How does language influence logic? In Gilboa’s canonical Theory of Decision Under Uncertainty, he describes the challenges to the theory of induction, which is a cornerstone of the “frequentist” approach to handling probabilities. The concept of induction is this: Suppose you want to determine the chance of an event occurring (say, the chance of a coin landing on heads when flipped). If you could flip the coin exactly the same way sufficiently enough times, and record the percentage of flips landing on heads, eventually (as the number of flips get really really high), that percentage would reflect the true probability of the coin landing on heads.
Induction hinges on the assumption that past occurrences can be used to inform us of the potential likelihood of future events. And, like it or not, we do exactly this all the time. Gilboa:
Every scientific field attempts to make generalizations and to predict the future based on the past. As such, it is engaging in inductive reasoning. Similarly, when I sit down on a chair, trusting that it will not break under the weight of my body, I use inductive reasoning, even if I tested the chair just a minute before sitting down.
Sounds reasonable enough, no? Well, one could very easily be concerned that we may not have any reason to expect events in the future to behave with the same consistency that they did in the past – the famous Hume’s Critique.
But another complication arises when the hypothesis of our observations depends heavily on the language we choose to describe it. Enter Goodman’s Paradox:
Assume that a scientist wishes to test the theory that emeralds are green, contrasted with the theory that they are blue. Testing one emerald after the other, she concludes that emeralds are indeed green.
Next assume that another scientist comes along and wants to test whether emeralds are “grue” as opposed to “bleen”. “Grue” emeralds are emeralds that appear to our eyes green if tested until time T, but appear blue if they are tested after time T. “Bleen” emeralds are defined symmetrically. Choose a time T in the future and observe that the scientist will find all emeralds to be grue. She will therefore conclude that after time T, all new emeralds to be tested, which will most likely be true as those tested up to T, will appear blue to our eyes.
Testing the hypotheses that emeralds are green versus blue seems perfectly reasonable. Testing for grueness versus bleenness appears weird at best. Yet, how can we explain our different reactions? The two procedures seem to be following the same logical structure. They both appear to be what the scientific method suggests. Why do we accept one and reject the other?
The two inductive conlcusions (one suggesting all emeralds are green and the second suggesting all emeralds are grue) are, by definition, contradictory. They cannot both be true. After time T, either they will continue to be green (and appear green to us), or they will continue to be grue (and appear blue to us). Yet, there is no clear way to choose one hypothesis as the “acceptable” approach. Gilboa:
A common reaction [to the paradox] is to say that if emeralds are tested and found green, there is no reason to suppose that, all of the sudden, new emeralds will start appearing blue. But one may claim, by the same token, that after all tested emeralds appeared grue, there is no reason for new emeralds to switch, at time T, and become bleen.
While “grue” and “bleen” seem odd, there is no inherent reason for them to be treated differently than “green” and “blue”. In other words, language seems to matter. Even the simple scientific approach of induction is influenced by our choice of language:
Goodman’s paradox has shown that not only does inductive reasoning lack logical foundations, it is not even clear what is meant by this process. Even if we are willing to accept the claim that “The thing that hath been, it is that which shall be,” it is not clear what future corresponds to a given past as its “natural” continuation.
From a practical perspective, perhaps it is much too convenient to use induction for sitting down on coffee shop chairs with a sufficient degree of confidence. But on a deeper level, the use of induction is a far subtler and messier scientific tool than at first glance.
[Epilogue: Since the emergence of Goodman’s Paradox, logicians and scientists have found ways to “solve” this concern, relying on simplicity as a metric for selecting the appropriate hypothesis (and therefore, preferring green/blue to grue/bleen). Gilboa’s discussion of this point is complete and worth reading.]