Call an explanation in which a non-mathematical fact is explained—in part or in whole—by mathematical facts: an extra-mathematical explanation. Such explanations have attracted a great deal of interest recently in arguments over mathematical realism. In this article, a theory of extra-mathematical explanation is developed. The theory is modelled on a deductive-nomological theory of scientific explanation. A basic DN account of extra-mathematical explanation is proposed and then redeveloped in the light of two difficulties that the basic theory faces. The (...) final view appeals to relevance logic and uses resources in information theory to understand the explanatory relationship between mathematical and physical facts. 1Introduction2Anchoring3The Basic Deductive-Mathematical Account4The Genuineness Problem5Irrelevance6Relevance and Information7Objections and Replies 7.1Against relevance logic7.2Too epistemic7.3Informational containment8Conclusion. (shrink)

The deductive nomological (DN) model has been the basis for discussions about scientific explanations for decades. The overcoming of the logical empiricist program together with the raise of several counter-examples to the DN model have progressively led to a renewal of the reflections on this topic. The first step of this paper is to clarify the framework in which the epistemological question of scientific explanation is adressed. We make a proposal for a universal structure of scientific models, which (...) constitute the basic epistemic unit of our analysis. It is then possible, within this framework, to clarify some discussions about scientific explanation, and to offer a new account of it. The latter tries to benefit from the advantages of the DN model, as resting on a law, together with neutralizing some of its typical criticisms. The work presented here is both abstract, exhibiting formal structures as general patterns of explanation, and concrete with real examples taken from actual scientific disciplines. (shrink)

According to the deductive-nomological account of ground, a fact A grounds another fact B in case the laws of metaphysics determine the existence of B on the basis of the existence of A. Accounts of grounding of this particular variety have already been developed in the literature. My aim in this paper is to sketch a new version of this account. My preferred account offers two main improvements over existing accounts. First, the present account is able to deal (...) with necessitarian as well as non-necessitarian cases of grounding by acknowledging the existence of two types of metaphysical laws. I will argue that we should assume that metaphysical laws come in the necessitarian as well as in the non-necessitarian varietyclosely paralleling the distinction between strict and non-strict laws in the philosophy of science. The second main improvement of the present account is that this account is able to provide an explanation of why the laws of metaphysics have a direction built into them. I will argue that we should characterize metaphysical laws with the help of Theodore Sider’s (2011) notion of structure, which is a descendent of David Lewis’s (1983) notion of naturalness. According to the account of metaphysical laws developed in this paper, metaphysical laws express in their antecedents either perfectly structural truths or more structural truths than in their consequents. Since on Sider’s account structural features of reality are fundamental features of reality, the account is able to explain as to why the laws of metaphysics take us from the fundamental to the derivative. (shrink)

Opinionated state of the art paper on scientific explanation. Analysis and discussion of the most relevant models and theories in the contemporary literature, namely, the deductive-nomological model, the models of inductive-statistical and statistical relevance, the pragmatic theory of why questions, the unifying theory of standard arguments, and the causal/non-causal counterfactual theory.

An account of distinctively mathematical explanation (DME) should satisfy three desiderata: it should account for the modal import of some DMEs; it should distinguish uses of mathematics in explanation that are distinctively mathematical from those that are not (Baron [2016]); and it should also account for the directionality of DMEs (Craver and Povich [2017]). Baron’s (forthcoming) deductive-mathematical account, because it is modelled on the deductive-nomological account, is unlikely to satisfy these desiderata. I provide a counterfactual account of (...) DME, the Narrow Ontic Counterfactual Account (NOCA), that can satisfy all three desiderata. NOCA appeals to ontic considerations to account for explanatory asymmetry and ground the relevant counterfactuals. NOCA provides a unification of the causal and the non-causal, the ontic and the modal, by identifying a common core that all explanations share and in virtue of which they are explanatory. (shrink)

ACTION-THEORETICALLY EXPLANATORY INTERPRETATIONS AS A MEANS OF SEMANTIC MEANING ANALYSIS The article first develops a general procedure for semantic meaning analysis in difficult cases where the meaning is very uncertain. The procedure consists of searching for one or more possible hypothetical causal explanations of the text, these explanations containing, among other things, the semantic intention of the author, his subjective reasons for this meaning and for the writing down of the text, but also the path of transmission of the text (...) from the writing down to the respective present text version. These hypothetical explanations (interpretations) are deductive-nomological or probabilistic, therefore presuppose corresponding general law hypotheses. If there is only one possible (hypothetical) explanation of the text at hand, this hypothetical explanation is also true. Very often, however, there are several possible explanations. In this case, the probabilities of these possible explanations must be determined; these probabilities then carry over to the individual hypotheses in the explanation (according to Bayes' law). This general procedure of meaning analysis is used in the second part of the article to reconstruct two famous interpretations (by Wapnewski and Hahn) of a poem by Walther von der Vogelweide ("Nemt, frowe, disen kranz"). -/- GERMAN ABSTRACT In dem Artikel wird zunächst ein allgemeines Verfahren zur semantischen Bedeutungsanalyse in schwierigen Fällen entwickelt, in denen die Bedeutung sehr unsicher ist. Das Verfahren besteht darin, daß eine oder mehrere mögliche hypothetische kausale Erklärungen des Textes gesucht werden, wobei diese Erklärungen u.a. die semantische Intention des Autors enthalten, seine subjektiven Gründe für diese Bedeutung und für die Niederschrift des Textes, aber auch den Weg der Tradierung des Textes von der Niederschrift bis zur jeweils vorliegenden Textversion. Diese hypothetischen Erklärungen (Deutungen) sind deduktiv-nomologisch oder probabilistisch, setzen deshalb entsprechende allgemeine Gesetzeshypothesen voraus. Wenn es nur eine mögliche (hypothetische) Erklärung des vorliegenden Textes gibt, ist diese hypothetische Erklärung auch wahr. Sehr häufig gibt es aber mehrere mögliche Erklärungen. In diesem Fall müssen die Wahrscheinlichkeiten dieser möglichen Erklärungen bestimmt werden; diese Wahrscheinlichkeiten übertragen sich dann auch auf die einzelnen Hypothesen in der Erklärung (nach dem Bayesschen Gesetz). Dieses allgemeine Verfahren der Bedeutungsanalyse wird im zweiten Teil des Artikels zur Rekonstruktion zweier berühmter Interpretationen (von Wapnewski und Hahn) eines Gedichts von Walther von der Vogelweide („Nemt, frowe, disen kranz“) verwendet. (shrink)

In recent years, New Mechanism has become one of the most popular and widely discussed philosophical accounts of scientific explanation. Some of its proponents see it as a successor to traditional deductive nomological and statistical approaches to the philosophy of explanation. New Mechanists thus argue for the generality of their approach as a model of scientific explanation. Here we will show that the generality of NME as an account of scientific explanation is restricted. Most significantly, NME is widely (...) recognized to have trouble in accounting for explanation in traditional physics. Yet, in the face of obstacles to NME in the physical sciences, New Mechanists have yet to provide a convincing defense of the generality of their view. In this paper, I (1) summarize when and how NME works, (2) describe briefly what NME is not and can’t be, (3) outline some areas of contemporary scientific explanation where New Mechanism misses the mark; and (4) demonstrate that New Mechanism does not constitute a general theory of scientific explanation but rather covers a restricted scope of explanations. (shrink)

In this article, I explore the compatibility of inference to the best explanation (IBE) with several influential models and accounts of scientific explanation. First, I explore the different conceptions of IBE and limit my discussion to two: the heuristic conception and the objective Bayesian conception. Next, I discuss five models of scientific explanation with regard to each model’s compatibility with IBE. I argue that Kitcher’s unificationist account supports IBE; Railton’s deductive–nomological–probabilistic model, Salmon’s statistical-relevance model, and van Fraassen’s erotetic (...) account are incompatible with IBE; and Salmon’s causal–mechanical model is merely consistent with IBE. In short, many influential models of scientific explanation do not support IBE. I end by outlining three possible conclusions to draw: (1) either philosophers of science or defenders of IBE have seriously misconstrued the concept of explanation, (2) philosophers of science and defenders of IBE do not use the term ‘explanation’ univocally, and (3) the ampliative conception of IBE, which is compatible with any model of scientific explanation, deserves a closer look. (shrink)

Husserl’s early picture of explanation in the sciences has never been completely provided. This lack represents an oversight, which we here redress. In contrast to currently accepted interpretations, we demonstrate that Husserl does not adhere to the much maligned deductive-nomological (DN) model of scientific explanation. Instead, via a close reading of early Husserlian texts, we reveal that he presents a unificationist account of scientific explanation. By doing so, we disclose that Husserl’s philosophy of scientific explanation is no mere (...) anachronism. It is, instead, tenable and relevant. We discuss how Husserl and other contemporary thinkers draw theoretical inspiration from the same source—namely, Bernard Bolzano. Husserl’s theory of scientific explanation shares a common language and discusses the same themes as, for example, Phillip Kitcher and Kit Fine. To advance our novel reading, we discuss Husserl’s investigations of grounding, inter-lawful explanation, intra-mathematical explanation, and scientific unification. (shrink)

In a recent article, Erasmus, Brunet, and Fisher (2021) argue that Artificial Neural Networks (ANNs) are explainable. They survey four influential accounts of explanation: the Deductive-Nomological model, the Inductive-Statistical model, the Causal-Mechanical model, and the New-Mechanist model. They argue that, on each of these accounts, the features that make something an explanation is invariant with regard to the complexity of the explanans and the explanandum. Therefore, they conclude, the complexity of ANNs (and other Machine Learning models) does not (...) make them less explainable. In this reply, it is argued that Erasmus et al. left out one influential account of explanation from their discussion: the Unificationist model. It is argued that, on the Unificationist model, the features that makes something an explanation is sensitive to complexity. Therefore, on the Unificationist model, ANNs (and other Machine Learning models) are not explainable. It is emphasized that Erasmus et al.’s general strategy is correct. The literature on explainable Artificial Intelligence can benefit by drawing from philosophical accounts of explanation. However, philosophical accounts of explanation do not settle the problem of whether ANNs are explainable because they do not unanimously declare that explanation is invariant with regard to complexity. (shrink)

This paper addresses a fundamental line of research in neuroscience: the identification of a putative neural processing core of the cerebral cortex, often claimed to be “canonical”. This “canonical” core would be shared by the entire cortex, and would explain why it is so powerful and diversified in tasks and functions, yet so uniform in architecture. The purpose of this paper is to analyze the search for canonical explanations over the past 40 years, discussing the theoretical frameworks informing this research. (...) It will highlight a bias that, in my opinion, has limited the success of this research project, that of overlooking the dimension of cortical development. The earliest explanation of the cerebral cortex as canonical was attempted by David Marr, deriving putative cortical circuits from general mathematical laws, loosely following a deductive-nomological account. Although Marr’s theory turned out to be incorrect, one of its merits was to have put the issue of cortical circuit development at the top of his agenda. This aspect has been largely neglected in much of the research on canonical models that has followed. Models proposed in the 1980s were conceived as mechanistic. They identified a small number of components that interacted as a basic circuit, with each component defined as a function. More recent models have been presented as idealized canonical computations, distinct from mechanistic explanations, due to the lack of identifiable cortical components. Currently, the entire enterprise of coming up with a single canonical explanation has been criticized as being misguided, and the premise of the uniformity of the cortex has been strongly challenged. This debate is analyzed here. The legacy of the canonical circuit concept is reflected in both positive and negative ways in recent large-scale brain projects, such as the Human Brain Project. One positive aspect is that these projects might achieve the aim of producing detailed simulations of cortical electrical activity, a negative one regards whether they will be able to find ways of simulating how circuits actually develop. (shrink)

In a recent paper, Erasmus et al. (2021) defend the idea that the ambiguity of the term “explanation” in explainable AI (XAI) can be solved by adopting any of four different extant accounts of explanation in the philosophy of science: the Deductive Nomological, Inductive Statistical, Causal Mechanical, and New Mechanist models. In this chapter, I show that the authors’ claim that these accounts can be applied to deep neural networks as they would to any natural phenomenon is mistaken. (...) I also provide a more general argument as to why the notion of explainability as it is currently used in the XAI literature bears little resemblance to the traditional concept of scientific explanation. It would be more fruitful to use the label “understandable AI” to avoid the confusion that surrounds the goal and purposes of XAI. In the second half of the chapter, I argue for a pragmatic conception of understanding that is better suited to play the central role attributed to explanation in XAI. Following De Regt (2017) and Kuorikoski and Ylikoski (2015), the conditions of satisfaction for understanding an ML system are fleshed out in terms of an agent’s success in using the system. (shrink)

1. Introduction: a look back at the reasons vs. causes debate. 2. The interventionist account of causation. 3. Four objections to interventionism. 4. The counterfactual analysis of event causation. 5. The role of free agency. 6. Causality in the human sciences. -- The reasons vs. causes debate reached its peak about 40 years ago. Hempel and Dray had debated the nature of historical explanation and the broader issue of whether explanations that cite an agent’s reasons are causal or not. Melden, (...) Peters, Winch, Kenny and Anscombe had contributed their anticausal conceptions. The neo-Wittgensteinians seemed to be winning the day when in 1963 Donald Davidson published his seminal paper “Actions, Reasons, and Causes”. Davidson’s paper devastated the Wittgensteinian camp. It contained, among other things, a powerful attack on the logical connection argument. Davidson argued that the existence of a logical or conceptual connection between descriptions can never eliminate a causal relation, which holds between events simpliciter, not between events under certain descriptions. Davidson maintained that in a way, reasons can be causes. When somebody acts for a certain reason, his intentional attitudes, or rather changes in his attitudes, cause his bodily movements. Davidson also argued that rationalization is a species of causal explanation. For the definition of action, he argued that intentional actions are bodily movements caused in the right way by beliefs and desires that rationalize them. Davidson’s paper paved the way for causal theories of action, which superseded neo-Wittgensteinian analyses in the following decades. The causal theory was rapidly adopted by Alvin Goldman, David Armstrong, Paul Churchland, Myles Brand and many others, entering the mainstream and dominating the philosophy of action to this very day. In 1971 Georg Henrik von Wright published his book "Explanation and Understanding". The second chapter did not deal with agency, but with causation. It developed a new account of causation, the interventionist or experimentalist account. Focusing on causation, von Wright remedied a major shortcoming of the reasons vs. causes debate. The concept of causality, and the nature of the causal relation, received little attention in this debate, a fact that holds true for both camps. Mostly it was simply taken for granted that, as Hempel had declared, “causal explanation is a special type of deductive-nomological explanation”. One camp then aligned intentional explanations with D-N explanations, while the other camp insisted on their disparity. So strictly speaking, the label “reasons/causes debate” was a misnomer. The controversy dealt primarily with the question as to whether intentional explanations can take the form of D-N explanations, while the notion of causation, and the metaphysics of the causal relation, were left obscured. With von Wright’s new approach, the situation changed. Von Wright was primarily concerned with causation, but his approach contained an implicit attack on the causal theory of action as well. His core idea was that the notion of causality is intimately linked with, or even derived from, the notion of intentionally making something happen. Other philosophers, even Hume, had considered such a connection before, but often just to reject this view, regarding it as a kind of myth belonging to the infancy of the human mind. Von Wright took the idea seriously. He submitted the analysis that p is the cause of q if and only if by doing p we could bring about q. The causal theory of action was also concerned with the relation between causation and agency, to which its name bears witness. The causal theory of action holds that actions are bodily movements with a certain causal history. This is why von Wright’s account constituted a momentous challenge to the causal theory: it reversed the direction of conceptual dependency between both notions. Davidson and his followers tried to define what an intentional action is by using the notion of causation. The causal condition which the causal theory sets is part of the definition of “doing something intentionally”. Von Wright claimed that the conceptual dependency is the other way round. He used the notions of doing, and bringing about, to explain what causal relations are. So, instead of a causal theory of action, he advocated an agency theory of causation, as it may be dubbed. It is remarkable how seldom this clash of opinions about conceptual primacy is reflected in the literature. There are few exceptions: Fred Stoutland noticed the conflict, and he published a number of papers in which he compared Davidson’s and von Wright’s views. Von Wright’s book "Explanation and Understanding" was widely read and discussed in the seventies, especially in Europe. But it strikes me that especially in North America, where the causal theory of action became the orthodoxy of the day, von Wright’s challenge went largely unnoticed. Even Davidson did not seem to take it seriously. He nowhere takes notice of the interventionist theory of causation, while he does discuss von Wright’s earlier book "Norm and Action". As is well-known, Davidson favoured an alternative account of causation, based on “the principle of the nomological character of causality”, as he somewhat clumsily called it, or, later and less clumsily, “the cause-law thesis”. Davidson’s firm adherence to a nomological theory of causality may explain why he did not take much interest in alternative accounts. [...] -/- . 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In an earlier discussion, I argued that Kant's moral theory satisfies some of the basic criteria for being a genuine theory: it includes testable hypotheses, nomological higher-and lower-level laws, theoretical constructs, internal principles, and bridge principles. I tried to show that Kant's moral theory is an ideal, descriptive deductive-nomological theory that explains the behavior of a fully rational being and generates testable hypotheses about the moral behavior of actual agents whom we initially assume to conform to its (...) theoretical constructs. I argued that the moral "ought" is best understood as the "ought" of tentative prediction expressed in the range of uses of the German sollen; and that the degree to which such a theory is well-confirmed is a function of the degree to which we actually judge individual human agents, on a case-by-case basis, to be motivated by rationality, stupidity, or moral corruption in their actions. I assume that a similar case could be made for other major contenders, such as Utilitarianism or Aristotelianism. But there still remains unanswered the question of which of these theories is the best among the available alternatives. To answer this question, further criteria of selection must be invoked. Among these are structural elegance and explanatory simplicity, but even these do not exhaust the desiderata for an adequate moral theory. More pressing in the case of moral theory is the requirement that the theory enable us to understand all the available data of moral experience; that the theory be sufficiently inclusive that in the formulation of its descriptive laws and practical principles, it be capable of identifying as morally significant all the behavior to which moral praise, condemnation, or acquittal is a relevant and appropriate response. (shrink)

(1) In the first part of this paper, I review Chomsky's meandering journey from the formalism/mentalism of Syntactic Structures, through several methodological positions, to the minimalist theory of his latest work. Infected with mentalism from first to last, each and every position vitiates Chomsky's repeated claims that his theories will provide useful guidance to later theories in such fields as cognitive psychology and cognitive neuroscience. With the guidance of his insights, he claims, psychologists and neuroscientists will be able to avoid (...) costly dead-end lines of research. -/- (2) This never happened. As I have shown, this never could have happened. (See Johnston, 2018). What has happened, instead, is that current neurolinguistic research (with the arguable exception of the now-dated Lemma Model of Willem Levelt) proceeds without reference to Chomsky. It also wholeheartedly rejects the mentalism of the associated Language of Thought theory of Jerry Fodor. (See Johnston, 2018). -/- (3) I make this argument in the first part of this paper. I would also like to point out that most of my argument was developed in 1972, when I was a graduate student. I know of no other sustained criticisms of Chomsky at that time, and certainly none along the lines I had developed back then. -/- (4) In the second part of this paper, I present my own account of the methodology of science. When I was a graduate student, philosophy of science was dominated by an attempt to describe a methodology common to all the specific sciences, i.e. Hempel’s deductive-nomological model. These days, Hempel’s emphasis on the methodological unity of science has been rejected by such “dis-unity” philosophers of science as Ian Hacking, Patrick Suppes and Nancy Cartwright (see Cat, 2021). -/- (5) I view this change as the swing of a pendulum or, to change the metaphor, a journey from one end point of a continuum to another. As the level of abstraction at which one tries to describe scientific method is raised, the descriptions become increasingly general. Whether or not unity-of-science theories become so general as to be vacuous, is ultimately a subjective judgment. And so I expect that philosophers will eventually become tired of increasingly specific “close to the workbench” descriptions of how scientists work, and begin to turn back to methodological “big pictures”, finding in them powerful abstractions rather than empty irrelevancies. -/- (6) In the second part of this paper, I present my own account of the methodology of science, which I would situate somewhere between the “unity” and “dis-unity” accounts. However, I am not a scientist. My own views about scientific method have three origins: -/- (6a) my work as a graduate student from 1966 until I passed my comprehensive exams in 1973 (at a different university); -/- (6b) reading every issue of Scientific American from 1972 until nearly 2000, (at which point I continued to read it only sporadically, since I concluded that, around that time, it had evolved from a serious science magazine to a popular science magazine); and -/- (6c) my three-year immersion in the cognitive neuroscience of language after I retired, based on repeated study of and note-taking for (Banesh & Compton, 2018), (Kemmerer, 2015), several other books and, finally, numerous articles not hidden behind a paywall. -/- So, as always with my writings: caveat emptor. (shrink)

Primitive, unanalysable grounding relations are considered by many to be indispensable constituents of the metaphysician’s toolkit. Yet, as a primitive ontological posit, grounding must earn its keep by explaining features of the world not explained by other tools already at our disposal. Those who defend grounding contend that grounding is required to play two interconnected roles: accounting for widespread intuitions regarding what is ontologically prior to what, and forming the backbone of a theory of metaphysical explanation, in much the same (...) way that causal relations have been thought to underpin theories of scientific explanation. This thesis undermines the need to posit grounding relations to perform either of these jobs. With regard to the first, it is argued that a pair of human psychological mechanisms—for which there is substantial empirical support—can provide a more theoretically virtuous explanation of why we have the intuitions that we do. With regard to the second, I begin by considering what we want from a theory of explanation, and go on to develop three attractive (yet grounding-free) theories of metaphysical explanation. I offer: i) a psychologistic theory that calls upon the aforementioned psychological mechanisms, as well as the modal relations of necessitation and supervenience, ii) a metaphysical variant of the deductive-nomological theory of scientific explanation, and iii) a metaphysical variant of the unificationist theory of scientific explanation. Furthermore, these theories draw upon mechanisms and relations (both logical and ontological) to which we are already committed. Thus, to posit grounding relations in order to explain our priority intuitions, or in order to develop a theory of metaphysical explanation, is ontologically profligate. I conclude that we should not posit relations of ground. (shrink)

In his bold and excellent book on Hume's scientific methodology, David Landy positions himself between the "Deductive-Nomological" reading, which explains particular phenomena in terms of empirical regularities, and the "New Hume" position, which considers empirical regularities to be the explananda and unknowable essences the explanans. Landy sides with the New Humeans, except that for him the essences, or "theoretical posits," are knowable. These essences become knowable, despite their being in principle unobservable, through the tool of "perceptible models." Landy (...) argues that "theoretical posits are only possible via the deployment of a perceptible model," and he explains that perceptual models allow us to... (shrink)

In this dissertation I achieve the following: (1) I present motivating criteria for a general comprehensive theory of scientific explanation. I review historical approaches to modeling explanation in light of these criteria. (2) I present New Mechanist Explanation ("NME") as the leading candidate for a contemporary, complete theory of scientific explanation. (3) I present constraints on the applicability of New Mechanism in modeling biology, chemistry, and physics. I argue for the unsuitability of NME as a candidate for a general theory (...) of explanation due to its significant omissions of explanatory content and subsequent incompleteness. (4) I present arguments for the indispensable explanatory power of mathematics in science, noting that much of NME's failures are owed to its poor integration of mathematical explanation as applied to science. (5) I propose a Rational-Relational theory of explanation(“RRE”). I present RRE in detail and demonstrate it as a true working general theory of explanation which best meets the motivating criteria presented in (1). I disambiguate RRE from structural explanation ("SE"). I show that historical models of explanation, including the deductive nomological (DNE) model, statistical relevance ("SRE") model, structural model (SE) and unification model ("UE") may be successfully treated as sub-classes of explanation within the scope of RRE. I provide a rubric for selecting amongst sub-theories of explanation under RRE in the purpose of highlighting particular features of relational explanations. (6) In conclusion, I argue that philosophers of science should move forward using RRE as a general model of scientific explanation as a working alternative to the incomplete model of NME. (shrink)

I defend an account of explanatory depth according to which explanations in the non-fundamental sciences can be deeper than explanations in fundamental physics.

David Friedrich Strauss is best known for his mythical interpretation of the Gospel narratives. He opposed both the supernaturalists (who regarded the Gospel stories as reliable) and the rationalists (who offered natural explanations of purportedly supernatural events). His mythical interpretation suggests that many of the stories about Jesus were woven out of pre-existing messianic beliefs and expectations. Picking up this suggestion, I argue that the Gospel writers thought paradigmatically rather than historically. A paradigmatic explanation assimilates the event-to-be- explained to what (...) is thought to be a prototypical instance of divine action. It differs from a historical or scientific explanation insofar as it does not specify the conditions under which it should be applied. It is, therefore, a wonderfully flexible way to understand the present in the light of the past. (shrink)

Deductionism assimilates nature to conceptual artifacts (models, equations), and tacitly holds that real physical systems are such artifacts. Some physical concepts represent properties of deductive systems rather than of nature. Properties of mathematical or deductive systems can thereby sometimes falsely be ascribed to natural systems.

According to the Best System Account of lawhood, laws of nature are theorems of the deductive systems that best balance simplicity and strength. In this paper, I advocate a different account of lawhood which is related, in spirit, to the BSA: according to my account, laws are theorems of deductive systems that best balance simplicity, strength, and also calculational tractability. I discuss two problems that the BSA faces, and I show that my account solves them. I also use (...) my account to illuminate the nomological character of special science laws. (shrink)

G. A. Cohen defends and Jon Elster criticizes Marxist use of functional explanation. But Elster's mechanical conception of explanation is, contrary to Elster's claims, a better basis for vindication of functional explanation than Cohen's nomological conception, which cannot provide an adequate account of functional explanation. Elster also objects that functional explanation commits us to metaphysically bizarre collective subjects, but his argument requires an implausible reading of methodological individualism which involves an unattractive eliminativism about social phenomena.

According to the Nomological Argument, observed regularities in nature are best explained by an appeal to a supernatural being. A successful explanation must avoid two perils. Some explanations provide too little structure, predicting a universe without regularities. Others provide too much structure, thereby precluding an explanation of certain types of lawlike regularities featured in modern scientific theories. We argue that an explanation based in the creative, intentional action of a supernatural being avoids these two perils whereas leading competitors do (...) not. Although our argument falls short of a full defense, it does suggest that the Nomological Argument is worthy of philosophical attention. (shrink)

A deduction is an inference that aims for validity and can be either valid or invalid. An invalid deduction can never be valid, because if an inference is valid in one possible world, it must be valid in all. One possible world where an inference is valid implies that there are no worlds where the inference is invalid. If the only genuine deductions are the valid ones, then our talk about deduction is an indirect way of referring to validity rather (...) than an inference type. There is simply no deduction to speak of—only validity. But then again, it’s clear that validity is an attribute that some inferences possess, while others do not. This is a paradox. (shrink)

Deductive Cogency holds that the set of propositions towards which one has, or is prepared to have, a given type of propositional attitude should be consistent and closed under logical consequence. While there are many propositional attitudes that are not subject to this requirement, e.g. hoping and imagining, it is at least prima facie plausible that Deductive Cogency applies to the doxastic attitude involved in propositional knowledge, viz. belief. However, this thought is undermined by the well-known preface paradox, (...) leading a number of philosophers to conclude that Deductive Cogency has at best a very limited role to play in our epistemic lives. I argue here that Deductive Cogency is still an important epistemic requirement, albeit not as a requirement on belief. Instead, building on a distinction between belief and acceptance introduced by Jonathan Cohen and recent developments in the epistemology of understanding, I propose that Deductive Cogency applies to the attitude of treating propositions as given in the context of attempting to understand a given phenomenon. I then argue that this simultaneously accounts for the plausibility of the considerations in favor of Deductive Cogency and avoids the problematic consequences of the preface paradox. (shrink)

From the second half of the last century, there has been a widespread view in the Anglophone world that Kant’s transcendental deduction (aka TD) aims to vindicate our common-sense view of the world as composed of public and objective particulars against some unqualified forms of skepticism. This widespread assumption has raised serious doubt not only about the success of TD but also about the very nature of its argument in both editions of the Critique. Yet, if there is a connection (...) between TD and global skepticism, the intriguing question is: Who is this skeptic? According to Strawson, “the skeptic” is a hypothesis of a purely sense-datum experience. In contrast, the fact that TD turns on the key notion of self-consciousness has induced several other scholars to assume that the skeptic is no none but a Cartesian external-world skeptic. None of those readings find textual support or are compatible with the very structure of the first Critique. The question is: Does this mean that TD aims only to undermine empiricism, as Guyer suggests? I do not believe so. I am pretty convinced that TD addresses a peculiar form of global skepticism, namely “Hume’s challenge to reason.” Assuming that we cognize (erkennen) and experience (erfahren) appearances as objects as a requirement of Newtonian physics, TD aims to provide a justification of the principle that nature is uniform that is superior to Hume’s justification. (shrink)

Since causal processes can be prevented and interfered with, law-governed causation is a challenge for necessitarian theories of laws of nature. To show that there is a problematic friction between necessity and interference, I focus on David Armstrong's theory; with one proviso, his lawmaker, nomological necessity, is supposed to be instantiated as the causation of the law's second relatum whenever its first relatum is instantiated. His proviso is supposed to handle interference cases, but fails to do so. In order (...) to be able to handle interferences, any theory which utilizes a kind of necessitation as lawmaker has to downgrade what it treats as necessity to something more akin to (Newtonian) forces. (shrink)

In this paper, I consider a family of three-valued regular logics: the well-known strong and weak S.C. Kleene’s logics and two intermedi- ate logics, where one was discovered by M. Fitting and the other one by E. Komendantskaya. All these systems were originally presented in the semantical way and based on the theory of recursion. However, the proof theory of them still is not fully developed. Thus, natural deduction sys- tems are built only for strong Kleene’s logic both with one (...) (A. Urquhart, G. Priest, A. Tamminga) and two designated values (G. Priest, B. Kooi, A. Tamminga). The purpose of this paper is to provide natural deduction systems for weak and intermediate regular logics both with one and two designated values. (shrink)

In mathematics, the deductive method reigns. Without proof, a claim remains unsolved, a mere conjecture, not something that can be simply assumed; when a proof is found, the problem is solved, it turns into a “result,” something that can be relied on. So mathematicians think. But is there more to mathematical justification than proof? -/- The answer is an emphatic yes, as I explain in this article. I argue that non-deductive justification is in fact pervasive in mathematics, and (...) that it is in good epistemic standing. (shrink)

Our universe features a harmonious match between laws and states: applying its laws to its states generates other states. This is a striking fact. Matters might have been otherwise. The universe might have been stillborn in a state unengaged by its laws. The problem of nomological harmony is that of explaining the noted striking fact. After introducing and developing this problem, we canvass candidate solutions and identify some of their virtues and vices. Candidate solutions invoke the likes of a (...) designer, axiarchic meta-laws, multiverses, essential causal powers, and Humean laws. (shrink)

Knowledge representation (KR) is actually more than representation: It involves also inference, namely inference of “new” knowledge, i.e. new facts. Logic programming is a suitable KR medium, but more often than not discussions on this programming paradigm focus on aspects other than KR. In this paper, I elaborate on the general theory of logic programming and give the essentials of two of its main implementations, to wit, Prolog and Datalog, from the viewpoint of deductive computing over knowledge bases, which (...) includes deductive programming. (shrink)

The deduction of categories in the 1781 edition of the Critique of the Pure Reason (A Deduction) has “two sides”—the “objective deduction” and the “subjective deduction”. Kant seems ambivalent about the latter deduction. I treat it as a significant episode of Kant’s thinking about categories that extended from the early 1770s to around 1790. It contains his most detailed answer to the question about the origin of categories that he formulated in the 1772 letter to Marcus Herz. The answer is (...) that categories are generated a priori through a kind of intellectual “epigenesis”. This account leaves unexplained why precisely such and such categories should be generated. While this observation caused Kant to worry about the hypothetical status of the subjective deduction in 1781, he would come to acquiesce in the recognition that the ground of the possibility of categories is itself inscrutable. I call this his “methodological skepticism”. (shrink)

The definitions of ‘deduction’ found in virtually every introductory logic textbook would encourage us to believe that the inductive/deductive distinction is a distinction among kinds of arguments and that the extension of ‘deduction’ is a determinate class of arguments. In this paper, we argue that that this approach is mistaken. Specifically, we defend the claim that typical definitions of ‘deduction’ operative in attempts to get at the induction/deduction distinction are either too narrow or insufficiently precise. We conclude by presenting (...) a deflationary understanding of the inductive/deductive distinction; in our view, its content is nothing over and above the answers to two fundamental sorts of questions central to critical thinking. (shrink)

Methods available for the axiomatization of arbitrary finite-valued logics can be applied to obtain sound and complete intelim rules for all truth-functional connectives of classical logic including the Sheffer stroke and Peirce’s arrow. The restriction to a single conclusion in standard systems of natural deduction requires the introduction of additional rules to make the resulting systems complete; these rules are nevertheless still simple and correspond straightforwardly to the classical absurdity rule. Omitting these rules results in systems for intuitionistic versions of (...) the connectives in question. (shrink)

The present article illustrates a conflict between the claim that rational belief sets are closed under deductive consequences, and a very inclusive claim about the factors that are sufficient to determine whether it is rational to believe respective propositions. Inasmuch as it is implausible to hold that the factors listed here are insufficient to determine whether it is rational to believe respective propositions, we have good reason to deny that rational belief sets are closed under deductive consequences.

We present a sound and complete Fitch-style natural deduction system for an S5 modal logic containing an actuality operator, a diagonal necessity operator, and a diagonal possibility operator. The logic is two-dimensional, where we evaluate sentences with respect to both an actual world (first dimension) and a world of evaluation (second dimension). The diagonal necessity operator behaves as a quantifier over every point on the diagonal between actual worlds and worlds of evaluation, while the diagonal possibility quantifies over some point (...) on the diagonal. Thus, they are just like the epistemic operators for apriority and its dual. We take this extension of Fitch’s familiar derivation system to be a very natural one, since the new rules and labeled lines hereby introduced preserve the structure of Fitch’s own rules for the modal case. (shrink)

What is an inference and when is an inference deductive rather than inductive, abductive, etc. The goal of this paper is precisely to determine what is that we, humans, do when we engage in deduction, i.e., whether there is something that satisfies both our pre-theoretical intuitions and theoretical presuppositions about deduction, as a cognitive process. The paper is structured in two parts: the first one deals with the issue of what is an inference. There, I will defend the hypothesis (...) that an inference is a causal process where an initial cognitive state causes another in such a way that the causal translation depends essentially on the content of the involved states. In other words, in an inference, a mental state A causes another B, partially because they have the content they have. Based on the conclusion from the first part, I will develop an account of deductive inference in the second part of the article. In order to achieve this, I will start by setting the stage and developing four constraints that an adequate account of deductive inference ought to satisfy. Then, I will present six different hypotheses that one might propose to characterize inferences of this kind. I have divided the proposals in two groups, the first one – which I will call the “logical proposals” – assume that whether an inference is a deduction or not depends exclusively on the content of the mental states involved in the inference, while the second – which I will call the “cognitive proposals” – reject the aforementioned assumption and instead recognize that extra cognitive information is required to determine whether an inference is a deduction or not. I will evaluate each proposal in turn, according to whether they satisfy the constraints introduced at the beginning of this second part. As a result of such evaluation, I will conclude that only one of the cognitive proposals actually satisfies the four constraints and therefore should be considered to give an adequate characterization of deductive inference. (shrink)

The zombie, mocking all nomological arguments, gives rise to axiological considerations that also result in a vindication of the nomological paradigm. So the ‘philosophical benefit of zombies’ ultimately proves to be that they lead to an understanding they were originally invented to refute.

This essay presents deductive arguments to an introductory-level audience via a discussion of Aristotle's three types of rhetoric, the goals of and differences between deductive and non-deductive arguments, and the major features of deductive arguments (e.g., validity and soundness).

The deduction of categories in the 1781 edition of the Critique of the Pure Reason (A Deduction) has “two sides”—the “objective deduction” and the “subjective deduction”. Kant seems ambivalent about the latter deduction. I treat it as a significant episode of Kant’s thinking about categories that extended from the early 1770s to around 1790. It contains his most detailed answer to the question about the origin of categories that he formulated in the 1772 letter to Marcus Herz. The answer is (...) that categories are generated a priori through a kind of intellectual “epigenesis”. This account leaves unexplained why precisely such and such categories should be generated. While this observation caused Kant to worry about the hypothetical status of the subjective deduction in 1781, he would come to acquiesce in the recognition that the ground of the possibility of categories is itself inscrutable. I call this his “methodological skepticism”. (shrink)

In attempt to provide an answer to the question of origin of deductive proofs, I argue that Aristotle’s philosophy of math is more accurate opposed to a Platonic philosophy of math, given the evidence of how mathematics began. Aristotle says that mathematical knowledge is a posteriori, known through induction; but once knowledge has become unqualified it can grow into deduction. Two pieces of recent scholarship on Greek mathematics propose new ways of thinking about how mathematics began in the Greek (...) culture. Both claimed there was a close relationship between the culture and mathematicians; mathematics was understood through imaginative processes, experiencing the proofs in tangible ways, and establishing a consistent unified form of argumentation. These pieces of evidence provide the context in which Aristotle worked and their contributions lend support to the argument that mathematical premises as inductively available is a better way of understanding the origins of deductive practices, opposed to the Platonic tradition. (shrink)

For deductive reasoning to be justified, it must be guaranteed to preserve truth from premises to conclusion; and for it to be useful to us, it must be capable of informing us of something. How can we capture this notion of information content, whilst respecting the fact that the content of the premises, if true, already secures the truth of the conclusion? This is the problem I address here. I begin by considering and rejecting several accounts of informational content. (...) I then develop an account on which informational contents are indeterminate in their membership. This allows there to be cases in which it is indeterminate whether a given deduction is informative. Nevertheless, on the picture I present, there are determinate cases of informative (and determinate cases of uninformative) inferences. I argue that the model I offer is the best way for an account of content to respect the meaning of the logical constants and the inference rules associated with them without collapsing into a classical picture of content, unable to account for informative deductive inferences. (shrink)

In section 1, I develop epistemic communism, my view of the function of epistemically evaluative terms such as ‘rational’. The function is to support the coordination of our belief-forming rules, which in turn supports the reliable acquisition of beliefs through testimony. This view is motivated by the existence of valid inferences that we hesitate to call rational. I defend the view against the worry that it fails to account for a function of evaluations within first-personal deliberation. In the rest of (...) the paper, I then argue, on the basis of epistemic communism, for a view about rationality itself. I set up the argument in section 2 by saying what a theory of rational deduction is supposed to do. I claim that such a theory would provide a necessary, sufficient, and explanatorily unifying condition for being a rational rule for inferring deductive consequences. I argue in section 3 that, given epistemic communism and the conventionality that it entails, there is no such theory. Nothing explains why certain rules for deductive reasoning are rational. (shrink)

Harold Hodes in [1] introduces an extension of first-order modal logic featuring a backtracking operator, and provides a possible worlds semantics, according to which the operator is a kind of device for ‘world travel’; he does not provide a proof theory. In this paper, I provide a natural deduction system for modal logic featuring this operator, and argue that the system can be motivated in terms of a reading of the backtracking operator whereby it serves to indicate modal scope. I (...) prove soundness and completeness theorems with respect to Hodes’ semantics, as well as semantics with fewer restrictions on the accessibility relation. (shrink)

This article examines how deduction preserves certainty and how much certainty it can preserve according to Descartes’s Rules for the Direction of the Mind. I argue that the certainty of a deduction is a matter of four conditions for Descartes. First, certainty depends on whether the conjunction of simple propositions is composed with necessity or contingency. Second, a deduction approaches the certainty of an intuition depending on how many “acts of conceiving” it requires and—third—the complexity or difficulty of the acts (...) of thinking, which is determined by the content of the thoughts and on external factors. Fourth, certainty depends on the intellectual aptitude of the person using the deduction. A deduction lacks certainty when it relies on memory such that it is not apprehended with immediacy. However, the mental capacity and speed of a mind can be increased by training the special mental faculties of perspicacity and discernment. Increasing one’s intellectual aptitude allows for more steps of a deduction to be inferred in fewer acts of conceiving, thereby helping preserve the certainty of a deduction. (shrink)

In this paper I investigate, within the framework of realistic interpretations of the wave function in nonrelativistic quantum mechanics, the mathematical and physical nature of the wave function. I argue against the view that mathematically the wave function is a two-component scalar field on configuration space. First, I review how this view makes quantum mechanics non- Galilei invariant and yields the wrong classical limit. Moreover, I argue that interpreting the wave function as a ray, in agreement many physicists, Galilei invariance (...) is preserved. In addition, I discuss how the wave function behaves more similarly to a gauge potential than to a field. Finally I show how this favors a nomological rather than an ontological view of the wave function. (shrink)

This presentation of Aristotle's natural deduction system supplements earlier presentations and gives more historical evidence. Some fine-tunings resulted from conversations with Timothy Smiley, Charles Kahn, Josiah Gould, John Kearns,John Glanvillle, and William Parry.The criticism of Aristotle's theory of propositions found at the end of this 1974 presentation was retracted in Corcoran's 2009 HPL article "Aristotle's demonstrative logic".

James Van Cleve has argued that Kant’s Transcendental Deduction of the categories shows, at most, that we must apply the categories to experience. And this falls short of Kant’s aim, which is to show that they must so apply. In this discussion I argue that once we have noted the differences between the first and second editions of the Deduction, this objection is less telling. But Van Cleve’s objection can help illuminate the structure of the B Deduction, and it suggests (...) an interesting reason why the rewriting might have been thought necessary. (shrink)

Juhani Yli-Vakkuri has argued that the Twin Earth thought experiments offered in favour of semantic externalism can be replaced by a straightforward deductive argument from premisses widely accepted by both internalists and externalists alike. The deductive argument depends, however, on premisses that, on standard formulations of internalism, cannot be satisfied by a single belief simultaneously. It does not therefore, constitute a proof of externalism. The aim of this article is to explain why.