Fact, Fiction and Forecast is one of the seminal texts of 20th century philosophy: you may know it from the famous “grue/bleen” example. The text deals principally with two problems, the meaning of counterfactuals and a “new riddle” of induction, where the first is essential for any social scientist to understand, and the second has, I think, some interesting implications for decision theory. I will discuss each in turn. My notes are from the 4th edition, including the foreword by the legendary Hilary Putnam.
The first involves counterfactual conditionals, or sentences of the type “If X were true, then Y would obtain” along with the fact that X is not actually true. Counterfactual conditionals are both the focus of a huge number of economics paper (“If the Fed had done X, then GDP would have done Y”, “If deworming had been expanded to 100% in this village, school attendance would have been Y”, etc.). Counterfactuals are also, I would argue, the concept which has been forefront in the minds of the world’s leading philosophers over the past 60 years.
When economists use counterfactuals, I think they are naively trying to say something like “If the world is precisely the same, except that also X is true, then Y would hold.” There are a ton of problems with this. First, if everything in the world is precisely the same, then Not X is true, and since X and Not X are both true, by the principle of explosion, everything is true, including Not Y. So we must mean that everything in the world is precisely the same, except that X holds and Not X does not. Call the counterfactual set of true statements S’. But here we have more problems: S’ may contain a logical inconsistency, in that X may deductively imply some statement Z which is logically incompatible with something in S. Getting around that problem presents even more difficulties; David Lewis has the most famous resolution with his possible worlds logic, but even that is far from unproblematic.
Ignoring this basic problem of what is meant by a counterfactual, it is not well-known among social scientists that counterfactual conditionals are absolutely not strictly defined by their logical content, in the way that standard deductive logic is. That is, consider the statement If A then B, where A is a counterfactual. Let A’ be logically equivalent to A. It is easy to construct an example where you intuitively accept that A implies B, but not that A’ implies B. For instance, let A be “Bill Clinton were the same person as Julius Caesar,” A’ be “Julius Caesar were the same person as Bill Clinton” and B be “Bill Clinton would be emperor of Rome.” Given the importance of counterfactual logic to economics, there is a lot to be gained for our science from a better understanding of the philosophic issues here.
The more interesting point in Goodman for the decision theorist concerns induction. Hume showed in the 18th century why induction is invalid; the validity of induction involves assuming some sort of continuity of nature, and such an assumption is an induction itself. Even probabilistic induction – “The sun has risen every day, so I think it probable the sun will rise tomorrow” – is invalid for the same reason. There are many arguments contra Hume, but I hope you’ll take my word that they have all failed, and that the validity of induction is no longer an open question. That said, the wisdom of induction certainly is. Though we know induction is invalid reasoning, we nonetheless rely on it trivially every day (I get on a bus going north to my office, and not south, on the inductive assumption that my office is still north of my apartment) and less trivially on important policy issues (acceptance of “science” as a valid method for learning truth, rather than reading sacred books, is implicitly an acceptance of the wisdom of induction). What exactly do we mean when we say induction is wise? We mean that, there exist regularities for which the past existence of the regularity is evidence that we should expect the regularity in the future.
What Goodman points out is that the interesting question is not whether induction is valid – it isn’t – but rather what do we mean by a “regularity” anyway? This problem of induction is precisely the same to a problem in counterfactuals. Consider the regularity that every object is my pocket is a coin made of metal. I have investigated this many times, and every object I check is a metal coin. Consider the counterfactual “If I put a piece of chocolate in my pocket” or the induction on objects in my pocket where the only thing in my pocket today is a chocolate. Surely we don’t think we should induct that the chocolate will be a metal coin when I take it from my pocket. Alternatively, consider the regularity that all metal coins conduct electricity. I have investigated this many times also, and every metal coin I check conducts. If I check another coin, I do believe it will conduct. What is the difference between the chocolate example and the coin example? It is that I trust induction when I believe a law holds for some regularity, and do not trust induction when I believe past draws are simply random. The “grue/bleen” example, if you know it, is even stronger: I interpret it to mean that whatever rationale we use to delineate coincidences from regularities depends on more than how we selected instances in the past, or on the type of the property (say, color, or conductivity) we are examining. Goodman proposes some thoughts on how we know what histories are evidence of laws and what aren’t, but the exact delineation remains controversial.
So what does this mean for decision theory? Decision theory is heavily influenced by de Finetti and Savage, and somewhat by Carnap, and less so by other massive philosophy figures in this literature like Ayer, Goodman, Putnam, and Quine. That is, we conceive of the world as having states over which agents have a prior, and evidence changing that prior according to Bayes’ rule. Let Ω(₶) be the state space, where states are a countably infinite product space of potential observations. Let a “lawlike” set of hypotheses be a set of (infinite-length) observations that are compatible with some law, where the nature of possible laws is given exogenously. For instance, a lawlike set might be “all metals conduct” and the state space simply made up of tests of conductivity of various metals in each period plus a draw from the set {0,1}. The nature of the set of possible laws in the prior is that either all metals conduct, or the conductivity properties of various metals is not linked. Imagine in periods 1 and 2 that all metals conduct and we draw a 0 each time, and that in a second possible world, in periods 1 and 2 all metals conduct except copper in period 2, and we draw a 0 each time. What can we conclude as a Savage-style Bayesian? Think about what conditions on the prior are imposed.
There is one further worry for the standard econ model. How we induct in Goodman depends on what predicates we have as potential sources of laws: how ought we set up the state space? If we, say, put 0 prior on the world where all emeralds are grue, and positive prior on the world where all emeralds are green – and the standard model of state space means that we must include both possibilities as states – then we are violating Carnap’s “principle of total evidence” since we rule of grue before even seeing any evidence, and we are violating any of the standard rationales for putting positive probability on all possible states in the prior.
http://books.google.com/books?hl=en&lr=&id=i97_LdPXwrAC (The Google Books preview contains the entire introduction plus the foreword by Putnam, which should give a good taste of the content. Among economists, Itzhak Gilboa seems to have done the most work on expanding Goodman-style ideas to decision theory.)
A Fine Theorem 