Logics in scientific discovery

ATOCHA ALISEDA LOGICS IN SCIENTIFIC DISCOVERY ABSTRACT. In this paper I argue for a place for logic in scientific methodology, at the same level as that of computational and historical approaches. While it is well known that a a whole generation of philosophers dismissed Logical Positivism (not just for the logic though), there are at least two reasons to reconsider logical approaches in the philosophy of science. On the one hand, the present situation in logical research has gone far beyond the formal developments that deductive logic reached last century, and new research includes the formalization of several other types of reasoning, like induction and abduction. On the other hand, we call for a balanced Philosophy of Science, one in which both methods, the formal and the historical may be complementary, together providing a pluralistic view of science, in which no method is the predominant one. KEY WORDS: abduction, heuristics, logic, scientific discovery 1. INTRODUCTION A central theme in the study of human reasoning is the construction of explanations that give us an understanding of the world we live in. Broadly speaking, abduction is a reasoning process invoked to explain a puzzling observation. If we wake up, and the lawn is wet, we might explain this observation by assuming that it must have rained, or by assuming that the sprinklers have been on. This is a practical setting found in our day-to-day common sense reasoning. Abduction also occurs in more theoretical scientific contexts. For instance, it has been claimed that Johannes Kepler’s great discovery that the orbit of the planets is elliptical rather than circular was a prime piece of abductive reasoning (Hanson, 1961; Peirce 2.623, 1958). What initially led to this discovery was his observation that the longitudes of Mars did not fit circular orbits. However, before even dreaming that the best explanation involved ellipses instead of circles, he tried several other forms. Moreover, Kepler had to make several other assumptions about the planetary system, without which his discovery does not work. His heliocentric view allowed Foundation of Science 9: 339–363, 2004. © 2004 Kluwer Academic Publishers. Printed in the Netherlands. 340 ATOCHA ALISEDA him to think that the sun, so near to the center of the planetary system, and so large, must somehow cause the planets to move as they do. In addition to this strong conjecture, he also had to generalize his findings for Mars to all the other planets, by assuming that the same physical conditions obtained throughout the solar system. This whole process of explanation took many years. Abduction is thinking from evidence to explanation, a type of reasoning characteristic of many different situations with incomplete information. Note that the word explanation – which we treat as largely synonymous with abduction – is a noun which denotes either an activity, indicated by its corresponding verb, or the result of that activity. That is, it may be used both to refer to a finished product, the explanation of a phenomenon, or to an activity, the process that led to that explanation. These two uses are closely related. The process of explanation produces explanations as its product, but the two are not the same. One can relate this distinction to more traditional ones. An example is Reichenbach’s (1938) well-known opposition of ‘context of discovery’ versus ‘context of justification’, which served as a basis for logical positivism. In this tradition, a fundamental assumption is that the methodology of science has logic as its tool and that it deals only with aspects of justification in scientific research. Kepler’s explanation-product “the orbits of the planets are elliptical”, which justifies the observed facts, does not include the explanation-process of how he came to make this discovery. The context of discovery is taken to be purely psychological. This situation is not particular to the philosophy of science. The high level of formal rigour that logical research reached in the 20th century, is due on the one hand to its divorce from Psychologism and on the other hand in compliance with the deductive method inherited from Euclides, in which all (geometrical) truths are derived from a few basic axioms and rules of inference. Studies of the ways in which proofs are generated or new axioms invented, has been totally outside formal analysis. If something is ever mentioned, it is only part of the history of mathematics. Our claim, however, is that the study of the context of discovery allows, to some extent, for a precise formal treatment. Cognitive psychologists study mental patterns of discovery, learning theorists LOGICS IN SCIENTIFIC DISCOVERY 341 in artificial intelligence study formal hypothesis formation, and one can even work with concrete computational algorithms that produce explanations. Reinchenbach’s distinction between the contexts of justification and of discovery has left out of its analysis – especially from a formal point of view – a very important part of scientific practice, that which includes issues related to the generation of new theories and scientific explanations, concept formation as well as aspects of progress and discovery in science. Section two deals with the contexts of research in the methodology of science in philosophy as well as in artificial intelligence and cognitive psychology. Section three argues for the role of logics in scientific methodology, and section four gives a general account of the logic of abduction. Finally, section five presents further challenges and suggests directions for future research. 2. CONTEXTS OF RESEARCH IN THE METHODOLOGY OF SCIENCE 2.1. Context of Justification The dominant trend in philosophy has focused on abduction as product rather than as process, just as it has done for other epistemic notions. Aristotle, Mill, and in the XXth century, the influential philosopher of science Carl Hempel, all based their accounts of explanation on proposing criteria to characterize its products. These accounts generally may be classified as argumentative and nonargumentative types of explanation (Ruben, 1990; Salmon, 1990). Of particular importance is the ‘argumentative’ Hempelian tradition. Its followers aim to model empirical why-questions, whose answers are scientific explanations in the form of arguments. In these arguments, the ‘explanandum’ (the fact to be explained) is derived (deductively or inductively) from the ‘explanans’ (that which does the explaining) supplemented with relevant ‘laws’ (general or statistical) and ‘initial conditions’. For instance, the fact that an explosion occurred may be explained by my lighting the match, given the laws of physics, and initial conditions to the effect that oxygen was present, the match was not wet, etcetera. Therefore, explanations are public objects of ‘justification’, that can be checked and tested by independent logical criteria. 342 ATOCHA ALISEDA In its deductive version, the Hempelian account, found in many standard texts on the philosophy of science (e.g. Salmon, 1990) is called deductive-nomological, for obvious reasons. But its engine is not just standard deduction. Additional restrictions must be imposed on the relation between explanans and explanandum, as neither deduction nor induction is a sufficient condition for genuine explanation. To mention a simple example, every formula is derivable from itself (E ⊢ E), but it seems counterintuitive, or at least very uninformative, to explain anything by itself. Other, non-deductive approaches to explanation exist in the literature. For instance, Ruben (1990) points out these two: Salmon (1977, p. 159) takes them to be: “an assemblage of factors that are statistically relevant . . .”, while van Frassen (1980, p. 134) makes them simply: “an answer”. For Salmon, the question is not how probable the explanans renders the explanandum, but rather whether the facts adduced make a difference to the probability of the explanandum. Moreover, this relationship need not be in the form of an argument. For van Fraassen, a representative of pragmatic approaches to explanation, the explanandum is a contrastive whyquestion. Thus, rather than asking “why E?”, one asks “why E rather than G?”. The pragmatic view seems closer to abduction as a process, and indeed, the focus on questions introduces some dynamics of explaining. Still, it does not tell us how to produce explanations. There are also alternative deductive approaches. Another, and rather famous deductivist tradition is Popper’s logic of scientific discovery (1958). Its method of conjectures and refutations proposes the testing of hypotheses, by attempting to refute them: The actual procedure of science is to operate with conjectures: to jump to conclusions – often after one single observation. (Popper, 1963, p. 53) Thus science starts from problems, and not from observations; though observations may give rise to a problem, specially if they are unexpected; that is to say, if they clash with our expectations or theories. (Popper, 1963, p. 222) Popper’s deductive focus is on refutation of falsehoods, rather than explanation of truths. One might speculate about a similar ‘negative’ counterpart to abduction. Although Popper’s method claims to be a logic of scientific discovery, he views the actual construction of explanations as an exclusive matter for psychology – and hence his LOGICS IN SCIENTIFIC DISCOVERY 343 ‘trial and error’ philosophy offers no further clues for abduction as a logic of discovery. What is common to all these approaches in the philosophy of science is the conception of scientific practice as a finished product and not as a process in itself. They characterize notions like explanation through logical, statistical or pragmatic criteria, but do not describe how is it that explanations are constructed. However, they do highlight the importance of a hidden parameter in abduction. Whether with Hempel, Salmon, or Popper, scientific explanation never takes place in isolation, but always in the context of some background theory. 2.2. Context of Discovery It is well-known that great philosophers and mathematicians have been brilliant exceptions in the study of discovery in science, and that their non conventional contributions to this field, although great inspirations, have not set new paradigms in the methodology of science. An example is the work of Rescher (1978), which introduces a direction of thought. Interestingly, this establishes a temporal distinction between ‘prediction’ and ‘retroduction’ (another term for abduction), by marking the precedence of the explanandum over the hypothesis in the latter case. Another author emphasizing explanation as a process of discovery is Hanson (1961), who gave an account of patterns of discovery, recognizing a central role for retroduction. Also relevant here is the work by Lakatos (1976), a critical response to Popper’s logic of scientific discovery: There is no infallibilist logic of scientific discovery leading infallibly to results, but there is a fallibilistic logic of discovery which is the logic of scientific progress. But Popper, who has laid the basis for this logic of discovery was not interested in the meta-question of what is the nature of this investigation, so he did not realize that it is neither psychology nor logic, but an independent field, the logic of discovery, heuristics. (Lakatos, 1976, p. 167. My emphasis.) It is interesting to note that Lakatos was greatly inspired by the history of mathematics, paying particular attention to processes that created new concepts – often referring to G. Polya as the founding father of heuristics in mathematical discovery. What all these examples reveal is that in science, explanation involves the invention of new concepts, just as much as the positing 344 ATOCHA ALISEDA of new statements (in some fixed conceptual framework). So far, this has not led to extensive formal studies of concept formation, similar to what is known about deductive logic. (Exceptions that prove the rule are occasional uses of Beth’s Definability Theorem in the philosophical literature. A similar lacuna vis-à-vis concept revision exists in the current theory of belief revision in Artificial Intelligence (Thagard, 1992; Kuipers, 1999; Aliseda, 2000).) Several other exceptions in the study of the context of discovery worthy of mention include the pioneering work of Herbert Simon and his team (Simon et al., 1981; Langley et al., 1987) who conceives scientific reasoning as ‘problem solving’ and proposes to use the machinery of his programs based on heuristic search (e.g. General Problem Solver) to develop programs that simulate the discovery of quantitative laws (the famous BACON) in Physics and qualitative laws (GLAUBER) in chemistry. In the same spirit, Paul Thagard proposes a new field of research, “Computational Philosophy of Science” (Thagard, 1988) and puts forward the computational program PI (Processes of Induction) to model some aspects of scientific practice, such as concept formation and theory building. Even though these approaches come from apparently distant disciplines to philosophy of science, namely cognitive psychology and artificial intelligence, they are proposals which suggest the inclusion of computational tools in the philosophy of science research methodology and by so doing claim to reincorporate aspects from the context of discovery within its agenda. The key concept in all this is that of heuristics, the guide in scientific discovery which is neither totally rational nor absolutely blind. However, despite the fact that computational approaches propose a dynamic view, in which notions like explanation and theory building are not regarded as finished products but rather as processes in themselves, the role of logic seems to be foreign to their methodology. Their systems are based on concepts like heuristic search in artificial intelligence, but the proposed computational systems lack logical foundation (we will argue for a logic with heuristics in the last section). Researchers in this field resemble historians rather than (post)positivists with computers. The fact that they use computational methods seems to place them far from the LOGICS IN SCIENTIFIC DISCOVERY 345 historical approach and close to the formal one, but a quick view to their programs shows that their program design is inspired by historical reconstructions of the scientific discoveries they simulate. There are fundamental differences in the assumptions they take for their implementations, showing in many cases that their discrepancies are of historical and not of computational nature. For instance, there is no agreement in the relevant assumptions that Kepler uses for his discovery that the orbits of the planets are elliptical rather than circular. While for Thagard (1992) the main assumption is that the earth is stationary and the sun moves around it, Simon et al. (1987) and his team together with Hanson (1961), think that Kepler’s discovery would have been impossible without an heliocentric view of the planetary system. 3. LOGIC(S) FOR SCIENTIFIC METHODOLOGY Logic, classical or otherwise, in philosophy of science is nowadays, to put it simply, out of fashion. In fact, although classical logic is still part of the curricula in philosophy of science graduate programs, students soon learn that a whole generation of philosophers regarded logical positivism as a failed attempt (though not just for the logic), claiming that scientific practice does not follow logical patterns of reasoning, many of which favoured studies of science based on historical cases. So, why bother about the place of logic in scientific research? On the one hand, the present situation in logical research has gone far beyond the formal developments that deductive logic reached last century, and new research includes the formalization of several other types or reasoning, like induction and abduction. On the other hand, we claim for a balanced philosophy of science, one in which both methods, the formal and the historical may be complementary, together providing a pluralistic view of science, in which no method is the predominant one. 3.1. Logical Research in the XIX Century As for the first claim, one can place logical research in this century in a broader setting for general logic (van Benthem, 1996), looking back at the original program of Bernard Bolzano (1781–1848), in his 346 ATOCHA ALISEDA “Wissenschaftslehre” (Bolzano, 1973), in which he engaged (among other things) in the study of different varieties of inference. Bolzano’s notion of deducibility (Ableitbarkeit) has long been recognized as a predecessor of Tarski’s notion of logical consequence (Corcoran, 1975). However, the two differ in several respects and in our broader view of logic they even appear radically different. These differences have been studied both from a philosophical (Thompson, 1981) and from a logical point of view (van Benthem, 1984). One of Bolzano’s goals was to show why the claims of science form a theory as opposed to an arbitrary set of propositions. For this purpose, he defines his notion of deducibility as a logical relationship extracting conclusions from premises forming compatible propositions, those for which some set of ideas make all propositions true when uniformly substituted throughout. In addition, compatible propositions must share common ideas. Bolzano’s use of ‘substitutions’ is of interest by itself, but for our purposes here, we will identify these (somewhat roughly) with the logicomathematical use of ‘models’. Thompson attributes the difference between Bolzano’s consequence and Tarski’s to the fact that the former notion is epistemic while the latter is ontological. These differences have pronounced technical effects. With Bolzano, the premises must be consistent (sharing at least one model), with Tarski, they need not. Therefore, from a contradiction, everything follows for Tarski, and nothing for Bolzano. Restated in model-theoretic terms, Bolzano’s notion of deducibility reads as follows (cf. van Benthem, 1984): T, C ⇒ E if (1) The conjunction of T and C is consistent. (2) Every model for T plus C verifies E. Therefore, Bolzano’s notion may be seen (anachronistically) as Tarski’s consequence plus the additional condition of consistency. Bolzano does not stop here. A finer grain to deducibility occurs in his notion of exact deducibility which imposes greater requirements of ‘relevance’. A modern version, involving inclusion-minimality for sets of abducibles, may be transcribed (again, with some historical injustice) as: T, C ⇒+ E if LOGICS IN SCIENTIFIC DISCOVERY 347 (1) T, C ⇒ E (2) There is no proper subset of C, C′ , such that T, C′ ⇒ E. That is, in addition to consistency with the background theory, the premise set C must be ‘fully explanatory’ in that no subpart of it would suffice for the derivation. Notice that this leads to non-monotonicity.1 Here is an example: T, a → b, a ⇒+ b T, a → b, a, b → c
+ b Bolzano’s agenda for logic is relevant to the study of abductive reasoning (and the study of general non-monotonic consequence relations) for several reasons. It suggests the methodological point that what we need is not so much proliferation of different logics as a better grasp of different styles of consequence. Moreover, his work supports the claim we present in what follows, that truth is not all there is to understanding explanatory reasoning. 3.2. Logical Research in the XX Century and Beyond In the first half of the XX century, Tarski’s notion of truth gave meaning to the notion of logical consequence. A conclusion follows if it is true in all models where the premises are true. But the contemporary philosophical and computational traditions have shown that natural notions of inference may need more than truth in the above sense, or may even hinge on different properties altogether. For example, among the candidates which revolve around truth, statistical inference requires not total inclusion of premise models in conclusion models, but only a significant overlap, resulting in a high degree of certainty. Other approaches introduce new semantic primitives. Notably, Shoham’s notion of causal and default reasoning (Shoham, 1988) introduces a preference order on models, requiring only that the most preferred models of the premises be included in the models of the conclusion. More radically, dynamic semantics replaces the notion of truth by that of information change, aiming to model the flow of information. This move leads to a redesign for Tarski semantics, with e.g. quantifiers becoming actions on assignments (van Benthem and Cepparello, 1994). This logical paradigm has room for many different inferential notions (Groeneveld, 1995). An example is update- 348 ATOCHA ALISEDA to-test-consequence: “Process the successive premises, thereby absorbing their informational content into the initial information state. At the end, check if the resulting state is rich enough to satisfy a conclusion”. Informational content rather than truth is also the key semantic property in situation theory (Perry and Barwise, 1983). In addition to truth-based and information-based approaches, there are, of course, also various proof-theoretic variations on standard consequence. Examples are deafult reasoning: “E is provable unless and until E is disproved” (Reiter, 1980), and indeed Hempel’s hypothetico-deductive model of scientific inference itself. There are new notions of derivability as, for example, the one that dictates that a conclusion is followed only with a high degree of certainty (e.g. statistical inference). There are also more dynamic versions in which the concern is to incorporate a new piece of information into a database, a scientific theory, or a set of common sense beliefs (Gärdenfors, 1988). In this approach (a work whose roots lie in the philosophy of science), three main types of belief change are operations of ‘expansion’, ‘contraction’, and ‘revision’. A theory may be expanded by adding new formulas, contracted by deleting existing formulas, or revised by first being contracted and then expanded. These operations are defined in such a way as to ensure that the theory or belief system remains consistent and suitably ‘closed’ when incorporating new information. In connection to philosophy of science, there are some proposals that lay bridges between non-monotonic logics and Hempel’s models for explanation. In Tan (1992) the inductive statistical model is constructed based on Reiter’s default logic and in (Aliseda, 1997) two models of scientific explanation (deductive and statistical) are presented as cases of abductive logic, thus claiming that these models no dot follow the canons of classical logic. All these logics evolved within the last two decades of the XX century, seeking to model common sense as well as scientific reasoning. Therefore they need to allow for a richer language and logical formats than the one developed in classical logic for mathematical reasoning. All above alternatives agree with our view of abduction. On our view, abduction is not a new notion of inference. It is rather a topic-dependent practice of explanatory reasoning, which can LOGICS IN SCIENTIFIC DISCOVERY 349 be supported by various members of the above family. In fact, it is appealing to think of abductive inference in several respects, as inference involving preservation of both truth and explanatory power. In fact, appropriately defined, both might turn out to be equivalent. It has also been argued that since abduction is a form of reversed deduction, just as deduction is truth-preserving, abduction must be falsity-preserving (Michalski, 1994). However, Flach (1995) gives convincing arguments against this particular move. Moreover, as we have already discussed intuitively, abduction is not just deduction in reverse. As stated earlier, we argue for a balanced philosophy of science, one in which we take advantage of a variety of methodologies, such as logical, computational and even historical, all together giving a broad view of science. I think present research has finally overcome the view that a single treatment in scientific methodology is enough to give us an understanding of scientific practice, a view that was already fostered by Pat Suppes (1979) in the late sixties. We know at present that logical models (classical or otherwise) are insufficient to completely characterize notions like explanation, confirmation or falsification in philosophy of science, but this fact does not exclude that some problems in the history of science may be tackled from a formal point of view. For instance, “some claims about scientific revolutions, seem to require statistical and quantitative data analysis, if there is some serious pretension to regard them with the same status as other claims about social or natural phenomena” (Suppes, 1979, p. 97). In fact, “Computational Philosophy of Science” may be regarded as a successful marriage between historical and formal approaches. It is argued that although several heuristic rules have been derived from historical reconstructions in Science, they are proposed to be used for future research (Meheus and Nickles, 1999). This is not to say however, that historical analysis of scientific practice could be done in a formal fashion, or that logical treatment should care for some kind of “historic parameter” in its methodology, but we claim instead that these two views should share their insights and findings in order to complement each other. After all, recall that part of Hempel’s inspiration to device a “logic of 350 ATOCHA ALISEDA explanation” were his studies on the logic for ‘drawing documents from history’. 4. LOGIC(S) FOR ABDUCTION So far I have argued for a place for logic in scientific methodology, at the same level of computational and historical approaches. However, I have not proposed a particular logical enterprise with a specific application to philosophy of science. Instead of giving the precise details of any single existing proposal, I will sketch the general problems and solutions toward a logic for abductive explanations. In the next section I briefly summarize several other proposals. To start with, the construction of an explanation depends on what we take to be the scientific background theory. But even this is not the only parameter that plays a role, there are still further complicating factors. Sometimes, no single obvious explanation is available, but rather several competing ones – and we have to select. Sometimes, the explanation involves not just advancing facts or rules in our current conceptual frame, but rather the creation of new concepts, that allow for new description of the relevant phenomena. Evidently, we must draw a line somewhere for a formal study of abductive reasoning. Abductive reasoning is not a case of ordinary deductive reasoning, and this for a number of reasons. In particular, the explanations produced might be defeated. Maybe the lawn is wet (cf. Introduction) because children have been playing with water. Co-occurrence of clouds and the lawn being wet does not necessarily link them in a causal way. What we learn subsequently can invalidate an earlier abductive conclusion. Moreover, the reasoning involved in this type of thinking seems to go in the reverse order, as compared with deduction, as all these cases run from evidence to hypothesis, and not from premises to conclusion, as is usual in deductive patterns. Finally, describing the way in which an explanation is found, does not seem to follow specific rules. Indeed, the precise nature of Kepler’s ‘discovery’ remains under intensive debate. In what follows, we shall describe abduction in artificial intelligence. Research in this field dates back to the seventies (Pople, LOGICS IN SCIENTIFIC DISCOVERY 351 1973), but it is only fairly recently that it has attracted great interest, in areas like logic programming, knowledge assimilation (Kakas et al., 1993), and diagnosis (Poole, 1987), to name just a few. Abduction is also coming up in the context of data bases and knowledge bases, that is, in mainstream computer science. In this setting, the product-process distinction has a natural counterpart, namely, in logic-based vs computational-based approaches to abduction. While the former focuses on giving a semantics to the logic of abduction, usually defined as ‘backwards deduction plus additional conditions’, the latter is concerned with providing algorithms to produce abductions. It is impossible to give an overview here of this exploding field. Therefore, we limit ourselves to a brief description of abduction as logical inference. There is much more in this field of potential philosophical interest, however. For abduction in Bayesian networks, connectionism, and many other angles, the reader is advised to consult Josephson (1994), Konolige (1996) and Paul (1993). 4.1. Abduction as Logical Inference The general trend in logic based approaches to abduction in AI interprets abduction as backwards deduction plus additional conditions. This brings it very close to deductive-nomological explanation in the Hempel style, witness the following format. What follows is the standard version of abduction as deduction via some consistent additional assumption, satisfying certain extra conditions. It combines some common requirements from the literature (cf. Konolige, 1990; Kakaes et al., 1993; Mayer and Pirri, 1993; and Aliseda, 1997 for further motivation): Given a theory T (a set of formulae) and a formula E, C is an explanation if (1) T ∪ C |= E (2) C is consistent with T (3) C is ‘minimal’ (there are several ways to characterize minimality, cf. Aliseda, 1997) (4) C has some restricted syntactical form (usually an atomic formula or a conjunction of them). An additional condition not always made explicit is that T | = E. This says that the fact to be explained should not already follow from the 352 ATOCHA ALISEDA background theory alone. Sometimes, the latter condition figures as a precondition for an abductive problem. What can one say in general about the properties of such an ‘enriched’ notion of consequence? A logical trend in AI studies variations of classical consequence via their ‘structural rules’, which govern the combination of basic inferences, without referring to any special logical connectives. (Cf. the analysis of non-monotonic consequence relations in AI of Gabbay (1994), Kraus et al. (1990), and the analysis of dynamic styles of inference in linguistics and cognition in van Benthem (1996).) Perhaps the first example of this approach in abduction is the work in Flach (1995) – and indeed our analysis in Aliseda (1997) follows this same pattern. 4.2. A Taxonomy for Abduction What we have seen so far may be summarized as follows. Abduction is a general process of explanation, whose products are specific explanations, with a certain inferential structure. We consider these two aspects to be of equal importance. Moreover, on the process side, we distinguished between constructing possible explanations and selecting the best one amongst these. As for the logical form of abduction, we have found that it may be viewed as a threefold relation: T, C ⇒ E between an observation E, an abduced item C, and a background theory T. (Other parameters are possible here, such as a preference ranking – but these would rather concern the further selection process.) Against this background, we propose three main parameters that determine types of abduction. (i) An ‘inferential parameter’ (⇒) sets some suitable logical relationship among explananda, background theory, and explanandum. (ii) Next, ‘triggers’ determine what kind of abduction is to be performed: E may be a novel phenomenon, or it may be in conflict with the theory T. (iii) Finally, ‘outcomes’ (C) are the various products of an abductive process: facts, rules, or even new theories. In the above schema, the notion of explanatory inference ⇒ is not fixed. It can be classical derivability ⊢ or semantic entailment |=, but it does not have to be. Instead, we regard it as a parameter which can be set independently. It ranges over such diverse values LOGICS IN SCIENTIFIC DISCOVERY 353 as probable inference (T, C ⇒probable E), in which the explanans renders the explanandum only highly probable, or as the inferential mechanism of logic programming (T, C ⇒prolog E). Further interpretations include dynamic inference (T, C ⇒dynamic E, cf. van Benthem, 1996), replacing truth by information change potential along the lines of belief update or revision. Our point here is that abduction is not one specific non-standard logical inference mechanism, but rather a way of using any one of these. According to Peirce, abductive reasoning is triggered by a surprising phenomenon. The notion of surprise, however, is a relative one, for a fact E is surprising only with respect to some backgound theory T providing ‘expectations’. What is surprising to me (e.g. that the canal bridge floor goes up from time to time) might not be surprising to a Dutch person. We interpret a surprising fact as one which needs an explanation. From a logical point of view, this assumes that the fact is not already explained by the background theory T: T
E. Moreover, our claim is that one also needs to consider the status of the negation of E. Does the theory explain the negation of observation instead (T ⇒ ¬E)? Thus, we identify at least two triggers for abduction: novelty and anomaly: Abductive Novelty: T
E, T
¬E E is novel. It cannot be explained (T
E), but it is consistent with the theory (T
¬E). Abductive Anomaly: T
E, T ⇒ ¬E E is anomalous. The theory explains rather its negation (T ⇒ ¬E). In the computational literature on abduction, novelty is the condition for an abductive problem (Kakas et al., 1993). My suggestion is to incorporate anomaly as a second basic type. Of course, non-surprising facts (where T ⇒ E) should not be candidates for explanation. Even so, one might speculate if facts which are merely probable on the basis of T might still need explanation of some sort to further cement their status. Abducibles themselves come in various forms: facts, rules, or even theories. Sometimes one simple fact suffices to explain a surprising phenomenon, such as rain explaining why the lawn is wet. 354 ATOCHA ALISEDA In other cases, a rule establishing a causal connection might serve as an explanation, as in our case connecting cloud types with rainfall. And many cases of abduction in science provide new theories to explain surprising facts. These different options may sometimes exist for the same observation, depending on how seriously we want to take it. Moreover, we are aware of the fact that genuine explanations sometimes introduce new concepts, over and above the given vocabulary. (For instance, the eventual explanation of planetary motion was not Kepler’s, but Newton’s, who introduced a new notion of ‘force’ – and then derived elliptic motion via the Law of Gravity.) Abduction via new concepts will be outside the scope of our analysis. Once the above parameters get set, several kinds of abductive processes arise. For example, abduction triggered by novelty with an underlying deductive inference, calls for a process by which the theory is expanded with an explanation. The fact to be explained is consistent with the theory, so an explanation added to the theory accounts deductively for the fact. However, when the underlying inference is statistical, in a case of novelty, theory expansion might not be enough. The added statement might lead to a ‘marginally consistent’ theory with low probability, which would not yield a strong explanation for the observed fact. In such a case, theory revision is needed (i.e. removing some data from the theory) to account for the observed fact with high probability. Our aim is not to classify abductive processes, but rather to point out that several kinds of them are used for different combinations of the above parameters (in Aliseda (1997) we explore in detail some procedures for computing different types of outcomes in a deductive format). This taxonomy gives us the big picture of abductive reasoning. We can now see the patterns in clearer focus. Varying the inferential parameter, we cover not only cases of deduction but also statistical inferences. Thus, Hempel’s statistical model of explanation also becomes a case of abduction. As for triggers, novelty drives the rain examples and trigger by anomaly occurs in the Kepler example, since his initial observation of the longitudes of Mars contradicted the previous rule of circular orbits of the planets. Different forms LOGICS IN SCIENTIFIC DISCOVERY 355 of outcomes will play a role in different types of procedures for producing explanations. In computer science jargon, triggers and outcomes are, respectively, preconditions and outputs of abductive devices, whether these be computational procedures or inferential ones. 4.3. Abductive Logic: Inference + Search Strategy Classical logical systems have two components: a semantics and a proof theory. The former aims at characterizing what it is for a formula to be true in a model, and it is based on the notions of truth and interpretation. The latter characterizes what counts as a valid proof for a formula, by providing the inference rules of the system; having for its main notions proof and derivability. These two formats can be studied independently, but they are closely connected. At least in classical (first-order) logic, the completeness theorem tells us that all true formulas have a proof, and the converse claim made by the Soundness theorem. Many logical systems have been proposed that follow this pattern: propositional logic, predicate logic, modal logic, and various typed logics. From a modern perspective, however, there is much more to reasoning than this. Computer science has posed a new challenge to logic; that of providing automatic procedures to operate logical systems. This requires a further fine-structure of reasoning. In fact, recent studies in AI give precedence to a control strategy for a logic over its complete proof theory. In particular, the heart of logic programming lies in its control strategies, which lead to much greater sensitivity as to the order in which premises are given, the avoidance of search loops, or the possibility to cut proof searches (using the extra-logical operator!) when a solution has been found. These features are extralogical from a classical perspective, but they do have a clear formal structure, which can be brought out, and has independent interest as a formal model for broader patterns of argumentation (cf. van Benthem, 1996; Kalsbeek, 1995; Kowalski, 1979). Several contemporary authors stress the importance of control strategies, and a more finely-structured algorithmic description of logics. This concern is found both in the logical tradition (Gabbay, 1994; van Benthem, 1996), and in the philosophical tradition 356 ATOCHA ALISEDA (Gillies, 1996), the latter arguing for a conception of logic as: inference + control. (Note the shift here away from Kowalski’s famous dictum “Algorithm = Logic + Control”.) In line with this philosophy, we wish to approach abduction with two things in mind. First, there is the inference parameter, already discussed, which may have several interpretations. But given any specific choice, there is still a significant issue of a suitable search strategy over this inference, which models some particular abductive practice. The former parameter may be defined in semantic or proof-theoretic terms. The search procedure, on the other hand, deals with concrete mechanisms for producing valid inferences. It is then possible to control which kinds of outcome are produced with a certain efficiency. In particular, in abduction, we may want to produce only ‘useful’ or ‘interesting’ formulas, preferably even just some ‘minimal set’ of these. In this light, the aim of an abductive search procedure is not necessarily completeness with respect to some semantics. A procedure that generates all possible explanations might be of no practical use, and might also miss important features of human abductive activity. 5. DISCUSSION AND FURTHER CHALLENGES Our positional claim for this paper has been that the study of the context of discovery allows for a precise formal treatment. The dominant trend in contemporary philosophy of science neglected the study of the processes of theory discovery in science, partly because other questions were on focus, still we knew nothing about the canons of justification, and the historical approach to scientific practice was making its way into methodological issues. It was research on scientific discovery as problem solving by computational means that really brought back issues of discovery to the philosophy of science agenda. And the field was ready for it, as philosophy of science questions had also evolved. Issues of theory building, concept formation, theory evaluation and scientific progress were at issue. It is now time to unravel these questions with the hand of logical analysis. And here again, we get inspiration, or rather the techniques, from work in logics for common sense reasoning and scientific reasoning developed for artificial intelligence. The LOGICS IN SCIENTIFIC DISCOVERY 357 present situation resembles the time of pre-Fregean logic, when the boundaries between logic and general methodology were still rather fluid, as witnessed by the work of Bolzano in his several conceptions of consequence relations, and later by Charles Peirce’s investigations into styles of reasoning (deduction, induction, abduction) from a logical perspective. In our view, current post-Fregean logical research is slowly moving back towards this same broader agenda. Our claim is that we must bring back logic into scientific methodology, on a par with computational and historical approaches. This move would complement the existing historical account, and as far as its relation to computational approaches in scientific discovery is concerned, it may bring benefits in regard to giving a foundation to the heuristics, to the logic(s) behind the (human) processes of discovery in science. Let me now answer the following question (raised by Pat Langley at the time of the conference): how to integrate a logic in the algorithmic design for scientific discovery? Well, we must deal with a logic which besides (or even aside from) its semantics and proof theory, gives an account of the search strategy tied to a discovery system. That is, we must provide the automatic procedures to operate a logic, its control strategy, its procedures to acquire new information without disturbing its coherence and hopefully, achieve some learning in the end. By putting forward a logic for scientific discovery we claim no lack of rigour. But what is clear is that standard deductive logic cannot account for abductive or inductive types of reasoning. And the field is now ready to use new logics for this purpose. An impressive body of results has been obtained about deduction – and a general goal could be the study the wider field of human reasoning while hanging on to these standards of rigour and clarity. In order to glance at the logical structure of abductive reasoning, we have proposed a general taxonomy. Several forms of abduction are obtained by instantiating three parameters: the kind of reasoning involved (e.g., deductive, statistical), the kind of observation triggering the abduction (novelty or anomaly with respect to some background theory), and the kind of explanations produced (facts, rules, or theories). But there is much more to the logic of abduction than what we have been able to describe in this paper. The 358 ATOCHA ALISEDA product-process distinction in abduction is given by defining it as a logical inference on the one hand, and by giving search strategies to produce explanations on the other. However, there is no single way to define either its inferential pattern or its search strategy (cf. Aliseda (1997) for several characterizations of abductive inference as well as for algorithms to produce abductions). The fact of the matter is that abduction is a non-standard logical system: it is not a single phenomenon but rather a general practice of explanatory reasoning resulting in a family of logics. There are still many challenges ahead for the formal study of reasoning in scientific discovery, such as giving an integrated account of deductive, inductive, abductive and analogical styles of inference, the use of diagrams by logical means, and in general the device of logical operations for theory building and change. Already new logical research (some of it presented at MBR’01) is moving into these directions. In Burger and Heidema (forthcoming) degrees of ‘abductive boldness’ are proposed as spectrum for inferential strength, ranging from cases with poor background information to those with (almost) complete information. Systems dealing with several notions of derivability all at once have also been proposed. A formula may be ‘unconditionally derived’ or ‘conditionally derived’, the latter case occurring when a line in a proof asserts a formula which depends on hypotheses which may be later falsified, thus pointing to a notion of proof which allows for addition of lines which are non-deductively derived as well as for deletion of them when falsifying instances occur. This account is found in the framework of (ampliative) adaptive logics, a natural home for abductive inference (cf. Meheus, forthcoming) and for (enumerative) induction inference (cf. Batens, forthcoming) alike. The formalization of analogical reasoning is still a new area of research (I do not know of any rigorous account for it), without even a clear idea of what exactly an analogy amounts to. Perhaps research on mathematical analogy (Polya, 1945) or investigations into analogical argumentation theory recently proposed for abduction (Gabbay and Woods, forthcoming), may serve to guide research in this direction. Finally, the study of ‘diagrammatic reasoning’ is a research field in its own right (Barwise and Etchemendy, 1995), showing that the logical language is not restricted to the two- LOGICS IN SCIENTIFIC DISCOVERY 359 dimensional left to right syntactic representation, but its agenda still needs to be expanded on research for non-deductive logics. As for formal approaches for theory building and change applied to philosophy of science, my present research in this direction concerns the extension of a classical logical method to model empirical progress in science, as conceived by Theo Kuipers (1999, 2000). In particular, the goal is to operationalize the task of instrumentalist abduction, that is, theory revision aiming at empirical progress. My account shows that evaluation and improvement of a theory can be modeled by (an extension of) the framework of Semantic Tableaux (Aliseda, forthcoming). All the above suggests that the use of non-standard logics to model processes in scientific practice, such as confirmation, falsification, explanation building and theory improvement is, after all, a feasible project. Nevertheless, this claim requires a broad conception of what logic is. Still, many logicians remain doubtful, and withhold the status of bona fide ‘logical inference’ to the products of non-standard styles. This situation is somewhat reminiscent of the emergence of non-Euclidean geometries in the nineteenth century. Euclidean geometry was thought of as the one and only geometry until the fifth postulate (the parallel axiom) was rejected, giving rise to new geometries. Most prominently are the one by Lobachevsky which admits of more than one parallel, and the one by Riemann admitting none.2 The legitimacy of these geometries was initially doubted but their impact gradually emerged. The analogy with logic can be carried out even further, as these new geometries were sometimes labeled ‘meta-geometries’ (cf. Aliseda, 1997). In our view, all these alternative logics might get their empirical vindication, too – as reflecting different modes of human reasoning. Whether non-classical modes of reasoning are really logical is like asking if non-Euclidean geometries are really geometries. 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Instituto de Investigaciones Filosóficas UNAM, México E-mail: atocha@filosoficas.unam.mx and Faculteit der Wijsbegeerte RuG, Groningen The Netherlands E-mail: atocha@philos.rug.nl