The idea is to say when we "test" scientific theories, and then form opinions regarding those theories based on such tests. What does scientific investigation consist in, and what are the rules governing it?
Induction - Scientific investigation is a type of inductive process, where we increase the evidential basis for or against a particular theory without the evidence conclusively establishing the theory. "Induction" has had various meanings in the past: in the Renaissance, it was thought that the way to develop scientific theories was to examine all the evidence you could and then extrapolate from that to form theories. (This was a method of developing theories as well as a method of justifying them.) This was in contrast to positing "hypotheses" about unobservable entities to explain the phenomena. (Indeed, Newton was criticized for formulating such "hypotheses.") This view did not survive, however, since it became apparent that you can't form theories in this way. Thus, we have to understand "induction" differently (supposing that it is a useful concept at all).
Carnap is an inductivist, and in this respect he differs from Popper. However, both agree (taking inspiration from Hume) that there is a serious problem with the justification of "inductive inference." Carnap discusses it in terms of a puzzle about how we arrive at and form opinions regarding laws. (Note that notion of a law that Carnap assumes is similar to Hempel's.) Laws are universal statements (at least), hence apply to an at least potentially infinite domain. However, our empirical data is always finite. (Consider ideal gas law.) What does deductive logic give us to evaluate theories?
Suppose that h -> e1, e2, e3,.... If we show the truth any finite number of the ei, we haven't established that h; however, if we show that one of these is false, h must also be false.
Thus deductive logic cannot give us the tools to establish a scientific theory; we cannot "infer" from evidence to theory. There are different ways in which this can be so. Carnap distinguishes between empirical and theoretical laws. The latter refer to unobservable entities or properties; theoretical laws are absolutely necessary in science, but they cannot simply be derived from large bodies of research. Science postulates new categories of things not just generalizations about regularities of things we can observe. (Consider the kinetic theory of gases.) The upshot is that theories can always be wrong, no matter how much evidence we've found that is consistent with those theories. Scientific theories are not "proven" in the sense that given that a certain body of empirical data they are immune from all refutation. Q: How can we form an opinion regarding theories? Moreover, what kind of opinion should we form?
Both of these authors assume that the answer to this question will be in the form of a "logic of scientific discovery." (Logic = formal system of implication, concerned with relations of implication between statements, but not relations of statements to the world, i.e., whether they are true or false). Indeed, this is the title of Popper's famous book. The point on which they differ is the following: Is deductive logic all that we're limited to in science? (Popper - yes, no inductive logic; Carnap - No, there's an "inductive logic" too, which is relevant to scientific investigation.)
Popper and Carnap also agree on is that there is a distinction between the contexts of justification and discovery. This is a contentious view, as we'll see when we start looking at Kuhn and Laudan. The traditional approach to induction assumed that gathering evidence would enable one to formulate and justify a theory. (Consider Popper's example of a person who gave all his "observations" to the Royal Society. Also, consider the injunction to "Observe!") Both Carnap and Popper distinguish the two: what they were concerned with is not how to come up with scientific hypotheses--this is a creative process (e.g., the formulation of a theoretical framework or language), not bound by rules of logic but only (perhaps) laws of psychology--but how to justify our hypotheses once we come up with them. Carnap's answer is that we can't "prove" our hypotheses but we can increase (or decrease) their probability by gathering evidence. (This is inductive since you proceed to partial belief in a theory by examining evidence.)
Let's make this a bit more precise: is induction a kind of argument? That is, is there such a thing as "inductive inference?" This is one view of inductive logic: a formal theory about relations of (partial) implication between statements. Can be thought of in two ways. (1) You accept a conclusion (all-or-nothing) based on evidence that confirms a theory to a certain degree (e.g., if sufficiently high). (2) You accept a conclusion to a certain degree (i.e., as more or less probable) based on certain evidence. The latter is what Carnap's approach involves. The basic notion there is degree of confirmation. What science does when it "justifies" a theory (or tests it) is to provide evidence that confirms or disconfirms it to a certain degree, i.e., makes it more or less probable than it was before the evidence was considered.
We've already made informal use of the notion of probability. However, the precise sense of the term turns out to be of importance when thinking about induction in Carnap's sense.
Carnap thought that the Logical notion was the one operative in scientific reasoning. Analogy with deductive logic: formal, relation between statements because of their formal properties (i.e., irrespective of what facts obtain). Disanalogy: No acceptance or belief: pr(h|e) = x means that e partially implies h, to degree x. Only partial belief, guided by the logical probabilities. The latter is a matter of the logical relationship between the two statements; it should guide our opinion (degrees of belief), but it does not on Carnap's view reduce to it. Scientific reasoning is then the formulation of a broad framework (language) in which theories can be expressed; inherent in that framework will be relations of partial implication (conditional logical probabilities) between evidence and hypotheses, i.e., pr(h|e). Evidence confirms the theory if pr(h|e) > pr(h) (and if pr(h|e) is sufficiently high). Then making an observation involves determining whether e is true; we seek to develop and confirm theories in this way.
Popper rejected this framework altogether. Popper thought that Carnap's inductive logic--indeed, the very idea of inductive logic--was fundamentally mistaken. (How do we establish our starting point in Carnap's theory, i.e., the logical probabilities? It didn't work out very well.) There is no such thing: the only logic available to science is deductive logic. (Note that this doesn't mean that you can't use statistics. Indeed, you'll use it frequently, though the relationship between statistical laws and statistical data will be deductive.) What is available then? Well, we can't verify a theory (in the sense of justifying belief or partial belief in a hypothesis by verifying its predictions), but we can certainly falsify a theory using purely deductive logic. If h Þ e1, e2, e3, ..., then if even one of e1, e2, e3, ... turn out to be false, the theory as a whole is falsified, and must be rejected (unless it can be amended to account for the falsity of the falsifying evidence). Thus, Popper argues that science proceeds ("advances") by a process of conjecture and refutation. He summarizes several important features of this process as follows (page 141).
The process is then to start with a conjecture and try to falsify it; it that succeeds, move on to the next conjecture, and so on, until you find a conjecture that you do not falsify. Keep on trying, though. If you have trouble falsifying it, you say that it has been "corroborated." This does not mean that it has a high degree of probability, however. It still may be improbable, given the evidence at hand. (Indeed, it should be improbable if it says anything of interest.) We only tentatively accept scientific theories, while continuing to try to refute them. (Here "tentative acceptance" does not mean to believe that they are true, or even to be highly confident in their truth.)
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