## 1. Biographical Sketch



Carl G(ustav) Hempel (1905-97), known as "Peter" to
his friends, was born near Berlin, Germany, on January 8, 1905. He
studied philosophy, physics and mathematics at the Universities of
Gottingen and Heidelberg before coming to the University of
Berlin in 1925, where he studied with Hans Reichenbach. Impressed by
the work of David Hilbert and Paul Bernays on the foundations of
mathematics and introduced to the studies of Rudolf Carnap by
Reichenbach, Hempel came to believe that the application of symbolic
logic held the key to resolving a broad range of problems in
philosophy, including that of separating genuine problems from merely
apparent ones. Hempel's commitment to rigorous explications of
the nature of cognitive significance, of scientific explanation, and
of scientific rationality would become the hallmark of his research,
which exerted great influence on professional philosophers, especially
during the middle decades of the 20th Century.



In 1929, at Reichenbach's suggestion, Hempel spent the fall
semester at the University of Vienna, where he studied with Carnap,
Moritz Schlick, and Frederick Waismann, who were advocates of logical
positivism and members of (what came to be known as) "the Vienna
Circle". It would fall to Hempel to become perhaps the most
astute critic of that movement and to contribute to its refinement as
logical empiricism. As Hitler increased his power in Germany, Hempel,
who was not Jewish but did not support the Nazi regime, moved to
Brussels and began collaborating with Paul Oppenheim, which would
result in several classic papers, including "Studies in the
Logic of Explanation", which appeared in 1948 (Rescher 2005:
Chs. 8 and 9). Hempel would also visit the United States
twice--the University of Chicago in 1937-38 and then the
City College of New York in 1939-40, where he held his first
academic position--and eventually became a naturalized
citizen.



He was productive throughout his career, publishing such important
papers as "The Function of General Laws in History" (1942)
and "Studies in the Logic of Confirmation",
"Geometry and Empirical Science", and "The Nature of
Mathematical Truth" (all in 1945), before leaving City College
for Yale. While there, Hempel would publish "Problems and
Changes in the Empiricist Criterion of Meaning" (1950) and
"The Concept of Cognitive Significance: A Reconsideration"
(1951), as well as his first book, a volume in the International
Encyclopedia of Unified Science, *Fundamentals of Concept Formation
in Empirical Science* (1952). Hempel moved to Princeton in 1955,
where his research program flourished and his influence upon
professional philosophers became immense.



During his two decades at Princeton, Hempel's approach dominated
the philosophy of science. His major articles during this interval
included "The Theoretician's Dilemma" (1958),
"Inductive Inconsistencies" (1960), "Rational
Action" (1961), and "Deductive-Nomological vs. Statistical
Explanation" and "Explanation in Science and in
History" and (both in 1962). A classic collection of his
studies, *Aspects of Scientific Explanation* (1965c), became a
scholar's bible for generations of graduate students. His
introductory text, *Philosophy of Natural Science* (1966a),
would be translated into ten languages. Other articles he published
thereafter included "Recent Problems of Induction" (1966b)
and "Maximal Specificity and Lawlikeness in Probabilistic
Explanation" (1968) as well as a series of studies that included
"On the 'Standard Conception' of Scientific
Theories" (1970).



At the University of Pittsburgh following his mandatory retirement
from Princeton in 1973, he continued to publish significant articles,
including studies of the nature of scientific rationality,
"Scientific Rationality: Analytic vs. Pragmatic
Perspectives" (1979), and "Turns in the Evolution of the
Problem of Induction" (1981), and on the structure of scientific
theories, including, most importantly, "Limits of a Deductive
Construal of the Function of Scientific Theories" (1988a) as
well as "Provisos: A Problem Concerning the Inferential Function
of Scientific Theories" (1988b), further enhancing his
reputation by his willingness to reconsider earlier positions. After
his death in 1997, new collections of his papers appeared (Jeffrey
2000; Fetzer 2001), which complemented studies of his research
(Rescher 1969; Esler et al. 1985; Kitcher & Salmon 1989; Fetzer
2000b).


## 2. The Critique of Logical Positivism



However surprising it may initially seem, contemporary developments in
the philosophy of science can only be properly appreciated in relation
to the historical background of logical positivism. Hempel himself
attained a certain degree of prominence as a critic of this movement.
*Language, Truth and Logic* (1936; 2nd edition,
1946), authored by A.J. Ayer, offers a lucid exposition of the
movement, which was--with certain variations--based upon the
analytic/synthetic distinction, the observational/theoretical
distinction, and the verifiability criterion of meaningfulness. A
fundamental desideratum motivating its members was to establish
standards for separating genuine questions for which answers might be
found from pseudo-questions for which no answers could be found.



According to the first principle, sentences are *analytic*
relative to a language framework \(\mathbf{L}\) when their truth
follows from its grammar and vocabulary alone. In English,
"Bachelors are unmarried" cannot be false, since
"bachelor \(=\_{\df}\) unmarried, adult male". Sentences of
this kind make no claims about the world, but instead reflect features
of the linguistic framework as syntactical or semantic truths in
\(\mathbf{L}\). And sentences are *synthetic* when they make
claims about the world. Their truth in \(\mathbf{L}\) does not follow
from its grammar and vocabulary alone but hinges upon properties of
the world and its history. According to logical positivism, all such
claims about the world have to be evaluated on the basis of
experience, which means the kind of knowledge they display is *a
posteriori*. But kinds of knowledge whose truth can be established
independently of experience are *a priori*.



Logical positivism affirmed that, given a language \(\mathbf{L}\), all
*a priori* knowledge is analytic and all synthetic knowledge is
*a posteriori*, thus denying the existence of knowledge that is
both synthetic and *a priori*. Indeed, the denial of the
existence of synthetic *a priori* knowledge is commonly assumed
to define the position known as "Empiricism", while the
affirmation of its existence defines "Rationalism". Figure
1 thus reflects the intersection of kinds of sentences and kinds of
knowledge on the Empiricist approach:





|  |  |  |
| --- | --- | --- |
|  | *A Priori* Knowledge | *A Posteriori* Knowledge |
| Synthetic Sentences | No | Yes |
| Analytic Sentences | Yes | ? |



Figure 1. The Empiricist Position




The category for sentences that are analytic and yet represent *a
posteriori* knowledge deserves discussion. The empirical study of
the use of language within language-using communities by field
linguists involves establishing the grammar and the vocabulary
employed within each such community. Their empirical research yields
theories of the languages, \(\mathbf{L}\), used in those communities
and affords a basis for distinguishing between which sentences are
analytic-in-\(\mathbf{L}\) and which are synthetic-in-\(\mathbf{L}\).
The kind of knowledge they acquire about specific sentences based on
empirical procedures thus assumes the form, "Sentence \(S\)
is analytic-in-\(\mathbf{L}\)", when that is
true of sentence \(S\), which is *a
posteriori*.



One of Hempel's early influential articles was a defense of
logicism, according to which mathematics--with the notable
exception of geometry, which he addressed separately--can be
reduced to logic (for Hempel, including set theory) as its foundation
(Hempel 1945c). Mathematics thus becomes an exemplar of analytic *a
priori* knowledge. Two subtheses should be distinguished: (i) that
all mathematical *concept*s can be defined by means of basic
logical concepts; and (ii) that all mathematical *theorems* can
be deduced from basic logical truths. In order to distinguish logicism
from formalism, however, the former maintains that there is *one
system of logic that is fundamental to all inquiries*, where all
mathematical terms are reducible to logical terms, and all
mathematical axioms are derivable from logical ones, which formalism
denies (Rech 2004).



The tenability of logicism has been disputed on multiple grounds, the
most prominent of which is that the notion of *membership*
fundamental to the theory of sets is not a logical notion but rather a
symbol that must be added to first-order logic to formalize what is
properly understood as a *non-logical theory*. Nor would
philosophers today accept the conception of the axioms of set theory
as logical axioms, since there exist alternatives. So even if
mathematics were reducible to set theory, these considerations
undermine Hempel's claim that mathematics is thereby reducible
to logic (cf. Benacerraf 1981 and Linsky & Zalta 2006, which
provides an extensive bibliography). Hempel's views about
geometry, in retrospect, thus appear to have been the better
founded.


### 2.1 The Analytic/Synthetic Distinction



The analytic/synthetic distinction and the observational/theoretical
distinction were tied together by *the verifiability criterion of
meaningfulness*, according to which, in relation to a given
language, \(\mathbf{L}\), a sentence \(S\) is
meaningful if and only if it is either analytic-in-\(\mathbf{L}\) or
synthetic-in-\(\mathbf{L}\) as an observation sentence or a sentence
whose truth follows from a finite set of observation sentences. By
this standard, sentences that are non-analytic but also
non-verifiable, including various theological or metaphysical
assertions concerning God or The Absolute, qualify as cognitively
meaningless. This was viewed as a desirable result. But, as Hempel
would demonstrate, its scope was far too sweeping, since it also
rendered meaningless the distinctively scientific assertions made by
laws and theories.



From an historical perspective, logical positivism represents a
linguistic version of the empiricist epistemology of David Hume
(1711-76). It refines his crucial distinctions of
"relations between ideas" and "matters of
fact" by redefining them relative to a language \(\mathbf{L}\)
as sentences that are analytic-in-\(\mathbf{L}\) and
synthetic-in-\(\mathbf{L}\), respectively. His condition that
significant ideas are those which can be traced back to impressions in
experience that gave rise to them now became the claim that synthetic
sentences have to be justified by derivability from finite classes of
observation sentences. Hume applied this criterion to exclude the idea
of *necessary connections*, which are not observable, from
significant causal claims, which were thereby reduced to relations of
regular association, spatial contiguity, and temporal succession. And
logical positivism followed Hume's lead.



Empiricism historically stands in opposition to Rationalism, which is
represented most prominently by Immanuel Kant, who argued that the
mind, in processing experiences, imposes certain properties on
whatever we experience, including what he called Forms of Intuition
and Categories of Understanding. The Forms of Intuition impose
Euclidean spatial relations and Newtonian temporal relations; the
Categories of Understanding require objects to be interpreted as
substances and causes as inherently deterministic. Several
developments in the history of science, such as the emergence of the
theory of relativity and of quantum mechanics, undermine Kant's
position by introducing the role of frames of reference and of
probabilistic causation. Newer versions are associated with Noam
Chomsky and with Jerry Fodor, who have championed the ideas of an
innate syntax and innate semantics, respectively (Chomsky 1957; Fodor
1975; Chomsky 1986)



Indeed, according to the computational theory of the mind, human
minds, like computing machines, are special kinds of formal systems.
Since deviations from formal systems of language in practice can be
considerable, Chomsky introduced a distinction between
*competence* and *performance*, where the former models
the formal system and various explanations are advanced for deviations
from that model in practice, similar to differences between the fall
of bodies in a vacuum and in air, which raises questions about
testability that parallel those for scientific theories, in general.
If languages are not best understood as formal systems, however, or if
syntax and semantics are not innate, then Chomsky and Fodor's
views are as vulnerable as those of Kant. If syntax is an emergent
property of semantic complexity, for example, then grammar is not
innate; and if mentality has and continues to evolve, Chomsky and
Fodor are wrong (Schoenemann 1999; Fetzer 2005).



In his study of formal systems for geometry (Hempel 1945b), Hempel
discusses the existence of alternatives based upon different axioms,
which differentiate Euclidean geometry from its non-Euclidean rivals.
According to Euclid, for example, the sum of the interior angles of a
triangle must equal 180deg and, in relation to a point separate from
a given line, one and only one parallel line passes through it. The
alternatives advanced by Lobachevsky (hyperbolic) and by Riemann
(elliptical), however, which represent the surface of a sphere and of
a saddle, respectively, violate both of those conditions, albeit in
different ways. Hempel emphasized that all three, as formal systems,
are on a par, where the most appropriate choice to describe the
geometry of space depends on the outcome of empirical studies. As it
happened, Einstein would adopt a generalized form of Riemannian
geometry in his general theory of relativity.



Hempel accordingly drew a distinction of fundamental importance
between *pure* and *applied* mathematics, which he
emphasized by using a quotation from Einstein, who had observed,
"As far as the laws of mathematics refer to reality, they are
not certain; and as far as they are certain, they do not refer to
reality" (1921). The existence of alternative and incompatible
formal systems, moreover, appears to affect Hempel's defense of
logicism from another direction. If mathematics is supposed to be
reducible to logic and logic is supposed to be consistent, then how
can alternative geometries be consistently reducible to logic? No one
would dispute that they exist as distinct formal systems with their
own axioms and primitives, but if these geometries are jointly
reducible to logic only if logic is inconsistent, their existence
suggests that, perhaps, as formalism claims, it is not the case there
is one system of logic that is fundamental to all inquiries.


#### 2.1.1 Quine's Complaints



The analytic/synthetic distinction took a decided hit when the noted
logician, Willard van Orman Quine, published "Two Dogmas of
Empiricism" (1953), challenging its adequacy. Quine argued that
the notion of analyticity presupposes the notion of synonymy-in-use,
which in turn presupposes understanding
inter-substitutability-while-preserving-truth. He claimed none of the
notions can be understood without the other, creating a circular
relation between them. Thus, matching up a *definiens* with a
*definiendum* could only be done if we already understood that
the definiens specifies the meaning of the word that is being
defined--a variation of "the paradox of analysis",
according to which we either already know the meaning of words (in
which case analysis is unnecessary) or we do not (in which case it is
impossible). The idea of analyticity appeared to have been
deposed.



The paper created a sensation and has been the most influential
philosophical article of the past 100 years. But Quine explicitly
allowed for the existence of the class of logical truths (such as
"No unmarried man is married") as *narrowly*
analytic sentences and their relationship to *broadly* analytic
sentences (such as "No bachelor is married"), when the
definitional relationship between them has been suitably stipulated
(as in "bachelor \(=\_{\df}\) unmarried, adult male" in a
language framework \(\mathbf{L})\). In cases of this kind, he
conceded, inter-substitutability, synonymy, and analyticity are
related in an unproblematic way. It would have been odd for a logician
to deny the existence of logical truths or the role of stipulations,
which are basic to the construction of formal systems, which suggests
that he may not have actually defeated the idea of analyticity, after
all (Fetzer 1993).



Indeed, Carnap (1939) had explained that the process of constructing a
vocabulary and a grammar for a language-in-use involves several
stages, including observation of the use of language by members of the
community, formulating hypotheses regarding the meaning of its phrases
and expressions, and drawing inferences about the underlying grammar.
These are pragmatic, semantic, and syntactical procedures,
respectively, and decisions have to be made in arriving at a theory
about the language as the outcome of empirical research. The
construction of formal systems thus provides an illustration of the
elements of artificial languages, where accounts of natural language
counterparts can be subject to further testing and refinement. The
linguistic practices of a specific community can thus be modeled and
thereby overcome Quine's professed objections.


#### 2.1.2 Hempel's Concerns



Moreover, in *Fundamentals of Concept Formation in Empirical
Science* (1952), Hempel had endorsed *explication* as a
method of definition analogous to theory construction by taking words
and phrases that are somewhat vague and ambiguous and subjecting them
to a process of clarification and disambiguation. Adequate
explications are required to satisfy criteria of syntactical
determinacy, semantic relevance, and pragmatic benefit (by clarifying
and illuminating the meaning of those words and phrases in specific
contexts). The word "probability" is vague and ambiguous,
for example, but various contexts of its usage can be distinguished
and theories of evidential support, relative frequency, and causal
propensity can be advanced. Hempel, following Carnap (1950), had
accordingly advanced a methodology that dealt with the paradox of
analysis before "Two Dogmas".



About the same time, however, Hempel found reasons of his own for
doubting that the analytic/synthetic distinction was tenable, which
surfaced in his study of dispositional predicates, such as
"malleable", "soluble" and
"magnetic". They designate, not directly observable
properties, but (in the case of "magnetic") tendencies on
the part of some kinds of things to display specific reactions (such
as attracting small iron objects) under suitable conditions (such as
the presence of small iron objects in the vicinity). It is very
tempting to define them using the *material conditional*,
"\_\_\_ \(\supset \ldots\)" by definitions like,



(D1)
\(x\) is magnetic at \(t =\_{\df}\) if, at \(t\), a small iron
object is close to \(x\), then it moves toward \(x\)


which could then be formalized by means of the horseshoe and suitable
abbreviations,

 

\[\tag{D2} Mxt =\_{\df} Sxt \supset Txt\* \]


where "\(t^\*\)" is equal to or later than "\(t\)"
and reflects that effects brought about by
causes entail changes across time. Since the material "if \_\_\_
then ..." is equivalent to "either not-\_\_\_ or
...", however, the meaning of (D2) turns out to be
logically equivalent to,

 

\[\tag{D3} Mxt =\_{\df} \neg Sxt \vee Txt\* \]


which means that anything not subject to the test, such as a brown
cow, is magnetic. Hempel acknowledged that the use of the
*subjunctive conditional*, say, "\_\_\_ \(\rightarrow
\ldots\)", formalizing *what would be the case ... if
something \_\_\_ were the case*, in this case,

 

\[\tag{D4} Mxt =\_{\df} Sxt \rightarrow Txt\* \]


"if, at \(t\), as small iron object were close to
\(x\), then it would move toward \(x\)
at \(t^\*\)" (which assumes the satisfaction of the test
condition) would avoid the problem, because these conditionals take
for granted their antecedents are satisfied (that
"*Sxt*" is true). But while acknowledging the
importance of subjunctive conditionals for an understanding of both
counterfactual conditionals and lawlike sentences, Hempel regarded
their explication as not fully satisfactory and their use as "a
program, rather than a solution" (Hempel 1952: 25).



He therefore adopted a method from Carnap to overcome this difficulty,
where, instead of attributing the property in question to anything not
subject to the test, the predicate is partially defined by means of a
*reduction sentence*, such as "if, at \(t\),
a small iron object is close to \(x\),
then \(x\) is magnetic at \(t\)
if and only if it moves toward \(x\)
at \(t\)" or symbolically,

 

\[\tag{D5} Sxt \supset (Mxt \equiv Txt\*)\]


where a biconditional, "\_\_\_ \(\equiv \ldots\)", is true
when "\_\_\_" and "..." have the same truth
value and otherwise is false. This solved one problem by abandoning
the goal of defining the predicate for a partial specification of
meaning. But it created another, insofar as, if there should be more
than one test/response for a property--such as that, "if
\(x\) moves through a closed wire loop at \(t\),
then \(x\) is magnetic at \(t\)
if and only if an electric current flows in the loop
at \(t^\*\)"--in conjunction they jointly imply that any
object \(x\) that is near small iron objects and moves
through a closed wire loop will generate a current in the loop if and
only if it attracts those iron objects. But this no longer has the
character of even a partial definition but instead that of an
empirical law. The prospect that analytic sentences might have
synthetic consequences was not a welcome result (Hempel 1952).


#### 2.1.3 Intensional Methodology



Carnap (1963) was receptive to the adoption of an intensional
methodology that went beyond the constraints of extensional logic,
which Hempel (1965b) would consider but leave for others to pursue
(Fetzer 1981, 1993). The distinction can be elaborated with respect to
the difference between the actual world and alternative possible
worlds as sequences of events that diverge from those that define the
history of the actual world. If specific conditions that obtained at a
specific time had been different, for example, the course of ensuing
events would have changed. These intensional methods can be applied to
the problems of defining dispositions and the nature of laws by
employing descriptions of possible worlds as variations on the actual,
not as alternatives that are "as real as" the
actual--as David Lewis (2001a,b) has proposed--but as a
means for formally specifying the semantic content of subjunctives and
counterfactuals (where counterfactuals are subjunctives with false
antecedents), using an alternative calculus.



There appear to be two broad kinds of justification for subjunctive
conditionals, which are logical and ontological, where logical
justifications are derived from the grammar and vocabulary of a
specific language, such as English. The subjunctive, "If John
were a bachelor, then John would be unmarried", for example,
follows from the definition of "bachelor" as
"unmarried, adult male". Analogously, the subjunctive,
"If this were gold, then it would be malleable", could be
justified on ontological grounds if being malleable were (let us call
it) a *permanent attribute* of being gold as a reference
property, where attributes are "permanent" when there is
no process or procedure, natural or contrived, by means of which
things having those reference properties could lose those attributes
except by no longer possessing those reference properties, even though
that is not a logical consequence of their respective definitions
(Fetzer 1977). The approach appeals to necessary connections, which
are unobservable and therefore unacceptable to Hume. As we shall
discover, that they are unobservable doesn't mean they are
empirically untestable.



The elaboration of a possible-worlds formal semantics that might be
suitable for this purpose, however, requires differentiating between
familiar *minimal-change* semantics, where the world remains
the same except for specified changes, and a *maximal-change*
semantics, in which everything can change except for specified
properties, which is the ingredient that seems to be required to
satisfy the constraints of scientific inquiries as opposed to
conversational discourse (Fetzer & Nute 1979, 1980). In the 1950s
and 60s, however, Nelson Goodman (1955) and Karl Popper (1965) were
attempting to sort out the linkage between dispositions, subjunctives,
and laws from distinctive points of view. Hempel's commitments
to extensional logic and to Humean constraints would lead him to
endorse an account of laws that was strongly influenced by Goodman and
to embrace a pragmatic account that was both epistemically and
contextually-dependent.


### 2.2 The Observational/Theoretical Distinction



While the analytic/synthetic distinction appears to be justifiable in
modeling important properties of languages, the
observational/theoretical distinction does not fare equally well.
Within logical positivism, *observation language* was assumed
to consist of names and predicates whose applicability or not can be
ascertained, under suitable conditions, by means of direct observation
(such as using names and predicates for colors, shapes, sounds) or
relatively simple measurement (names and predicates for heights,
weights, and sizes, for example). This was an epistemic position, of
course, since it was sorting them out based upon their accessibility
by means of experience. Both theoretical and dispositional predicates,
which refer to non-observables, posed serious problems for the
positivist position, since the verifiability criterion implies they
must be reducible to observables or are empirically meaningless. Karl
Popper (1965, 1968), however, would carry the argument in a different
direction by looking at the ontic nature of properties.



Popper has emphasized that we are theorizing all the time. Consider
the observation of a glass full of clear liquid. Suppose it's
water. Then it quenches thirst and extinguishes fires and nourishes
plants. But what if it's alcohol instead? Just describing it as
"water" entails innumerable subjunctives about the kinds
of responses it would display under a wide variety of test conditions.
They are *the would be's* of things of that kind.
Consider the differences between basket balls, billiard balls, and
tennis balls. Things of different kinds can do different things. Even
the seemingly simplest observation of a rabbit in the backyard, for
example, implies that it is going to display rabbit-like behavior,
including eating carrots when my wife puts them out. It is going to
hop around and create more rabbits. If it's a rabbit, it is
going to have rabbit DNA. It will not turn out to be stuffed. And this
suggested that observational properties and predicates are
dispositional, too.



From the Humean epistemic perspective, observational, dispositional,
and theoretical predicates are successively more and more
problematical in relation to their accessibility via experience. The
*observational* describe observable properties of observable
entities; the *dispositional*, unobservable properties of
observable entities; and the *theoretical*, unobservable
properties of unobservable entities. Popper suggested that
observational and theoretical properties (gravitational strengths
electromagnet fields, and such) are ontologically dispositional, too
(Popper 1965: 425). But if *universals* as properties that can
be attributed to any member of any possible world are dispositions and
the kind of property dispositions are does not depend upon the ease
with which their presence or absence can be ascertained, then
*nomological* subjunctives and counterfactuals--taken as
instantiations of *lawlike* generalizations for specific
individuals, places, and times--might be explicable as displays
of dispositions and of natural necessities (Fetzer 1981).


### 2.3 The Verifiability Criterion of Cognitive Significance



Hempel (1950, 1951), meanwhile, demonstrated that the verifiability
criterion could not be sustained. Since it restricts empirical
knowledge to observation sentences and their deductive consequences,
scientific theories are reduced to logical constructions from
observables. In a series of studies about cognitive significance and
empirical testability, he demonstrated that the verifiability
criterion implies that existential generalizations are meaningful, but
that universal generalizations are not, even though they include
general laws, the principal objects of scientific discovery.
Hypotheses about relative frequencies in finite sequences are
meaningful, but hypotheses concerning limits in infinite sequences are
not. The verifiability criterion thus imposed a standard that was too
strong to accommodate the characteristic claims of science and was not
justifiable.



Indeed, on the assumption that a sentence \(S\) is
meaningful if and only if its negation is meaningful, Hempel
demonstrated that the criterion produced consequences that were
counterintuitive if not logically inconsistent. The sentence,
"At least one stork is red-legged", for example, is
meaningful because it can be verified by observing one red-legged
stork; yet its negation, "It is not the case that even one stork
is red-legged", cannot be shown to be true by observing any
finite number of red-legged storks and is therefore not meaningful.
Assertions about God or The Absolute were meaningless by this
criterion, since they are not observation statements or deducible from
them. They concern entities that are non-observable. That was a
desirable result. But by the same standard, claims that were made by
scientific laws and theories were also meaningless.



Indeed, scientific theories affirming the existence of gravitational
attractions and of electromagnetic fields were thus rendered
comparable to beliefs about transcendent entities such as an
omnipotent, omniscient, and omni-benevolent God, for example, because
no finite sets of observation sentences are sufficient to deduce the
existence of entities of those kinds. These considerations suggested
that the logical relationship between scientific theories and
empirical evidence cannot be exhausted by means of observation
sentences and their *deductive* consequences alone, but needs
to include observation sentences and their *inductive*
consequences as well (Hempel 1958). More attention would now be
devoted to the notions of testability and of confirmation and
disconfirmation as forms of partial verification and partial
falsification, where Hempel would recommend an alternative to the
standard conception of scientific theories to overcome otherwise
intractable problems with the observational/theoretical
distinction.


## 3. Scientific Reasoning



The need to dismantle the verifiability criterion of meaningfulness
together with the demise of the observational/theoretical distinction
meant that logical positivism no longer represented a rationally
defensible position. At least two of its defining tenets had been
shown to be without merit. Since most philosophers believed that Quine
had shown the analytic/synthetic distinction was also untenable,
moreover, many concluded that the enterprise had been a total failure.
Among the important benefits of Hempel's critique, however, was
the production of more general and flexible criteria of cognitive
significance in Hempel (1965b), included in a famous collection of his
studies, *Aspects of Scientific Explanation* (1965d). There he
proposed that cognitive significance could not be adequately captured
by means of principles of verification or falsification, whose defects
were parallel, but instead required a far more subtle and nuanced
approach.



Hempel suggested multiple criteria for assessing the cognitive
significance of different theoretical systems, where significance is
not categorical but rather a matter of degree:



> 
> *Significant systems range from those whose entire extralogical
> vocabulary consists of observation terms, through theories whose
> formulation relies heavily on theoretical constructs, on to systems
> with hardly any bearing on potential empirical findings.* (Hempel
> 1965b: 117, italics added)
> 



The criteria Hempel offered for evaluating the "degrees of
significance" of theoretical systems (as conjunctions of
hypotheses, definitions, and auxiliary claims) were (a) the clarity
and precision with which they are formulated, including explicit
connections to observational language; (b) the
systematic--explanatory and predictive--power of such a
system, in relation to observable phenomena; (c) the formal simplicity
of the systems with which a certain degree of systematic power is
attained; and (d) the extent to which those systems have been
confirmed by experimental evidence (Hempel 1965b). The elegance of
Hempel's study laid to rest any lingering aspirations for simple
criteria of cognitive significance and signaled the demise of logical
positivism as a philosophical movement.



Precisely what remained, however, was in doubt. Presumably, anyone who
rejected one or more of the three principles defining
positivism--the analytic/synthetic distinction, the
observational/theoretical distinction, and the verifiability criterion
of significance--was not a logical positivist. The precise
outlines of its philosophical successor, which would be known as
"logical empiricism", were not entirely evident. Perhaps
this study came the closest to defining its intellectual core. Those
who accepted Hempel's four criteria and viewed cognitive
significance as a matter of degree were members, at least in spirit.
But some new problems were beginning to surface with respect to
Hempel's covering-law explication of explanation and old
problems remained from his studies of induction, the most remarkable
of which was known as "the paradox of confirmation".


### 3.1 The Paradox of Confirmation



Hempel's most controversial argument appeared in an article
about induction entitled "Studies in the Logic of
Confirmation" (1945a), where he evaluates the conditions under
which an empirical generalization would be confirmed or disconfirmed
by instances or non-instances of its antecedent and consequent. He
focused on universally quantified material conditionals, exemplified
by sentences of the form, "\((x)(Rx \supset Bx)\)". With
"\(Rx\)" interpreted as "\(x\) is a
raven" and "\(Bx\)" as "\(x\)
is black", this schema represents, in first-order symbolic
logic, the claim, "All ravens are black". He also
considered sentences of more complex logical structures, but nothing
hinges upon their use that cannot be addressed relative to an example
of the simplest possible kind. And, indeed, Hempel took sentences of
this kind as exemplars of "lawlike sentences", which
combine purely *universal form* with what he called *purely
qualitative* predicates. So lawlike sentences that are true as
extensional generalizations are "laws" (Hempel &
Oppenheim 1948).



Hempel applied "Nicod's criterion" to this example,
where Nicod had proposed that, in relation to conditional hypotheses,
instances of their antecedents that are also instances of their
consequents confirm them; instances of their antecedents that are not
instances of their consequents disconfirm them; and non-instantiations
of their antecedents are neutral, neither confirming nor
disconfirming. Applied to the raven hypothesis, this means that, given
a thing named "\(c\)", the truth of
"\(Rc\)" and "\(Bc\)" confirms it; the truth
of "\(Rc\)" and "\(\neg Bc\)" disconfirms it;
and the truth of "\(\neg Rc\)" neither confirms nor
disconfirms it, but remains evidentially neutral, regardless of the
truth value of "\(Bc\)". To these highly intuitive
conditions, Hempel added that, since logically equivalent hypotheses
have the same empirical content, whatever confirms one member of a set
of logically equivalent hypotheses must also confirm the others, which
he called "the equivalence condition".



No matter how intuitive, Hempel proceeded to demonstrate that this
creates a paradox. By Nicod's criterion, "\((x)(Rx \supset
Bx)\)" is confirmed by ravens that are black. But, by that same
standard, "\((x)(\neg Bx \supset \neg Rx)\)" is confirmed
by non-black non-ravens, such as white shoes! Since these are
logically equivalent and have the same empirical content, they must be
confirmed or disconfirmed by all and only the same instances. This
means that--no matter how counter-intuitive--the lawlike
hypothesis, "All ravens are black", is confirmed by
observations of white shoes! Since these hypotheses are also
equivalent to "\((x)(\neg Rx \vee Bx)\)", which asserts
that everything either is not a raven or is black, it is also the case
that observations of non-ravens confirms the hypothesis, regardless of
their color. Observations of non-ravens, however, unlike those of
black ravens, confirm alternative hypotheses, like "All ravens
are blue" and "All ravens are green". Hempel's
point was that the application of Nicod's criterion means that,
since even observations of non-ravens are confirmatory, the class of
neutral instances in fact has no members.


### 3.2 The Equivalence Condition



Few papers in the philosophy of science have produced such a
voluminous literature. In a "Postscript", Hempel addressed
a few of the suggestions that have been advanced to analyze the
paradoxical quality of the argument (Hempel 1965a). Several were
intent upon explaining them quantitatively on the ground that, for
example, there are many more non-black things than black things or
that the probability of being non-black is much great than the
probability of being a raven, whereas others appealed to Bayesian
principles and suggested that the prior probability for ravens being
black makes testing non-ravens of considerably less relative risk.
Hempel replied that even the existence of a quantitative measure of
evidential support poses no challenge to his conclusion that the
paradoxical cases--non-black non-ravens, such as white
shoes--are confirmatory.



Hempel acknowledges that an explanation for why the paradoxical cases
appear to be non-confirmatory may have something to do with fashioning
hypotheses about classes that are affected by their relative size.
Since the class of non-ravens is so much larger than the class of
ravens, where what we are interested in, by hypothesis, is the color
of ravens, instances of non-black non-ravens might count as
confirmatory but to a lesser degree than instances of black ravens. He
allows the possibility that perhaps the theory of confirmation should
proceed quantitatively, which might provide a less perplexing basis
for assessments of this kind. But he steadfastly maintains that the
consequences he identified following from the application of the
principles he employed are logically impeccable, no matter how
psychologically surprising they may seem (Hempel 1960).



The most important claim of Hempel (1965a) is that confirmation cannot
be adequately defined by linguistic means alone. Here he cites Goodman
(1955) to demonstrate that some hypotheses of the form,
"\((x)(Fx \supset Gx)\)", are not confirmable even by
instances of the kind "\(Fc\)" and "\(Gc\)".
If "\(Rx\)" stands for "\(x\) is a
raven" and "\(Bx\)" for "\(x\)
is blite" (where \(x\) is blite when it has been
examined before time \(t\) and is black or has not been
examined before \(t\) and is white), then any raven
observed before \(t\) and found to be black confirms
the hypothesis, "\((x)(Rx \supset Bx)\)"; yet this
hypothesis implies that all ravens not examined before \(t\)
are white, a consequence that, in Hempel's
language, "must surely count as disconfirmed rather than as
confirmed". And he endorses Goodman's suggestion that
whether a universal conditional is capable of being confirmed by its
positive instances turns out to depend upon the character of its
constituent predicates and their past use.


### 3.3 The Limits of Extensionality



Goodman (1955) draws a distinction between universal conditional
generalizations that can be confirmed by their instances and those
that cannot, where the former are said to be
"projectible". Those that can be projected from examined
cases to unexamined ones are those that have a history of past
projections. Thus, the predicate "black" has many past
projections, but the predicate "blite" does not. Since a
history of past projections enhances the projectibility of predicates
only when they are successful, the measure of a predicate's
degree of projectibility, which Goodman calls its *degree of
entrenchment*, is the relative frequency of its successful use in
past predictions. Hume observed that, no matter how consistently a
regularity has held in the past, that provides no guarantee it will
continue to hold in the future. Goodman nevertheless adopted the past
as a guide, which qualifies as his solution to the linguistic version
of Hume's problem of induction.



Since Goodman is offering a pragmatic solution for (what most would
assume to be) a syntactical and semantical problem, it may be useful
to consider whether or not there might be a more promising approach.
In embracing Goodman's approach, Hempel was modifying his
conception of *lawlike sentences* as extensional
generalizations which are restricted to purely qualitative predicates
which make no mention of specific individuals, specific places or
specific times, but which might, in special cases, reference samples
or exemplars, such as the standard meter or the atomic clock (Hempel
1965d: 269, where he also mentions Popper's notion of universal
predicates). Goodman's account does not actually capture the
concept of laws but the rather restricted notion of hypotheses that
have been projected and are supposed to be laws. Laws themselves,
after all, exist even when they have not yet been discovered, as in
the case of Archimedes' principle before Archimedes,
Snell's law before Snell, and Newton's before Newton. A
more promising approach lay in the direction of universals in
Popper's sense, which are dispositions with subjunctive force,
where laws can be characterized as unrestricted intensional
conditionals.



Popper (1965, 1968) championed falsifiability as a criterion of
demarcation that is more appropriate than verifiability as a criterion
of meaningfulness, on the ground that what we need is a basis for
distinguishing scientific from nonscientific statements, where the
latter can still be meaningful, even when they are not scientific. He
suggested that laws should be understood as having *the force of
prohibitions,* which are empirically testable by attempts to
falsify them. And he observed there are no "paradoxes of
falsification" to parallel the "paradoxes of
confirmation". Even in the case of material conditionals, the
only falsifying instances are those that combine the truth of their
antecedents with the falsity of their consequents. Relative to
"\((x)(Rx \supset Bx)\)", for example, the only potential
falsifiers are instances of "\(Rx\)" that are instances of
"\(\neg Bx\)", and similarly for "\((x)(\neg Bx
\supset \neg Rx)\)" and for "\((x)(\neg Rx \vee
Bx)\)", not to mention its subjunctive counterpart,
"\((x)(Rx \rightarrow Bx)\)". The absence of paradox
suggested that Popper's approach to problems of demarcation and
of cognitive significance might possess substantial advantages over
the alternatives.


## 4. Scientific Explanations



Hempel's most important contributions to the theory of science
have been a series of explications of the structure of scientific
explanations. His methodology was such that, following the preliminary
examination of the linguistic and physical phenomena under
consideration, he would advance a semi-formal characterization, one
which he would subsequently subject to formal characterization using
the resources of symbolic logic. The overarching theme of his work was
the conception of explanation by subsumption, where specific events
are subsumed by corresponding laws (of physics, of chemistry, of
biology, and so forth). The conception of explanation by subsumption
is rather ancient in its lineage, but Hempel advanced explicit
formulations that drew distinctions between explanations of different
kinds, especially those that invoke universal (or deterministic) laws,
statistical (or probabilistic) laws, and the explanation of laws by
means of theories.



Hempel implemented the conception of subsumption by presuming that
explanations explain the occurrence of singular events by deriving
their descriptions from premises that include at least one lawlike
sentence, which thereby displays (what he called) their *nomic
expectability*. In the simplest cases, explanations assume the
following form:





|  |  |
| --- | --- |
| Premises: | \(L\_1, L\_2,\ldots L\_k\) |
| \(C\_1,
C\_2,\ldots C\_r\) |
| Conclusion: | \(E\) |



Figure 2. An Explanation Schema




Thus, in relation to Figure 2, \(C\_1\), \(C\_2\),..., \(C\_r\)
describe specific conditions (referred to as "initial" or
"antecedent") and \(L\_1\), \(L\_2\),..., \(L\_k\)
general laws, where "\(E\)" describes the
event to be explained. The explanation takes the form of an inductive
or deductive argument, where the premises are called the
"explanans" and the conclusion the
"explanandum".



Richard Jeffrey (1969) noted that Hempel's conception harmonizes
well with Aristotle's definition of (what he called)
*unqualified scientific knowledge*,



> 
> 
> [where] the premisses of demonstrated knowledge must be true, primary,
> immediate, better known than and prior to the conclusion, which is
> further related to them as effect to cause. (*Posterior
> Analytics* 1.71-2, as quoted in Jeffrey 1969: 104)
> 
> 
> 



A potentially even more illuminating comparison emerges from the
perspective of Aristotle's theory of the four causes, where the
laws are the material cause, the antecedent conditions the efficient
cause, the logical relation between the explanans and the explanandum
the formal cause and the explanandum the final cause (Fetzer 2000a:
113-114). There were important differences in their conceptions
of law. Aristotle's general premises were definitional,
necessarily, whereas Hempel's were not. The following figure
shows the parallel.





|  |  |  |
| --- | --- | --- |
| *The Material Cause:* | The Covering Law(s) |  |
| *The Efficient Cause:* | The Initial Conditions |  |
|  | ********************************** |  | *The Formal Cause* |
| *The Final Cause:* | The Explanandum |



  

Figure 3. Hempelian Explanation in
Aristotelian Perspective




In the above figure, the use of the dotted line is meant to convey
that the analogy applies equally to deductive arguments (which would
be represented by a solid line) and inductive arguments (which would
be represented by a double line with a probability value
attached).


### 4.1 The Semi-Formal Explication



For Aristotle, the general premises of scientific explanations are
generalizations that describe *commensurately* universal
properties of things of a subject kind, \(K\). Merely
universal properties are ones that everything of kind \(K\)
has but could be without and remain a thing of that
kind, just as every Honda might have Michelin tires. Aristotle
referred to such properties as "accidental".
Commensurately universal properties are ones that belong to everything
of the kind as necessary attributes that they cannot be without. A
triangle, for example, has three lines and angles. Aristotle referred
to them as "essential". Because generalizations about the
essential properties of things of a kind are "proper
definitions", they provide the basis for explanations that
qualify as analytic. Hempel encompassed analytic generalizations
within the scope of "fundamental laws" as defined in
Hempel and Oppenheim (1948), but he focused on those that were
synthetic.



Analytic explanations are common in the currents of daily life,
especially in the context of explaining how you "know
that" something is the case. A mother might explain to her
single daughter that she knows that John must be unmarried, because a
friend told her that John is a bachelor. Similar cases of analytic
explanations can occur in scientific contexts, such as knowing that
the element they are dealing with is gold because it has atomic number
79, when gold is defined by its atomic number. Knowing why John is a
bachelor, however, is another matter. Indeed, in Hempel (1965c), he
would distinguish between *reason-seeking why-questions* and
*explanation-seeking why-questions*, where the former seek
reasons that justify believing that something is the case, as opposed
to the latter, which are usually motivated by knowledge that a
specific event has occurred.



In his semi-formal explication of the requirements for adequate
scientific explanations, Hempel specified four conditions of adequacy
(CA) that have to be satisfied, namely:



(CA-1)
The explanandum must be a deductive consequence of the explanans;

(CA-2)
The explanans must contain general laws, which are required to
satisfy (CA-1);
(CA-3)
The explanans must have empirical content and must be capable of
test; and, 
(CA-4)
The sentences of the explanans must be true. (Hempel &
Oppenheim 1948)


These conditions are intended to serve as requirements whose
satisfaction guarantees that a proposed explanation is adequate.
Hempel drew several distinctions over time between *potential*
scientific explanations (which satisfy the first three conditions, but
possibly not the fourth) and *confirmed* scientific
explanations (which are believed to be true but might turn out to be
false). Hempel recognized that (CA-3) was a redundant condition, since
it would have to be satisfied by any explanation that satisfied (CA-1)
and (CA-2). Insofar as the explanandum describes an event that
occurred during the history of the world, its derivation thereby
implies the explanans has empirical content.


### 4.2 Ontic vs. Epistemic Conceptions



Hempel's conditions had many virtues, not least of which was
that they appeared to fit many familiar examples of scientific
explanations, such as explaining why a coin expanded when it was
heated by invoking the law that copper expands when heated and noting
that the coin was copper. Hempel doesn't specify the form laws
may take, which could be simple or complex. Most of his examples were
simple, such as "All ravens are black", "All gold is
malleable", and so on. Others were quantitative, such as
Archimedes' principle (A body totally immersed in a
fluid--a liquid or a gas--experiences an apparent loss in
weight equal to the weight of the fluid it displaces), Snell's
law (During refraction of light, the ratio of the sines of the angles
of incidence and of refraction is a constant equal to the refractive
index of the medium), and Newton's law of gravitation (Any two
bodies attract each other with a force proportional to the product of
their masses and inversely proportional to the square of the distance
between them), which he discussed.



More complicated examples can be found in standard sources, such as
*The Feynman Lectures on Physics* (Feynman et al.
1963-65), the first volume of which is devoted to mechanics,
radiation, and heat; the second, to electromagnetism and matter; and
the third, to quantum mechanics. Hempel explored the implications of
 (CA-4)
 for laws:



> 
> 
> The requirement of truth for laws has the consequence that a given
> empirical statement \(S\) can never be *definitely
> known* to be a law; for the sentence affirming the truth of \(S\)
> is tantamount to \(S\) and is
> therefore capable only of acquiring a more or less high probability,
> or degree of confirmation, relative to the experimental evidence
> available at any given time (Hempel and Oppenheim 1948: note 22,
> italics added).
> 
> 
> 



Hempel considered weakening condition
 (CA-4)
 to one of high confirmation instead of truth, but concluded that it
would be awkward to have "adequate explanations" that are
later superseded by *different* "adequate
explanations" with the acquisition of additional evidence and
alternative hypotheses across time. He therefore retained the
condition of truth. Whether or not the conditions of explanatory
adequacy should be relative to an epistemic context of confirmation
rather than to an ontic context of truth would become an important
question in coping with the requirements for probabilistic
explanations.



No aspect of Hempel's position generated more controversy than
*the symmetry thesis*, which holds that, for any adequate
explanation, had its premises--its initial conditions and
covering laws--been taken into account at a suitable prior time,
then a deductive prediction of the occurrence of the explanandum event
would have been possible, and conversely (Hempel & Oppenheim
1948). Inferences from adequate explanations to potential predictions
were generally accepted, but not the converse. Critics, such as
Michael Scriven (1962), advanced counter-examples that were based on
correlations or "common causes": the occurrence of a storm
might be predicted when cows lie down in their fields, for example,
yet their behavior does not explain why the storm occurs. Sylvain
Bromberger offered the example of the length of the shadow cast by a
flagpole, which is sufficient to deduce the height of the flagpole and
thus satisfies Hempel's conditions but does not explain why the
flagpole has that height (Bromberger 1966). Since logical relations
are non-temporal, Hempel may have taken the symmetry thesis to be a
trivial consequence of his account, but deeper issues are involved
here.


### 4.3 The Problem of Lawlikeness



Hempel observed that universal generalizations of the form,
"\((x)(Fx \supset\) *Gx*)", are true if and only if
logically equivalent generalizations of the form, "\((x)(\neg Fx
\vee Gx)\)", are true. But if "\(Fx\)" stands for
*an uninstantiated property*, such as being a vampire, then,
since "\(\neg Fx\)" is satisfied by everything, regardless
of the attribute "\(Gx\)", "\(\neg Gx\)",
etc., that hypothesis--and its logical equivalents--are not
only universally confirmed but even qualify as "laws".
This is not surprising, from a logical point of view, since
extensional logic is purely truth functional, where the truth value of
molecular sentences is a function of the truth values of its atomic
constituents. But it implies that an empirical generalization that
might be true of the world's history may or may not be
"lawful", since it could be the case that its truth was
merely "accidental". Hempel's endorsement of
Goodman's approach to select generalizations that support
subjunctive conditionals, therefore, was not only consistent with the
tradition of Hume--who believed that attributions of natural
necessity in causal relations are merely "habits of mind",
psychologically irresistible, perhaps, but logically
unwarranted--but circumvented a disconcerting consequence of his
explication.



There is a fundamental difference between sentences that
"support subjunctives" and those that "entail
subjunctives", when the selection of those that support
subjunctives is made on pragmatic grounds. Popper captured the crucial
difference between material (extensional) generalizations and
subjunctive (intensional) generalizations as follows:



> 
> 
> A statement may be said to be *naturally or physically
> necessary* if, and only if, it is deducible from a statement
> function which is satisfied in all worlds that differ from our own, if
> at all, only with respect to initial conditions. (Popper 1965:
> 433)
> 
> 
> 



Indeed, the existence of irreducibly ontic probabilistic phenomena,
where more than one outcome is possible under the same initial
conditions, would mean that the history of an indeterministic world
might be identical with the history of a deterministic one, where
their differences were concealed "by chance". Even a
complete description of the history of the actual world might not
suffice to distinguish between them, where fundamental aspects of
their causal structure would remain beyond empirical detection.



The difference between Popper's universal predicates and
Goodman's paradoxical ones may derive from the consideration
that incorporating specific times into his definitions entails
reference to specific moments of time \(t\), such as
*midnight tonight*, during the history of the actual world, a
point to which we are going to return. It follows that they cannot be
"universal" in Popper's sense and do not even
qualify as "purely qualitative" in Hempel's, either.
Reliance upon material conditionals within first-order symbolic logic,
moreover, forfeits the benefits of synthetic subjunctives. If the
hypothesis, "All ravens are black", is more adequately
formalized as a subjunctive of the form, "\((x)(Rx \rightarrow
Bx)\)", where the truth of a subjunctive hypothetically assumes
that its antecedent condition is satisfied, that obviates the paradox:
white shoes are not ravens, and Nicod's criteria apply Instances
of "\(Rx\)" and "\(Bx\)" thereby confirm the
hypothesis; instances of "\(Rx\)" and "\(\neg
Bx\)" disconfirm it; and instances of "\(\neg Rx\)"
are epistemically neutral, precisely as Nicod said.


## 5. Explanation, Prediction, Retrodiction



Hempel's critics--both early and late--have not always
shown an appreciation for the full dimensions of his explication.
Michael Scriven (1959), for example, once objected to "the
deductive model, with its syllogistic form, where no student of
elementary logic could fail to complete the inference, given the
premise" (1959: 462), when no restriction is placed on the
complexity of the relevant laws or the intricacy of these deductions
(Hempel 1965c: 339). Similar objections have been lodged more recently
by William Bechtel and Adele Abrahamsen (2005), for example, who
claim, in defense of "the mechanist alternative", that,
among its many benefits,



> 
> 
> investigators are not limited to linguistic representations and
> logical inference in presenting explanations but frequently employ
> diagrams and reasoning about mechanisms by simulating them. (2005:
> 421)
> 
> 
> 



Hempel's conditions of adequacy, however, are capable of
satisfaction without their presentation as formal arguments.



Hempel remarks that his model of explanation does not directly apply
to the wordless gesticulations of a Yugoslavian automobile mechanic or
guarantee that explanations that are adequate are invariably
subjectively satisfying. His purpose was to formalize the conditions
that must be satisfied when an explanation is adequate without denying
that background knowledge and prior beliefs frequently make a
difference in ordinary conversational contexts. Jan might not know
some of the antecedent conditions, Jim might have misunderstood the
general laws, or features of the explanandum event may have been
missed by them both. Even when information is conveyed using diagrams
and simulations, for example, as long as it satisfies conditions
 (CA-1)
 through
 (CA-4)--no
 matter whether those conditions are satisfied implicitly or
explicitly--an adequate scientific explanation is at hand. But it
has to satisfy all four of those requirements.



Demonstrating that an adequate scientific explanation is at hand,
however, imposes demands beyond the acquisition of information about
initial conditions and laws. In the *examiner's sense of
knowledge* Ian Hacking identified (Hacking 1967: 319), to prove an
adequate scientific explanation is available, it would be necessary to
show that each of Hempel's adequacy conditions has been
satisfied. It would not be enough to show, say, that one person is
knowledgeable about the initial conditions, another about the covering
laws, and a third about the explanandum. It would be necessary to show
that the explanandum can be derived from the explanans, that those
laws were required for the derivation, and that the initial conditions
were those present on that occasion.
 (CA-1)-(CA-4)
 qualifies as a "check list" to insure that a scientific
explanation is adequate.


### 5.1 The Symmetry Thesis



Students of Hempel have found it very difficult to avoid the
impression that Hempel was not only defending the position that every
adequate scientific explanation is potentially predictive but also the
position that every adequate scientific prediction is potentially
explanatory. This impression is powerfully reinforced in "The
Theoretician's Dilemma" (Hempel 1958), where, in the
course of demonstrating that the function of theories goes beyond
merely establishing connections between observables, he offers this
passage:



> 
> 
> Scientific explanations, predictions, and postdictions all have the
> same logical character: they show that the fact under consideration
> can be inferred from certain other facts by means of specified general
> laws. In the simplest case, the type of argument may be schematized as
> a deductive inference of the following form [here substituting the
> "simplest case" for the abstract schemata presented in the
> original]:
> 
> 
> 
> 
> 
> |  |  |
> | --- | --- |
> | \((x)(Rx \supset Bx)\) | Explanans |
> | \(Rc\) |
> | \(Bc\) | Explanandum |
> 
> 
> 
> Figure 4. A Covering-Law Explanation
> 
> 
> 
> 
> ... While explanation, prediction, and postdiction are alike in
> their logical structure, they differ in certain other respects. For
> example, an argument [like Figure 4 above] will qualify as a
> prediction only if [its explanandum] refers to an occurrence at a time
> later than that at which the argument is offered; in the case of
> postdiction, the event must occur before the presentation of the
> argument. These differences, however, require no further study here,
> for the purpose of the preceding discussion was simply to point out
> the role of general laws in scientific explanation, prediction, and
> postdiction (Hempel 1958: 37-38).
> 
> 
> 



What is crucial about this passage is Hempel's emphasis on the
pragmatic consideration of the time at which *the argument is
presented* *in relation to the time of occurrence of the
explanandum event itself*. Let us take this to be the observation
that \(c\) is black and assume that occurs at a
specific time \(t\_1\). If the argument is presented prior to time
\(t\_1\) (before observing that \(c\) is black), then it
is a prediction. If the argument is presented after time \(t\_1\)
(after observing that \(c\) is black), then it is a
postdiction. Since explanations are usually advanced after the time of
the occurrence of the event to be explained, they usually occur as
postdictions. But there is nothing about their form that requires
that.



As Hempel uses the term, reasoning is "scientific" when it
involves inferences with laws. Indeed, this appears to be the
principal ground upon which he wants to maintain that explanations,
predictions, and postdictions share the same logical form. His
position here makes it logically impossible for explanations,
predictions and postdictions to not have the same logical form. If
there are modes of prediction that are non-explanatory, such as might
be the case when they are based upon accidental generalizations, then
they would not thereby demonstrate that the symmetry thesis does not
hold for the kinds of distinctively "scientific"
explanations that are the focus of his work. Scriven's cow
example, for example, does not show that the symmetry thesis does not
hold for *scientific* explanations and *scientific*
predictions, even if it shows that some *ordinary* predictions
are not also *ordinary* explanations. But other
counter-examples posed greater threats.


### 5.2 The Paradoxes of Explanation



Indeed, in the process of drawing the distinction between
explanation-seeking and reason-seeking why-questions, Hempel (1965c)
proposed a different kind of symmetry thesis, where adequate answers
to explanation-seeking why-questions also provide adequate answers to
reason-seeking why-questions, but not conversely. That was very
appropriate, since the purpose of reason-seeking why-questions is to
establish grounds for believing that the event described by the
explanandum sentence has taken (or will take) place, while
explanation-seeking why-questions typically take for granted that we
know their explanandum events have already occurred. It follows that,
while Hempel's conditions are meant to specify *when we know
why*, they also entail *how we know that*.



Insofar as Hempel focused on logically equivalent formulations of
lawlike sentences in addressing the paradoxes of confirmation, some
may find it remarkable that he does not explore the consequences of
logically equivalent formulations in relation to explanations. We tend
to assume we can explain why a specific thing \(c\) is
black on the grounds that it is a raven and all ravens are black. Yet
we would hesitate to explain why a specific thing is not a raven on
the ground that it is not black. We may refer to this as *the
paradox of transposition*. Notice, the contention that one member
of a class of logically equivalent hypotheses is explanatory but
others are not generates "paradoxes of explanation" that
parallel the "paradoxes of confirmation" (Fetzer 2000a).
Figure 5 offers an illustration:


\[ \frac{\begin{gathered} (x)(\neg Bx \supset \neg Rx)\\ \neg Bc\end{gathered}}{\neg Rc} \]


Figure 5. The Transposition Paradox




Even if we accept the transposed form of "\((x)(Rx \supset
Bx)\)" as the basis for an answer to a reason-seeking
why-question--in this case, *how we know* *that*
\(c\) is no raven--Figure 5 surely does not
explain *why that is the case*, which presumably has to do with
the genes of entity \(c\) and its developmental history
as a living thing. If logically equivalent forms equally qualify as
lawlike--there's no basis in extensional logic to deny
it--then Hempel confronts a dilemma. In fact, by his own
conditions of adequacy, this argument is not only explanatory but also
predictive. It is *not* a "postdiction" as he has
defined it above, but it appears to qualify as a
"retrodiction" that does not explain its explanandum
event.



Suppose Hempel were to resort to Goodman's approach and deny
that the negation of a purely qualitative predicates is also a purely
qualitative predicate. That would mean that sentences logically
equivalent to lawlike sentences--which have the same
content--are not therefore also lawlike by invoking pragmatic
differences in their degree of entrenchment and projectibility of
their constituent predicates. An appeal to pragmatic considerations
within this context has a decidedly *ad hoc* quality about it,
which appears to contravene the spirit of the paradoxes of
confirmation, where Hempel insists that the counterintuitive cases are
confirmatory and their paradoxical character is merely psychological.
Even if Hempel were to adopt this position and take for granted that
one member of a class of logically equivalent sentences can be lawlike
while the others are not, another difficulty arises from the use of
*modus tollens* in lieu of *modus ponens*, as Figure 6
exemplifies:


\[\frac{\begin{gathered} (x)(Rx \supset Bx)\\ \neg Bc \end{gathered}} {\neg Rc} \]


Figure 6. The Modus Tollens Paradox




Here we have a more threatening problem, since there is no apparent
basis for denying that an argument having this form satisfies
Hempel's criteria and therefore ought to be *both explanatory
and predictive*. Similarly for temporally quantified arguments,
where the fact that a match of kind \(K\) has not
lighted at \(t^\*\) would certainly not be adequate to explain *why
it was not struck at t*, even though every match of kind \(K\)
will light when it is struck, under suitable
conditions! Something serious thus appears to be wrong (Fetzer
2000a).


## 6. Inductive-Statistical Explanation



In his studies of inductive reasoning, Hempel (1960, 1962a, 1966b)
discusses the ambiguity of induction, which arises because
incompatible conclusions can appear to be equally-well supported by
inductive arguments, where all the premises of both arguments are
true. This problem has no analogue for deductive arguments:
inconsistent conclusions cannot be validly derived from consistent
premises, even if they are false. Inconsistent conclusions can receive
inductive support from consistent premises, however, even when they
are all true. Consider Hempel's illustration of conflicting
predictions for a patient's recovery:



(e1)
Jones, a patient with a sound heart, has just had an appendectomy,
and of all persons with sound hearts who underwent appendectomy in the
past decade, 93% had an uneventfully recovery.


This information, taken by itself, would surely lend strong support to
the hypothesis,



(h1)
Jones will have an uneventful recovery. 


But suppose that we also have the information:



(e2)
Jones is an octogenarian with serious kidney failure; he just had
an appendectomy after his appendix had ruptured; and in the past
decade, of all cases of appendectomy after rupture of the appendix
among octogenarians with serious kidney failure, only 8% had an
uneventful recovery.


This information by itself would lend strong support to the
contradictory of (h1):



\((\neg\)h1)
Jones will not have an uneventful recovery. (Hempel
1966b) 


Thus, Hempel observed, (e1) and (e2) are logically compatible and
could all be part of the evidence available when Jones'
prognosis is being considered. The solution Hempel endorsed was
*the requirement of total evidence*, according to which, in
reasoning about the world, arguments must be based upon all the
available evidence, though he noted that evidence can be omitted when
it is irrelevant and its omission does not affect the level of
support. Here, when (e1) is combined with (e2), \((\neg\)h1) is better
supported than (h1).


### 6.1 The Requirement of Maximal Specificity



Hempel referred to statistical explanations as
"inductive-statistical" in contrast with his prior
discussion of "deductive-nomological" explanations (Hempel
1958). Because generalizations of both kinds are supposed to be
lawlike, however, more appropriate names for them might have been
"universal-deductive" and
"statistical-inductive" (or, perhaps,
"statistical-probabilistic", respectively (Fetzer 1974).
In formalizing these arguments, Hempel symbolized statistical premises
attributing a certain probability to outcome \(G\),
under conditions \(F\), by "\(P(G, F) =
r\)", where these explanations then assume this form:





|  |
| --- |
| \(P(G,F) = r\) |
| \(Fc\) | \([r]\) |
| \(Gc\) |



Figure 7. An I-S Explanation Schema




While Hempel initially interpreted the bracketed variable [\(r\)]
as the measure of inductive support which the
explanans confers upon the explanandum, \(Gi\), (in conformity with
the requirement of total evidence)--where the value of [\(r\)]
equals that of \(r\)--it
occurred to him that, since inductive and deductive explanations are
typically offered on the basis of the knowledge that their explanandum
events have already occurred, viewing statistical explanations as
inductive arguments whose purpose is to establish how we know that
they have occurred (or that they will occur) is not the correct
perspective for understanding them:



> 
> 
> *the point of an explanation is not to provide evidence for the
> occurrence of the explanandum, but to exhibit it as nomically
> expectable*. And the probability attached to an I-S explanation is
> the probability of the conclusion relative to the explanatory
> premises, not relative to the total class \(\mathbf{K}\) [representing
> our total knowledge at that time]. (Hempel 1968, original
> emphasis)
> 
> 
> 



In order to formalize this conception, he advanced the concept of
*a maximally specific predicate related to "Gi" in*
\(\mathbf{K},\) where, when "\(P(G, F) = r\)", then, if
the premises include a predicate stronger than "\(F\)",
say, "\(M\)", which implies the presence of more
properties than does "\(F\)", then \(P(G, M) = r,\) where,
as before, \(r = P(G,F).\) Thus additional predicates could be
included in the explanans of an adequate scientific explanation,
provided that they made no difference to the nomic expectability of
the explanandum. Including the position of the moon or the day of the
week a match is struck, for example, might be irrelevant, but his
condition did not exclude them. Wesley C. Salmon (1970), however,
would pursue this issue by arguing that Hempel had embraced the wrong
standard upon which to base explanatory relevance relations, where
explanations, in Salmon's view, strictly speaking, do not even
qualify as arguments.


### 6.2 Statistical Relevance vs. Causal Relevance



Salmon offered a series of counterexamples to Hempel's approach.
In the case of I-S explanations, Hempel required that the nomic
expectability of the explanandum must be equal to or greater than .5,
which preserved the symmetry thesis for explanations of this kind.
This implied that events with low probability could not be explained.
There were persuasive illustrations, moreover, demonstrating that
explanations could satisfy Hempel's criteria, yet not explain
why their explanandum events occur. For example,



(CE-1)
John Jones was almost certain to recover from his cold within a
week, because he took vitamin C, and almost all colds clear up within
a week after administration of vitamin C. 
(CE-2)
John Jones experienced significant remission of his neurotic
symptoms, for he underwent extensive psychoanalytic treatment and a
substantial percentage of those who undergo psychoanalytic treatment
experience significant remission of neurotic symptoms. 


Most colds clear up within a week, with or without vitamin C, and
similarly for neurotic symptoms. Salmon thought that this problem was
peculiar to statistical explanations, but was corrected by Henry
Kyburg (1965), who offered examples of the following kind:



(CE-3)
This sample of table salt dissolves in water, for it has had a
dissolving spell cast upon it, and all samples of table salt that have
had dissolving spells cast upon them dissolve in water. 
(CE-4)
John Jones avoided becoming pregnant during the past year, for he
has taken his wife's birth control pills regularly, and every
man who regularly takes birth control pills avoids
pregnancy. 


For Hempel, a property \(F\) is *explanatorily
relevant* to the occurrence of an outcome \(G\) if
there is a lawful relationship that relates \(F\) to
\(G\). If table salt dissolves in water, so does
Morton's table salt, Morton's finest table salt,
Morton's finest table salt bought on sale, and so on. Salmon
concluded that Hempel's conception of I-S explanation was
wrong.



A student of Reichenbach (1938, 1949), Salmon had adopted and defended
the limiting frequency interpretation of physical probability, where
"\(P(G/F) = r\)" means that \(G\) occurs
with a limiting frequency \(r\) in a reference class of
instances of \(F\), which has to be infinite for limits
to exist (Salmon 1967, 1971). On this approach, a property \(H\)
is *explanatorily relevant* to the occurrence
of an attribute \(G\) within a reference class \(F\)
just in case (SR):



(SR)
\(P(G/F \amp H) = m\) and \(P(G/F \amp \neg H) = n\), where \(m
\ne n\);


that is, the limiting frequency for \(G\) in \(F\)-and-\(H\)
differs from the limiting
frequency for \(G\) in \(F\)-and-not-\(H\).
Properties whose presence does not make a difference
to the limiting frequency for the occurrence of \(G\)
in \(F\) would therefore qualify as statistically
irrelevant. Relative frequencies were typically relied upon in
practice, but in principle they had to be limiting.



Salmon also rejected Hempel's requirement that the nomic
expectability of a statistical explanation must be equal to or greater
than .5, which led him to abandon the notion of explanations as
arguments. Even events of low probability were explainable within the
context of Salmon's approach, which he compared with
Hempel's approach as follows:



> 
> 
> I-S model (Hempel): an explanation is an *argument* that
> renders the explanandum *highly probable*;
> 
> 
> 
> S-R model (Salmon): an explanation is an *assembly of facts
> statistically relevant* to the explanandum *regardless of the
> degree of probability* that results. (Salmon 1971, original
> emphasis)
> 
> 
> 



Salmon's work created a sensation, since Hempel's
dominance of the philosophy of science, especially in relation to the
theory of explanation, now had a significant rival. The students of
explanation who recognized that probabilities as properties of
infinite classes posed substantial problems were few, and those who
realized that Salmon's position confronted difficulties of its
own fewer still.



Salmon was ingenious in devising "screening off" criteria
to insure that the properties cited in an S-R explanation were
restricted to those that were statistically relevant. He may have
appreciated that the available evidence was restricted to relative
frequencies in finite sequences, but that could be discounted as
typical of the distinction between a context \(\mathbf{K}\)
representing our total knowledge at that time and the truth condition,
since scientific knowledge is always tentative and fallible. A deeper
problem arose from the existence of properties that were statistically
relevant but not explanatorily relevant, however. If women whose
middle initials began with vowels, for example, experienced
miscarriages with a different frequency than women whose middle
initials began with consonants--even after other properties were
taken into account--then that property would have to qualify as
statistically relevant and *therefore* explanatorily relevant
(Fetzer 1974, 1981). This result implied that statistical relevance
cannot adequately define explanatory relevance, and Salmon would
gradually move toward the propensity approach in Salmon (1980,
1989).


### 6.3 Revising Hempel's Criteria of Adequacy



The advantage of propensities over frequencies are considerable,
since, on the propensity account, a probabilistic law no longer simply
affirms that a certain percentage of the reference class belongs to
the attribute class. What it asserts instead is that every member of
the reference class possesses a certain disposition, which in the case
of statistical laws is of probabilistic strength and in the case of
universal laws of universal strength. On its single-case formulation,
moreover, short runs and long runs are simply finite and infinite
sequences of single cases, where each trial has propensities that are
equal and independent from trial to trial and classic theorems of
statistical inference apply.



Even this intensional explication would still be vulnerable to the
problems of irrelevant properties and the *modus tollens*
paradox but for the adoption of a condition to exclude the occurrence
of predicates that are not nomically relevant to the explanandum event
from the explanans of an adequate scientific explanation. This
criterion, known as *the requirement of strict maximal
specificity*, permits the reformulation of Hempel's four
conditions of adequacy by replacing the redundant empirical content
condition with this new requirement (Fetzer 1981; Salmon 1989).
Explanations not only display the *nomic expectability* of
their explanandum events--which, in the case of those that occur
with high probability, would enable them to have been predicted, as
Hempel proposed but--more importantly--explain them by
specifying all and only those properties *nomically
responsible* for their occurrence, even when they occur with low
probability (Fetzer 1992).



Bromberger's flagpole counterexample provides a severe test of
this requirement. The reason why the inference from the length of the
shadow to the height of the flagpole is non-explanatory is because the
length of the shadow is not *nomically responsible* for the
flagpole's height. Hempel's original conditions could not
cope with situations of this kind, where inferences using laws support
predictions and retrodictions that are not also explanations, even
though they satisfy all four. This alternative condition thus requires
that the properties cited in the antecedent of the lawlike premise(s)
must be nomically relevant to the explanandum or may not be included
there (Fetzer 1981, 1992). The height of the flagpole, but not the
length of the shadow, qualifies as nomically relevant, which resolves
the quandary--at the expense of acknowledging classes of
arguments that are *predictive or retrodictive but not
explanatory,* even when they involve inferences from law(s) that
satisfy Hempel's original criteria. The reformulated conditions
are:



(CA-1\*)
The explanandum must be a suitable logical consequence of the
explanans; 
(CA-2)
The explanans must contain general laws, which are required to
satisfy (CA-1\*);
(CA-3\*)
The explanans must satisfy the requirement of strict maximal
specificity; and,
(CA-4\*)
The sentences--both of the explanans and of the
explanandum--must be true.


By formulating (CA-1\*) in this way, the covering law conception of the
subsumption of singular events by general laws is preserved; but
abandonment of the high-probability requirement led both Salmon (1971)
and Alberto Coffa (1973) to question whether or not explanations still
properly qualify as "arguments". At the least, however,
they would appear to be special kinds of "explanatory
arguments", even when they involve low probabilities.



These revised conditions implicitly require abandoning Hempel's
commitment to extensional methodology to capture the notion of nomic
responsibility, but the benefits of an intensional approach appear to
be profound. As Salmon has observed,



> 
> 
> [on such an explication] the distinction between *description and
> prediction*, on the one hand, and *explanation*, on the
> other, is that the former can proceed in an extensional language
> framework, while the latter demands an intensional language framework.
> It remains to be seen whether the intensional logic can be
> satisfactorily formulated (Salmon 1989: 172, original emphasis).
> 
> 
> 



This approach to explanation incorporates the causal relevance
criterion, according to which a property \(H\) is
*causally relevant* to the occurrence of an attribute \(G\)
relative to a reference property \(F\)--within
the context of a causal
explanation--just in case (CR):



(CR)
\((Fxt \amp Hxt) =m \Rightarrow Gxt\*\) and \((Fxt \amp \neg Hxt) =n
\Rightarrow Gxt\*\), where \(m \ne n\);


that is, the propensity for \(G\), given \(F \amp H\), differs from
the propensity for \(G\), given \(F \amp \neg H\), where the presence
or absence of \(H\) affects the single-case causal tendency for
\(G\). The universal generalization of sentential functions like these
thus produces *lawlike sentences*, while their instantiation to
individual constants or to ambiguous names produces (what are known
as) *nomological conditionals* (Fetzer 1981: 49-54). The
introduction of the probabilistic causal calculus \(C\), moreover,
responds to Salmon's concerns by providing a formalization
within intensional logic that resolves them (Fetzer & Nute 1979,
1980). Paul Humphreys (1985, 1989) has argued that "propensities
cannot be probabilities" because their *causal
directedness* precludes satisfying symmetry conditions requiring
probabilities from effects to causes as well as from causes to effects
(on the plausible assumption that there are no propensities for
effects to bring about their causes). A more linguistically precise
characterization would be that propensities cannot be 'standard
probabilities', which has no effect on their role in explanatory
contexts.


## 7. The Problem of Provisos



What may come as some surprise is that Hempel exposed yet another
significant problem confronting the theory of scientific explanation.
One of the most remarkable features of his career is that he continued
to publish original and innovative studies well into his eightieth
decade. Rather surprisingly, he authored a series of articles that
moved away from the influential conception of scientific theories as
*formal calculi* combined with *empirical
interpretations* that had been characteristic of logical
empiricism. In Hempel (1966a), at the time, the most widely adopted
introduction to the philosophy of science (which has been translated
into ten other languages), he advanced the novel explication of
scientific theories as consisting of *internal principles* and
*bridge principles*, where the lawlike hypotheses that
distinguish theories are linked to observation, measurement, and
experiment by principles expressed in various mixtures of ordinary and
of technical language. Now antecedent understanding replaces explicit
definability, which seems to have been, in part, a response to the
demise of the observational/theoretical distinction.



Even more strikingly, Hempel (1988a,b) observed that the application
of scientific theories presupposes the absence of factors that might
affect the internal principles of the theory, which goes beyond the
content of the theory itself. Deriving predictions and explanations
from classical mechanics presupposes that bodies are being acted upon
exclusively by gravitational forces, for example, where the presence
of electromagnetic forces would invalidate those derivations. This is
not simply a matter of conducting tests "under suitable
conditions", where the malleability of gold differs when struck
by a hammer when at extremely cold temperatures (where the
combustibility of paper differs when the paper is wet, and so on),
which are familiar examples of the various specific conditions that
must be identified in complete definitions of dispositional
predicates. What Hempel noticed is that these properties may not only
be affected by conditions covered by the theory being applied but
involve entirely different theories. He took this to mean that the
application of theories has to be accompanied by
"provisos" affirming that no properties beyond those
specified by the theory are present in a specific case.



The function of these provisos means that instrumentalist
constructions of scientific theories as mere calculating devices and
programs for the elimination of theoretical language by reduction to
observational language alone are misguided and cannot be sustained.
And this is because, if other theoretical properties make a difference
to the application of a particular theory, then the observational
consequences of that theory cannot be assumed to obtain in any
instance without provisos about other theoretical properties beyond
those specified by the theory, which requires separate investigation
on the basis of observation, measurement, and experiment. The
conditions for testing, confirming or falsifying alternative
hypotheses and theories is thereby rendered vastly more complex than
had previously been supposed. Strikingly, as an essential element of
an adequate explanation, Coffa (1973) had advanced "extremal
clauses" that assert that no factors other than those specified
by the initial conditions of the explanans are relevant to the
occurrence of the explanandum event. Indeed, Salmon points out that,
since the application of every law involves a tacit extremal (or
"ceteris paribus") clause, any law can be protected from
refutation by discounting disconfirming evidence on the expedient of
claiming that, because the clause was not satisfied, the law is still
correct, which is another way to accent the source of Hempel's
concern (Salmon 1989: 84-85).



These observations are related to the claim that "the laws of
physics lie", which has been suggested by Nancy Cartwright
(1983). She has claimed that there are theoretical laws, which are
supposed to be "simple" but rarely if ever instantiated,
and phenomenological laws, which are frequently instantiated but
typically if not always complex. Armstrong (1983: 137-140) draws
a distinction between laws that are "iron" and have no
exceptions and those that are "oaken" and have no
exceptions *as long as no interfering conditions are present*.
In his language, unless Cartwright has overlooked the possibility that
some laws might be both complex and true, she seems to be saying that
theoretical laws are "oaken" laws for which there are no
"iron" counterparts. If that is right, then she has failed
to appreciate the distinction between counterfactual conditionals
(which may be true *in spite of* having no instances) and mere
indicative conditionals (which may be true *because* they have
no instances). The "provisos" problem implies that
satisfying the requirement of maximal specificity may be more
demanding than has been generally understood in the past. Either way,
Cartwright's theses appear to trade on an equivocation between
the non-existence of "iron" counterparts and their
non-availability: even if the "iron" laws within a domain
are not yet known, it does not follow that those laws do not
exist.


## 8. Reflections on Rationality



The publication of Thomas S. Kuhn's *The Structure of
Scientific Revolutions* (1962), ironically, was among the
historical developments that contributed to a loss of confidence in
science. Kuhn's work, which turned "paradigm" into a
household word, was widely regarded as assimilating revolutions in
science to revolutions in politics, where one theory succeeds another
only upon the death of its adherents. It was interpreted as having
destroyed the myth that philosophers possess some special kind of
wisdom or insight in relation to the nature of science or the thought
processes of scientists with respect to their rationality, almost as
though every opinion were on a par with every other. A close reading
of Kuhn's work shows that these were not his own conclusions,
but they were enormously influential. And among the public at large
and many social scientists, the tendency to no longer hold science in
high esteem or to be affected by its findings has induced political
ramifications that are inimical to the general good. When our beliefs
are not well founded, actions we base upon them are unlikely to
succeed, often with unforeseen effects that are harmful. Rational
actions ought to be based upon rational beliefs, where science has
proven to be the most reliable method for acquiring knowledge about
ourselves and the world around us.



This study supports the conclusion that Hempel's conception of
scientific explanations as involving the subsumption of singular
events by means of covering laws was well-founded, even though his
commitment to an extensional methodology inhibited him from embracing
a more adequate account of natural laws. The symmetry thesis turns out
to require qualification, not only with respect to predictions for
events that occur only with low probability, but also for
retrodictions derived by *modus tollens*. The link that most
accurately embodies the relationship between Hempel's work on
explanation and decision-and-inference, as we have seen, appears to be
reflected by the *Principal Principle,* which formalizes the
recognition that subjective (or personal) probabilities, as degrees of
belief, should have the same values as corresponding objective
probabilities, when they are known. The values of propensities as
properties of laws, no doubt, should be given precedence over the
values of frequencies, insofar as laws, unlike frequencies, cannot be
violated and cannot be changed and thus provide the more reliable
guide.



A recent trend presumes that the philosophy of science has been
misconceived and requires "a naturalistic turn" as a kind
of science of science more akin to history or to sociology than to
philosophy. In studies published in the twilight of his career, Hempel
demonstrated that, without standards to separate science from
pseudo-science, it would be impossible to distinguish frauds,
charlatans, and quacks from the real thing (Hempel 1979, 1983).
Without knowing "the standards of science", we would not
know which of those who call themselves "scientists" are
scientists and which methods are "scientific", where the
place of explanation remains central within the context of inference
to the best explanation (Fetzer 2002). The philosophy of science,
therefore, cannot be displaced by history or by sociology. Not least
among the important lessons of Hempel's enduring legacy is the
realization that the standards of science cannot be derived from mere
descriptions of its practice alone but require rational justification
in the form of explications satisfying the highest standards of
philosophical rigor.