Some remarks and references on selected topics in the philosophy of science
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Context of discovery vs. context of justification
The explicit distinction between the context of discovery vs. the context of justification (German: Entdeckung vs. Rechtfertigung) was introduced by Reichenbach in 1938. In some sense the distinction defines the "division of labor" between the "history of science" and the "philosophy of science". While the former should be concerned with the task to investigate the historical events which led to, say, the discovery of a law, the later should analyze its justification, especially by means of a logical analysis.
This distinction is clearly rooted in the logic-empirical view on science (and philosophy!) which regarded that the primary goal of philosophy is the logical analysis of the results produced by the material sciences. So it comes as no surprise that the validity of this distinction has been challenged in the meantime.
In "Context of discovery versus context of justification and Thomas Kuhn" Paul Hoyningen-Huene (2006) introduces various formulations of the DJ distinction which are usually conflated. He argues that there is a core of the concept which has survived any attack on this distinction, especially the famous challenge by Thomas Kuhn.
The problem of induction
The (rather trivial) observation that inductive reasoning (i.e. the move from the special to the general) can not claim logical necessity is usually traced back to Hume and called the "problem of induction". Indeed, we routinely rely on beliefs and expectations which are based on logically incomplete evidence. More specifically the "problem of induction" is often conceived as the question how this may be justified.
Rules of inductive acceptance
Hempel (1981) investigates some more recent turns in the evolution of this problem. First he notes, that inductive acceptance can not be based on a rule such as "All examined instances of A have been B supports the claim that all A are B". The adoption of this formulation would lead to situation in which the same body of evidence supports incompatible hypotheses. The example is familiar: through a given set of data points different curves can be drawn. These functions may represent the different hypotheses regarding e.g. the results of future measurements. One can modify the rule of inductive acceptance to avoid these situations. One only has to assume that inductive reasoning does not yield "inductive conclusions" (which in turn might be incompatible although based on the same evidence) but only assigns a certain "rational credibility" to a given hypothesis relative to the evidence available. Indeed, this "rational credibility" is something like the probability of the hypothesis to be true. This is the basic idea behind Carnap's inductive logic and lives on in the Bayesian approach to statistics and data analysis.
Viewed this way, the problem splits into two issues: (i) finding rules for the probability assignment and (ii) finding rules of acceptance based on this probability. The first question gave rise to many technical investigations in the field of probability. E.g. Carnap's "Logical Foundation of Probability" aims at a formal theory of how to assign probabilities to hypotheses given certain evidence. While many detailed aspects of such a program remain disputed and while presumably such a complex program may never be accomplished to everyone's satisfaction, we may set aside this question and assume for the moment that problem (i) can be solved. Interestingly, the much more innocent looking problem (ii) poses a serious problem. Naively one may accept a hypothesis if the assigned probability exceeds an arbitrary threshold (say, 0.5 or 0.99). However, the so-called lottery-paradox shows that certain types of evidence would force one to adopt logically incompatible hypotheses. But Hempel points to a strategy to overcome this problem.
Induction and value judgment
The key idea comes from decision theory and utilizes the concept of the "value" attached to avoiding the mistake of (i) rejecting a hypothesis although it is true or (ii) accepting the hypothesis while it is false. That is, not only the available data but also the impact of the decision has to be taken into account. The standard example is the following: in testing a medicament one reasonably requires more evidential support for the hypothesis that it is safe and effective than when dealing with the question if a "large quantity of machine-stamped belt buckles are non-defective" (Hempel p.394). Hempel discusses some objections against this proposal by Rudner (1954), especially the question whether a scientist is actually concerned with a value judgment of a moral kind sketched above. Hempel suggests that the judgments of a scientist are of a different kind, namely dealing with epistemic values.
Induction and Kuhn's scientific revolutions
The idea of an (epistemic-)value judgment in the course of scientific work allows Hempel to make contact with Kuhn's ideas on theory change. First, theory change can be viewed as an instance of inductive reasoning. Given the experimental evidence one needs to decide if (and which) rival theory is to be preferred. Second, according to Kuhn, this process is guided (or at least strongly influenced) by "cognitive values", i.e. that a theory should be simple, explanatory, consistent with the empirical data, of large scope etc.pp. Such values may change over time and in addition individual scientist may value them more or less. But at a given time they represent the prevailing value system of the specific scientific community (see Hoyningen-Huene 1992 for some elaboration). It is presumably impossible to formulate these values explicitly, i.e. according to Kuhn theory change can not be characterized by the application of strict rules. However, given that the acceptance of a new theory (i.e. the end of a "scientific revolution") is guided by this commitment to the "improvement of scientific knowledge", there is rationality in the process of theory change nevertheless (this remark is important since Kuhn is constantly accused of irrationality). Hempel does not side with Kuhn's position entirely. For example he remarks that the alleged rationality is not of the kind he would require. However, Hempel admits Kuhn's value as an important corrective for the approach who pictures science as an entirely rule-driven business (e.g. the analytic-empirical school).
Theory underdetermination by evidence
That a given body of evidence does not determine a theory (which is supposed to "explain" this evidence) uniquely can be viewed as a special case of the "problem of induction".
The theory-ladenness of observation
We have seen that the "problem of induction" challenges the justification of our habit to adopt beliefs and expectations on logically incomplete evidence. Applied to the relation between observations and theories we are faced with the problem of "Theory underdetermination by evidence". Short but to the point: the development of general theories which can account for experimental observations is not driven by logical necessity. However, it gets even worse! Up to now we were at least assuming "observations" and "theories" to be logically independent and that the main problem was the only finite number of possible observations which could not support any universal quantification. However, it has been challenged that "observations" are independent of "theories" at all; rather observations should be viewed as "theory-laden". http://www.galilean-library.org/theory.html
Hoyningen-Huene (2005) compares Kuhn and Feyerabend with respect to their notion of "incommensurability".