Carl G. Hempel (1905–1997) was the principal proponent of the“covering law” theory of explanation and the paradoxes ofconfirmation as basic elements of the theory of science. A master ofphilosophical methodology, Hempel pursued explications of initiallyvague and ambiguous concepts, which were required to satisfy veryspecific criteria of adequacy. With Rudolf Carnap and HansReichenbach, he was instrumental in the transformation of the dominantphilosophical movement of the 1930s and 40s, which was known as“logical positivism”, into the more nuanced position knownas “logical empiricism”. His studies of induction,explanation, and rationality in science exerted a profound influenceupon more than a generation of philosophers of science, many of whombecame leaders of the discipline in their own right.

• 1. Biographical Sketch
• 2. The Critique of Logical Positivism
  • 2.1 The Analytic/Synthetic Distinction
    • 2.1.1 Quine’s Complaints
    • 2.1.2 Hempel’s Concerns
    • 2.1.3 Intensional Methodology
  • 2.2 The Observational/Theoretical Distinction
  • 2.3 The Verifiability Criterion of Cognitive Significance
• 3. Scientific Reasoning
  • 3.1 The Paradox of Confirmation
  • 3.2 The Equivalence Condition
  • 3.3 The Limits of Extensionality
• 4. Scientific Explanations
  • 4.1 The Semi-Formal Explication
  • 4.2 Ontic vs. Epistemic Conceptions
  • 4.3 The Problem of Lawlikeness
• 5. Explanation, Prediction, Retrodiction
  • 5.1 The Symmetry Thesis
  • 5.2 The Paradoxes of Explanation
• 6. Inductive-Statistical Explanation
  • 6.1 The Requirement of Maximal Specificity
  • 6.2 Statistical Relevance vs. Causal Relevance
  • 6.3 Revising Hempel’s Criteria of Adequacy
• 7. The Problem of Provisos
• 8. Reflections on Rationality
• Bibliography
  • Primary Sources: Works by Hempel
  • Primary Sources: Coauthored Works
  • Secondary Literature
• Academic Tools
• Other Internet Resources
• Related Entries

1. Biographical Sketch

Carl G(ustav) Hempel (1905–97), known as “Peter” tohis friends, was born near Berlin, Germany, on January 8, 1905. Hestudied philosophy, physics and mathematics at the Universities ofGöttingen and Heidelberg before coming to the University ofBerlin in 1925, where he studied with Hans Reichenbach. Impressed bythe work of David Hilbert and Paul Bernays on the foundations ofmathematics and introduced to the studies of Rudolf Carnap byReichenbach, Hempel came to believe that the application of symboliclogic held the key to resolving a broad range of problems inphilosophy, including that of separating genuine problems from merelyapparent ones. Hempel’s commitment to rigorous explications ofthe nature of cognitive significance, of scientific explanation, andof scientific rationality would become the hallmark of his research,which exerted great influence on professional philosophers, especiallyduring the middle decades of the 20^th Century.

In 1929, at Reichenbach’s suggestion, Hempel spent the fallsemester at the University of Vienna, where he studied with Carnap,Moritz Schlick, and Frederick Waismann, who were advocates of logicalpositivism and members of (what came to be known as) “the ViennaCircle”. It would fall to Hempel to become perhaps the mostastute critic of that movement and to contribute to its refinement aslogical empiricism. As Hitler increased his power in Germany, Hempel,who was not Jewish but did not support the Nazi regime, moved toBrussels and began collaborating with Paul Oppenheim, which wouldresult in several classic papers, including “Studies in theLogic of Explanation”, which appeared in 1948 (Rescher 2005:Chs. 8 and 9). Hempel would also visit the United Statestwice—the University of Chicago in 1937–38 and then theCity College of New York in 1939–40, where he held his firstacademic position—and eventually became a naturalizedcitizen.

He was productive throughout his career, publishing such importantpapers as “The Function of General Laws in History” (1942)and “Studies in the Logic of Confirmation”,“Geometry and Empirical Science”, and “The Nature ofMathematical Truth” (all in 1945), before leaving City Collegefor Yale. While there, Hempel would publish “Problems andChanges 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 InternationalEncyclopedia of Unified Science,Fundamentals of Concept Formationin Empirical Science (1952). Hempel moved to Princeton in 1955,where his research program flourished and his influence uponprofessional philosophers became immense.

During his two decades at Princeton, Hempel’s approach dominatedthe philosophy of science. His major articles during this intervalincluded “The Theoretician’s Dilemma” (1958),“Inductive Inconsistencies” (1960), “RationalAction” (1961), and “Deductive-Nomological vs. StatisticalExplanation” and “Explanation in Science and inHistory” and (both in 1962). A classic collection of hisstudies,Aspects of Scientific Explanation (1965c), became ascholar’s bible for generations of graduate students. Hisintroductory text,Philosophy of Natural Science (1966a),would be translated into ten languages. Other articles he publishedthereafter included “Recent Problems of Induction” (1966b)and “Maximal Specificity and Lawlikeness in ProbabilisticExplanation” (1968) as well as a series of studies that included“On the ‘Standard Conception’ of ScientificTheories” (1970).

At the University of Pittsburgh following his mandatory retirementfrom Princeton in 1973, he continued to publish significant articles,including studies of the nature of scientific rationality,“Scientific Rationality: Analytic vs. PragmaticPerspectives” (1979), and “Turns in the Evolution of theProblem of Induction” (1981), and on the structure of scientifictheories, including, most importantly, “Limits of a DeductiveConstrual of the Function of Scientific Theories” (1988a) aswell as “Provisos: A Problem Concerning the Inferential Functionof Scientific Theories” (1988b), further enhancing hisreputation by his willingness to reconsider earlier positions. Afterhis death in 1997, new collections of his papers appeared (Jeffrey2000; Fetzer 2001), which complemented studies of his research(Rescher 1969; Esler et al. 1985; Kitcher & Salmon 1989; Fetzer2000b).

2. The Critique of Logical Positivism

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

According to the first principle, sentences areanalyticrelative to a language framework \(\mathbf{L}\) when their truthfollows from its grammar and vocabulary alone. In English,“Bachelors are unmarried” cannot be false, since“bachelor \(=_{\df}\) unmarried, adult male”. Sentences ofthis kind make no claims about the world, but instead reflect featuresof the linguistic framework as syntactical or semantic truths in\(\mathbf{L}\). And sentences aresynthetic when they makeclaims about the world. Their truth in \(\mathbf{L}\) does not followfrom its grammar and vocabulary alone but hinges upon properties ofthe world and its history. According to logical positivism, all suchclaims about the world have to be evaluated on the basis ofexperience, which means the kind of knowledge they display isaposteriori. But kinds of knowledge whose truth can be establishedindependently of experience area priori.

Logical positivism affirmed that, given a language \(\mathbf{L}\), alla priori knowledge is analytic and all synthetic knowledge isa posteriori, thus denying the existence of knowledge that isboth synthetic anda priori. Indeed, the denial of theexistence of synthetica priori knowledge is commonly assumedto define the position known as “Empiricism”, while theaffirmation of its existence defines “Rationalism”. Figure1 thus reflects the intersection of kinds of sentences and kinds ofknowledge on the Empiricist approach:

The category for sentences that are analytic and yet representaposteriori knowledge deserves discussion. The empirical study ofthe use of language within language-using communities by fieldlinguists involves establishing the grammar and the vocabularyemployed within each such community. Their empirical research yieldstheories of the languages, \(\mathbf{L}\), used in those communitiesand affords a basis for distinguishing between which sentences areanalytic-in-\(\mathbf{L}\) and which are synthetic-in-\(\mathbf{L}\).The kind of knowledge they acquire about specific sentences based onempirical procedures thus assumes the form, “Sentence \(S\)is analytic-in-\(\mathbf{L}\)”, when that istrue of sentence \(S\), which isaposteriori.

One of Hempel’s early influential articles was a defense oflogicism, according to which mathematics—with the notableexception of geometry, which he addressed separately—can bereduced to logic (for Hempel, including set theory) as its foundation(Hempel 1945c). Mathematics thus becomes an exemplar of analyticapriori knowledge. Two subtheses should be distinguished: (i) thatall mathematicalconcepts can be defined by means of basiclogical concepts; and (ii) that all mathematicaltheorems canbe deduced from basic logical truths. In order to distinguish logicismfrom formalism, however, the former maintains that there isonesystem of logic that is fundamental to all inquiries, where allmathematical terms are reducible to logical terms, and allmathematical axioms are derivable from logical ones, which formalismdenies (Rech 2004).

The tenability of logicism has been disputed on multiple grounds, themost prominent of which is that the notion ofmembershipfundamental to the theory of sets is not a logical notion but rather asymbol that must be added to first-order logic to formalize what isproperly understood as anon-logical theory. Nor wouldphilosophers today accept the conception of the axioms of set theoryas logical axioms, since there exist alternatives. So even ifmathematics were reducible to set theory, these considerationsundermine Hempel’s claim that mathematics is thereby reducibleto logic (cf. Benacerraf 1981 and Linsky & Zalta 2006, whichprovides an extensive bibliography). Hempel’s views aboutgeometry, in retrospect, thus appear to have been the betterfounded.

2.1 The Analytic/Synthetic Distinction

The analytic/synthetic distinction and the observational/theoreticaldistinction were tied together bythe verifiability criterion ofmeaningfulness, according to which, in relation to a givenlanguage, \(\mathbf{L}\), a sentence \(S\) ismeaningful if and only if it is either analytic-in-\(\mathbf{L}\) orsynthetic-in-\(\mathbf{L}\) as an observation sentence or a sentencewhose truth follows from a finite set of observation sentences. Bythis standard, sentences that are non-analytic but alsonon-verifiable, including various theological or metaphysicalassertions concerning God or The Absolute, qualify as cognitivelymeaningless. This was viewed as a desirable result. But, as Hempelwould demonstrate, its scope was far too sweeping, since it alsorendered meaningless the distinctively scientific assertions made bylaws and theories.

From an historical perspective, logical positivism represents alinguistic version of the empiricist epistemology of David Hume(1711–76). It refines his crucial distinctions of“relations between ideas” and “matters offact” by redefining them relative to a language \(\mathbf{L}\)as sentences that are analytic-in-\(\mathbf{L}\) andsynthetic-in-\(\mathbf{L}\), respectively. His condition thatsignificant ideas are those which can be traced back to impressions inexperience that gave rise to them now became the claim that syntheticsentences have to be justified by derivability from finite classes ofobservation sentences. Hume applied this criterion to exclude the ideaofnecessary connections, which are not observable, fromsignificant causal claims, which were thereby reduced to relations ofregular association, spatial contiguity, and temporal succession. Andlogical positivism followed Hume’s lead.

Empiricism historically stands in opposition to Rationalism, which isrepresented most prominently by Immanuel Kant, who argued that themind, in processing experiences, imposes certain properties onwhatever we experience, including what he called Forms of Intuitionand Categories of Understanding. The Forms of Intuition imposeEuclidean spatial relations and Newtonian temporal relations; theCategories of Understanding require objects to be interpreted assubstances and causes as inherently deterministic. Severaldevelopments in the history of science, such as the emergence of thetheory of relativity and of quantum mechanics, undermine Kant’sposition by introducing the role of frames of reference and ofprobabilistic causation. Newer versions are associated with NoamChomsky and with Jerry Fodor, who have championed the ideas of aninnate syntax and innate semantics, respectively (Chomsky 1957; Fodor1975; Chomsky 1986)

Indeed, according to the computational theory of the mind, humanminds, like computing machines, are special kinds of formal systems.Since deviations from formal systems of language in practice can beconsiderable, Chomsky introduced a distinction betweencompetence andperformance, where the former modelsthe formal system and various explanations are advanced for deviationsfrom that model in practice, similar to differences between the fallof bodies in a vacuum and in air, which raises questions abouttestability that parallel those for scientific theories, in general.If languages are not best understood as formal systems, however, or ifsyntax and semantics are not innate, then Chomsky and Fodor’sviews are as vulnerable as those of Kant. If syntax is an emergentproperty of semantic complexity, for example, then grammar is notinnate; and if mentality has and continues to evolve, Chomsky andFodor are wrong (Schoenemann 1999; Fetzer 2005).

In his study of formal systems for geometry (Hempel 1945b), Hempeldiscusses 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 atriangle must equal 180° and, in relation to a point separate froma given line, one and only one parallel line passes through it. Thealternatives advanced by Lobachevsky (hyperbolic) and by Riemann(elliptical), however, which represent the surface of a sphere and ofa saddle, respectively, violate both of those conditions, albeit indifferent ways. Hempel emphasized that all three, as formal systems,are on a par, where the most appropriate choice to describe thegeometry of space depends on the outcome of empirical studies. As ithappened, Einstein would adopt a generalized form of Riemanniangeometry in his general theory of relativity.

Hempel accordingly drew a distinction of fundamental importancebetweenpure andapplied mathematics, which heemphasized by using a quotation from Einstein, who had observed,“As far as the laws of mathematics refer to reality, they arenot certain; and as far as they are certain, they do not refer toreality” (1921). The existence of alternative and incompatibleformal systems, moreover, appears to affect Hempel’s defense oflogicism from another direction. If mathematics is supposed to bereducible to logic and logic is supposed to be consistent, then howcan alternative geometries be consistently reducible to logic? No onewould dispute that they exist as distinct formal systems with theirown axioms and primitives, but if these geometries are jointlyreducible to logic only if logic is inconsistent, their existencesuggests that, perhaps, as formalism claims, it is not the case thereis 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 notedlogician, Willard van Orman Quine, published “Two Dogmas ofEmpiricism” (1953), challenging its adequacy. Quine argued thatthe notion of analyticity presupposes the notion of synonymy-in-use,which in turn presupposes understandinginter-substitutability-while-preserving-truth. He claimed none of thenotions can be understood without the other, creating a circularrelation between them. Thus, matching up adefiniens with adefiniendum could only be done if we already understood thatthe definiens specifies the meaning of the word that is beingdefined—a variation of “the paradox of analysis”,according to which we either already know the meaning of words (inwhich case analysis is unnecessary) or we do not (in which case it isimpossible). The idea of analyticity appeared to have beendeposed.

The paper created a sensation and has been the most influentialphilosophical article of the past 100 years. But Quine explicitlyallowed for the existence of the class of logical truths (such as“No unmarried man is married”) asnarrowlyanalytic sentences and their relationship tobroadly analyticsentences (such as “No bachelor is married”), when thedefinitional relationship between them has been suitably stipulated(as in “bachelor \(=_{\df}\) unmarried, adult male” in alanguage framework \(\mathbf{L})\). In cases of this kind, heconceded, inter-substitutability, synonymy, and analyticity arerelated in an unproblematic way. It would have been odd for a logicianto deny the existence of logical truths or the role of stipulations,which are basic to the construction of formal systems, which suggeststhat he may not have actually defeated the idea of analyticity, afterall (Fetzer 1993).

Indeed, Carnap (1939) had explained that the process of constructing avocabulary and a grammar for a language-in-use involves severalstages, including observation of the use of language by members of thecommunity, formulating hypotheses regarding the meaning of its phrasesand 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 theoryabout the language as the outcome of empirical research. Theconstruction of formal systems thus provides an illustration of theelements of artificial languages, where accounts of natural languagecounterparts can be subject to further testing and refinement. Thelinguistic practices of a specific community can thus be modeled andthereby overcome Quine’s professed objections.

2.1.2 Hempel’s Concerns

Moreover, inFundamentals of Concept Formation in EmpiricalScience (1952), Hempel had endorsedexplication as amethod of definition analogous to theory construction by taking wordsand phrases that are somewhat vague and ambiguous and subjecting themto a process of clarification and disambiguation. Adequateexplications are required to satisfy criteria of syntacticaldeterminacy, semantic relevance, and pragmatic benefit (by clarifyingand illuminating the meaning of those words and phrases in specificcontexts). The word “probability” is vague and ambiguous,for example, but various contexts of its usage can be distinguishedand theories of evidential support, relative frequency, and causalpropensity can be advanced. Hempel, following Carnap (1950), hadaccordingly advanced a methodology that dealt with the paradox ofanalysis before “Two Dogmas”.

About the same time, however, Hempel found reasons of his own fordoubting that the analytic/synthetic distinction was tenable, whichsurfaced in his study of dispositional predicates, such as“malleable”, “soluble” and“magnetic”. They designate, not directly observableproperties, but (in the case of “magnetic”) tendencies onthe part of some kinds of things to display specific reactions (suchas attracting small iron objects) under suitable conditions (such asthe presence of small iron objects in the vicinity). It is verytempting to define them using thematerial conditional,“___ \(\supset \ldots\)” by definitions like,

(D1)

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

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

where “\(t^*\)” is equal to or later than “\(t\)”and reflects that effects brought about bycauses entail changes across time. Since the material “if ___then …” is equivalent to “either not-___ or…”, however, the meaning of (D2) turns out to belogically equivalent to,

which means that anything not subject to the test, such as a browncow, is magnetic. Hempel acknowledged that the use of thesubjunctive conditional, say, “___ \(\rightarrow\ldots\)”, formalizingwhat would be the case … ifsomething ___ were the case, in this case,

“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 testcondition) would avoid the problem, because these conditionals takefor granted their antecedents are satisfied (that“Sxt” is true). But while acknowledging theimportance of subjunctive conditionals for an understanding of bothcounterfactual conditionals and lawlike sentences, Hempel regardedtheir explication as not fully satisfactory and their use as “aprogram, 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 notsubject to the test, the predicate is partially defined by means of areduction 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,

where a biconditional, “___ \(\equiv \ldots\)”, is truewhen “___” and “…” have the same truthvalue and otherwise is false. This solved one problem by abandoningthe goal of defining the predicate for a partial specification ofmeaning. But it created another, insofar as, if there should be morethan 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 loopat \(t^*\)”—in conjunction they jointly imply that anyobject \(x\) that is near small iron objects and movesthrough a closed wire loop will generate a current in the loop if andonly if it attracts those iron objects. But this no longer has thecharacter of even a partial definition but instead that of anempirical law. The prospect that analytic sentences might havesynthetic consequences was not a welcome result (Hempel 1952).

2.1.3 Intensional Methodology

Carnap (1963) was receptive to the adoption of an intensionalmethodology 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 tothe difference between the actual world and alternative possibleworlds as sequences of events that diverge from those that define thehistory of the actual world. If specific conditions that obtained at aspecific time had been different, for example, the course of ensuingevents would have changed. These intensional methods can be applied tothe problems of defining dispositions and the nature of laws byemploying descriptions of possible worlds as variations on the actual,not as alternatives that are “as real as” theactual—as David Lewis (2001a,b) has proposed—but as ameans for formally specifying the semantic content of subjunctives andcounterfactuals (where counterfactuals are subjunctives with falseantecedents), using an alternative calculus.

There appear to be two broad kinds of justification for subjunctiveconditionals, which are logical and ontological, where logicaljustifications are derived from the grammar and vocabulary of aspecific language, such as English. The subjunctive, “If Johnwere 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 bejustified on ontological grounds if being malleable were (let us callit) apermanent attribute of being gold as a referenceproperty, where attributes are “permanent” when there isno process or procedure, natural or contrived, by means of whichthings having those reference properties could lose those attributesexcept by no longer possessing those reference properties, even thoughthat is not a logical consequence of their respective definitions(Fetzer 1977). The approach appeals to necessary connections, whichare unobservable and therefore unacceptable to Hume. As we shalldiscover, that they are unobservable doesn’t mean they areempirically untestable.

The elaboration of a possible-worlds formal semantics that might besuitable for this purpose, however, requires differentiating betweenfamiliarminimal-change semantics, where the world remainsthe same except for specified changes, and amaximal-changesemantics, in which everything can change except for specifiedproperties, which is the ingredient that seems to be required tosatisfy the constraints of scientific inquiries as opposed toconversational discourse (Fetzer & Nute 1979, 1980). In the 1950sand 60s, however, Nelson Goodman (1955) and Karl Popper (1965) wereattempting to sort out the linkage between dispositions, subjunctives,and laws from distinctive points of view. Hempel’s commitmentsto extensional logic and to Humean constraints would lead him toendorse an account of laws that was strongly influenced by Goodman andto embrace a pragmatic account that was both epistemically andcontextually-dependent.

2.2 The Observational/Theoretical Distinction

While the analytic/synthetic distinction appears to be justifiable inmodeling important properties of languages, theobservational/theoretical distinction does not fare equally well.Within logical positivism,observation language was assumedto consist of names and predicates whose applicability or not can beascertained, under suitable conditions, by means of direct observation(such as using names and predicates for colors, shapes, sounds) orrelatively simple measurement (names and predicates for heights,weights, and sizes, for example). This was an epistemic position, ofcourse, since it was sorting them out based upon their accessibilityby means of experience. Both theoretical and dispositional predicates,which refer to non-observables, posed serious problems for thepositivist position, since the verifiability criterion implies theymust be reducible to observables or are empirically meaningless. KarlPopper (1965, 1968), however, would carry the argument in a differentdirection by looking at the ontic nature of properties.

Popper has emphasized that we are theorizing all the time. Considerthe observation of a glass full of clear liquid. Suppose it’swater. Then it quenches thirst and extinguishes fires and nourishesplants. But what if it’s alcohol instead? Just describing it as“water” entails innumerable subjunctives about the kindsof responses it would display under a wide variety of test conditions.They arethe would be’s of things of that kind.Consider the differences between basket balls, billiard balls, andtennis balls. Things of different kinds can do different things. Eventhe seemingly simplest observation of a rabbit in the backyard, forexample, implies that it is going to display rabbit-like behavior,including eating carrots when my wife puts them out. It is going tohop around and create more rabbits. If it’s a rabbit, it isgoing to have rabbit DNA. It will not turn out to be stuffed. And thissuggested that observational properties and predicates aredispositional, too.

From the Humean epistemic perspective, observational, dispositional,and theoretical predicates are successively more and moreproblematical in relation to their accessibility via experience. Theobservational describe observable properties of observableentities; thedispositional, unobservable properties ofobservable entities; and thetheoretical, unobservableproperties of unobservable entities. Popper suggested thatobservational and theoretical properties (gravitational strengthselectromagnet fields, and such) are ontologically dispositional, too(Popper 1965: 425). But ifuniversals as properties that canbe attributed to any member of any possible world are dispositions andthe kind of property dispositions are does not depend upon the easewith which their presence or absence can be ascertained, thennomological subjunctives and counterfactuals—taken asinstantiations oflawlike generalizations for specificindividuals, places, and times—might be explicable as displaysof dispositions and of natural necessities (Fetzer 1981).

2.3 The Verifiability Criterion of Cognitive Significance

Hempel (1950, 1951), meanwhile, demonstrated that the verifiabilitycriterion could not be sustained. Since it restricts empiricalknowledge to observation sentences and their deductive consequences,scientific theories are reduced to logical constructions fromobservables. In a series of studies about cognitive significance andempirical testability, he demonstrated that the verifiabilitycriterion implies that existential generalizations are meaningful, butthat universal generalizations are not, even though they includegeneral laws, the principal objects of scientific discovery.Hypotheses about relative frequencies in finite sequences aremeaningful, but hypotheses concerning limits in infinite sequences arenot. The verifiability criterion thus imposed a standard that was toostrong to accommodate the characteristic claims of science and was notjustifiable.

Indeed, on the assumption that a sentence \(S\) ismeaningful if and only if its negation is meaningful, Hempeldemonstrated that the criterion produced consequences that werecounterintuitive if not logically inconsistent. The sentence,“At least one stork is red-legged”, for example, ismeaningful because it can be verified by observing one red-leggedstork; yet its negation, “It is not the case that even one storkis red-legged”, cannot be shown to be true by observing anyfinite number of red-legged storks and is therefore not meaningful.Assertions about God or The Absolute were meaningless by thiscriterion, since they are not observation statements or deducible fromthem. They concern entities that are non-observable. That was adesirable result. But by the same standard, claims that were made byscientific laws and theories were also meaningless.

Indeed, scientific theories affirming the existence of gravitationalattractions and of electromagnetic fields were thus renderedcomparable to beliefs about transcendent entities such as anomnipotent, omniscient, and omni-benevolent God, for example, becauseno finite sets of observation sentences are sufficient to deduce theexistence of entities of those kinds. These considerations suggestedthat the logical relationship between scientific theories andempirical evidence cannot be exhausted by means of observationsentences and theirdeductive consequences alone, but needsto include observation sentences and theirinductiveconsequences as well (Hempel 1958). More attention would now bedevoted to the notions of testability and of confirmation anddisconfirmation as forms of partial verification and partialfalsification, where Hempel would recommend an alternative to thestandard conception of scientific theories to overcome otherwiseintractable problems with the observational/theoreticaldistinction.

3. Scientific Reasoning

The need to dismantle the verifiability criterion of meaningfulnesstogether with the demise of the observational/theoretical distinctionmeant that logical positivism no longer represented a rationallydefensible position. At least two of its defining tenets had beenshown to be without merit. Since most philosophers believed that Quinehad 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, wasthe production of more general and flexible criteria of cognitivesignificance in Hempel (1965b), included in a famous collection of hisstudies,Aspects of Scientific Explanation (1965d). There heproposed that cognitive significance could not be adequately capturedby means of principles of verification or falsification, whose defectswere parallel, but instead required a far more subtle and nuancedapproach.

Hempel suggested multiple criteria for assessing the cognitivesignificance of different theoretical systems, where significance isnot categorical but rather a matter of degree:

Significant systems range from those whose entire extralogicalvocabulary consists of observation terms, through theories whoseformulation relies heavily on theoretical constructs, on to systemswith hardly any bearing on potential empirical findings. (Hempel1965b: 117, italics added)

The criteria Hempel offered for evaluating the “degrees ofsignificance” of theoretical systems (as conjunctions ofhypotheses, definitions, and auxiliary claims) were (a) the clarityand precision with which they are formulated, including explicitconnections to observational language; (b) thesystematic—explanatory and predictive—power of such asystem, in relation to observable phenomena; (c) the formal simplicityof the systems with which a certain degree of systematic power isattained; and (d) the extent to which those systems have beenconfirmed by experimental evidence (Hempel 1965b). The elegance ofHempel’s study laid to rest any lingering aspirations for simplecriteria of cognitive significance and signaled the demise of logicalpositivism as a philosophical movement.

Precisely what remained, however, was in doubt. Presumably, anyone whorejected one or more of the three principles definingpositivism—the analytic/synthetic distinction, theobservational/theoretical distinction, and the verifiability criterionof significance—was not a logical positivist. The preciseoutlines of its philosophical successor, which would be known as“logical empiricism”, were not entirely evident. Perhapsthis study came the closest to defining its intellectual core. Thosewho accepted Hempel’s four criteria and viewed cognitivesignificance as a matter of degree were members, at least in spirit.But some new problems were beginning to surface with respect toHempel’s covering-law explication of explanation and oldproblems remained from his studies of induction, the most remarkableof which was known as “the paradox of confirmation”.

3.1 The Paradox of Confirmation

Hempel’s most controversial argument appeared in an articleabout induction entitled “Studies in the Logic ofConfirmation” (1945a), where he evaluates the conditions underwhich an empirical generalization would be confirmed or disconfirmedby instances or non-instances of its antecedent and consequent. Hefocused on universally quantified material conditionals, exemplifiedby sentences of the form, “\((x)(Rx \supset Bx)\)”. With“\(Rx\)” interpreted as “\(x\) is araven” and “\(Bx\)” as “\(x\)is black”, this schema represents, in first-order symboliclogic, the claim, “All ravens are black”. He alsoconsidered sentences of more complex logical structures, but nothinghinges upon their use that cannot be addressed relative to an exampleof the simplest possible kind. And, indeed, Hempel took sentences ofthis kind as exemplars of “lawlike sentences”, whichcombine purelyuniversal form with what he calledpurelyqualitative predicates. So lawlike sentences that are true asextensional 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 theirconsequents confirm them; instances of their antecedents that are notinstances of their consequents disconfirm them; and non-instantiationsof their antecedents are neutral, neither confirming nordisconfirming. Applied to the raven hypothesis, this means that, givena thing named “\(c\)”, the truth of“\(Rc\)” and “\(Bc\)” confirms it; the truthof “\(Rc\)” and “\(\neg Bc\)” disconfirms it;and the truth of “\(\neg Rc\)” neither confirms nordisconfirms it, but remains evidentially neutral, regardless of thetruth value of “\(Bc\)”. To these highly intuitiveconditions, Hempel added that, since logically equivalent hypotheseshave the same empirical content, whatever confirms one member of a setof logically equivalent hypotheses must also confirm the others, whichhe called “the equivalence condition”.

No matter how intuitive, Hempel proceeded to demonstrate that thiscreates a paradox. By Nicod’s criterion, “\((x)(Rx \supsetBx)\)” is confirmed by ravens that are black. But, by that samestandard, “\((x)(\neg Bx \supset \neg Rx)\)” is confirmedby non-black non-ravens, such as white shoes! Since these arelogically equivalent and have the same empirical content, they must beconfirmed or disconfirmed by all and only the same instances. Thismeans that—no matter how counter-intuitive—the lawlikehypothesis, “All ravens are black”, is confirmed byobservations of white shoes! Since these hypotheses are alsoequivalent to “\((x)(\neg Rx \vee Bx)\)”, which assertsthat everything either is not a raven or is black, it is also the casethat observations of non-ravens confirms the hypothesis, regardless oftheir color. Observations of non-ravens, however, unlike those ofblack ravens, confirm alternative hypotheses, like “All ravensare blue” and “All ravens are green”. Hempel’spoint was that the application of Nicod’s criterion means that,since even observations of non-ravens are confirmatory, the class ofneutral instances in fact has no members.

3.2 The Equivalence Condition

Few papers in the philosophy of science have produced such avoluminous literature. In a “Postscript”, Hempel addresseda few of the suggestions that have been advanced to analyze theparadoxical quality of the argument (Hempel 1965a). Several wereintent upon explaining them quantitatively on the ground that, forexample, there are many more non-black things than black things orthat the probability of being non-black is much great than theprobability of being a raven, whereas others appealed to Bayesianprinciples and suggested that the prior probability for ravens beingblack makes testing non-ravens of considerably less relative risk.Hempel replied that even the existence of a quantitative measure ofevidential support poses no challenge to his conclusion that theparadoxical cases—non-black non-ravens, such as whiteshoes—are confirmatory.

Hempel acknowledges that an explanation for why the paradoxical casesappear to be non-confirmatory may have something to do with fashioninghypotheses about classes that are affected by their relative size.Since the class of non-ravens is so much larger than the class ofravens, where what we are interested in, by hypothesis, is the colorof ravens, instances of non-black non-ravens might count asconfirmatory but to a lesser degree than instances of black ravens. Heallows the possibility that perhaps the theory of confirmation shouldproceed quantitatively, which might provide a less perplexing basisfor assessments of this kind. But he steadfastly maintains that theconsequences he identified following from the application of theprinciples he employed are logically impeccable, no matter howpsychologically surprising they may seem (Hempel 1960).

The most important claim of Hempel (1965a) is that confirmation cannotbe 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 byinstances of the kind “\(Fc\)” and “\(Gc\)”.If “\(Rx\)” stands for “\(x\) is araven” and “\(Bx\)” for “\(x\)is blite” (where \(x\) is blite when it has beenexamined before time \(t\) and is black or has not beenexamined before \(t\) and is white), then any ravenobserved before \(t\) and found to be black confirmsthe hypothesis, “\((x)(Rx \supset Bx)\)”; yet thishypothesis implies that all ravens not examined before \(t\)are white, a consequence that, in Hempel’slanguage, “must surely count as disconfirmed rather than asconfirmed”. And he endorses Goodman’s suggestion thatwhether a universal conditional is capable of being confirmed by itspositive instances turns out to depend upon the character of itsconstituent predicates and their past use.

3.3 The Limits of Extensionality

Goodman (1955) draws a distinction between universal conditionalgeneralizations that can be confirmed by their instances and thosethat cannot, where the former are said to be“projectible”. Those that can be projected from examinedcases to unexamined ones are those that have a history of pastprojections. Thus, the predicate “black” has many pastprojections, but the predicate “blite” does not. Since ahistory of past projections enhances the projectibility of predicatesonly when they are successful, the measure of a predicate’sdegree of projectibility, which Goodman calls itsdegree ofentrenchment, is the relative frequency of its successful use inpast predictions. Hume observed that, no matter how consistently aregularity has held in the past, that provides no guarantee it willcontinue to hold in the future. Goodman nevertheless adopted the pastas a guide, which qualifies as his solution to the linguistic versionof Hume’s problem of induction.

Since Goodman is offering a pragmatic solution for (what most wouldassume to be) a syntactical and semantical problem, it may be usefulto consider whether or not there might be a more promising approach.In embracing Goodman’s approach, Hempel was modifying hisconception oflawlike sentences as extensionalgeneralizations which are restricted to purely qualitative predicateswhich make no mention of specific individuals, specific places orspecific times, but which might, in special cases, reference samplesor exemplars, such as the standard meter or the atomic clock (Hempel1965d: 269, where he also mentions Popper’s notion of universalpredicates). Goodman’s account does not actually capture theconcept of laws but the rather restricted notion of hypotheses thathave been projected and are supposed to be laws. Laws themselves,after all, exist even when they have not yet been discovered, as inthe case of Archimedes’ principle before Archimedes,Snell’s law before Snell, and Newton’s before Newton. Amore promising approach lay in the direction of universals inPopper’s sense, which are dispositions with subjunctive force,where laws can be characterized as unrestricted intensionalconditionals.

Popper (1965, 1968) championed falsifiability as a criterion ofdemarcation that is more appropriate than verifiability as a criterionof meaningfulness, on the ground that what we need is a basis fordistinguishing scientific from nonscientific statements, where thelatter can still be meaningful, even when they are not scientific. Hesuggested that laws should be understood as havingthe force ofprohibitions, which are empirically testable by attempts tofalsify them. And he observed there are no “paradoxes offalsification” to parallel the “paradoxes ofconfirmation”. Even in the case of material conditionals, theonly falsifying instances are those that combine the truth of theirantecedents with the falsity of their consequents. Relative to“\((x)(Rx \supset Bx)\)”, for example, the only potentialfalsifiers are instances of “\(Rx\)” that are instances of“\(\neg Bx\)”, and similarly for “\((x)(\neg Bx\supset \neg Rx)\)” and for “\((x)(\neg Rx \veeBx)\)”, not to mention its subjunctive counterpart,“\((x)(Rx \rightarrow Bx)\)”. The absence of paradoxsuggested that Popper’s approach to problems of demarcation andof cognitive significance might possess substantial advantages overthe alternatives.

4. Scientific Explanations

Hempel’s most important contributions to the theory of sciencehave been a series of explications of the structure of scientificexplanations. His methodology was such that, following the preliminaryexamination of the linguistic and physical phenomena underconsideration, he would advance a semi-formal characterization, onewhich he would subsequently subject to formal characterization usingthe resources of symbolic logic. The overarching theme of his work wasthe conception of explanation by subsumption, where specific eventsare subsumed by corresponding laws (of physics, of chemistry, ofbiology, and so forth). The conception of explanation by subsumptionis rather ancient in its lineage, but Hempel advanced explicitformulations that drew distinctions between explanations of differentkinds, especially those that invoke universal (or deterministic) laws,statistical (or probabilistic) laws, and the explanation of laws bymeans of theories.

Hempel implemented the conception of subsumption by presuming thatexplanations explain the occurrence of singular events by derivingtheir descriptions from premises that include at least one lawlikesentence, which thereby displays (what he called) theirnomicexpectability. In the simplest cases, explanations assume thefollowing form:

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 theevent to be explained. The explanation takes the form of an inductiveor deductive argument, where the premises are called the“explanans” and the conclusion the“explanandum”.

Richard Jeffrey (1969) noted that Hempel’s conception harmonizeswell 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 isfurther related to them as effect to cause. (PosteriorAnalytics 1.71–2, as quoted in Jeffrey 1969: 104)

A potentially even more illuminating comparison emerges from theperspective of Aristotle’s theory of the four causes, where thelaws are the material cause, the antecedent conditions the efficientcause, the logical relation between the explanans and the explanandumthe formal cause and the explanandum the final cause (Fetzer 2000a:113–114). There were important differences in their conceptionsof law. Aristotle’s general premises were definitional,necessarily, whereas Hempel’s were not. The following figureshows the parallel.

In the above figure, the use of the dotted line is meant to conveythat the analogy applies equally to deductive arguments (which wouldbe represented by a solid line) and inductive arguments (which wouldbe represented by a double line with a probability valueattached).

4.1 The Semi-Formal Explication

For Aristotle, the general premises of scientific explanations aregeneralizations that describecommensurately universalproperties of things of a subject kind, \(K\). Merelyuniversal properties are ones that everything of kind \(K\)has but could be without and remain a thing of thatkind, just as every Honda might have Michelin tires. Aristotlereferred to such properties as “accidental”.Commensurately universal properties are ones that belong to everythingof the kind as necessary attributes that they cannot be without. Atriangle, for example, has three lines and angles. Aristotle referredto them as “essential”. Because generalizations about theessential properties of things of a kind are “properdefinitions”, they provide the basis for explanations thatqualify as analytic. Hempel encompassed analytic generalizationswithin the scope of “fundamental laws” as defined inHempel and Oppenheim (1948), but he focused on those that weresynthetic.

Analytic explanations are common in the currents of daily life,especially in the context of explaining how you “knowthat” something is the case. A mother might explain to hersingle daughter that she knows that John must be unmarried, because afriend told her that John is a bachelor. Similar cases of analyticexplanations can occur in scientific contexts, such as knowing thatthe element they are dealing with is gold because it has atomic number79, when gold is defined by its atomic number. Knowing why John is abachelor, however, is another matter. Indeed, in Hempel (1965c), hewould distinguish betweenreason-seeking why-questions andexplanation-seeking why-questions, where the former seekreasons that justify believing that something is the case, as opposedto the latter, which are usually motivated by knowledge that aspecific event has occurred.

In his semi-formal explication of the requirements for adequatescientific 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 tosatisfy (CA-1);

(CA-3)

The explanans must have empirical content and must be capable oftest; and,

(CA-4)

The sentences of the explanans must be true. (Hempel &Oppenheim 1948)

These conditions are intended to serve as requirements whosesatisfaction guarantees that a proposed explanation is adequate.Hempel drew several distinctions over time betweenpotentialscientific explanations (which satisfy the first three conditions, butpossibly not the fourth) andconfirmed scientificexplanations (which are believed to be true but might turn out to befalse). Hempel recognized that (CA-3) was a redundant condition, sinceit would have to be satisfied by any explanation that satisfied (CA-1)and (CA-2). Insofar as the explanandum describes an event thatoccurred during the history of the world, its derivation therebyimplies the explanans has empirical content.

4.2 Ontic vs. Epistemic Conceptions

Hempel’s conditions had many virtues, not least of which wasthat they appeared to fit many familiar examples of scientificexplanations, such as explaining why a coin expanded when it washeated by invoking the law that copper expands when heated and notingthat the coin was copper. Hempel doesn’t specify the form lawsmay take, which could be simple or complex. Most of his examples weresimple, such as “All ravens are black”, “All gold ismalleable”, and so on. Others were quantitative, such asArchimedes’ principle (A body totally immersed in afluid—a liquid or a gas—experiences an apparent loss inweight equal to the weight of the fluid it displaces), Snell’slaw (During refraction of light, the ratio of the sines of the anglesof incidence and of refraction is a constant equal to the refractiveindex of the medium), and Newton’s law of gravitation (Any twobodies attract each other with a force proportional to the product oftheir masses and inversely proportional to the square of the distancebetween them), which he discussed.

More complicated examples can be found in standard sources, such asThe 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; andthe third, to quantum mechanics. Hempel explored the implications of(CA-4) for laws:

The requirement of truth for laws has the consequence that a givenempirical statement \(S\) can never bedefinitelyknown to be a law; for the sentence affirming the truth of \(S\)is tantamount to \(S\) and istherefore capable only of acquiring a more or less high probability,or degree of confirmation, relative to the experimental evidenceavailable 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 itwould be awkward to have “adequate explanations” that arelater superseded bydifferent “adequateexplanations” with the acquisition of additional evidence andalternative hypotheses across time. He therefore retained thecondition of truth. Whether or not the conditions of explanatoryadequacy should be relative to an epistemic context of confirmationrather than to an ontic context of truth would become an importantquestion in coping with the requirements for probabilisticexplanations.

No aspect of Hempel’s position generated more controversy thanthe symmetry thesis, which holds that, for any adequateexplanation, had its premises—its initial conditions andcovering laws—been taken into account at a suitable prior time,then a deductive prediction of the occurrence of the explanandum eventwould have been possible, and conversely (Hempel & Oppenheim1948). Inferences from adequate explanations to potential predictionswere generally accepted, but not the converse. Critics, such asMichael Scriven (1962), advanced counter-examples that were based oncorrelations or “common causes”: the occurrence of a stormmight be predicted when cows lie down in their fields, for example,yet their behavior does not explain why the storm occurs. SylvainBromberger offered the example of the length of the shadow cast by aflagpole, which is sufficient to deduce the height of the flagpole andthus satisfies Hempel’s conditions but does not explain why theflagpole has that height (Bromberger 1966). Since logical relationsare non-temporal, Hempel may have taken the symmetry thesis to be atrivial consequence of his account, but deeper issues are involvedhere.

4.3 The Problem of Lawlikeness

Hempel observed that universal generalizations of the form,“\((x)(Fx \supset\) Gx)”, are true if and only iflogically equivalent generalizations of the form, “\((x)(\neg Fx\vee Gx)\)”, are true. But if “\(Fx\)” stands foran uninstantiated property, such as being a vampire, then,since “\(\neg Fx\)” is satisfied by everything, regardlessof the attribute “\(Gx\)”, “\(\neg Gx\)”,etc., that hypothesis—and its logical equivalents—are notonly universally confirmed but even qualify as “laws”.This is not surprising, from a logical point of view, sinceextensional logic is purely truth functional, where the truth value ofmolecular sentences is a function of the truth values of its atomicconstituents. But it implies that an empirical generalization thatmight be true of the world’s history may or may not be“lawful”, since it could be the case that its truth wasmerely “accidental”. Hempel’s endorsement ofGoodman’s approach to select generalizations that supportsubjunctive conditionals, therefore, was not only consistent with thetradition of Hume—who believed that attributions of naturalnecessity in causal relations are merely “habits of mind”,psychologically irresistible, perhaps, but logicallyunwarranted—but circumvented a disconcerting consequence of hisexplication.

There is a fundamental difference between sentences that“support subjunctives” and those that “entailsubjunctives”, when the selection of those that supportsubjunctives is made on pragmatic grounds. Popper captured the crucialdifference between material (extensional) generalizations andsubjunctive (intensional) generalizations as follows:

A statement may be said to benaturally or physicallynecessary if, and only if, it is deducible from a statementfunction which is satisfied in all worlds that differ from our own, ifat 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 initialconditions, would mean that the history of an indeterministic worldmight be identical with the history of a deterministic one, wheretheir differences were concealed “by chance”. Even acomplete description of the history of the actual world might notsuffice to distinguish between them, where fundamental aspects oftheir causal structure would remain beyond empirical detection.

The difference between Popper’s universal predicates andGoodman’s paradoxical ones may derive from the considerationthat incorporating specific times into his definitions entailsreference to specific moments of time \(t\), such asmidnight tonight, during the history of the actual world, apoint to which we are going to return. It follows that they cannot be“universal” in Popper’s sense and do not evenqualify as “purely qualitative” in Hempel’s, either.Reliance upon material conditionals within first-order symbolic logic,moreover, forfeits the benefits of synthetic subjunctives. If thehypothesis, “All ravens are black”, is more adequatelyformalized as a subjunctive of the form, “\((x)(Rx \rightarrowBx)\)”, where the truth of a subjunctive hypothetically assumesthat its antecedent condition is satisfied, that obviates the paradox:white shoes are not ravens, and Nicod’s criteria apply Instancesof “\(Rx\)” and “\(Bx\)” thereby confirm thehypothesis; instances of “\(Rx\)” and “\(\negBx\)” 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 alwaysshown an appreciation for the full dimensions of his explication.Michael Scriven (1959), for example, once objected to “thedeductive model, with its syllogistic form, where no student ofelementary logic could fail to complete the inference, given thepremise” (1959: 462), when no restriction is placed on thecomplexity of the relevant laws or the intricacy of these deductions(Hempel 1965c: 339). Similar objections have been lodged more recentlyby William Bechtel and Adele Abrahamsen (2005), for example, whoclaim, in defense of “the mechanist alternative”, that,among its many benefits,

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

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

Hempel remarks that his model of explanation does not directly applyto the wordless gesticulations of a Yugoslavian automobile mechanic orguarantee that explanations that are adequate are invariablysubjectively satisfying. His purpose was to formalize the conditionsthat must be satisfied when an explanation is adequate without denyingthat background knowledge and prior beliefs frequently make adifference in ordinary conversational contexts. Jan might not knowsome of the antecedent conditions, Jim might have misunderstood thegeneral laws, or features of the explanandum event may have beenmissed by them both. Even when information is conveyed using diagramsand simulations, for example, as long as it satisfies conditions(CA-1) through(CA-4)—no matter whether those conditions are satisfied implicitly orexplicitly—an adequate scientific explanation is at hand. But ithas to satisfy all four of those requirements.

Demonstrating that an adequate scientific explanation is at hand,however, imposes demands beyond the acquisition of information aboutinitial conditions and laws. In theexaminer’s sense ofknowledge Ian Hacking identified (Hacking 1967: 319), to prove anadequate scientific explanation is available, it would be necessary toshow that each of Hempel’s adequacy conditions has beensatisfied. It would not be enough to show, say, that one person isknowledgeable about the initial conditions, another about the coveringlaws, and a third about the explanandum. It would be necessary to showthat the explanandum can be derived from the explanans, that thoselaws were required for the derivation, and that the initial conditionswere those present on that occasion.(CA-1)–(CA-4) qualifies as a “check list” to insure that a scientificexplanation is adequate.

5.1 The Symmetry Thesis

Students of Hempel have found it very difficult to avoid theimpression that Hempel was not only defending the position that everyadequate scientific explanation is potentially predictive but also theposition that every adequate scientific prediction is potentiallyexplanatory. This impression is powerfully reinforced in “TheTheoretician’s Dilemma” (Hempel 1958), where, in thecourse of demonstrating that the function of theories goes beyondmerely establishing connections between observables, he offers thispassage:

Scientific explanations, predictions, and postdictions all have thesame logical character: they show that the fact under considerationcan be inferred from certain other facts by means of specified generallaws. In the simplest case, the type of argument may be schematized asa deductive inference of the following form [here substituting the“simplest case” for the abstract schemata presented in theoriginal]:

\((x)(Rx \supset Bx)\)	Explanans
\(Rc\)
\(Bc\)	Explanandum

Figure 4. A Covering-Law Explanation

… While explanation, prediction, and postdiction are alike intheir logical structure, they differ in certain other respects. Forexample, an argument [like Figure 4 above] will qualify as aprediction only if [its explanandum] refers to an occurrence at a timelater than that at which the argument is offered; in the case ofpostdiction, the event must occur before the presentation of theargument. These differences, however, require no further study here,for the purpose of the preceding discussion was simply to point outthe role of general laws in scientific explanation, prediction, andpostdiction (Hempel 1958: 37–38).

What is crucial about this passage is Hempel’s emphasis on thepragmatic consideration of the time at whichthe argument ispresentedin relation to the time of occurrence of theexplanandum event itself. Let us take this to be the observationthat \(c\) is black and assume that occurs at aspecific time \(t_1\). If the argument is presented prior to time\(t_1\) (before observing that \(c\) is black), then itis a prediction. If the argument is presented after time \(t_1\)(after observing that \(c\) is black), then it is apostdiction. Since explanations are usually advanced after the time ofthe occurrence of the event to be explained, they usually occur aspostdictions. But there is nothing about their form that requiresthat.

As Hempel uses the term, reasoning is “scientific” when itinvolves inferences with laws. Indeed, this appears to be theprincipal ground upon which he wants to maintain that explanations,predictions, and postdictions share the same logical form. Hisposition here makes it logically impossible for explanations,predictions and postdictions to not have the same logical form. Ifthere are modes of prediction that are non-explanatory, such as mightbe the case when they are based upon accidental generalizations, thenthey would not thereby demonstrate that the symmetry thesis does nothold for the kinds of distinctively “scientific”explanations that are the focus of his work. Scriven’s cowexample, for example, does not show that the symmetry thesis does nothold forscientific explanations andscientificpredictions, even if it shows that someordinary predictionsare not alsoordinary explanations. But othercounter-examples posed greater threats.

5.2 The Paradoxes of Explanation

Indeed, in the process of drawing the distinction betweenexplanation-seeking and reason-seeking why-questions, Hempel (1965c)proposed a different kind of symmetry thesis, where adequate answersto explanation-seeking why-questions also provide adequate answers toreason-seeking why-questions, but not conversely. That was veryappropriate, since the purpose of reason-seeking why-questions is toestablish grounds for believing that the event described by theexplanandum sentence has taken (or will take) place, whileexplanation-seeking why-questions typically take for granted that weknow their explanandum events have already occurred. It follows that,while Hempel’s conditions are meant to specifywhen we knowwhy, they also entailhow we know that.

Insofar as Hempel focused on logically equivalent formulations oflawlike sentences in addressing the paradoxes of confirmation, somemay find it remarkable that he does not explore the consequences oflogically equivalent formulations in relation to explanations. We tendto assume we can explain why a specific thing \(c\) isblack on the grounds that it is a raven and all ravens are black. Yetwe would hesitate to explain why a specific thing is not a raven onthe ground that it is not black. We may refer to this astheparadox of transposition. Notice, the contention that one memberof a class of logically equivalent hypotheses is explanatory butothers are not generates “paradoxes of explanation” thatparallel the “paradoxes of confirmation” (Fetzer 2000a).Figure 5 offers an illustration:

Even if we accept the transposed form of “\((x)(Rx \supsetBx)\)” as the basis for an answer to a reason-seekingwhy-question—in this case,how we knowthat\(c\) is no raven—Figure 5 surely does notexplainwhy that is the case, which presumably has to do withthe genes of entity \(c\) and its developmental historyas a living thing. If logically equivalent forms equally qualify aslawlike—there’s no basis in extensional logic to denyit—then Hempel confronts a dilemma. In fact, by his ownconditions of adequacy, this argument is not only explanatory but alsopredictive. It isnot a “postdiction” as he hasdefined it above, but it appears to qualify as a“retrodiction” that does not explain its explanandumevent.

Suppose Hempel were to resort to Goodman’s approach and denythat the negation of a purely qualitative predicates is also a purelyqualitative predicate. That would mean that sentences logicallyequivalent to lawlike sentences—which have the samecontent—are not therefore also lawlike by invoking pragmaticdifferences in their degree of entrenchment and projectibility oftheir constituent predicates. An appeal to pragmatic considerationswithin this context has a decidedlyad hoc quality about it,which appears to contravene the spirit of the paradoxes ofconfirmation, where Hempel insists that the counterintuitive cases areconfirmatory and their paradoxical character is merely psychological.Even if Hempel were to adopt this position and take for granted thatone member of a class of logically equivalent sentences can be lawlikewhile the others are not, another difficulty arises from the use ofmodus tollens in lieu ofmodus ponens, as Figure 6exemplifies:

Here we have a more threatening problem, since there is no apparentbasis for denying that an argument having this form satisfiesHempel’s criteria and therefore ought to beboth explanatoryand predictive. Similarly for temporally quantified arguments,where the fact that a match of kind \(K\) has notlighted at \(t^*\) would certainly not be adequate to explainwhyit was not struck at t, even though every match of kind \(K\)will light when it is struck, under suitableconditions! Something serious thus appears to be wrong (Fetzer2000a).

6. Inductive-Statistical Explanation

In his studies of inductive reasoning, Hempel (1960, 1962a, 1966b)discusses the ambiguity of induction, which arises becauseincompatible conclusions can appear to be equally-well supported byinductive arguments, where all the premises of both arguments aretrue. This problem has no analogue for deductive arguments:inconsistent conclusions cannot be validly derived from consistentpremises, even if they are false. Inconsistent conclusions can receiveinductive support from consistent premises, however, even when theyare all true. Consider Hempel’s illustration of conflictingpredictions 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 thepast decade, 93% had an uneventfully recovery.

This information, taken by itself, would surely lend strong support tothe 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 hadan appendectomy after his appendix had ruptured; and in the pastdecade, of all cases of appendectomy after rupture of the appendixamong octogenarians with serious kidney failure, only 8% had anuneventful recovery.

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

\((\neg\)h1)

Jones will not have an uneventful recovery. (Hempel1966b)

Thus, Hempel observed, (e1) and (e2) are logically compatible andcould all be part of the evidence available when Jones’prognosis is being considered. The solution Hempel endorsed wasthe requirement of total evidence, according to which, inreasoning about the world, arguments must be based upon all theavailable evidence, though he noted that evidence can be omitted whenit is irrelevant and its omission does not affect the level ofsupport. Here, when (e1) is combined with (e2), \((\neg\)h1) is bettersupported than (h1).

6.1 The Requirement of Maximal Specificity

Hempel referred to statistical explanations as“inductive-statistical” in contrast with his priordiscussion of “deductive-nomological” explanations (Hempel1958). Because generalizations of both kinds are supposed to belawlike, 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 premisesattributing a certain probability to outcome \(G\),under conditions \(F\), by “\(P(G, F) =r\)”, where these explanations then assume this form:

While Hempel initially interpreted the bracketed variable [\(r\)]as the measure of inductive support which theexplanans confers upon the explanandum, \(Gi\), (in conformity withthe requirement of total evidence)—where the value of [\(r\)]equals that of \(r\)—itoccurred to him that, since inductive and deductive explanations aretypically offered on the basis of the knowledge that their explanandumevents have already occurred, viewing statistical explanations asinductive arguments whose purpose is to establish how we know thatthey have occurred (or that they will occur) is not the correctperspective for understanding them:

the point of an explanation is not to provide evidence for theoccurrence of the explanandum, but to exhibit it as nomicallyexpectable. And the probability attached to an I-S explanation isthe probability of the conclusion relative to the explanatorypremises, not relative to the total class \(\mathbf{K}\) [representingour total knowledge at that time]. (Hempel 1968, originalemphasis)

In order to formalize this conception, he advanced the concept ofa maximally specific predicate related to “Gi” in\(\mathbf{K},\) where, when “\(P(G, F) = r\)”, then, ifthe premises include a predicate stronger than “\(F\)”,say, “\(M\)”, which implies the presence of moreproperties than does “\(F\)”, then \(P(G, M) = r,\) where,as before, \(r = P(G,F).\) Thus additional predicates could beincluded in the explanans of an adequate scientific explanation,provided that they made no difference to the nomic expectability ofthe explanandum. Including the position of the moon or the day of theweek a match is struck, for example, might be irrelevant, but hiscondition did not exclude them. Wesley C. Salmon (1970), however,would pursue this issue by arguing that Hempel had embraced the wrongstandard upon which to base explanatory relevance relations, whereexplanations, in Salmon’s view, strictly speaking, do not evenqualify 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 nomicexpectability 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 thatexplanations could satisfy Hempel’s criteria, yet not explainwhy their explanandum events occur. For example,

(CE-1)

John Jones was almost certain to recover from his cold within aweek, because he took vitamin C, and almost all colds clear up withina week after administration of vitamin C.

(CE-2)

John Jones experienced significant remission of his neuroticsymptoms, for he underwent extensive psychoanalytic treatment and asubstantial percentage of those who undergo psychoanalytic treatmentexperience significant remission of neurotic symptoms.

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

(CE-3)

This sample of table salt dissolves in water, for it has had adissolving spell cast upon it, and all samples of table salt that havehad dissolving spells cast upon them dissolve in water.

(CE-4)

John Jones avoided becoming pregnant during the past year, for hehas taken his wife’s birth control pills regularly, and everyman who regularly takes birth control pills avoidspregnancy.

For Hempel, a property \(F\) isexplanatorilyrelevant to the occurrence of an outcome \(G\) ifthere is a lawful relationship that relates \(F\) to\(G\). If table salt dissolves in water, so doesMorton’s table salt, Morton’s finest table salt,Morton’s finest table salt bought on sale, and so on. Salmonconcluded that Hempel’s conception of I-S explanation waswrong.

A student of Reichenbach (1938, 1949), Salmon had adopted and defendedthe limiting frequency interpretation of physical probability, where“\(P(G/F) = r\)” means that \(G\) occurswith a limiting frequency \(r\) in a reference class ofinstances of \(F\), which has to be infinite for limitsto exist (Salmon 1967, 1971). On this approach, a property \(H\)isexplanatorily relevant to the occurrenceof 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 limitingfrequency for \(G\) in \(F\)-and-not-\(H\).Properties whose presence does not make a differenceto the limiting frequency for the occurrence of \(G\)in \(F\) would therefore qualify as statisticallyirrelevant. Relative frequencies were typically relied upon inpractice, but in principle they had to be limiting.

Salmon also rejected Hempel’s requirement that the nomicexpectability of a statistical explanation must be equal to or greaterthan .5, which led him to abandon the notion of explanations asarguments. Even events of low probability were explainable within thecontext of Salmon’s approach, which he compared withHempel’s approach as follows:

I-S model (Hempel): an explanation is anargument thatrenders the explanandumhighly probable;

S-R model (Salmon): an explanation is anassembly of factsstatistically relevant to the explanandumregardless of thedegree of probability that results. (Salmon 1971, originalemphasis)

Salmon’s work created a sensation, since Hempel’sdominance of the philosophy of science, especially in relation to thetheory of explanation, now had a significant rival. The students ofexplanation who recognized that probabilities as properties ofinfinite classes posed substantial problems were few, and those whorealized that Salmon’s position confronted difficulties of itsown fewer still.

Salmon was ingenious in devising “screening off” criteriato insure that the properties cited in an S-R explanation wererestricted to those that were statistically relevant. He may haveappreciated that the available evidence was restricted to relativefrequencies in finite sequences, but that could be discounted astypical 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 deeperproblem arose from the existence of properties that were statisticallyrelevant but not explanatorily relevant, however. If women whosemiddle initials began with vowels, for example, experiencedmiscarriages with a different frequency than women whose middleinitials began with consonants—even after other properties weretaken into account—then that property would have to qualify asstatistically relevant andtherefore explanatorily relevant(Fetzer 1974, 1981). This result implied that statistical relevancecannot adequately define explanatory relevance, and Salmon wouldgradually 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 simplyaffirms that a certain percentage of the reference class belongs tothe attribute class. What it asserts instead is that every member ofthe reference class possesses a certain disposition, which in the caseof statistical laws is of probabilistic strength and in the case ofuniversal laws of universal strength. On its single-case formulation,moreover, short runs and long runs are simply finite and infinitesequences of single cases, where each trial has propensities that areequal and independent from trial to trial and classic theorems ofstatistical inference apply.

Even this intensional explication would still be vulnerable to theproblems of irrelevant properties and themodus tollensparadox but for the adoption of a condition to exclude the occurrenceof predicates that are not nomically relevant to the explanandum eventfrom the explanans of an adequate scientific explanation. Thiscriterion, known asthe requirement of strict maximalspecificity, permits the reformulation of Hempel’s fourconditions of adequacy by replacing the redundant empirical contentcondition with this new requirement (Fetzer 1981; Salmon 1989).Explanations not only display thenomic expectability oftheir explanandum events—which, in the case of those that occurwith high probability, would enable them to have been predicted, asHempel proposed but—more importantly—explain them byspecifying all and only those propertiesnomicallyresponsible for their occurrence, even when they occur with lowprobability (Fetzer 1992).

Bromberger’s flagpole counterexample provides a severe test ofthis requirement. The reason why the inference from the length of theshadow to the height of the flagpole is non-explanatory is because thelength of the shadow is notnomically responsible for theflagpole’s height. Hempel’s original conditions could notcope with situations of this kind, where inferences using laws supportpredictions and retrodictions that are not also explanations, eventhough they satisfy all four. This alternative condition thus requiresthat the properties cited in the antecedent of the lawlike premise(s)must be nomically relevant to the explanandum or may not be includedthere (Fetzer 1981, 1992). The height of the flagpole, but not thelength of the shadow, qualifies as nomically relevant, which resolvesthe quandary—at the expense of acknowledging classes ofarguments that arepredictive or retrodictive but notexplanatory, even when they involve inferences from law(s) thatsatisfy Hempel’s original criteria. The reformulated conditionsare:

(CA-1*)

The explanandum must be a suitable logical consequence of theexplanans;

(CA-2)

The explanans must contain general laws, which are required tosatisfy (CA-1*);

(CA-3*)

The explanans must satisfy the requirement of strict maximalspecificity; and,

(CA-4*)

The sentences—both of the explanans and of theexplanandum—must be true.

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

These revised conditions implicitly require abandoning Hempel’scommitment to extensional methodology to capture the notion of nomicresponsibility, but the benefits of an intensional approach appear tobe profound. As Salmon has observed,

[on such an explication] the distinction betweendescription andprediction, on the one hand, andexplanation, on theother, is that the former can proceed in an extensional languageframework, while the latter demands an intensional language framework.It remains to be seen whether the intensional logic can besatisfactorily formulated (Salmon 1989: 172, original emphasis).

This approach to explanation incorporates the causal relevancecriterion, according to which a property \(H\) iscausally relevant to the occurrence of an attribute \(G\)relative to a reference property \(F\)—withinthe context of a causalexplanation—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 fromthe propensity for \(G\), given \(F \amp \neg H\), where the presenceor absence of \(H\) affects the single-case causal tendency for\(G\). The universal generalization of sentential functions like thesethus produceslawlike sentences, while their instantiation toindividual constants or to ambiguous names produces (what are knownas)nomological conditionals (Fetzer 1981: 49–54). Theintroduction of the probabilistic causal calculus \(C\), moreover,responds to Salmon’s concerns by providing a formalizationwithin intensional logic that resolves them (Fetzer & Nute 1979,1980). Paul Humphreys (1985, 1989) has argued that “propensitiescannot be probabilities” because theircausaldirectedness precludes satisfying symmetry conditions requiringprobabilities from effects to causes as well as from causes to effects(on the plausible assumption that there are no propensities foreffects to bring about their causes). A more linguistically precisecharacterization would be that propensities cannot be ‘standardprobabilities’, which has no effect on their role in explanatorycontexts.

7. The Problem of Provisos

What may come as some surprise is that Hempel exposed yet anothersignificant problem confronting the theory of scientific explanation.One of the most remarkable features of his career is that he continuedto publish original and innovative studies well into his eightiethdecade. Rather surprisingly, he authored a series of articles thatmoved away from the influential conception of scientific theories asformal calculi combined withempiricalinterpretations that had been characteristic of logicalempiricism. In Hempel (1966a), at the time, the most widely adoptedintroduction to the philosophy of science (which has been translatedinto ten other languages), he advanced the novel explication ofscientific theories as consisting ofinternal principles andbridge principles, where the lawlike hypotheses thatdistinguish theories are linked to observation, measurement, andexperiment by principles expressed in various mixtures of ordinary andof technical language. Now antecedent understanding replaces explicitdefinability, which seems to have been, in part, a response to thedemise of the observational/theoretical distinction.

Even more strikingly, Hempel (1988a,b) observed that the applicationof scientific theories presupposes the absence of factors that mightaffect the internal principles of the theory, which goes beyond thecontent of the theory itself. Deriving predictions and explanationsfrom classical mechanics presupposes that bodies are being acted uponexclusively by gravitational forces, for example, where the presenceof electromagnetic forces would invalidate those derivations. This isnot simply a matter of conducting tests “under suitableconditions”, where the malleability of gold differs when struckby a hammer when at extremely cold temperatures (where thecombustibility of paper differs when the paper is wet, and so on),which are familiar examples of the various specific conditions thatmust be identified in complete definitions of dispositionalpredicates. What Hempel noticed is that these properties may not onlybe affected by conditions covered by the theory being applied butinvolve entirely different theories. He took this to mean that theapplication of theories has to be accompanied by“provisos” affirming that no properties beyond thosespecified by the theory are present in a specific case.

The function of these provisos means that instrumentalistconstructions of scientific theories as mere calculating devices andprograms for the elimination of theoretical language by reduction toobservational language alone are misguided and cannot be sustained.And this is because, if other theoretical properties make a differenceto the application of a particular theory, then the observationalconsequences of that theory cannot be assumed to obtain in anyinstance without provisos about other theoretical properties beyondthose specified by the theory, which requires separate investigationon the basis of observation, measurement, and experiment. Theconditions for testing, confirming or falsifying alternativehypotheses and theories is thereby rendered vastly more complex thanhad previously been supposed. Strikingly, as an essential element ofan adequate explanation, Coffa (1973) had advanced “extremalclauses” that assert that no factors other than those specifiedby the initial conditions of the explanans are relevant to theoccurrence 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 fromrefutation by discounting disconfirming evidence on the expedient ofclaiming that, because the clause was not satisfied, the law is stillcorrect, which is another way to accent the source of Hempel’sconcern (Salmon 1989: 84–85).

These observations are related to the claim that “the laws ofphysics lie”, which has been suggested by Nancy Cartwright(1983). She has claimed that there are theoretical laws, which aresupposed to be “simple” but rarely if ever instantiated,and phenomenological laws, which are frequently instantiated buttypically if not always complex. Armstrong (1983: 137–140) drawsa distinction between laws that are “iron” and have noexceptions and those that are “oaken” and have noexceptionsas long as no interfering conditions are present.In his language, unless Cartwright has overlooked the possibility thatsome laws might be both complex and true, she seems to be saying thattheoretical laws are “oaken” laws for which there are no“iron” counterparts. If that is right, then she has failedto appreciate the distinction between counterfactual conditionals(which may be truein spite of having no instances) and mereindicative conditionals (which may be truebecause they haveno instances). The “provisos” problem implies thatsatisfying the requirement of maximal specificity may be moredemanding than has been generally understood in the past. Either way,Cartwright’s theses appear to trade on an equivocation betweenthe non-existence of “iron” counterparts and theirnon-availability: even if the “iron” laws within a domainare not yet known, it does not follow that those laws do notexist.

8. Reflections on Rationality

The publication of Thomas S. Kuhn’sThe Structure ofScientific Revolutions (1962), ironically, was among thehistorical developments that contributed to a loss of confidence inscience. Kuhn’s work, which turned “paradigm” into ahousehold word, was widely regarded as assimilating revolutions inscience to revolutions in politics, where one theory succeeds anotheronly upon the death of its adherents. It was interpreted as havingdestroyed the myth that philosophers possess some special kind ofwisdom or insight in relation to the nature of science or the thoughtprocesses of scientists with respect to their rationality, almost asthough every opinion were on a par with every other. A close readingof Kuhn’s work shows that these were not his own conclusions,but they were enormously influential. And among the public at largeand many social scientists, the tendency to no longer hold science inhigh esteem or to be affected by its findings has induced politicalramifications that are inimical to the general good. When our beliefsare not well founded, actions we base upon them are unlikely tosucceed, often with unforeseen effects that are harmful. Rationalactions ought to be based upon rational beliefs, where science hasproven to be the most reliable method for acquiring knowledge aboutourselves and the world around us.

This study supports the conclusion that Hempel’s conception ofscientific explanations as involving the subsumption of singularevents by means of covering laws was well-founded, even though hiscommitment to an extensional methodology inhibited him from embracinga more adequate account of natural laws. The symmetry thesis turns outto require qualification, not only with respect to predictions forevents that occur only with low probability, but also forretrodictions derived bymodus tollens. The link that mostaccurately embodies the relationship between Hempel’s work onexplanation and decision-and-inference, as we have seen, appears to bereflected by thePrincipal Principle, which formalizes therecognition that subjective (or personal) probabilities, as degrees ofbelief, should have the same values as corresponding objectiveprobabilities, when they are known. The values of propensities asproperties of laws, no doubt, should be given precedence over thevalues of frequencies, insofar as laws, unlike frequencies, cannot beviolated and cannot be changed and thus provide the more reliableguide.

A recent trend presumes that the philosophy of science has beenmisconceived and requires “a naturalistic turn” as a kindof science of science more akin to history or to sociology than tophilosophy. In studies published in the twilight of his career, Hempeldemonstrated that, without standards to separate science frompseudo-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 notknow which of those who call themselves “scientists” arescientists and which methods are “scientific”, where theplace of explanation remains central within the context of inferenceto the best explanation (Fetzer 2002). The philosophy of science,therefore, cannot be displaced by history or by sociology. Not leastamong the important lessons of Hempel’s enduring legacy is therealization that the standards of science cannot be derived from meredescriptions of its practice alone but require rational justificationin the form of explications satisfying the highest standards ofphilosophical rigor.

Bibliography

Primary Sources: Works by Hempel

• 1942, “The Function of General Laws in History”,Journal of Philosophy, 39(2): 35–48.doi:10.2307/2017635
• 1945a, “Studies in the Logic of Confirmation”,Mind, 54(213): 1–26 and 54(214):97–121.doi:10.1093/mind/LIV.213.1 doi:10.1093/mind/LIV.214.97
• 1945b, “Geometry and Empirical Science”,AmericanMathematical Monthly, 52(1): 7–17. doi:10.2307/2304830
• 1945c, “On the Nature of Mathematical Truth”,American Mathematical Monthly, 52(10): 543–556.doi:10.2307/2306103
• 1950, “Problems and Changes in the Empiricist Criterion ofMeaning”Revue Internationale de Philosophie, 41(11):41–63.
• 1951, “The Concept of Cognitive Significance: AReconsideration”,Proceedings of the American Academy ofArts and Sciences, 80(1): 61–77. doi:10.2307/20023635
• 1952,Fundamentals of Concept Formation in EmpiricalScience, Chicago: University of Chicago Press.
• 1958, “The Theoretician’s Dilemma”, inConcepts, Theories, and the Mind-Body Problem, (MinnesotaStudies in the Philosophy of Science, vol. II), Herbert Feigl, MichaelScriven, and Grover Maxwell (eds.), Minneapolis: University ofMinnesota Press, pp. 37–98. [Hempel 1958 available online]
• 1960, “Inductive Inconsistencies”,Synthese,12(4): 439–469. doi:10.1007/BF00485428
• 1961, “Rational Action”,Proceedings and Addressesof the American Philosophical Association, 35: 5–23.doi:10.2307/3129344
• 1962a, “Deductive-Nomological vs. StatisticalExplanation”, inScientific Explanation, Space &Time, (Minnesota Studies in the Philosophy of Science, vol. III),Herbert Feigl and Gordon Maxwell (eds.), Minneapolis: University ofMinnesota Press, pp. 98–169. [Hempel 1962a available online]
• 1962b, “Explanation in Science and in History”, inFrontiers of Science and Philosophy, Robert Garland Colodney(ed.), Pittsburgh, PA: University of Pittsburgh Press, pp.9–33.
• 1965a, “Postscript (1964)”, in Hempel 1965d:47–51.
• 1965b, “Empiricist Criteria of Cognitive Significance:Problems and Changes”, in Hempel 1965d: 101–119.
• 1965c, “Aspects of Scientific Explanation”, in Hempel1965d: 331–496.
• 1965d,Aspects of Scientific Explanation, New York, NY:Free Press.
• 1966a,Philosophy of Natural Science, Englewood Cliffs,NJ: Prentice-Hall.
• 1966b, “Recent Problems of Induction”, in Colodny1966: 112–134.
• 1968, “Maximal Specificity and Lawlikeness in ProbabilisticExplanation”,Philosophy of Science, 35(2):116–133. doi:10.1086/288197
• 1970, “On the ‘Standard Conception’ ofScientific Theories”, inTheories & Methods of Physicsand Psychology, (Minnesota Studies in the Philosophy of Science,vol. IV), Michael Radner and Stephen Winokur (eds.), Minneapolis, MN:University of Minnesota Press, 142–163. [Hempel 1970 available online]
• 1979, “Scientific Rationality: Analytic vs. PragmaticPerspectives”, in Théodore F. Geraets (ed.),Rationality To-Day, Ottawa, Canada: The University of OttowaPress, 46–58.
• 1981, “Turns in the Evolution of the Problem ofInduction”,Synthese, 46(3): 193–404.doi:10.1007/BF01130050
• 1983, “Valuation and Objectivity in Science”, inPhysics, Philosophy, and Psychoanalysis: Essays in Honor of AdolfGrunbaum, Robert S. Cohen and Larry Laudan (eds.), Dordrecht, TheNetherlands: Kluwer, 73–100.
• 1988a, “Limits of a Deductive Construal of the Function ofScientific Theories”, in E. Ullmann-Margalit (ed.),Sciencein Reflection. The Israeli Colloquium, (Volume 3), Dordrecht, TheNetherlands: Kluwer, 1–15.
• 1988b, “Provisos: A Problem Concerning the InferentialFunction of Scientific Theories”,Erkenntnis, 28(2):147–164. doi:10.1007/BF00166441

Primary Sources: Coauthored Works

• Hempel, Carl G. and Paul Oppenheim, 1945, “A Definition of‘Degree of Confirmation’”,Philosophy ofScience, 12(2): 98–115. doi:10.1086/286852
• –––, 1948, “Studies in the Logic ofExplanation”,Philosophy of Science, 15(2):135–175. doi:10.1086/286983

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Other Internet Resources

• Burkhart, Ford, “Carl G. Hempel Dies at 92; Applied Science to Philosophy”,The New York Times (November 23, 1997).
• “Carl Gustav Hempel Papers”,University of Pittsburgh, University Library System.
• “Hempel, Carl Gustav”,New World Encyclopedia.
• Murzi, Mauro, “Carl Gustav Hempel (1905–1997)”,Internet Encyclopedia of Philosophy

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