## The Problem with Inductive Arguments

All induction problems may be phrased in a way that depicts a sequence of predictions.  Inductive problems will contain a previous indicator or explanans for the explanandum.  For instance, Carl Hempel’s example of Jones’ infection:

Where j is Jones, p is the probability, Sj is Jones’ infection, Pj is he being treated with penicillin, and Rj is his recovery.  If the probability of observing R at any time given the past observations of S&P1S&P2S&Pn (the probability of the set meeting R is m) where R was close to 1 then a predictive explanans (the S&Pn ) can be made for future instances of m using an inductive-statistical explanation.  For if the probability m(S&Pn | S&P1S&P2 …) is a computable function, the range of data is finite then a posterior predication M can be made from mM can be legitimately referred to as a universal predictor in cases of m. This is where Hempel rejects the requirement of maximal specificity (RMS), contra Rudolph Carnap, in which the RMS is a maxim for inductive logic that states that this is a necessary condition for rationality of any given knowledge situation K.  Let K represent the set of data known in m.  According to Hempel we cannot have all the material for K For any future time when the explanandum is in sync with the explanans of K, in this case, Rj, may be different when factoring different data at different times. It may be the case that future data that was impossible to consider may bring about ~Rj.  I believe Carnap’s RMS should be understood as a principle rather than an axiom for inductive logic.  It seems RMS is an attempt to make inductive arguments like deductive arguments.  So, instead of using M as a universal instantiation of future m, M may simply be a categorical similarity of m as a mere prediction and only a prediction because it is tentative to future variations of like-conditions in future explanandums.  I know Carnap would suggest that in his system of inductive logic there can be a degree of confirmation of statements which assign an inductive probability to a hypothesis about a particular event relative to the evidence statements about other particular events and that no universal generalizations are involved.

Carl G. Hempel, “Inductive-Statistical Explanation” in Philosophy of Science. Eds. Martin Curd and J.A. Cover (New York: Norton, 1998), 706-708. Marcus Hutter, “On the Existence and Convergence of Computable Universal Priors,” Technical Report 5 no. 3 (2003): 1-3. Wesley Salmon, “Hempel’s Conception of Inductive Inferences in Inductive-Statistical Explanation,” Philosophy of Science 44 no. 2 (1977): 183.