Models of scientific inquiry

Scientific inquiry has two functions, first, to provide a descriptive account of how scientific inquiry is carried out in practice, second, to provide an explanatory account of why scientific inquiry succeeds as well as it appears to do in arriving at genuine knowledge of its objects.

The search for scientific knowledge extends far back into antiquity. At some point in the past, at least by the time of Aristotle, philosophers recognized that a fundamental distinction should be drawn between two kinds of scientific knowledge — roughly, knowledge that and knowledge why. It is one thing to know that each planet periodically reverses the direction of its motion with respect to the background of fixed stars; it is quite a different matter to know why. Knowledge of the former type is descriptive; knowledge of the latter type is explanatory. It is explanatory knowledge that provides scientific understanding of the world. (Salmon, 1990)

Classical model
The classical model of scientific inquiry derives from Aristotle, who distinguished the forms of approximate and exact reasoning, set out the threefold scheme of abductive, deductive, and inductive inference, and also treated the compound forms such as reasoning by analogy.

Logical empiricism
Wesley Salmon (1990) began his historical survey of scientific explanation with what he called the received view, as it was received from Hempel and Oppenheim in the years beginning with their "Studies in the Logic of Explanation" (1948) and culminating in Hempel's "Aspects of Scientific Explanation" (1965). Salmon summed up his analysis of these developments by means of the following Table.

In this classification, a deductive-nomological (D-N) explanation of an occurrence is a valid deduction whose conclusion states that the outcome to be explained did in fact occur. The deductive argument is called an explanation, its premisses are called the explanans (L: explaining) and the conclusion is called the explanandum (L: to be explained). Depending on a number of additional qualifications, an explanation may be ranked on a scale from potential to true.

Not all explanations in science are of the D-N type, however. An inductive-statistical (I-S) explanation accounts for an occurrence by subsuming it under statistical laws, rather than categorical or universal laws, and the mode of subsumption is itself inductive instead of deductive. The D-N type can be seen as a limiting case of the more general I-S type, the measure of certainty involved being complete, or probability 1, in the former case, whereas it is less than complete, probability &lt; 1, in the latter case.

In this view, the D-N mode of reasoning, in addition to being used to explain particular occurrences, can also be used to explain general regularities, simply by deducing them from still more general laws.

Finally, the deductive-statistical (D-S) type of explanation, properly regarded as a subclass of the D-N type, explains statistical regularities by deduction from more comprehensive statistical laws. (Salmon 1990, pp. 8-9).

Such was the received view of scientific explanation from the point of view of logical empiricism, that Salmon says "held sway" during the third quarter of the last century (Salmon, p. 10).

Deduction and induction
Deductive logic and inductive logic are quite different in their approaches.

Deductive logic is the reasoning of proof, or logical implication. It is the logic used in mathematics and other axiom based systems. In a deductive system, there will be axioms (postulates) which are not proven. Indeed, they cannot be proven without circularity. There will also be primitive terms which are not defined. They cannot be defined without circularity. For example, one can define a line as a set of points, but to then define a point as the intersection of two lines would be circular. Because of these interesting characteristics of deductive systems, Bertrand Russell humorously referred to mathematics as “the field where we don’t know what we are talking about, nor whether or not what we say is true”. All theorems and corollaries are proven by exploring the implications of the axioms and other theorems that have previously been developed. New terms are defined using the primitive terms and other derived definitions based on those primitive terms.

In a deductive system, one can correctly use the term proof, as applying to a new theorem. To say that a theorem is proven means that it is impossible for the axioms to be true and the theorem to be false. (We are assuming the law of the excluded middle which is the basis of Western thought) For example, we could do a simple syllogism such as the following:

(1) Arches National Park lies entirely within the state of Utah.

(2) I am standing in Arches National Park.

(3) Therefore, I am standing in the state of Utah.

Notice that it is not possible (assuming all of the trivial qualifying criteria are supplied) to be in Arches and not be in Utah. However, one can be in Utah while not in Arches National Park. The implication only works in one direction. Statements (1) and (2) taken together imply statement (3). Statement (3) does not imply anything about statements (1) or (2). Notice that we have not proven statement (3), but we have shown that statements (1) and (2) together imply statement (3). In mathematics, what is proven is not the truth of a particular theorem, but that the axioms of the system imply the theorem. In other words, it is impossible for the axioms to be true and the theorem to be false. The strength of deductive systems is that they are sure of their results. The weakness is that they are abstract constructs which are, unfortunately, one step removed from the physical world. They are very useful however, as mathematics has provided great insights into natural science by providing useful models of natural phenomena. One result is the development of products and processes that benefit mankind.

The physical world requires the use of inductive logic. Inductive logic is the logic of theory building. It is useful in such widely divergent enterprises as science and crime scene detective work. One makes a set of observations, and seeks to explain what one sees. The observer may form a theory in an attempt to explain what he/she has observed. The theory will have implications, which will point to certain other observations that would naturally result from either a repeat of the experiment or making more observations from a slightly different set of circumstances. If the predicted observations hold true, one feels excitement that they may be on the right track. However, the theory has not been proven. The theory implies that certain observations should follow, but all of the sustaining observations in the world do not imply the theory. They only make it more believable. It is quite possible that some other theory could also account for the known observations, and may do better with future experiments. The implication flows in only one direction, as in the syllogism used in the discussion on deduction. Therefore, it is never correct to say that a scientific principle or theory has been proven. (At least not in the iron clad sense of proof used in deductive systems)

A classic example of this is the law of gravitation. Newton formed a law for gravitation stating that the force of gravitation is directly proportional to the product of the two masses, and inversely proportional to the square of the distance between them. For over 170 years, all observations seemed to validate his equation. However, telescopes eventually became powerful enough to see a slight discrepancy in the orbit of Mercury. Scientists tried everything imaginable to explain the discrepancy, but they could not do so using the objects that would bear on the orbit of Mercury. Eventually Einstein developed his theory of General Relativity and it explained the orbit of Mercury and all other known observations dealing with gravitation. During the long period of time when scientists were making observations that seemed to validate Newton’s theory, they did not in fact prove his theory to be true. However, it must have seemed at the time that they did. It only took one counter-example (Mercury’s orbit) to prove that there was something wrong with his theory. This is typical of inductive logic. All of the observations that seem to validate the theory, do not prove its truth. But one counter-example can prove it false. Einstein’s theory of General Relativity has been supported by many observations using the best scientific instruments and well constructed experiments. However, his theory now has the same status as Newton’s theory of gravitation prior to seeing the problems in the orbit of Mercury. It is highly credible and validated with all we know, but it is not proven. It is only the best we have at this point in time.

Arguments from analogy are one type of inductive reasoning. In arguing from analogy, one infers that since two things are alike in several respects, they are likely to be alike in another respect. This is, of course, an assumption. It is natural to attempt to find similarities between two phenomena and wonder what one can learn from those similarities. However, to notice that two things share attributes in several respects does not imply any similarities in other respects. It is possible that the observer has already noticed all of the attributes that are shared and any other attributes will be distinct. Argument from analogy is an unsure method of reasoning and can lead to erroneous conclusions. To use this type of reasoning to establish scientific facts, would be erroneous.