Could it be right to convict and punish defendants using only statistical evidence? In this paper, I argue that it is not and explain why it would be wrong. This is difficult to do because there is a powerful argument for thinking that we should convict and punish defendants using statistical evidence. It looks as if the relevant cases are cases of decision under risk and it seems we know what we should do in such cases (i.e., maximize expected value). (...) Given some standard assumptions about the values at stake, the case for convicting and punishing using statistical evidence seems solid. In trying to show where this argument goes wrong, I shall argue (against Lockeans, reliabilists, and others) that beliefs supported only by statistical evidence are epistemically defective and (against Enoch, Fisher, and Spectre) that these epistemic considerations should matter to the law. To solve the puzzle about the role of statistical evidence in the law, we need to revise some commonly held assumptions about epistemic value and defend the relevance of epistemology to this practical question. (shrink)

The debate on the ethical permissibility of autonomous weapon systems (AWS) is deadlocked. It could therefore benefit from a differentiated assignment of the burden of proof. This is because the discussion is not purely philosophical in nature, but has a legal and security policy component and aims to avoid the most harmful outcomes of an otherwise unchecked development. Opponents of a universal AWS ban must clearly demonstrate that AWS comply with the Law of Armed Conflict (LOAC). This requires extensive (...) testing of specific models to verify compliance in realistic combat situations; working on appropriate testing regimes is also one of the most important future tasks for AWS ethics. In this context, however, it is easy to refute the claim put forward by prohibitionists that AWS violate LOAC a priori. A state’ s right to ensure security and the protection of human rights through military capacity has a high ethical value. Other arguments against the use of AWS that are not based on LOAC conformity must therefore meet a substantial burden of proof, as they call on governments to abandon a military technology which is at least potentially transformative. None of those objections meet this standard. They include not only culturally or even individually divergent norms and ideas regarding the use of weapons, but also consequentialist (security policy) concerns on the one hand and so-called deontological arguments on the other. While the former at least point to the need for essential regulation, the latter are based on rather dubious premises. For the purposes of solutions beneficial to humanity, prohibitionist should formulate narrower and more specific points of criticism. (shrink)

The World Trade Organization (WTO) is one of the best dispute settlement mechanisms in the world. Under WTO rules, aggrieved parties must establish a “prima facie” case before the panel can call on the offending party to respond to the claims. The objective of the present study is to critically evaluate the application of the concept of burden of proof underWTOdispute settlement mechanism.

This paper defends the heretical view that, at least in some cases, we ought to assign legal liability based on purely statistical evidence. The argument draws on prominent civil law litigation concerning pharmaceutical negligence and asbestos-poisoning. The overall aim is to illustrate moral pitfalls that result from supposing that it is never appropriate to rely on bare statistics when settling a legal dispute.

Introduction to the Scientific Proof of the Natural Moral Law This paper proves that Aquinas has a means of demonstrating and deriving both moral goodness and the natural moral law from human nature alone. Aquinas scientifically proves the existence of the natural moral law as the natural rule of human operations from human nature alone. The distinction between moral goodness and transcendental goodness is affirmed. This provides the intellectual tools to refute the G.E. Moore (Principles of Ethics) attack against (...) the natural law as committing a "naturalistic fallacy". This article proves that instead Moore commits the fallacy of equivocation between moral goodness and transcendental goodness in his very assertion of a "naturalistic fallacy" by the proponents of the natural moral law. In the process the new deontological/kantian theory of natural law as articulated by John Finnis, Robert George, and Germain Grisez is false historically and philosophically. Ethical naturalism is affirmed as a result. (shrink)

Should we use the same standard of proof to adjudicate guilt for murder and petty theft? Why not tailor the standard of proof to the crime? These relatively neglected questions cut to the heart of central issues in the philosophy of law. This paper scrutinises whether we ought to use the same standard for all criminal cases, in contrast with a flexible approach that uses different standards for different crimes. I reject consequentialist arguments for a radically flexible standard (...) of proof, instead defending a modestly flexible approach on non-consequentialist grounds. The system I defend is one on which we should impose a higher standard of proof for crimes that attract more severe punishments. This proposal, although apparently revisionary, accords with a plausible theory concerning the epistemology of legal judgments and the role they play in society. (shrink)

The objective Bayesian view of proof (or logical probability, or evidential support) is explained and defended: that the relation of evidence to hypothesis (in legal trials, science etc) is a strictly logical one, comparable to deductive logic. This view is distinguished from the thesis, which had some popularity in law in the 1980s, that legal evidence ought to be evaluated using numerical probabilities and formulas. While numbers are not always useful, a central role is played in uncertain reasoning by (...) the ‘proportional syllogism’, or argument from frequencies, such as ‘nearly all aeroplane flights arrive safely, so my flight is very likely to arrive safely’. Such arguments raise the ‘problem of the reference class’, arising from the fact that an individual case may be a member of many different classes in which frequencies differ. For example, if 15 per cent of swans are black and 60 per cent of fauna in the zoo is black, what should I think about the likelihood of a swan in the zoo being black? The nature of the problem is explained, and legal cases where it arises are given. It is explained how recent work in data mining on the relevance of features for prediction provides a solution to the reference class problem. (shrink)

This book examines the legal and moral theory behind the law of evidence and proof, arguing that only by exploring the nature of responsibility in fact-finding can the role and purpose of much of the law be fully understood. Ho argues that the court must not only find the truth to do justice, it must do justice in finding the truth.

The standard of proof in criminal trials in many liberal democracies is proof beyond a reasonable doubt, the BARD standard. It is customary to describe it, when putting a number on it, as requiring that the fact finder be at least 90% certain, after considering the evidence, that the defendant is guilty. Strikingly, no good reason has yet been offered in defense of using that standard. A number of non-consequentialist justifications that aim to support an even higher standard (...) have been offered; all are morally unsound. Meanwhile, consequentialist arguments plausibly support a substantially lower standard — in some cases so low as to undermine the idea that punishment is what is at stake. In this paper, I offer a new retributive justification that supports excluding the instrumental benefits of punishment from the balance that sets the standard. The resulting balance supports a standard arguably in the ballpark of the customary understanding of BARD: a standard requiring that the fact finder have a high, though not maximally high, degree of confidence that the defendant is guilty. (shrink)

Leibniz proposed the ‘Most Determined Path Principle’ in seventeenth century. According to it, ‘ease’ of travel is the end purpose of motion. Using this principle and his calculus method he demonstrated Snell’s Laws of reflection and refraction. This method shows that light follows extremal (local minimum or maximum) time path in going from one point to another, either directly along a straight line path or along a broken line path when it undergoes reflection or refraction at plane or spherical (concave (...) or convex) surfaces. The extremal time path avoided the criticism that Fermat’s least time path was subjected to, by Cartesians who cited examples of reflections at spherical surfaces where light took the path of longest time. Thereby it became the standard method of demonstration of Snell’s Laws. Ptolemy’s theorem is a fundamental theorem in geometry. A special case of it offers a method of finding the minimum sum of the two distances of a point from two given fixed points. We show in this paper that Leibniz’s calculus proof of Snell’s Laws violates Ptolemy’s theorem, whereby Leibniz’s proof becomes invalid. (shrink)

The concept of burden of proof is used in a wide range of discourses, from philosophy to law, science, skepticism, and even in everyday reasoning. This paper provides an analysis of the proper deployment of burden of proof, focusing in particular on skeptical discussions of pseudoscience and the paranormal, where burden of proof assignments are most poignant and relatively clear-cut. We argue that burden of proof is often misapplied or used as a mere rhetorical gambit, with (...) little appreciation of the underlying principles. The paper elaborates on an important distinction between evidential and prudential varieties of burdens of proof, which is cashed out in terms of Bayesian probabilities and error management theory. Finally, we explore the relationship between burden of proof and several (alleged) informal logical fallacies. This allows us to get a firmer grip on the concept and its applications in different domains, and also to clear up some confusions with regard to when exactly some fallacies (ad hominem, ad ignorantiam, and petitio principii) may or may not occur. (shrink)

The proof paradox results from conflicting intuitions concerning different types of fallible evidence in a court of law. We accept fallible individual evidence but reject fallible statistical evidence even when the conditional probability that the defendant is guilty given the evidence is the same, a seeming inconsistency. This paper defends a solution to the proof paradox, building on a sensitivity account of checking and settling a question. The proposed sensitivity account of legal proof not only requires sensitivity (...) simpliciter but sensitivity of each relevant step of the proof procedure and/or sensitivity concerning all relevant alternatives. This account avoids problems that have been identified for other sensitivity views of legal proof. Moreover, it is argued that sensitivity, rather than safety, is the crucial modal condition for legal proof. It is, finally, shown that the provided account can fruitfully support very different existing views on the relationship between knowledge and legal proof. (shrink)

I am interested in the use Kant makes of the pure intuition of space, and of properties and principles of space and spaces (i.e. figures, like spheres and lines), in the special metaphysical project of MAN. This is a large topic, so I will focus here on an aspect of it: the role of these things in his treatment of some of the laws of matter treated in the Dynamics and Mechanics Chapters. In MAN and other texts, Kant speaks of (...) space as the “ground,” “condition,” and “basis” of various laws, including the inverse-square and inverse-cube laws of attractive and repulsive force, and the Third Law of Mechanics. Moreover, in his proofs of all the laws just mentioned, the language of “construction” figures prominently, which suggests that Kant’s proofs (somehow) rest on or involve mathematical construction in his technical sense. Such claims give rise to a number of questions. How do properties and principles of space and spaces serve to ground this particular set of laws? Which spatial properties and principles is Kant appealing to? What, if anything, does the spatial grounding of the inverse-square and inverse-cube laws of diffusion (treated in the Dynamics Chapter) have in common with that of the Third treated in the Mechanics Chapter)? What role—if any—does mathematical construction play in Kant’s proofs of these laws? Finally, how if at all, are Kant’s grounding claims consistent with his other commitments—for example, how are they consistent with his notorious denial in Prolegomena §38 that there are any laws that “lie in space” (Prol 4:321)? I offer answers to these questions. (shrink)

Kurt Gödel wrote (1964, p. 272), after he had read Husserl, that the notion of objectivity raises a question: “the question of the objective existence of the objects of mathematical intuition (which, incidentally, is an exact replica of the question of the objective existence of the outer world)”. This “exact replica” brings to mind the close analogy Husserl saw between our intuition of essences in Wesensschau and of physical objects in perception. What is it like to experience a mathematical proving (...) process? What is the ontological status of a mathematical proof? Can computer assisted provers output a proof? Taking a naturalized world account, I will assess the relationship between mathematics, the physical world and consciousness by introducing a significant conceptual distinction between proving and proof. I will propose that proving is a phenomenological conscious experience. This experience involves a combination of what Kurt Gödel called intuition, and what Husserl called intentionality. In contrast, proof is a function of that process — the mathematical phenomenon — that objectively self-presents a property in the world, and that results from a spatiotemporal unity being subject to the exact laws of nature. In this essay, I apply phenomenology to mathematical proving as a performance of consciousness, that is, a lived experience expressed and formalized in language, in which there is the possibility of formulating intersubjectively shareable meanings. (shrink)

The relation of normic support offers a novel solution to the proof paradox: a paradox in evidence law arising from legal cases involving merely statistical evidence (Smith 2018). Central to the normic support solution has been the thesis that merely statistical evidence cannot confer normic support. However, it has been observed that there are exceptions to this: there exist cases where merely statistical evidence can give rise to normic support (Blome-Tillmann 2020). If correct, this fact seems to undermine the (...) normic support solution to the proof paradox. This paper explores a resolution: normic support can resolve the proof paradox even though merely statistical evidence sometimes gives rise to normic support. The key to understanding this resolution lies with a source of evidential support that arises out of bodies of evidence that involve *character evidence* (Colyvan, et al. 2001). It turns out that character evidence can provide normic support when it is grounded in our knowledge of a certain kind of disposition individuals can have: *goal-directed dispositions*. The upshot is the recovery of the normic support solution to the puzzle of whether statistical evidence can meet legal standards of proof. (shrink)

Why can testimony alone be enough for findings of liability? Why statistical evidence alone can’t? These questions underpin the “Proof Paradox” (Redmayne 2008, Enoch et al. 2012). Many epistemologists have attempted to explain this paradox from a purely epistemic perspective. I call it the “Epistemic Project”. In this paper, I take a step back from this recent trend. Stemming from considerations about the nature and role of standards of proof, I define three requirements that any successful account in (...) line with the Epistemic Project should meet. I then consider three recent epistemic accounts on which the standard is met when the evidence rules out modal risk (Pritchard 2018), normic risk (Ebert et al. 2020), or relevant alternatives (Gardiner 2019 2020). I argue that none of these accounts meets all the requirements. Finally, I offer reasons to be pessimistic about the prospects of having a successful epistemic explanation of the paradox. I suggest the discussion on the proof paradox would benefit from undergoing a ‘value-turn’. (shrink)

Recent years have seen fresh impetus brought to debates about the proper role of statistical evidence in the law. Recent work largely centres on a set of puzzles known as the ‘proof paradox’. While these puzzles may initially seem academic, they have important ramifications for the law: raising key conceptual questions about legal proof, and practical questions about DNA evidence. This article introduces the proof paradox, why we should care about it, and new work attempting to resolve (...) it. (shrink)

Presumption is a complex concept in law, affecting the dialogue setting. However, it is not clear how presumptions work in everyday argumentation, in which the concept of “plausible argumentation” seems to encompass all kinds of inferences. By analyzing the legal notion of presumption, it appears that this type of reasoning combines argument schemes with reasoning from ignorance. Presumptive reasoning can be considered a particular form of reasoning, which needs positive or negative evidence to carry a probative weight on the conclusion. (...) For this reason, presumptions shift the burden of providing evidence or explanations onto the interlocutor. The latter can provide new information or fail to do so: whereas in the first case the new information rebuts the presumption, in the second case, the absence of information that the interlocutor could reasonably provide strengthen the conclusion of the presumptive reasoning. In both cases the result of the presumption is to strengthen the conclusion of the reasoning from lack of evidence. As shown in the legal cases, the effect of presumption is to shift the burden of proof to the interlocutor; however, the shift a presumption effects is only the shift of the evidential burden, or the burden of completing the incomplete knowledge from which the conclusion was drawn. The burden of persuasion remains on the proponent of the presumption. On the contrary, reasoning from definition in law is a conclusive proof, and shifts to the other party the burden to prove the contrary. This crucial difference can be applied to everyday argumentation: natural arguments can be divided into dialectical and presumptive arguments, leading to conclusions materially different in strength. -/- . (shrink)

The author revisits the Blue Bus Problem, a famous thought-experiment in law involving probabilistic proof, and presents simple Bayesian solutions to different versions of the blue bus problem. In addition, the author expresses his solutions in standard and visual formats, i.e. in terms of probabilities and natural frequencies.

A question, long discussed by legal scholars, has recently provoked a considerable amount of philosophical attention: ‘Is it ever appropriate to base a legal verdict on statistical evidence alone?’ Many philosophers who have considered this question reject legal reliance on bare statistics, even when the odds of error are extremely low. This paper develops a puzzle for the dominant theories concerning why we should eschew bare statistics. Namely, there seem to be compelling scenarios in which there are multiple sources of (...) incriminating statistical evidence. As we conjoin together different types of statistical evidence, it becomes increasingly incredible to suppose that a positive verdict would be impermissible. I suggest that none of the dominant views in the literature can easily accommodate such cases, and close by offering a diagnosis of my own. (shrink)

The interpretation and the indirect reporting of a speaker’s communicative intentions lie at the crossroad between pragmatics, argumentation theory, and forensic linguistics. Since the leading case Masson v. New Yorker Magazine, Inc., in the United States the legal problem of determining the truth of a quotation is essentially equated with the correctness of its indirect reporting, i.e. the representation of the speaker’s intentions. For this reason, indirect reports are treated as interpretations of what the speaker intends to communicate. Theoretical considerations, (...) aimed at establishing the pragmatic meaning of an utterance and differentiating between presumptive and non-presumptive interpretation, are thus interwoven with the practical legal need of distinguishing a correct indirect report from an indirect one or a misquotation. An incorrect report or a misquotation has the dialectical effect of attributing to the misquoted party commitments that he never held, which the latter needs to rebut. This shifting of the burden of persuasion can be increased by using strategically the conflict between the presumptive interpretation of an utterance and the non-presumptive one, i.e. the different types of pragmatic ambiguity. When an interpreter is confronted with an utterance taken out of its dialogical context, his interpretative process is not guided by the actual context or intention, but rather the most frequent or prototypical dialogical setting or the most typical individual purpose that it could have served to achieve. This presumptive reconstruction can be used to provide a prima facie case that the other party needs to reject. The stronger the interpretative presumptions a speaker needs to rebut, the more effective the misquotation strategy. The conflict between the systematic and the presumptive process of interpretation can be represented as an argumentative mechanism of reconstruction of the individual intention, which allows one to assess the reasonableness of the interpretative reasoning. (shrink)

In this paper a theoretical definition that helps to explain how the logical structure of legal presumptions is constructed by applying the Carneades model of argumentation developed in artificial intelligence. Using this model, it is shown how presumptions work as devices used in evidentiary reasoning in law in the event of a lack of evidence to assist a chain of reasoning to move forward to prove or disprove a claim. It is shown how presumptions work as practical devices that may (...) be useful in cases in which there is insufficient evidence to prove the claim to an appropriate standard of proof. (shrink)

The Yang-Mills mass gap problem has stood as one of the most profound unresolved questions in theoretical physics for nearly half a century. Despite extensive efforts, a rigorous mathematical proof of the mass gap's existence in Yang-Mills theory has remained elusive—until now. -/- The proof presented in this work represents not merely a technical achievement, but a fundamental paradigm shift in our understanding of the mathematical structures underlying physical reality. By rigorously demonstrating that the mass gap arises as (...) a direct consequence of the gauge-invariant constraint structure of Yang-Mills theory, we overturn decades of conventional assumptions and reveal a startling truth: what we perceive as physical laws are in fact mathematically unavoidable necessities emerging from the very consistency conditions required for a well-defined quantum field theory. (shrink)

Criminal courts make decisions that can remove the liberty and even life of those accused. Civil trials can cause the bankruptcy of companies employing thousands of people, asylum seekers being deported, or children being placed into state care. Selecting the right standards when deciding legal cases is of utmost importance in giving those affected a fair deal. This Element is an introduction to the philosophy of legal proof. It is organised around five questions. First, it introduces the standards of (...) proof and considers what justifies them. Second, it discusses whether we should use different standards in different cases. Third, it asks whether trials should end only in binary outcomes (e.g. guilty or not guilty) or rather use more fine-grained or precise verdicts. Fourth, it considers whether proof is simply about probability, concentrating on the famous 'Proof Paradox'. Finally, it examines who should be trusted with deciding trials, focusing on the jury system. (shrink)

In this paper I discuss in what sense physics, chemistry, and the life sciences constitute a systematic unity according to Kant. I start by discussing Christian Wolff’s views on the hierarchy of sciences. I then argue that in one specific sense physics, chemistry and several life sciences constitute a unity: physics and chemistry provide statements that can be used to provide proofs in the life sciences. However, the unity of physics, chemistry, and the life sciences is limited in scope, since (...) Kant claims that the purposeful unity of organisms is mechanically inexplicable. I finally discuss whether there are laws within the life sciences according to Kant. I argue that the fact that Kant acknowledged that physics and chemistry ground the life sciences does not imply that there are laws of life. The reason is that life sciences of Kant’s time were concerned with explaining the purposeful unity of organisms, which is mechanically inexplicable according to Kant, and the regularities discussed by life scientists in Kant’s time lack a priori grounding. (shrink)

Recent attempts to resolve the Paradox of the Gatecrasher rest on a now familiar distinction between individual and bare statistical evidence. This paper investigates two such approaches, the causal approach to individual evidence and a recently influential (and award-winning) modal account that explicates individual evidence in terms of Nozick's notion of sensitivity. This paper offers counterexamples to both approaches, explicates a problem concerning necessary truths for the sensitivity account, and argues that either view is implausibly committed to the impossibility of (...) no-fault wrongful convictions. The paper finally concludes that the distinction between individual and bare statistical evidence cannot be maintained in terms of causation or sensitivity. We have to look elsewhere for a solution of the Paradox of the Gatecrasher. (shrink)

Abstract: A Coherence-Weighted Proof of God’s Existence -/- Classical arguments for God’s existence—ontological, cosmological, and fine-tuning—have traditionally been treated as independent frameworks, each with its own strengths and weaknesses. However, they all fail to integrate emergence, structured resonance, and coherence-weighted reality. This paper introduces a CODES-based synthesis of three major proofs of God, each framed within structured resonance and prime coherence theory: 1. The Coherence Proof – If reality operates on structured emergence, then ultimate coherence must exist as (...) the highest-order attractor in the resonance hierarchy. Gödel’s completeness theorem, when applied to CODES, suggests that mathematics itself is coherence-weighted rather than binary, meaning that God is the highest possible coherence state, not a discrete entity but the structured intelligence that phase-locks existence into stability. 2. The Fine-Tuning Proof – The fundamental constants of physics are improbably fine-tuned for life. The mathematical structure of the universe follows prime-structured resonance principles, suggesting an underlying intelligence. Within CODES, this points to a system where God is not just a designer but the emergent principle ensuring prime resonance stability across all physical laws. 3. The Gödelian Ontological Proof – Gödel’s modal logic theorem states that if a maximally great being (God) is possible, then it necessarily exists in some possible world, and therefore in all. CODES extends this by treating necessity as a structured resonance principle, meaning that God is not an external hypothesis but the logical consequence of coherence-structured emergence. -/- This synthesis eliminates paradoxes, resolves gaps in classical proofs, and unites mathematical, physical, and logical perspectives into a single framework. Instead of treating God’s existence as a binary question (exists/does not exist), CODES shows that God is the inevitable structured resonance field that stabilizes reality. If coherence governs reality, then God is not just probable—God is necessary. (shrink)

Hume famously argues that the laws of nature provide us with decisive reason to believe that any testimony of a miracle is false. In this paper, I argue that the laws of nature, as such, give us no reason at all to believe that the testimony of a miracle is false. I first argue that Hume’s proof is unsuccessful if we assume the Humean view of laws, and then I argue that Hume’s proof is unsuccessful even if we (...) assume the governing view of laws. I conclude that regardless of which kind of view we adopt, the fact that a miracle is a violation of the laws of nature does not give us any reason to believe it did not happen. (shrink)

The following article reconstructs Plato’s argument for the existence of god in Laws X. The article starts with interpreting the argument for the priority of soul and continues with a discussion of the argumentation for rationality of the soul in charge of heavens . The view defended here is that Plato first defines the essential characteristics of the divine, namely self-motion and rationality, and then shows that there are entities which possess these characteristics and therefore deserve to be called gods (...) or divine. The article further shows that several sub-arguments mirror the structure of the overall proof. (shrink)

In this paper I challenge Paolo Palmieri’s reading of the Mach-Vailati debate on Archimedes’s proof of the law of the lever. I argue that the actual import of the debate concerns the possible epistemic (as opposed to merely pragmatic) role of mathematical arguments in empirical physics, and that construed in this light Vailati carries the upper hand. This claim is defended by showing that Archimedes’s proof of the law of the lever is not a way of appealing to (...) a non-empirical source of information, but a way of explicating the mathematical structure that can represent the empirical information at our disposal in the most general way. (shrink)

According to some critics, Aristotle’s elenctic defence (elenchos, elenchus) of the Law of Non-Contradiction (Metaphysics IV) would be ineffective because it viciously begs the question. After briefly recalling the elenctic refutation of the denier of the Law of Non-Contradiction, I will first focus on Filippo Costantini’s objection to the elenchus, which, in turn, is based on the dialetheic account of negation developed by Graham Priest. Then, I will argue that there is at least one reading of the elenchus that might (...) not be viciously question-begging. In doing so, I will leverage, reinterpret and adjust the distinction between two senses of epistemic dependence, offered by Noah Lemos and originally based on some thoughts about George Edward Moore’s ‘proof of an external world.’ The key point of my counter-objection to recover the elenchus is to use the distinction between a necessary-condition relation between propositions (p only if q) and a grounding relation between facts (the fact that an epistemic agent S believes that p is grounded in the fact that S believes that q), where p and q are the content of S’s beliefs. (shrink)

The first part of this paper surveys five perspectives in Kant’s philosophy on the quantity of retribution to be inflicted on wrongdoers, ordered by two dimensions of difference – whether they are theoretical or practical perspectives, and the quantity of retribution they prescribe: (1) theoretical zero, the perspective of theoretical philosophy; (2) practical infinity, the perspective of God and conscience; (3) practical equality, the perspective of punishment in public law; (4) practical degrees, the perspective we adopt in private relations to (...) others and (5) practical zero, a perspective I argue is entailed by Kant’s doctrine of strict right, which is his justification of coercing compliance with public law. Kant acknowledges 1–4 but not 5. The second part draws on Kant’s account of the burden of proof in criminal law to argue that Kant is wrong to adopt 3 in responding to criminals, and that we ought to adopt 5 instead. (shrink)

In the first half, I suggest that Kant’s conception of our moral life goes through a significant shift after 1793, with reverberations in his eschatology. The earlier account, based on the postulate of immortality, describes our moral life as an endless pursuit of the highest good, but all this changes in the later account, and I point out three possible reasons for this change of heart. In the second half, I explore how the considerations Kant brings up to argue for (...) his accounts can inform our process of formulating positions with respect to the afterlife. I argue that, in the absence of a convincing theoretical proof for or against the afterlife as well as apodictically certain knowledge of how demanding the moral law is, the Kantian strategy would be to ask which account of our moral life delivers the kind of contentment that can sustain our moral resolve. I also point out a way theists might be able to find contentment despite their moral failures by imagining God’s moral kenosis. (shrink)

In this paper, building on recent and longstanding work (Warren 2001, Friedman 2013, Glezer 2018), I investigate how the account of the essences or natures of material substances in the Metaphysical Foundations is related to Kant’s demand for the completeness of the system of nature. We must ascribe causal powers to material substances for the properties of those substances to be observable and knowable. But defining those causal powers requires admitting laws of nature, taken as axioms or principles of natural (...) science, which govern anything that can be constructed mathematically or “given as an object of experience” (MAN, 4:474–75). Presenting a complete system of nature requires adumbrating the properties of material substances that can be objects of experience. But it is not possible to show how those properties are involved in explanations of the phenomena without involving the laws that govern interactions between material substances. For Kant, or so I will argue, it is not possible to account for the natures of things, or for the features of laws of nature, independently of each other. The paper will substantiate the Finitist Account (FA) of Kant on laws. The Finitist Account has it that modal judgments about the possible proofs that can be made, or interactions that can be explained, are (1) based on concrete intuitive reasoning and motivated by the desire to avoid appeal to the actual infinite, (2) grounded in finite decision procedures, which are (3) based on reliable systems of axioms or rules of inference. (shrink)

Apart from his critique of Kant, Maimon’s significance for the history of philosophy lies in his crucial role in the rediscovery of Spinoza by the German Idealists. Specifically, Maimon initiated a change from the common eighteenth-century view of Spinoza as the great ‘atheist’ to the view of Spinoza as an ‘acosmist’, i.e., a thinker who propounded a deep, though unorthodox, religious view denying the reality of the world and taking God to be the only real being. I have discussed this (...) aspect of Maimon’s philosophy in other places, and though the topic of the current paper has an interesting relation to certain doctrines of Spinoza, I will not develop this issue here. Neither of these two issues -- Maimon’s criticism of Kant or his original interpretation of Spinoza -- was considered by Maimon as his main contribution to philosophy. There is little doubt that if Maimon were asked to point out his single most important innovation he would have picked his doctrine of the Principle of Determinability [Satz der Bestimmbarkeit]. Regarding this doctrine Maimon writes: ... [T]he principle of determinability laid down in this work is a principle of all objectively real thought, and consequently of philosophy as a whole too. All the propositions of philosophy can be derived from, and be determined by it [woraus sich alle Sätze herleiten und wodurch sie sich bestimmen lassen]. … I have made available a supreme principle of all objectively real thought, viz., the principle of determinability... and have established as the ground of the whole of pure philosophy -- a principle which, if it is ever grasped, will, I hope, withstand every scrutiny. These claims may strike the reader as somewhat presumptuous, to say the least. But, if we pay attention to the last sentence of the passage, we can see that Maimon doubts whether his great finding will ever be understood. It is not unlikely that in this phrase (“wenn er nur einmal eingesehen werden wird”) Maimon was reacting to his own repeatedly unsuccessful attempts to explain the principle. The fate of Maimon’s principle has not been much better in the few works written on Maimon’s philosophy, and though almost all commentators agree that the principle of determinability is the linchpin of the positive philosophy Maimon was trying to develop, we do not yet have a clear explanation of this principle, or of the reason why Maimon assigns such importance to it. Recently, Oded Schechter developed an excellent reading of this principle, and in most aspects my view agrees with his (primarily, in its rejecting the attempt to explain the principle as a version of Leibniz’s predicate-in-subject [Praedicatum inest subjecto] containment thesis). My paper consists of two parts. The first is expository in nature. In this part, I spell out briefly the main aspects of Maimon’s principle of determinability and its aims. In the second part, I examine Maimon’s surprising claim that once we accept the principle of determinability, we have to deny the possibility of two subjects sharing the same predicate. Maimon provides several proofs for this highly counterintuitive claim, and I will try to clarify and evaluate these proofs. (shrink)

In epistemology, the concept of knowledge is of distinctive interest. This fact is also reflected in the discussion of epistemic value, which focuses to a large extend on the value problem of knowledge. This discussion suggests that knowledge has an outstanding value among epistemic standings because its value exceeds the value of its constitutive parts. I will argue that the value of knowledge is not outstanding by presenting epistemic standings of checking, transferring knowledge, and proving in court, whose values exceed (...) the value of knowledge in certain contexts. Moreover, the values of these other epistemic standings do not always rely on the value of knowledge. In terms of value, knowledge is not an outstanding epistemic concept. Hence, in terms of value we cannot find support for the privileged position that knowledge enjoys in epistemology. (shrink)

An overview of what Frege accomplishes in Part II of Grundgesetze, which contains proofs of axioms for arithmetic and several additional results concerning the finite, the infinite, and the relationship between these notions. One might think of this paper as an extremely compressed form of Part II of my book Reading Frege's Grundgesetze.

Corcoran’s 27 entries in the 1999 second edition of Robert Audi’s Cambridge Dictionary of Philosophy [Cambridge: Cambridge UP]. -/- ancestral, axiomatic method, borderline case, categoricity, Church (Alonzo), conditional, convention T, converse (outer and inner), corresponding conditional, degenerate case, domain, De Morgan, ellipsis, laws of thought, limiting case, logical form, logical subject, material adequacy, mathematical analysis, omega, proof by recursion, recursive function theory, scheme, scope, Tarski (Alfred), tautology, universe of discourse. -/- The entire work is available online free at more (...) than one website. Paste the whole URL. http://archive.org/stream/RobertiAudi_The.Cambridge.Dictionary.of.Philosophy/Robert.Audi_The.Cambrid ge.Dictionary.of.Philosophy -/- The 2015 third edition will be available soon. Before you think of buying it read some reviews on Amazon and read reviews of its competition: For example, my review of the 2008 Oxford Companion to Philosophy, History and Philosophy of Logic,29:3,291-292. URL: http://dx.doi.org/10.1080/01445340701300429 -/- Some of the entries have already been found to be flawed. For example, Tarski’s expression ‘materially adequate’ was misinterpreted in at least one article and it was misused in another where ‘materially correct’ should have been used. The discussion provides an opportunity to bring more flaws to light. -/- Acknowledgements: Each of these entries was presented at meetings of The Buffalo Logic Dictionary Project sponsored by The Buffalo Logic Colloquium. The members of the colloquium read drafts before the meetings and were generous with corrections, objections, and suggestions. Usually one 90-minute meeting was devoted to one entry although in some cases, for example, “axiomatic method”, took more than one meeting. Moreover, about half of the entries are rewrites of similarly named entries in the 1995 first edition. Besides the help received from people in Buffalo, help from elsewhere was received by email. We gratefully acknowledge the following: José Miguel Sagüillo, John Zeis, Stewart Shapiro, Davis Plache, Joseph Ernst, Richard Hull, Concha Martinez, Laura Arcila, James Gasser, Barry Smith, Randall Dipert, Stanley Ziewacz, Gerald Rising, Leonard Jacuzzo, George Boger, William Demopolous, David Hitchcock, John Dawson, Daniel Halpern, William Lawvere, John Kearns, Ky Herreid, Nicolas Goodman, William Parry, Charles Lambros, Harvey Friedman, George Weaver, Hughes Leblanc, James Munz, Herbert Bohnert, Robert Tragesser, David Levin, Sriram Nambiar, and others. -/- . (shrink)

Kant’s views on the relation between freedom and moral law seem to undergo a major, unannounced shift. In the third section of the Groundwork, Kant seems to be using the fact that we must act under the idea of freedom as a foundation for the moral law. However, in the Critique of Practical Reason, Kant claims that our awareness of our freedom depends on our awareness of the moral law. I argue that the apparent conflict between the two texts depends (...) on a reading of the opening paragraphs of Groundwork III, and on an interpretation of Kant’s claim that we “act under the idea of freedom”, that is implausible on textual and on philosophical grounds. I then present an alternative interpretation of what Kant means by “acting under the idea of freedom” and of the opening paragraphs of Groundwork III. I argue that the only substantive conclusion of these paragraphs is that no theoretical proof of freedom is necessary. Moreover I argue that although these paragraphs raise concerns about the validity of the moral law, these concerns and Kant’s answers to them, do not give rise to any significant conflict with his views in the Critique of Practical Reason. (shrink)

A dogma accepted in many ethical, religious, and legal frameworks is that the reasons behind conscientious objection (CO) in healthcare cannot be evaluated or judged by any institution because conscience is individual and autonomous. This paper shows that this background view is mistaken: the requirement to reveal and explain the reasons for conscientious objection in healthcare is ethically justified and legally desirable. Referring to real healthcare cases and legal regulations, this paper argues that these reasons should be evaluated either ex (...) ante or ex post and defends novel conceptual claims that have not been analyzed in the debates on CO. First, a moral threshold requirement: CO is only justified if the reasons behind a refusal are of a moral nature and meet a certain threshold of moral importance. Second, this paper considers the rarely discussed conceptual similarities between CO in healthcare and the legal regulations concerning military refusals that place the burden of proof on conscientious objectors. This paper concludes that conscientious objection in healthcare can be accommodated only in some cases of destroying or killing human organisms. (shrink)

I argue that No-Ought-From-Is (in the sense that I believe it) is a relatively trivial affair. Of course, when people try to derive substantive or non-vacuous moral conclusions from non-moral premises, they are making a mistake. But No-Non-Vacuous-Ought-From-Is is meta-ethically inert. It tells us nothing about the nature of the moral concepts. It neither refutes naturalism nor supports non-cognitivism. And this is not very surprising since it is merely an instance of an updated version of the conservativeness of logic (in (...) a logically valid inference you don’t get out what you haven’t put in): so long as the expressions F are non-logical, you cannot get non-vacuous F-conclusions from non-F premises. However, the triviality of No-Non-Vacuous-Ought-From-Is is important and its non-profundity profound. No-Ought-From-Is is widely supposed to tell us something significant about the nature of the moral concepts. If, in fact, it tells us nothing, this is a point well worth shouting from the housetops. This brings me to my dispute with Gerhard Schurz who has proved a related version of No-Ought-From-Is, No-Ought-Relevant-Ought-From-Is, a proof which relaxes my assumption that ‘ought’ should not be treated as a logical constant. But if ought is not a logical expression then it does not really matter much that No-Ought-From-Is would be salvageable even if it were. Furthermore, Schurz’s proof depends on special features of the moral concepts and this might afford the basis for an abductive argument to something like non-cognitivism. As an error theorist, and therefore a cognitivist, I object. Finally I take a dim view of deontic logic. Many of its leading principles are false, bordering on the nonsensical, and even the reasonably plausible ones are subject to devastating counter-examples. (shrink)

In this paper, I treat the question of whether and in what regard Kant's theory of biology contains a version of the argument from design, which is the question of whether Kant considers the purposive order of organized nature as a physicotheological proof for the existence of God, and in turn, the existence of God as the supersensible ground for the teleological order of organized nature. As an introduction to the topic, I name traditional examples of the argument from (...) design (section 1). I then outline Kant's changing attitude towards the argument in his Theory of Heavens, his Argument essay and the three Critiques, highlighting Kant's return to the argument from design in the CPJ after examining and rejecting it in his earlier writings. I elaborate in detail Kant's different uses of the argument in the "Critique of the Teleological Power of Judgment" (section 2). In section 3, I develop a consistent reading of Kant's references to the physicotheological proof in the "Critique of the Teleological Power of Judgment": in the "Analytic" he develops a teleological account of nature that makes no use of the argument from design but is consistent with it. In the "Dialectic" and the "Methodology", however, Kant discusses more ambitious systematic questions: the unity of the theoretical laws of nature and the unity of the natural and the supernatural moral laws. These are questions that require an explicit reference to the argument from design (section 3). (shrink)

Mathematicians often speak of conjectures as being confirmed by evidence that falls short of proof. For their own conjectures, evidence justifies further work in looking for a proof. Those conjectures of mathematics that have long resisted proof, such as the Riemann hypothesis, have had to be considered in terms of the evidence for and against them. In recent decades, massive increases in computer power have permitted the gathering of huge amounts of numerical evidence, both for conjectures in (...) pure mathematics and for the behavior of complex applied mathematical models and statistical algorithms. Mathematics has therefore become (among other things) an experimental science (though that has not diminished the importance of proof in the traditional style). We examine how the evaluation of evidence for conjectures works in mathematical practice. We explain the (objective) Bayesian view of probability, which gives a theoretical framework for unifying evidence evaluation in science and law as well as in mathematics. Numerical evidence in mathematics is related to the problem of induction; the occurrence of straightforward inductive reasoning in the purely logical material of pure mathematics casts light on the nature of induction as well as of mathematical reasoning. (shrink)

Medicine, like law, is a pragmatic, probabilistic activity. Both require that decisions be made on the basis of available evidence, within a limited time. In contrast to law, medicine, particularly evidence-based medicine as it is currently practiced, aspires to a scientific standard of proof, one that is more certain than the standards of proof courts apply in civil and criminal proceedings. But medicine, as Dr. William Osler put it, is an "art of probabilities," or at best, a "science (...) of uncertainty." One can better practice medicine by using other evidentiary standards in addition to the "scientific." To employ only the scientific standard of proof is inappropriate, if not impossible; furthermore, as this review will show, its application in medicine is fraught with bias. Evidence is information. It supports or undermines a proposition, whether a hypothesis in science, a diagnosis in medicine, or a fact or point in question in a legal investigation. In medicine, physicians marshal evidence to make decisions on how to best prevent, diagnose, and treat disease, and improve health. In law, courts decide the facts and render justice. Judges and juries assess evidence to establish liability, to settle custody and medical issues, and to determine a defendant's guilt or innocence. (shrink)

The aim of this study is to proof the argument – i.e. ‘there are significant linkages amongst tolerance, hybrid identities and migration.’ These linkages can be comprehended by means of conceptualising extensions of hybrid identities in aggregate trans/inter-migration processes. It can be put forward that arising hybrid identities are embedded in a blurring structure of thoughts, beliefs, states of affairs, facts, belongings and so forth. From multiculturalism and cosmopolitanism viewpoints, it is argued that tolerance and migration ought to be (...) analysed in frame of sociology of law, human rights, international migration law, and of course the European Union law. So far, normative arguments and soft law approaches are very much well integrated with social aspects of migration, tolerance, identity and culture. In this context, the study examines to what extent cultural and human components are protected by law in multiple levels from an interdisciplinary perspective. In this framework, it is crucial to raise the research enquiries: What are possible criteria for the limits of tolerance? To what extent tolerance is related to human rights and morality? What ought to be the limit of tolerance towards hybrid identities in multicultural and cosmopolitan societies? In a consistent manner, the criteria and standards developed by some leading scholars were reconstructed and discussed throughout this paper. These criteria and standards are both moral (part of a universal system of morality) and legal (international, supranational, or national – i.e. constitutional). In order to apply such standards their validity was discussed (i.e. the respective normative power has the norms at the various levels). In the case of morals, their difference to particular ethnic systems was established (i.e. criteria by which the two can be distinguished). Then the research argument was elaborated on whether and how the legal standards comply with the moral standards, how national standards comply with supranational or international standards. Likewise, the study highlights the crucial contributions of “World Society and World System Paradigms” that are associated with social space, global civil society, cosmopolitanism, ethnic diversity, cultural hybridity, human rights activism and public sphere. Recent debates in Refugee Studies (e.g. Syrian Refugee Crisis, Dual and Multiple Citizenship Issues and so forth) highlight the fact that the European Union needs to consider new aspects of tolerance for hybrid identities and tolerate cultural rights of hybrid identities, create cohesion in communities and establish intercultural dialogue amongst home-states and host-states. Sanguinely, the efforts of hybrid identities are strengthening the relations between home-host states and these ought to attract some considerable attention. The authors of this study hope that their endeavours may contribute somewhat towards that. (shrink)

This paper presents a formalization of the classical proof of completeness in Henkin-style developed by Troelstra and van Dalen for intuitionistic logic with respect to Kripke models. The completeness proof incorporates their insights in a fresh and elegant manner that is better suited for mechanization. We discuss details of our implementation in the Lean theorem prover with emphasis on the prime extension lemma and construction of the canonical model. Our implementation is restricted to a system of intuitionistic propositional (...) logic with implication, conjunction, disjunction, and falsity given in terms of a Hilbert-style axiomatization. As far as we know, our implementation is the first verified Henkin-style proof of completeness for intuitionistic logic following Troelstra and van Dalen's method in the literature. (shrink)

The first learning game to be developed to help students to develop and hone skills in constructing proofs in both the propositional and first-order predicate calculi. It comprises an autotelic (self-motivating) learning approach to assist students in developing skills and strategies of proof in the propositional and predicate calculus. The text of VALIDITY consists of a general introduction that describes earlier studies made of autotelic learning games, paying particular attention to work done at the Law School of Yale University, (...) called the ALL Project (Accelerated Learning of Logic). Following the introduction, the game of VALIDITY is described, first with reference to the propositional calculus, and then in connection with the first-order predicate calculus with identity. Sections in the text are devoted to discussions of the various rules of derivation employed in both calculi. Three appendices follow the main text; these provide a catalogue of sequents and theorems that have been proved for the propositional calculus and for the predicate calculus, and include suggestions for the classroom use of VALIDITY in university-level courses in mathematical logic. (shrink)

A close look at Frege's proof in "Foundations of Arithmetic" that every number has a successor. The examination reveals a surprising gap in the proof, one that Frege would later fill in "Basic Laws of Arithmetic".

Aristotle divided arguments that persuade into the rhetorical (which happen to persuade), the dialectical (which are strong so ought to persuade to some degree) and the demonstrative (which must persuade if rightly understood). Dialectical arguments were long neglected, partly because Aristotle did not write a book about them. But in the sixteenth and seventeenth century late scholastic authors such as Medina, Cano and Soto developed a sound theory of probable arguments, those that have logical and not merely psychological force but (...) fall short of demonstration. Informed by late medieval treatments of the law of evidence and problems in moral theology and aleatory contracts, they considered the reasons that could render legal, moral, theological, commercial and historical arguments strong though not demonstrative. At the same time, demonstrative arguments became better understood as Galileo and other figures of the Scientific Revolution used mathematical proof in arguments in physics. Galileo moved both dialectical and demonstrative arguments into mathematical territory. (shrink)

I have read many recent discussions of the limits of computation and the universe as computer, hoping to find some comments on the amazing work of polymath physicist and decision theorist David Wolpert but have not found a single citation and so I present this very brief summary. Wolpert proved some stunning impossibility or incompleteness theorems (1992 to 2008-see arxiv dot org) on the limits to inference (computation) that are so general they are independent of the device doing the computation, (...) and even independent of the laws of physics, so they apply across computers, physics, and human behavior. They make use of Cantor's diagonalization, the liar paradox and worldlines to provide what may be the ultimate theorem in Turing Machine Theory, and seemingly provide insights into impossibility, incompleteness, the limits of computation, and the universe as computer, in all possible universes and all beings or mechanisms, generating, among other things, a non- quantum mechanical uncertainty principle and a proof of monotheism. There are obvious connections to the classic work of Chaitin, Solomonoff, Komolgarov and Wittgenstein and to the notion that no program (and thus no device) can generate a sequence (or device) with greater complexity than it possesses. One might say this body of work implies atheism since there cannot be any entity more complex than the physical universe and from the Wittgensteinian viewpoint, ‘more complex’ is meaningless (has no conditions of satisfaction, i.e., truth-maker or test). Even a ‘God’ (i.e., a ‘device’with limitless time/space and energy) cannot determine whether a given ‘number’ is ‘random’, nor find a certain way to show that a given ‘formula’, ‘theorem’ or ‘sentence’ or ‘device’ (all these being complex language games) is part of a particular ‘system’. -/- Those wishing a comprehensive up to date framework for human behavior from the modern two systems view may consult my book ‘The Logical Structure of Philosophy, Psychology, Mind and Language in Ludwig Wittgenstein and John Searle’ 2nd ed (2019). Those interested in more of my writings may see ‘Talking Monkeys--Philosophy, Psychology, Science, Religion and Politics on a Doomed Planet--Articles and Reviews 2006-2019 2nd ed (2019) and Suicidal Utopian Delusions in the 21st Century 4th ed (2019) . (shrink)