Science, Logic, and Mathematics

Review of: "Zeno and Einstein".

1. Sometimes problems are encountered in a discipline that cannot be adequately solved within that discipline. For example, when climate scientists run models of the climate system to project chang...

1. The American logical empiricist Ernest Nagel's model of reduction has often acted as a start-point for philosophical discussions on reduction and reductionism. Regarding Nagel's model, however,...

Abductive problems and their solutions are presented by means of justification logic. We introduce additional meta-constructions in order to generate and compare different solutions to the same abductive problem. Our approach has three advantages: (i) it makes structurally explicit the solution to an abductive problem (as it has a syntactic nature); (ii) it gives a precise meaning to the notion of evidence; (iii) it provides clear definitions and procedures for the comparison of solutions that can be adapted to different needs.

In this paper, after presenting three challenges that any knowledge-based demarcation between science and non-science should meet, namely, the skeptical, triviality, and mimicry challenges, I show how a recent contender in epistemology, viz., presuppositional epistemic contextualism, allows these challenges to be met, hence pointing toward a novel solution to the perennial demarcation problem. Conceiving of scientific knowledge from the vantage point of contextualism forces us to consider science as being first and foremost a distinctive epistemological context, which has the peculiarity (...) of coming with a very high degree of stringency for the truth conditions of putative knowledge attributions. The fact that science imposes particularly stringent norms on knowledge is measured by the extension of the set of counterpossibilities that science is (i) in the business of eliminating on the basis of available evidence and (ii) ready to take seriously (insofar as they are consistent with the scientific community’s pragmatic presuppositions at a given time and place). (shrink)

Weak ontic necessity is the ontic necessity expressed by “should/ought to” in English. An example of it is “I should be dead by now”. A feature of this necessity is that whether it holds at the present world is irrelevant to whether its prejacent holds at the present world. In this paper, by combining premise semantics and update semantics for conditionals, we present a logical theory for conditional weak ontic necessity based on context update. A context is a set of (...) ordered defaults, determining expected possible states of the present world. Sentences are evaluated with respect to contexts. When evaluating the conditional weak ontic necessity with respect to a context, we first update the context with the antecedent, then check whether the consequent holds with respect to the updated context. We compare this theory to some related works. The logic’s expressivity is studied. (shrink)

I will analyse three fundamental ways of governing ignorance though abduction, which are essential from an eco-cognitive and eco-logical point of view, in which the central role in human cognition of natural and artefactual environment is taken into account. First of all, according to the so-called GW-schema, proposed by Gabbay and Woods, abduction presents an ignorance-preserving or (ignorance-mitigating) character: given the fact that the abduced hypotheses aim at becoming truths, the basic ignorance is neither solved nor left untouched. Second, I (...) will contend that abduction is not always ignorance preserving but also knowledge enhancing. Indeed, according to the eco-cognitive model of abduction, I myself have recently proposed, we can easily see that, occasionally, knowledge can be immediately increased (and ignorance ‘killed’) through abduction: in this perspective, abduction does not have to be rigidly conceptualized as an inference to the best explanation in the received sense, which is as an inference that, to grant truth, also takes advantage of an empirical evaluation phase or an inductive phase, as Peirce classified it. Finally, it is important to stress that abductive cognition is at the roots of two fundamental cognitive processes. On one side, abductive cognition constructs (and/or takes advantage of the exploitation of) embodiments, external representations, enactions, artefacts and props, which involve ignorant entities suitably transformed in cognitive representations and/or devices. On the other side, abductive cognition dominates those ‘special’ cognitive creative processes that lead to what I call the disseminated ‘computational domestication of ignorant entities’: I will illustrate the example of the recent unconventional computational domestication of ignorant entities, which creates novel cognitive embodiments (morphological computation). (shrink)

In this paper, it is argued that, contrary to some suggestions in the philosophical literature, dispositions cannot replace laws in the description of the physical world. If for a certain type of physical situation a well-working law-based account is available, then it is not possible to describe that situation equally well in terms of dispositions. Using an example consisting of four laws (Coulomb’s law, Newton’s law of gravitation, the rule for the composition of forces and Newton’s second law), it is (...) shown that when one attempts to replace a law-based account with a disposition-based account, the latter turns out to be either less informative or inferior with respect to theoretical virtues, such as simplicity, unification, non-triviality of predictions and explanatory power. Laws are also better suited to deal with interactions and their screening off. Using inference to the best explanation, one can conclude from this comparison that it is laws rather than dispositions that should be included in our ontological picture of nature. (shrink)

In this paper, the contributions to the account of meta-induction (Schurz 2019) collected in this volume are critically discussed and thereby, new insights are developed. How broad and expandable the program of meta-induction is can be learned from Ortner’s contribution. New insights about the transition from the a priori justification of meta-induction to the a posteriori justification of object-induction emerge from the reflection of Shogenji’s paper. How meta-induction may be applied also to religious prophecies and that their meta-inductive justification does (...) not fail for a priori reasons but because of missing evidence for predictive success is learned from the discussion of Pitts’ contribution. That meta-induction does not rely on a particular prior distribution, while the no free lunch theorem depends implicitly on a uniform prior, is the major conclusion drawn from the discussion of Wolpert’s article. How the problem of induction is treated in different versions of the Bayesian account is learned from the discussion of Willliamson’s paper. That meta-induction can also be employed for abduction, and that abductive theory-revision can offer meta-inductive aggregation methods is a new insight emerging from the reflection of Aliseda’s contribution. (shrink)

This is the introduction to the special issue “The Meta-Inductive Approach to Hume’s Problem”. The introduction includes introductory remarks and brief comments on each of the papers appearing in the special issue.

The important recent book by Schurz ( 2019 ) appreciates that the no-free-lunch theorems (NFL) have major implications for the problem of (meta) induction. Here I review the NFL theorems, emphasizing that they do not only concern the case where there is a uniform prior—they prove that there are “as many priors” (loosely speaking) for which any induction algorithm _A_ out-generalizes some induction algorithm _B_ as vice-versa. Importantly though, in addition to the NFL theorems, there are many _free lunch_ theorems. (...) In particular, the NFL theorems can only be used to compare the expected performance of an induction algorithm _A_, considered in isolation, with the expected performance of an induction algorithm _B_, considered in isolation. There is a rich set of free lunches which instead concern the statistical _correlations_ among the generalization errors of induction algorithms. As I describe, the meta-induction algorithms that Schurz advocates as a “solution to Hume’s problem” are simply examples of such a free lunch based on correlations among the generalization errors of induction algorithms. I end by pointing out that the prior that Schurz advocates, which is uniform over bit frequencies rather than bit patterns, is contradicted by thousands of experiments in statistical physics and by the great success of the maximum entropy procedure in inductive inference. (shrink)

This paper examines a twofold tension in Gerhard Schurz’s (2019) recent proposal to solve Hume’s problem of induction. Schurz concedes to the skeptic that there is no non-circular epistemic justification of the reliability of induction, but then argues for the optimality of meta-induction so that if any prediction method is reliable, then meta-induction is. There is a tension in this proposal between meta-induction and our inductive practice: Are we supposed to abandon our inductive practice in favor of meta-induction? Schurz claims (...) that given the actual way the world has been, the meta-inductive method supports the standard inductive method. There then arises a second tension: The challenges Schurz cites against the reliability of induction—such as the anti-inductive hypothesis and the state-uniform probability distribution—cast doubt on Schurz’s claim that meta-induction supports induction. The reasoning in this paper moves in the opposite direction: There are ways of answering these challenges in defense of Schurz’s claim that meta-induction supports induction, but these ways also point to ways of defending induction directly without the mediation of meta-induction. (shrink)

According to the Feigl–Reichenbach–Salmon–Schurz pragmatic justification of induction, no predictive method is guaranteed or even likely to work for predicting the future; but if anything will work, induction will work—at least when induction is employed at the meta-level of predictive methods in light of their track records. One entertains a priori all manner of esoteric prediction methods, and is said to arrive a posteriori at the conclusion, based on the actual past, that object-level induction is optimal. Schurz’s refinements largely solve (...) the notorious short-run problem. A difficulty is noted, however, related to short-run worries but based on localized disagreement about the past, a feature characteristic of real debates (especially early modern) involving induction in intellectual history. Given the evidence about past events, _unfiltered by induction_, meta-induction might support a partly non-inductive method—especially as judged by proponents of esoteric prediction methods, who presumably believe that their methods have worked. Thus induction is justified meta-inductively in contexts where it was uncontroversial, while not obviously justified in key contexts where it has been disputed. This objection, momentarily sensed by Reichenbach regarding clairvoyance, is borne out by the Stoics’ use of meta-induction to justify both science and divination and by ancient Hebrew examples of meta-induction. Schurz’s recently introduced criteria for acceptance of testimony play a crucial role in arriving at object-level induction using meta-induction, but one might question them. Given the need for judgment in accepting testimony, it is unclear that the subjectivity of Howson’s Bayesian answer to Hume’s problem is overcome. (shrink)

The aim of this paper is to show that the logical approach to philosophy of science could be further improved with tools like the ones put forward by Schurz in his proposal to model scientific theory development. Section 2 is a presentation of the basics in AGM epistemology of logical abduction and of their connection. In Sect. 3 several operations for theory change proposed by Schurz (2011; 2018) are presented, followed by my own proposal of a further case of hypothesis (...) refinement, consisting of the instantiation of an existential abduction. Section 4 offers a discussion and conclusions. (shrink)

Work in online learning traditionally considered induction-friendly (e.g. stochastic with a fixed distribution) and induction-hostile (adversarial) settings separately. While algorithms like Exp3 that have been developed for the adversarial setting are applicable to the stochastic setting as well, the guarantees that can be obtained are usually worse than those that are available for algorithms that are specifically designed for stochastic settings. Only recently, there is an increasing interest in algorithms that give (near-)optimal guarantees with respect to the underlying setting, even (...) in case its nature is unknown to the learner. In this paper, we review various online learning algorithms that are able to adapt to the hardness of the underlying problem setting. While our focus lies on the application of adaptive algorithms as meta-inductive methods that combine given base methods, concerning theoretical properties we are also interested in guarantees that go beyond a comparison to the best fixed base learner. (shrink)

When solving a problem with insight, people suddenly become aware of the solution (Mayer, 1995). Insight is often accompanied by a feeling of pleasure and certainty, known as an “Aha! moment” (Gick...

This paper proposes to recover the topic of the philosophy of scientific method from its late nineteenth-century roots. The subject matter of scientific method sprouted from key inferential ingredients identified by Charles Peirce. In this paper, the historical path is traversed from the viewpoint of contemporary Cognitive Structural Realism (CSR). Peirce’s semiotic theory of methods and practices of scientific inquiry prefigured CSR’s reliance on embodied informational structures and experimentation upon forms of relations that model generic scientific domains. Three results are (...) shown to follow from this convocation: (i) a naturalization of Peirce’s interconnected abductive, deductive and inductive stages of the logic of science, here characterized de novo in terms of CSR. (ii) a perspective to scientific modeling that incorporates processes of abstraction and generalization as originated from Peirce’s logic of science, and (iii) diagrammatic reasoning as a pivotal method in analyzing scientific reasoning in experimental practices. (shrink)

Explanation is commonly considered one of the central goals of science. Although computer simulations have become an important tool in many scientific areas, various philosophical concerns indicate that their explanatory power requires further scrutiny. We examine a case study in which atomistic simulations have been used to examine the factors responsible for the transport selectivity of certain channel proteins located at cell membranes. By elucidating how precisely atomistic simulations helped scientists draw inferences about the molecular system under investigation, we respond (...) to some concerns regarding their explanatory power. We argue that atomistic simulations can be tools for managing molecular complexity and for systematically assessing how the occurrence of the explanandum is sensitive to a range of factors. (shrink)

Arguments discrediting the value-free ideal of science have left us with the question of how to distinguish desirable values from biases that compromise the reliability of research. In this paper, I argue for a characterization of cognitive biases as deviations of thought processes that systematically lead scientists to the wrong conclusions. In particular, cognitive biases could help us understand a crucial issue in science today: how systematic error is introduced in research outcomes, even when research is evaluated as of good (...) quality. To conclude, I suggest that some debiasing mechanisms have great potential for countering implicit methodological biases in science. (shrink)

Hilary Putnam was one of the truly great philosophers of the twentieth century. In a memorial essay I published elsewhere, I wrote: Leading philosophy towards constant dynamic expeditions and holding on to an incredible style of self-critique, Hilary Putnam (1926–2016), over five decades, has been in the process of making laudable contributions to philosophy and philosophy of science by being a beacon to a series of philosophical generations. He was a profound scholar full of wisdom, morality, and love of humanity, (...) in a word a “Philosopher’s Philosopher.”1 Hilary Putnam, whom I called “Gurudev” (mentor), was a renaissance man of philosophy for his laudable and novel contribution in the fields of philosophy and philosophy of science. The impetus for bringing out this honorary volume on his fundamental contributions to philosophy and philosophy of science began after his peaceful death on March 13, 2016 in Arlington, Massachusetts. (shrink)