RECENT PROBLEMS OF INDUCTION[1]
• This lecture is based on an article entitled “Recent Problems of Induction[2]” by Carl Hempel.
• But one of the inherent problems with titling an article “Recent Problems… or “Modern Developments… etc. is that it quickly becomes dated.
• My point here is only that this was originally publishes in 1966. So…
• Nevertheless, the “problems” remains with us
Carl G. Hempel (1905-1997)
It is true that from truths we can conclude only truths; but there are certain falsehoods which are useful for finding the truth. -Leibniz, Letter to Canon Foucher (1692)
THE CLASSICAL PROBLEM OF INDUCTION
Referred to as the problem of induction.
As Hume puts is, it is problem of justification.
If successful, Hume’s “problem” undermines our ability to justify the principle of induction. Thus, we cannot “know” is nor anything based on it. This results in skepticism.[3]
More recently other problems of induction have been presented that are “no less perplexing and important than the classical one,” but which are logically prior to the classical problem presented by Hume..
“The classical problem cannot even be clearly stated-let alone solved-without some prior clarification of the new puzzles.”
Hempel points out that there are even more fundamental questions to be answered before one can resolve the problem of induction such as determining what, precisely, inductive reasoning is.
· More specifically, what are the rules or processes of inductive reasoning?
· How is inductive inference (induction) in science (or elsewhere) actually carried out exactly?
Hempel discusses some of these more recent problems of induction.
Demonstrative Inference: Effecting a transition from some set of propositions to further proposition which is in fact logically implied by the initial set. (Fancy term of “Deductive Reasoning)
Nondemonstrative Inference: Effecting a transition from some body of empirical information to a hypothesis which is not logically implied by it. (Fancy term for “Inductive Reasoning.”)
He refers to sentences specifying the evidence as the “premises” and to the hypothesis based on it as the “conclusion” of an "inductive inference." The simplest type of inductive inference: the evidence consists of a set of examined instances of a generalization, and the hypothesis is either:
1. the generalization itself or
a statement about some unexamined instances of it.
Some have characterized inductive generalization as going from particular instance to a universal claim. Others have suggested that inductive generalization as always involved in going from information gained from past and present experience to claims about the future.
Hempel points out that both of these characterizations are imprecise.
A standard example is the inference from the evidence statement “All ravens so far observed have been black.” We might conclude to the generalization that
1. All ravens are black.
Or to the prediction that
2. The birds now hatching in a given clutch of raven eggs will be black
Or to the retrodiction that
3. A raven whose skeleton was found at an archeological site was black.
As these examples show, induction does not always proceed from the particular to the general or from statements about the past or present to statements about the future. It can move from the particular to another particular (#2) or move from statements about the present to statements about the past (#3).
More complex and circumstantial kinds of nondemonstrative reasoning include making a
1. medical diagnosis on the basis of observed symptoms
2. in basing statements about remote historical events on presently available evidence
3. in establishing a theory on the basis of appropriate experimental data.
However, most of the problems the Hempel wished to consider in this article can be illustrated by inductions of the simple kind (i.e. from instances of a generalization)
THE NARROW
INDUCTIVIST VIEW OF SCIENTIFIC INQUIRY
Inductive inference is NOT an effective method of discovery, that is, it is not a mechanical procedure which leads from observational data to appropriate hypotheses or theories. This is a total misconception of how induction works in science in particular ad in life in general. Hempel terms this misconception as “The Narrow Inductivist View of Scientific Inquiry.” He outlines (below) what he believes to be the mistaken notion if “idealized” inductive inference to point out why it is a fantasy that could never really be realized, even by a superhuman mind.
“If we try to imagine how a mind of superhuman power and reach, but normal so far as the logical processes of its thought are concerned ... would use the scientific method, the process would be as follows: First, all facts would be observed and recorded, without selection or a priori guess as to their relative importance.
Second, the observed and recorded facts would be analyzed, compared, and classified, without hypothesis or postulates other than those necessarily involved in the logic of thought.
Third, from this analysis of the facts, generalization would be inductively drawn as to the relations, classificatory or causal, between them.
Fourth, further re~ search would be deductive as well as inductive, employing inferences from previously established generalizations.[4]
Hempel claims this account of induction in science is “highly fanciful”; many have argued against it as naive.
1. inquiry could never go beyond the first stage
“…for-presumably to safeguard scientific objectivity-no initial hypotheses about the mutual relevance and interconnections of facts are to be entertained in this stage, and as a result, there would be no criteria for the selection of the facts to be recorded. The initial stage would therefore degenerate into an indiscriminate and interminable gathering of data from an unlimited range of observable facts, and the inquiry would be totally without aim or direction.
In other word, if we do not start with a theory in mind, we wouldn’t know what fact to gather.
2. Could not go beyond the second stage either
“…if it could ever be reached-for the classification or comparison of data again requires criteria. These are normally suggested by hypotheses about the empirical connections between various features of the "facts" under study. But the conception just cited would prohibit the use of such hypotheses, and the second stage of inquiry as here envisaged would again lack aim and direction.
Ditto: Could this account be retained and suitably rectified?
Perhaps by adding
…the observation that any particular scientific investigation is aimed at solving a specified problem, and that the initial selection of data should therefore be limited to facts that are relevant to that problem.”
But the mere statement of a problem does not generally determine what kinds of data are relevant to its solution.
Consider the problem of determining the cause of lung cancer: what sorts of data would be relevant?
· differences in age
· occupation
· sex
· dietary habits
· dating habits
· taste in music
“The notion of "relevant" facts acquires a clear meaning only when some specific answer to the problem has been suggested, however tentatively, in the form of a hypothesis: an observed fact will then be favorably or unfavorably relevant to the hypothesis according as its occurrence is by implication affirmed or denied by the hypothesis.
For instance, we might hypothesize that smoking is a significant causative factor. This then would suggest what data are relevant. (e.g. A higher incidence of the disease among smokers than among nonsmokers would be a relevant fact.)
Data showing for a suitable group of subjects that this is the case or that it is not would therefore constitute favorably relevant (confirming) or unfavorably relevant (disconfirming) evidence for the hypothesis.
Conversely, data with respect to cancer rates and hair color, let’s say, would NOT be relevant (given the working hypothesis to be tested here) and so these need not be collected nor examined.
But were the hypothesis that hair color was a causative factor with respect to cancer, these data should be collected and would be potentially favorably relevant or unfavorably relevant.
Generally what data set is relevant can only be specified relative to the hypothesis to be confirmed or disconfirmed. So, you MUST begin with an hypothesis before you can begin your inductive investigation and perform your inductive inference.
3. Thus, contrary to the third stage of inquiry above, hypotheses cannot be inferred from empirical evidence by means of some set of mechanically applicable rules of induction.
There is no generally applicable mechanical routine of "inductive inference" which leads from a given set of data to a corresponding hypothesis or theory somewhat in the way in which the familiar routine of multiplication leads from any two given integers, by a finite number of mechanically performable steps, to the corresponding product.
Consider, for example the theory of gravitation.
Offered as a novel account for certain previously established empirical facts (e.g. as regularities of planetary motion)
The hypothesis was a novel interpretation of the facts…
“in the sense that they had played no role in the description of the empirical facts which the theory is designed to explain. And surely, no set of induction rules could be devised which would be generally applicable to just any set of empirical data (physical, chemical, biological, etc.) and which, in a sequence of mechanically performable steps, would generate appropriate novel concepts, functioning in an explanatory theory, on the basis of a description of the data.
Scientific hypotheses and theories, then, are not mechanically inferred from observed "facts"
Rather they are invented by an exercise of creative imagination.
Einstein, among others, often emphasized this point, and more than a century ago William Whewell presented the same basic view of induction.
“Whewell speaks of scientific discovery as a "process of invention, trial, and acceptance or rejection" of hypotheses and refers to great scientific advances as achieved by "Happy Guesses," by "felicitous and inexplicable strokes of inventive talent," and he adds: "No rules can ensure to us similar success in new cases; or can enable men who do not possess similar endowments, to make like advances in knowledge.'[5]
Karl Popper (1902 – 1994) claims they are the products of “conjectures” which must then be subjected to falsification.[6]
The chemist Friedrich August Kekulé’s most famous work was on the structure of benzene. In 1865 Kekulé published a paper in French (for he was then still in Francophone Belgium) suggesting that the structure contained a six-membered ring of carbon atoms with alternating single and double bonds. However, he claimed that his ring formula for the benzene molecule occurred to him as he dozed before his fireplace.
“Gazing into the flames, he seemed to see snakes dancing about; and suddenly one of them moved into the foreground and formed a ring by seizing hold of its own tail. Kekule does not tell us whether the snake was forming a hexagonal ring, but that was the structure he promptly ascribed to the benzene molecule.
No restrictions are imposed upon the invention of theories.
Scientific objectivity is achieved subsequently by careful testing (observational or experimental investigation)
If careful testing bears out the predicted consequences, the hypothesis is accordingly supported.
But here again occurred the problem of inductive inference:
But normally a scientific hypothesis asserts more than (i.e., cannot be inferred from) some finite set of consequences that may have been put to test, so that even strong evidential support affords no conclusive proof. It is precisely this fact, of course, that makes inductive "inference" nondemonstrative and gives rise to the classical problem of induction.
Karl Popper claims:
Inferences involved in scientific testing are deductive from the theory to implications about empirical facts, never in the opposite direction;
"Induction” in science, Popper claims is a myth.
It is neither a psychological fact, nor a fact of ordinary life, nor one of scientific procedure";[7] and it is essentially this observation which, he holds, "solves ... Hume's problem of induction."[8]
It is worth rehearsing here that Popper insists that we cannot confirm a theory inductively.
To reason:
• If H is true, then result R will occur.
• R occurs
Therefore
• H is true.
is fallacious. (i.e. the fallacy of affirming the consequent)
The fallacy of affirming the consequent
• P -> Q
• Q
Nothing follows from the conjunction of these two claims.
Thus we cannot confirm a theory inductively.
According to Popper, science must NOT try to verify theories, but rather science must seek to falsify theories.
• If H is true, then result R will occur.
• ~R occurs (R does NOT occur.)
Therefore
• ~H (H is not true.)
Theories which resist falsification gain our (tentative) acceptance. This is also why Popper claims sciences advances by deduction, not induction.
But Hempel point out..
While empirical science is not inductive in the narrow sense, it still may be said to be inductive in a wider sense. While scientific hypotheses and theories are not inferred from empirical data, they are, nevertheless, accepted on the basis of observational or experimental findings, findings which afford no deductively conclusive evidence for their truth.
In this sense, science does proceed by inductive (non-demonstrative) inference.
“Thus, the classical problem of induction retains its import: What justification is there for accepting hypotheses on the basis of incomplete evidence?
However, there arises the logically prior problem to the classical problem of induction, that of giving a more explicit characterization and precise criteria of what counts as acceptable inductive reasoning in science.
(Briefly: an analogous problem for deductive reasoning.)
DEDUCTION AND
INDUCTION; DISCOVERY AND VALIDATION
Deductive soundness:
Imprecisely characterized as “an argument is deductively valid if its 'premises and its conclusion are so related that if all the premises are true, then the conclusion cannot fail to be true as well.[9]
As for criteria of deductive validity, the theory of deductive logic specifies a variety of forms of inference which are deductively valid, such as, for example, modus ponens:
p -> q
p
q
This represents a sufficient, but not necessary condition of deductive validity. And this logical relation is expressible by reference to the syntactical structure (no reference to the meanings)
But of course, inductive “soundness” cannot be stated in purely syntactical terms, since soundness requires not only validity, but also the truth of the premise set. And truth is not a syntactical property of a premise or the premise set.
Therefore, the rules of induction cannot be expected to specify mechanical routines leading from empirical evidence to appropriate hypotheses.
But this may not be as big a difference between science on the one hand and math and logic on the other.
“A moment's reflection shows that no interesting theorem in these fields is discovered by a mechanical application of the rules of deductive inference. Unless a putative theorem has first been put forward, such application would lack direction. Discovery in logic and mathematics, no less than in empirical science, calls for imagination and invention, it does not follow any mechanical rules.
And even when a putative theorem has been proposed there exists no mechanical routine for proving or disproving it.
EX: Famous arithmetical conjectures of Goldbach and of Fermat
These were proposed centuries ago but have remained undecided to this day. The construction of a proof or a disproof even for a given logical or mathematical conjecture requires creative insight and ingenuity.
Nevertheless, once the a proof is proposed, then the rules of deductive logic afford a means of establishing the validity of the argument. If each step conforms to one of those rules - a matter which can be decided by mechanical check - then the argument is a valid proof of the proposed theorem.
So the formal rules of deductive inference are not rules of discovery, no more in deduction than in induction, but rather merely provide the criteria of validity.
So then the most we can hope for in inductive inference would be analogously, rule to provide the criteria of validation. These would serve to appraise the adequacy of the hypothesis on the basis of the evidence.
Inductive arguments might be thought of as taking one of these forms:
e –
= (i.e., evidence e supports hypothesis h)
h
e -
= [r] (I.e., evidence e supports hypothesis h to degree r)
h
Here, the double line is to indicate that the relation of e to h is not that of full deductive implication but that of partial inductive support.
The first schema treats inductive support or confirmation as a qualitative concept.
Thus rules would need specify conditions under which a given evidence sentence supports, or confirms, a given hypothesis.[10]
Hempel says in a footnote here:
“It seems to me, therefore, that Popper begs the question when he declares: "But it is obvious that this rule or craft of 'valid induction' ... simply does not exist. No rule can ever guarantee that a generalization inferred from true observations, however often repeated, is true" ("Philosophy of Science," p. 181). That inductive reasoning is not deductively valid is granted at the outset. The problem is that of constructing a concept of inductive validity. (My emphasis)
The second construes inductive support as a quantitative concept.
Thus rules would need to provide criteria for determining the degree of support
These criteria might even amount to a general definition assigning a definite value of r to any given e and h
This is one objective of Carnap's inductive logic. (Logical Foundations of Probability)
The formulation of rules of these or similar kinds will be required to explicate the concept of inductive inference in terms of which the classical problem of justification is formulated. And it is in this context of explication that the newer problems of induction arise. We now turn to one of those problems; it concerns the qualitative concept of confirmation.
THE PARADOXES OF
QUALITATIVE CONFIRMATION
The reason Popper was so adamant that induction plays not role in science is because, at least initially, to assert otherwise seems to commit the fallacy of affirming the consequent. If my hypothesis is that “All ravens are black.” then this hypothesis would, among other things, predict that the next raven I see will be black. Then I have an experience: The next raven I see is black. “OH,” I say, “this experience confirms my hypothesis.”
But this is fallacious. I am arguing:
If H then B
B
Therefore
H
But NOTHING follows from the conditional and the truth of
the consequent and to assert otherwise, according to Popper, is to commit the
fallacy of affirming the consequent. This is why Popper maintains that science is NOT in the
business of verifying theories, nor can it be.
It cannot verify theories; it can only falsify theories,
or attempt to. (The Verificationists were totally wrongheaded.) Falsification,
according to Popper is how science advances.
But much as Popper would protest, it does seem that the more a theory resists falsification, the more it gains in credence. So the question then becomes, what are the rules of induction inferences concerning generalizations.
Well, recall how they are structured:
All cases of F that have been found are also case of G.
Therefore:
"All F are G."
For example, the hypothesis "All ravens are black,"
“h” (our hypothesis)is = (x) (Rx -> Bx)
(For all x, If X is a raven, then X is also black.)
Is supported or confirmed by the individual instance “i” of a black raven.
Confirming Instance:
Ri & Bi (This this instance here is a raven, and this instance is also black.)
One this view, any evidence sentence of the form "Ri & Bi” lends inductive support our hypothesis.
Let us refer to such instances as positive instances of type I for h.
Conversely, h is disconfirmed by any evidence sentence of the form “Ri & ~ Bi.” (This this instance here is a raven, and this instance is NOT black.)
Advocated view of confirmation was suggested by French Logician and Philosopher Jean Nicod.
Nicod's Criterion: A test of the relevance of evidence for confirmation put forward by the French philosopher Jean Nicod, saying that for a generalization “All As are Bs,” an instance that an A is a B provides confirming evidence; an instance of an A that is not B disconfirms the generalization and justifies its rejection, and evidence of something which is neither A nor B is irrelevant, that is, it neither confirms nor disconfirms.
But Hempel points out that:
“h” is logically equivalent to the statement that “All nonblack things are non-ravens.”
Thus our initial hypothesis (h) is logically equivalent to a corollary one (h').
(x) (Rx -> Bx) = (x) (~Bx -> ~Rx)
h = h'
Thus we might render h' as (x) (~Bx -> ~Rx) (For all x, if x is not black, then x is not a raven.)
According to Nicod's criterion, this generalization is confirmed by its instances (i.e., by any individual instance, i, of (~Bx -> ~Rx) in other word any:
~Bi & ~Ri
But since h' expresses exactly the same assertion as h, any such individual will also confirm h. But this means that anything that is not black (~B) and also not a raven (~R) confirms the hypothesis that all ravens are black. Consequently, such things as a yellow rose, a green caterpillar, or a red herring confirm the generalization "All ravens are black," by virtue of being nonblack non-ravens. I will call such objects positive instances of type II for h.
Next, the hypothesis h is logically equivalent also to the, following statement:
(x) (Rx v ~Rx) -> (~Rx v Bx) (For all x, if (either x is a raven or x is not a raven), then (either x is not a raven or x is not black).
Thus h = hii
In words: Anything that is something is either a raven or it is not a raven- (i.e., anything at all), then it is either is not a raven or it is black. Confirmatory instances for this version which I will call positive instances of type III for h, consist of individuals k such that
(~Rk v Bk) -> (~Rk v Bk)
But this condition is met
1. by any object k that is not a raven (no matter whether it is black or not) and
2. by any object k that is black (no matter whether it is a raven).
Thus any instance of an object that is not a raven confirms the hypothesis that all ravens are black. And any instance of a black thing nay confirms the hypothesis that all ravens are black.
On this view of inductive confirmation, any such object affords a confirmatory instance in support of the hypothesis that all ravens are black. Thus, the hypothesis is confirmed by an oak tree (not a raven) and a black frying pan (something black). Any object at all confirms our generalizations that “All ravens are black.” so long as it is not a non-black raven.
So, it would seem that given Nicod's initially plausible criterion, the generalization “All ravens are black.” is given credence by instances of black ravens (though never fully verified), it is also given credence by instances of yellow roses, green caterpillars, red herrings, or the planet Venus. Clearly something troubling is going on.
hiii
Finally, the hypothesis h can be equivalently expressed by the sentence:
(x) [(Rx & ~BX) -> (Rx & ~Rx)}
In words: If (anything that is a raven and is not black), then (it is a raven and is not a raven).
For this formulation, nothing can possibly be a confirmatory instance in the sense of Nicod's criterion, since nothing can be both a raven and not a raven. (The consequent of the conditional can never be satisfied.)
Thus the paradox: we can and do have confirming instances of h, AND we do not and cannot have confirming instances of h’s logical equivalent.
Paradoxes of Confirmation
These peculiarities, and some related ones, of the notion of confirmatory instance of a generalization have come to be referred to as the paradoxes of confirmation. And indeed, at first glance they appear to be implausible and perhaps even logically unsound. But on further reflection one has to conclude, I think, that they are perfectly sound, that it is our intuition in the matter which leads us astray, so that the startling results are paradoxical only in a psychological, but not in a logical sense.
Hempel goes on the argue that thing my not be as bizarre as they appear however.
[1] From Robert G. Colodny (ed.), Mind and Cosmos, Essays in Contemporary Science and Philosophy (University of Pittsburgh Press, 1966), pp. 112-134. Reprinted by permission of the University of Pittsburgh Press
[3] Note: Hume is NOT suggesting the principle is false. It may indeed be true. It simply cannot be rationally justified according to Hume.
[4] A. B. Wolfe. "Functional Economics2 " The ·Trend 0/ Economics, ed. R. G. Tugwell (New York: Knopf, 1924), p. 450 (author's italics).
[5] William Whewell, The Philosophy of the Inductive Sciences, 2d ed. (London: John W. Parker, 1847), II, 41 (author's italics).
[6] See, for example, Popper's essay, “Science: Conjectures and Refutations," in his book, Conjectures and Refutations (New York and London: Basic Books, 1962).
[7] Karl Popper, "Philosophy of Science: A Personal Report," British Philosophy in the Mid-Century, ed. C. A. Mace (London: Allen and Unwin, 1957), pp. 155- 91, quotation from p. 181.
[8] Popper, "Philosophy of Science," p. 183
[9] Precise general characterizations of deductive validity, for arguments in languages of certain specified forms will be found, e.g., in W. V. O. Quine, Methods oj Logic, rev. ed. (New York: Holt, Rinehart & Winston, 1959).
[10] It seems to me, therefore, that Popper begs the question when he declares: "But it is obvious that this rule or craft of 'valid induction... simply does not exist. No rule can ever guarantee that a generalization inferred from true observations, however often repeated, is true" ("Philosophy of Science," p. 181). That inductive reasoning is not deductively valid is granted at the outset the problem is that of constructing a concept of inductive validity.