Conditional Probability Relations Represented by Popper Functions by James Allen Hawthorne
in Oxford Handbook of Probabilities and Philosophy, edited by Alan Hájek and Chris Hitchcock

Professor Hawthorne adopts essentially unchanged most of Koopman's axiomata from The Axioms and Algebra of Intuitive Probability, but sets aside Koopman's axiom of Subdivision, noting that it may not seem as intuitively compelling as the others. In my own paper, I showed that Koopman's axiom of Subdivision was a theorem of a much simpler, more general principle in combination with an axiom that is equivalent to two of the axiomata in Koopman's later revision of his system. (The article containing that revision is not listed in Hawthorne's bibliography.) I provided less radically simpler alternatives to other axiomata, and included axiomata that did not apply to Koopman's purposes in his paper but did to the purposes of a general theory of decision-making.

Nihil ex Nihilo

Tuesday, 6 December 2016

In his foundational work on probability,[1] Bernard Osgood Koopman would write something of form α /κ for a suggested observation α in the context of a presumption κ. That's not how I proceed, but I don't actively object to his having done so, and he had a reason for it. Though Koopman well understood that real-life rarely offered a basis for completely ordering such things by likelihood, let alone associating them with quantities, he was concerned to explore the cases in which quantification were possible, and he wanted his readers to see something rather like division there. Indeed, he would call the left-hand element α a numerator, and the right-hand element κ the denominator.

He would further use 0 to represent that which were impossible. This notation is usable, but I think that he got a bit lost because of it. In his presentation of axiomata, Osgood verbally imposes a tacit assumption that no denominator were 0. This attempt at assumption disturbs me, not because I think that a denominator could be 0, but because it doesn't bear assuming. And, as Koopman believed that probability theory were essentially a generalization of logic (as do I), I think that he should have seen that the proposition didn't bear assuming. Since Koopman was a logicist, the only thing that he should associate with a denominator of 0 would be a system of assumptions that entailed a self-contradiction; anything else is more plausible than that.

In formal logic, it is normally accepted that anything can follow if one allows a self-contradiction into a system, so that any conclusion as such is uninteresting. If faced by something such as X ∨ (Y ∧ ¬Y) (ie X or both Y and not-Y), one throws away the (Y ∧ ¬Y), leaving just the X; if faced with a conclusion Y ∧ ¬Y then one throws away whatever forced that awful thing upon one.[2] Thus, the formalist approach wouldn't so much forbid a denominator of 0 as declare everything that followed from it to be uninteresting, of no worth. A formal expression that no contradiction is entailed by the presumption κ would have the form ¬(κ ⇒ [(Y ∧ ¬Y)∃Y]) but this just dissolves uselessly ¬(¬κ ∨ [(Y ∧ ¬Y)∃Y])
¬¬κ ∧ ¬[(Y ∧ ¬Y)∃Y]
κ ∧ [¬(Y ∧ ¬Y)∀Y]
κ ∧ [(¬Y ∨ ¬¬Y)∀Y]
κ ∧ [(¬YY)∀Y]
(because (X ⇔ [X ∧ (Y ∨ ¬Y)∀Y])∀X).

In classical logic, the principle of non-contradiction is seen as the bedrock principle, not an assumption (tacit or otherwise), because no alternative can actually be assumed instead.[3]. From that perspective, one should call the absence of 0-valued denominators simply a principle.

[1] Koopman, Bernard Osgood; The Axioms and Algebra of Intuitive Probability, The Annals of Mathematics, Series 2 Vol 41 #2, pp 269-292; and The Bases of Probability, Bulletin of the American Mathematical Society, Vol 46 #10, pp 763-774.

[2] Indeed, that principle of rejection is the basis of proof by contradiction, which method baffles so many people!

[3] Aristoteles, The Metaphysics, Bk 4, Ch 3, 1005b15-22.