Message in a Bottle

Tuesday, July 20, 2004

Yet another Probability Paradox

Brian Weatherson from Thoughts, Arguments and Rants has just reminded us of the wonderful Monty Hall Problem. Here is a even stranger problem, taken from Raymon Smullyians wonderful Satan, Cantor and infinity: The Two Envelope Paradox. It goes like this: There are two envelopes, one contains twice as much as the other envelope. You see no difference between them from the outside. You choose one envelope and open it, it contains X>>0 dollars. Now you can either keep the money in that envelope or take the money from the other envelope. You are assumed to be risk averse, only caring for the expected amount of money you can get from a decision.

According to one train of thought, you should choose the other envelope since it contains 2X$ with a probability of 0.5 and 0.5X$ with the same probability. That gives you an expected net gain of 3/4X$, a strictly positive sum. Maybe you are convinced now, but there is another train of thought. Suppose you would have chosen the other envelope. You still would want to choose another envelope according to the argument above. But that means the new information you got doesn't influence your decision in any way. So why is that different from the initial position, where you could have chosen any envelope? So it doesn't matter whether you choose the other envelope or not.

The Baysian solution is very simple: You must first have a (subjective) probability distribution and calculate the expected gain of changing afterwards, which will probably lead to other probablities of the higher amount being in the other envelope than 0.5. A analysis for people with a background in basic probability heory can be found here.

Sunday, July 04, 2004

On the Popularity of Verificationism

Verificationism is generally believed to be dead, but is it? Samir Okasha thinks not. Here is a quote from his paper Verificationism, Realism And Scepticism, Erkenntnis Volume 55, Issue 3, Dec 2001, p 371-385:

According to the standard textbook account, verificationism is a doctrine that was popular back in the days of logical positivism. The positivists promised us a ‘verifiability criterion of meaningfulness’ which would help to distinguish ‘cognitively significant’ discourse from nonsense, but despite their best efforts, no workable criterion was ever produced. This led to verificationism being thoroughly discredited and consigned to the philosophical scrap-heap, along with most of the other doctrines characteristic of logical positivism.

Now this story is only partially accurate. For it ignores the fact that verificationist ideas, in one form or another, continue to find adherents among contemporary philosophers. Quine, Putnam and Dummett, to pick the three best-known cases, are all openly sympathetic to the basic verificationist idea that the content of a statement is closely tied to what would count as evidence for its truth. None of these philosophers believes in the ‘criterion of meaningfulness’ that the positivists hankered after, but the spirit of verificationism lives on in their work. Due in large part to the interest generated by Putnam’s and Dummett’s writings, verificationist ideas have enjoyed something of a resurgence in recent years, though not always under that name. In this paper, I examine one of the factors that lies behind the continued appeal of verificationism.

In contemporary discussions, verificationist ideas usually surface byway of opposition to some form of realism. Putnam explicitly contrasts his version of verificationism – which he calls ‘internal realism’ – with a position called ‘metaphysical realism’. Metaphysical realism holds that truth is a ‘radically non-epistemic’ notion, with no conceptual connection to justification or warranted assertion, while internal realism holds that truth is warranted assertibility in the ideal limit. Similarly Dummett contrasts the realist view that meaning is to explained in terms of ‘evidence transcendent’ truth-conditions, with his own anti-realist view that meaning should be explained in terms of verification-conditions, which cannot transcend our ability to detect them. For Putnam, Dummett and others, realism provides the foil for the development of their verificationist viewpoints.