Many philosophers are sceptical about the power of philosophy to refute commonsensical claims. They look at the famous attempts and judge them inconclusive. I prove that, even if those famous attempts are failures, there are alternative successful philosophical proofs against commonsensical claims. After presenting the proofs I briefly comment on their significance.

Many philosophers think that common sense knowledge survives sophisticated philosophical proofs against it. Recently, however, Bryan Frances (forthcoming) has advanced a philosophical proof that he thinks common sense can’t survive. Exploiting philosophical paradoxes like the Sorites, Frances attempts to show how common sense leads to paradox and therefore that common sense methodology is unstable. In this paper, we show how Frances’s proof fails and then present Frances with a dilemma.

The author argues that Plato’s “proof” that happiness follows justice has a fatal flaw – because the philosopher king in Plato’s Republic is itself a counter example.

This paper discusses proof-theoretic semantics, the project of specifying the meanings of the logical constants in terms of rules of inference governing them. I concentrate on Michael Dummett’s and Dag Prawitz’ philosophical motivations and give precise characterisations of the crucial notions of harmony and stability, placed in the context of proving normalisation results in systems of natural deduction. I point out a problem for defining the meaning of negation in this framework and prospects for an account of the (...) meanings of modal operators in terms of rules of inference. (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 suggestion of something akin to a ‘relativist solution to the Mind-Body problem’ has recently been held by some scientists and philosophers; either explicitly (Galadí, 2023; Lahav & Neemeh, 2022; Ludwig, 2015) or in more implicit terms (Solms, 2018; Velmans, 2002, 2008). In this paper I provide an argument in favor of a relativist approach to the Mind-Body problem, more specifically, an argument for ‘1st/3rd person relativism’, the claim that ‘The truth value of some sentences or propositions is relative to (...) 1st and 3rd person perspectives’. The argument for 1st/3rd person relativism is close to a forma proof. It is shown that, just by assuming the 1st/3rd person distinction itself and using first order logic and set theory, ‘1st/3rd person relativism’ follows as a theorem. Some consequences of ‘1st/3rd person relativism’ to the Mind-Body Problem are evaluated. It is shown that ‘1st/3rd person relativism’ predicts the existence of an (apparent) Explanatory Gap; explains why the Explanatory Gap is just apparent (and the origins of such illusion); dissolves the Hard-Problem; provides a possible solution the problem of Mental Causation; explains why Mental Causation looks like a problem in the first place and accurately predicts the actual empirically found correlation and covariation between conscious experiences and brain states. This explanatory power of ‘1st/3rd person relativism’ is particularly impressive since it was not designed as a possible solution to the Mind-Body problem in the first place. (shrink)

Prawitz conjectured that proof-theoretic validity offers a semantics for intuitionistic logic. This conjecture has recently been proven false by Piecha and Schroeder-Heister. This article resolves one of the questions left open by this recent result by showing the extensional alignment of proof-theoretic validity and general inquisitive logic. General inquisitive logic is a generalisation of inquisitive semantics, a uniform semantics for questions and assertions. The paper further defines a notion of quasi-proof-theoretic validity by restricting proof-theoretic validity to (...) allow double negation elimination for atomic formulas and proves the extensional alignment of quasi-proof-theoretic validity and inquisitive logic. (shrink)

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)

In mathematics, any form of probabilistic proof obtained through the application of a probabilistic method is not considered as a legitimate way of gaining mathematical knowledge. In a series of papers, Don Fallis has defended the thesis that there are no epistemic reasons justifying mathematicians’ rejection of probabilistic proofs. This paper identifies such an epistemic reason. More specifically, it is argued here that if one adopts a conception of mathematical knowledge in which an epistemic subject can know a mathematical (...) proposition based solely on a probabilistic proof, one is then forced to admit that such an epistemic subject can know several lottery propositions based solely on probabilistic evidence. Insofar as knowledge of lottery propositions on the basis of probabilistic evidence alone is denied by the vast majority of epistemologists, it is concluded that this constitutes an epistemic reason for rejecting probabilistic proofs as a means of acquiring mathematical knowledge. (shrink)

One of the most fundamental questions in the philosophy of mathematics concerns the relation between truth and formal proof. The position according to which the two concepts are the same is called deflationism, and the opposing viewpoint substantialism. In an important result of mathematical logic, Kurt Gödel proved in his first incompleteness theorem that all consistent formal systems containing arithmetic include sentences that can neither be proved nor disproved within that system. However, such undecidable Gödel sentences can be established (...) to be true once we expand the formal system with Alfred Tarski s semantical theory of truth, as shown by Stewart Shapiro and Jeffrey Ketland in their semantical arguments for the substantiality of truth. According to them, in Gödel sentences we have an explicit case of true but unprovable sentences, and hence deflationism is refuted. -/- Against that, Neil Tennant has shown that instead of Tarskian truth we can expand the formal system with a soundness principle, according to which all provable sentences are assertable, and the assertability of Gödel sentences follows. This way, the relevant question is not whether we can establish the truth of Gödel sentences, but whether Tarskian truth is a more plausible expansion than a soundness principle. In this work I will argue that this problem is best approached once we think of mathematics as the full human phenomenon, and not just consisting of formal systems. When pre-formal mathematical thinking is included in our account, we see that Tarskian truth is in fact not an expansion at all. I claim that what proof is to formal mathematics, truth is to pre-formal thinking, and the Tarskian account of semantical truth mirrors this relation accurately. -/- However, the introduction of pre-formal mathematics is vulnerable to the deflationist counterargument that while existing in practice, pre-formal thinking could still be philosophically superfluous if it does not refer to anything objective. Against this, I argue that all truly deflationist philosophical theories lead to arbitrariness of mathematics. In all other philosophical accounts of mathematics there is room for a reference of the pre-formal mathematics, and the expansion of Tarkian truth can be made naturally. Hence, if we reject the arbitrariness of mathematics, I argue in this work, we must accept the substantiality of truth. Related subjects such as neo-Fregeanism will also be covered, and shown not to change the need for Tarskian truth. -/- The only remaining route for the deflationist is to change the underlying logic so that our formal languages can include their own truth predicates, which Tarski showed to be impossible for classical first-order languages. With such logics we would have no need to expand the formal systems, and the above argument would fail. From the alternative approaches, in this work I focus mostly on the Independence Friendly (IF) logic of Jaakko Hintikka and Gabriel Sandu. Hintikka has claimed that an IF language can include its own adequate truth predicate. I argue that while this is indeed the case, we cannot recognize the truth predicate as such within the same IF language, and the need for Tarskian truth remains. In addition to IF logic, also second-order logic and Saul Kripke s approach using Kleenean logic will be shown to fail in a similar fashion. (shrink)

Proofs of God in Early Modern Europe offers a fascinating window into early modern efforts to prove God’s existence. Assembled here are twenty-two key texts, many translated into English for the first time, which illustrate the variety of arguments that philosophers of the seventeenth and eighteenth centuries offered for God. These selections feature traditional proofs—such as various ontological, cosmological, and design arguments—but also introduce more exotic proofs, such as the argument from eternal truths, the argument from universal aseity, and the (...) argument ex consensu gentium. Drawn from the work of eighteen philosophers, this book includes both canonical figures (such as Descartes, Spinoza, Newton, Leibniz, Locke, and Berkeley) and noncanonical thinkers (such as Norris, Fontenelle, Voltaire, Wolff, Du Châtelet, and Maupertuis). -/- Lloyd Strickland provides fresh translations of all selections not originally written in English and updates the spelling and grammar of those that were. Each selection is prefaced by a lengthy headnote, giving a biographical account of its author, an analysis of the main argument(s), and important details about the historical context. Strickland’s introductory essay provides further context, focusing on the various reasons that led so many thinkers of early modernity to develop proofs of God’s existence. -/- Proofs of God is perfect for both students and scholars of early modern philosophy and philosophy of religion. (shrink)

The “four-color” theorem seems to be generalizable as follows. The four-letter alphabet is sufficient to encode unambiguously any set of well-orderings including a geographical map or the “map” of any logic and thus that of all logics or the DNA plan of any alive being. Then the corresponding maximally generalizing conjecture would state: anything in the universe or mind can be encoded unambiguously by four letters. That admits to be formulated as a “four-letter theorem”, and thus one can search for (...) a properly mathematical proof of the statement. It would imply the “four colour theorem”, the proof of which many philosophers and mathematicians believe not to be entirely satisfactory for it is not a “human proof”, but intermediated by computers unavoidably since the necessary calculations exceed the human capabilities fundamentally. It is furthermore rather unsatisfactory because it consists in enumerating and proving all cases one by one. Sometimes, a more general theorem turns out to be much easier for proving including a general “human” method, and the particular and too difficult for proving theorem to be implied as a corollary in certain simple conditions. The same approach will be followed as to the four colour theorem, i.e. to be deduced more or less trivially from the “four-letter theorem” if the latter is proved. References are only classical and thus very well-known papers: their complete bibliographic description is omitted. (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 Four-Colour Theorem (4CT) proof, presented to the mathematical community in a pair of papers by Appel and Haken in the late 1970's, provoked a series of philosophical debates. Many conceptual points of these disputes still require some elucidation. After a brief presentation of the main ideas of Appel and Haken’s procedure for the proof and a reconstruction of Thomas Tymoczko’s argument for the novelty of 4CT’s proof, we shall formulate some questions regarding the connections between (...) the points raised by Tymoczko and some Wittgensteinian topics in the philosophy of mathematics such as the importance of the surveyability as a criterion for distinguishing mathematical proofs from empirical experiments. Our aim is to show that the “characteristic Wittgensteinian invention” (Mühlhölzer 2006) – the strong distinction between proofs and experiments – can shed some light in the conceptual confusions surrounding the Four-Colour Theorem. (shrink)

Mathematicians distinguish between proofs that explain their results and those that merely prove. This paper explores the nature of explanatory proofs, their role in mathematical practice, and some of the reasons why philosophers should care about them. Among the questions addressed are the following: what kinds of proofs are generally explanatory (or not)? What makes a proof explanatory? Do all mathematical explanations involve proof in an essential way? Are there really such things as explanatory proofs, and if so, (...) how do they relate to the sorts of explanation encountered in philosophy of science and metaphysics? (shrink)

Philosophical Issues, Volume 31, Issue 1, Page 355-373, October 2021.

I explore, from a proof-theoretic perspective, the hierarchy of classical and paraconsistent logics introduced by Barrio, Pailos and Szmuc in (Journal o f Philosophical Logic,49, 93-120, 2021). First, I provide sequent rules and axioms for all the logics in the hierarchy, for all inferential levels, and establish soundness and completeness results. Second, I show how to extend those systems with a corresponding hierarchy of validity predicates, each one of which is meant to capture “validity” at a different inferential (...) level. Then, I point out two potential philosophical implications of these results. (i) Since the logics in the hierarchy differ from one another on the rules, I argue that each such logic maintains its own distinct identity (contrary to arguments like the one given by Dicher and Paoli in 2019). (ii) Each validity predicate need not capture “validity” at more than one metainferential level. Hence, there are reasons to deny the thesis (put forward in Barrio, E., Rosenblatt, L. & Tajer, D. (Synthese, 2016)) that the validity predicate introduced in by Beall and Murzi in (Journal o f Philosophy,110(3), 143–165, 2013) has to express facts not only about what follows from what, but also about the metarules, etc. (shrink)

This chapter discusses Kant's 1763 "possibility proof" for the existence of God. I first provide a reconstruction of the proof in its two stages, and then revisit my earlier argument according to which the being the proof delivers threatens to be a Spinozistic-panentheistic God—a being whose properties include the entire spatio-temporal universe—rather than the traditional, ontologically distinct God of biblical monotheism. I go on to evaluate some recent alternative readings that have sought to avoid this result by (...) arguing that the relevant facts about real modality can be ultimately grounded in God’s powers or thoughts – or that Kant just leaves the grounding relations mysterious. I argue that the textual and philosophical costs of each of these alternative readings are formidable. The chapter concludes with a discussion of the fate of the proof in the critical period. Some commentators think that it disappears altogether, or that it is downgraded such that it produces a mere regulative idea of God as the most real being. I suggest that the proof survives but that the mode of assent it licenses towards its conclusion changes from knowledge to a certain kind of Belief (Glaube). (shrink)

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)

Most human actions are complex, but some of them are basic. Which are these? In this paper, I address this question by invoking slips, a common kind of mistake. The proposal is this: an action is basic if and only if it is not possible to slip in performing it. The argument discusses some well-established results from the psychology of language production in the context of a philosophical theory of action. In the end, the proposed criterion is applied to (...) discuss some well-known theories of basic actions. (shrink)

We investigated whether mathematicians typically agree about the qualities of mathematical proofs. Between-mathematician consensus in proof appraisals is an implicit assumption of many arguments made by philosophers of mathematics, but to our knowledge the issue has not previously been empirically investigated. We asked a group of mathematicians to assess a specific proof on four dimensions, using the framework identified by Inglis and Aberdein (2015). We found widespread disagreement between our participants about the aesthetics, intricacy, precision and utility of (...) the proof, suggesting that a priori assumptions about the consistency of mathematical proof appraisals are unreasonable. (shrink)

This paper considers logics which are formally dual to intuitionistic logic in order to investigate a co-constructive logic for proofs and refutations. This is philosophically motivated by a set of problems regarding the nature of constructive truth, and its relation to falsity. It is well known both that intuitionism can not deal constructively with negative information, and that defining falsity by means of intuitionistic negation leads, under widely-held assumptions, to a justification of bivalence. For example, we do not want to (...) equate falsity with the non-existence of a proof since this would render a statement such as “pi is transcendental” false prior to 1882. In addition, the intuitionist account of negation as shorthand for the derivation of absurdity is inadequate, particularly outside of purely mathematical contexts. To deal with these issues, I investigate the dual of intuitionistic logic, co-intuitionistic logic, as a logic of refutation, alongside intuitionistic logic of proofs. Direct proof and refutation are dual to each other, and are constructive, whilst there also exist syntactic, weak, negations within both logics. In this respect, the logic of refutation is weakly paraconsistent in the sense that it allows for statements for which, neither they, nor their negation, are refuted. I provide a proof theory for the co-constructive logic, a formal dualizing map between the logics, and a Kripke-style semantics. This is given an intuitive philosophical rendering in a re-interpretation of Kolmogorov’s logic of problems. (shrink)

In this article, I argue that it is impossible to complete infinitely many tasks in a finite time. A key premise in my argument is that the only way to get to 0 tasks remaining is from 1 task remaining, when tasks are done 1-by-1. I suggest that the only way to deny this premise is by begging the question, that is, by assuming that supertasks are possible. I go on to present one reason why this conclusion (that supertasks are (...) impossible) is important, namely that it implies a new verdict on a decision puzzle propounded by Jeffrey Barrett and Frank Arntzenius. (shrink)

Gaisi Takeuti extended Gentzen's work to higher-order case in 1950's–1960's and proved the consistency of impredicative subsystems of analysis. He has been chiefly known as a successor of Hilbert's school, but we pointed out in the previous paper that Takeuti's aimed to investigate the relationships between "minds" by carrying out his proof-theoretic project rather than proving the "reliability" of such impredicative subsystems of analysis. Moreover, as briefly explained there, his philosophical ideas can be traced back to Nishida's philosophy (...) in Kyoto's school. For the proving the consistency of such systems, it is crucial to prove the well-foundedness of ordinals called "ordinal diagrams" developed for it. Takeuti presented such arguments several times in order to show that they are admitted in his stand point. As a starting point of investigating his finitist stand point, we formulate the system of ordinal notations up to ε0 and reconstruct the well-foundedness arguments of them. (shrink)

Gaisi Takeuti (1926–2017) is one of the most distinguished logicians in proof theory after Hilbert and Gentzen. He extensively extended Hilbert's program in the sense that he formulated Gentzen's sequent calculus, conjectured that cut-elimination holds for it (Takeuti's conjecture), and obtained several stunning results in the 1950–60s towards the solution of his conjecture. Though he has been known chiefly as a great mathematician, he wrote many papers in English and Japanese where he expressed his philosophical thoughts. In particular, (...) he used several keywords such as "active intuition" and "self-reflection" from Nishida's philosophy. In this paper, we aim to describe a general outline of our project to investigate Takeuti's philosophy of mathematics. In particular, after reviewing Takeuti's proof-theoretic results briefly, we describe some key elements in Takeuti's texts. By explaining these texts, we point out the connection between Takeuti's proof theory and Nishida's philosophy and explain the future goals of our project. (shrink)

The problem of resurrection, one of the most important issues in the philosophy and theology. Some of Muslim philosophers and the vast majority of theologians always discussed about this topic. Some of Muslim philosophers accepted this problem and prove it, but, in quality of occurrence of them, they have differ believe from each other. However, some of Muslim theologians except those who believe in transmogrification, they consider the resurrection to be one of the principle of religion, beside monotheism and prophecy. (...) Moreover, they proved the principle of resurrection through reasons, and explanations them elaborates through the Holy Quran and Prophetic saying (Traditions- Hadith). Avicenna and Suhrawardi have been proving resurrection. Of course, for prove of this problem, they adopted of different manner from Muslim Theologians. Both of philosophers according to codify of ordinary psychological system, have proved and explanation this issue. Thus, for understand of Avicenna and Suhrawardi’s eschatology. First of all, we must understand of their psychological system. Furthermore, in this paper, we will discuss about survival of the soul according to immaterial soul and perfecting of soul as two premises for prove of resurrection. (shrink)

We argue that Wittgenstein’s philosophical perspective on Gödel’s most famous theorem is even more radical than has commonly been assumed. Wittgenstein shows in detail that there is no way that the Gödelian construct of a string of signs could be assigned a useful function within (ordinary) mathematics. — The focus is on Appendix III to Part I of Remarks on the Foundations of Mathematics. The present reading highlights the exceptional importance of this particular set of remarks and, more specifically, (...) emphasises its refined composition and rigorous internal structure. (shrink)

Transfinite ordinal numbers enter mathematical practice mainly via the method of definition by transfinite recursion. Outside of axiomatic set theory, there is a significant mathematical tradition in works recasting proofs by transfinite recursion in other terms, mostly with the intention of eliminating the ordinals from the proofs. Leaving aside the different motivations which lead each specific case, we investigate the mathematics of this action of proof transforming and we address the problem of formalising the philosophical notion of elimination (...) which characterises this move. (shrink)

Kant’s “moral proof” for the existence of God has been the subject of much criticism, even among his most sympathetic commentators. According to the critics, the primary problem is that the notion of the “highest good,” on which the moral proof depends, introduces an element of contingency and heteronomy into Kant’s otherwise strict, autonomy-based moral thinking. In this paper, I shall argue that Kant’s moral proof is not only more defensible than commentators have typically acknowledged, but also (...) has some very interesting implications (e.g. the moral proof is “circular” and thus implicitly self-validating). My account shall proceed in five stages: 1. Preliminary Discussion of the Moral Proof 2. the Argument of the Moral Proof 3. Critics of the Moral Proof 4. Defense of the Moral Proof 5. Implications of the Moral Proof: Circularity and Self-referentiality.” . (shrink)

Bilateral proof systems, which provide rules for both affirming and denying sentences, have been prominent in the development of proof-theoretic semantics for classical logic in recent years. However, such systems provide a substantial amount of freedom in the formulation of the rules, and, as a result, a number of different sets of rules have been put forward as definitive of the meanings of the classical connectives. In this paper, I argue that a single general schema for bilateral (...) class='Hi'>proof rules has a reasonable claim to inferentially articulating the core meaning of all of the classical connectives. I propose this schema in the context of a bilateral sequent calculus in which each connective is given exactly two rules: a rule for affirmation and a rule for denial. Positive and negative rules for all of the classical connectives are given by a single rule schema, harmony between these positive and negative rules is established at the schematic level by a pair of elimination theorems, and the truth-conditions for all of the classical connectives are read off at once from the schema itself. (shrink)

It is widely assumed among contemporary philosophers that Descartes’ version of ontological proof, among other weaknesses, makes an impossible and unjustified move from the mental world of concepts to the real world of things. Contrary to this opinion I will try to show that Descartes’ famous principle of clear and distinct perception suffices to find an adequate inferential connection between the contents of the human mind and extra-mental reality. In a clear and distinct way we cognitively grasp the concept (...) of supremely perfect being as the concept that we do not construct by an arbitrary definition of the word ”God’. (shrink)

Philosophical conceptual analysis is an experimental method. Focusing on this helps to justify it from the skepticism of experimental philosophers who follow Weinberg, Nichols & Stich. To explore the experimental aspect of philosophical conceptual analysis, I consider a simpler instance of the same activity: everyday linguistic interpretation. I argue that this, too, is experimental in nature. And in both conceptual analysis and linguistic interpretation, the intuitions considered problematic by experimental philosophers are necessary but epistemically irrelevant. They are like (...) variables introduced into mathematical proofs which drop out before the solution. Or better, they are like the hypotheses that drive science, which do not themselves need to be true. In other words, it does not matter whether or not intuitions are accurate as descriptions of the natural kinds that undergird philosophical concepts; the aims of conceptual analysis can still be met. (shrink)

After introductory reminder of and comments on Gödel’s ontological proof, we discuss the collapse of modalities, which is provable in Gödel’s ontological system GO. We argue that Gödel’s texts confirm modal collapse as intended consequence of his ontological system. Further, we aim to show that modal collapse properly fits into Gödel’s philosophical views, especially into his ontology of separation and union of force and fact, as well as into his cosmological theory of the nonobjectivity of the lapse of (...) time. As a result, modal collapse should not be conceived in Gödel so much as a deficit, but rather as a kind of the rise of modalities to the “perfect” being. We further show that, in accordance with Gödel’s ontology, the concepts of modality and time should be derived in terms of the “fundamental philosophical concept” of cause. To give an example of how a formalization of such causative Gödelian ontology and ontological proof might look, we propose the transformation of GO into a kind of causally re-interpreted justification logic. (shrink)

Mill’s Principle of Utility: Origins, Proof, and Implications (Leiden: Brill, 2022) is a scholarly monograph on John Stuart Mill’s utilitarianism with a particular emphasis on his proof of the principle of utility. Originally published as Mill’s Principle of Utility: A Defense of John Stuart Mill’s Notorious Proof (Amsterdam: Editions Rodopi, 1994), the present volume is a revised and enlarged edition with additional material, tighter arguments, crisper discussions, and updated references. The initiative is still principally an analysis, interpretation, (...) and defense of the controversial proof, which has yet to attract a scholarly consensus on how it works and whether it succeeds. Well over a century and a half after Mill’s contribution, the subject matter continues to figure prominently in classroom discussions, where the principle and its proof are presented and debated in connection with the critical baggage of logical problems and conceptual confusions wrongly attributed to Mill from the beginning. The overarching aim of the book, now supplemented by a comprehensive historical background and salient philosophical implications, remains the vindication of Mill’s reasoning in pursuit of what he promises as a proof of the principle of utility. (shrink)

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.

We develop a reading of Moore’s “Proof of an External World” that emphasizes the connections between this paper and Moore’s earlier concerns and strategies. Our reading has the benefit of explaining why the claims that Moore advances in “Proof of an External World” would have been of interest to him, and avoids attributing to him arguments that are either trivial or wildly unsuccessful. Part of the evidence for our view comes from unpublished drafts which, we believe, contain important (...) clues concerning Moore’s aims and intent. While our approach to PEW may be classified alongside other broadly "metaphysical" readings, we believe that a proper recognition of the continuity in Moore’s philosophical concerns and strategies across his philosophical career shows that the customary distinction between "epistemological" and "metaphysical" interpretative approaches to PEW is at best superficial. (shrink)

Hegel endorsed proofs of the existence of God, and also believed God to be a person. Some of his interpreters ignore these apparently retrograde tendencies, shunning them in favor of the philosopher's more forward-looking contributions. Others embrace Hegel's religious thought, but attempt to recast his views as less reactionary than they appear to be. Robert Williams's latest monograph belongs to a third category: he argues that Hegel's positions in philosophical theology are central to his philosophy writ large. The book (...) is diligently researched, and marshals an impressive amount of textual evidence concerning Hegel's view of the proofs, his theory of personhood, and his views on religious community.Many of... (shrink)

Not focusing on the history of classical logic, this book provides discussions and quotes central passages on its origins and development, namely from a philosophical perspective. Not being a book in mathematical logic, it takes formal logic from an essentially mathematical perspective. Biased towards a computational approach, with SAT and VAL as its backbone, this is an introduction to logic that covers essential aspects of the three branches of logic, to wit, philosophical, mathematical, and computational.

The subject of the thesis is the relationship between aesthetic and God. It aims to discuss whether the sense of beauty is proof of the existence of God and to determine the plausibility of aesthetic proof. As a matter of fact that reality and the perception of beauty point to two-way consciousness. In this context, it will be inevitable to mention God's relation with consciousness in the emergence of beauty. In the first part, the concepts of aesthetics will (...) be analyzed, and their philosophical discussions will be investigated. In the second part, the source of beauty will be questioned, and beauty will be grounded way of a priori and a posteriori with God. Afterward, aesthetic proofs will be presented and an evaluation will be made in the context of naturalistic claims and objections. Among the traditional arguments in the philosophy of religion, aesthetic proof offers a value-oriented approach rather than a mechanical view. It appreciates beauty from an inclusive perspective and has an existential aspect. The study concentrates on the nature of beauty and aesthetics, philosophical discussions, and the rational value of aesthetic proof and concludes that it is rational to suppose that the existence of God can be ontologically and epistemologically explanatory for aesthetic properties. (shrink)

* Special Issue on the Philosophical Dimensions of the Trial* This summarises and discusses the contributions.

This paper argues that we should reject G. E. Moore’s anti-skeptical argument as it is presented in “Proof of an External World.” However, the reason I offer is different from traditional objections. A proper understanding of Moore’s “proof” requires paying attention to an important distinction between two forms of skepticism. I call these Ontological Skepticism and Epistemic Skepticism. The former is skepticism about the ontological status of fundamental reality, while the latter is skepticism about our empirical knowledge. Philosophers (...) often assume that Moore’s response to “external world skepticism” deals exclusively with the former, not the latter. But this is a mistake. I shall argue that Moore’s anti-skeptical argument targets an ontological form of skepticism. Thus, the conclusion is an ontological claim about fundamental reality, while the premises are epistemic claims. If this is correct, then the conclusion outstrips the scope of its premises and proves too much. (shrink)

Georg Cantor's absolute infinity, the paradoxical Burali-Forti class Ω of all ordinals, is a monstrous non-entity for which being called a "class" is an undeserved dignity. This must be the ultimate vexation for mathematical philosophers who hold on to some residual sense of realism in set theory. By careful use of Ω, we can rescue Georg Cantor's 1899 "proof" sketch of the Well-Ordering Theorem––being generous, considering his declining health. We take the contrapositive of Cantor's suggestion and add Zermelo's choice (...) function. This results in a concise and uncomplicated proof of the Well-Ordering Theorem. (shrink)

Ideas on meaning, rules and mathematical proofs abound in Wittgenstein’s writings. The undeniable fact that they are present together, sometimes intertwined in the same passage of Philosophical Investigations or Remarks on the Foundations of Mathematics, does not show, however, that the connection between these ideas is necessary or inextricable. The possibility remains, and ought to be checked, that they can be plausibly and consistently separated. I am going to examine two views detectable in Wittgenstein’s works: one about proofs, the (...) other about meaning and rules. The first is the denial of the objectivity of proof. The second is a conception of meaning stemming from the rule-following considerations. I shall argue that, though Wittgenstein seems to conjoin the two views, they can be, and should be, separated1. (shrink)

In the opening chapter of the Monologion, Anselm offers an intriguing proof for the existence of a Platonic form of goodness. This proof is extremely interesting, both in itself and for its place in the broader argument for God’s existence that Anselm develops in the Monologion as a whole. Even so, it has yet to receive the scholarly attention that it deserves. My aim in this article is to begin correcting this state of affairs by examining Anslem’s (...) class='Hi'>proof in some detail. In particular, I aim to clarify the proof’s structure, motivate and explain its central premises, and begin the larger project of evaluating its overall success as an argument for Platonism about goodness. (shrink)

Peter van Inwagen's Direct Argument (DA) for incompatibilism purports to establish incompatibilism with respect to moral responsibility and determinism without appealing to assumptions that compatibilists usually consider controversial. Recently, Michael McKenna has presented a novel critique of DA. McKenna's critique raises important issues about philosophical dialectics. In this article, we address those issues and contend that his argument does not succeed.

Gödel's ontological argument is related to Gödel's view that causality is the fundamental concept in philosophy. This explicit philosophical intention is developed in the form of an onto-theological Gödelian system based on justification logic. An essentially richer language, so extended, offers the possibility to express new philosophical content. In particular, theorems on the existence of a universal cause on a causal "slingshot" are formulated.

One popular conception of natural theology holds that certain purely rational arguments are insulated from empirical inquiry and independently establish conclusions that provide evidence, justification, or proof of God’s existence. Yet, some raise suspicions that philosophers and theologians’ personal religious beliefs inappropriately affect these kinds of arguments. I present an experimental test of whether philosophers and theologians’ argument analysis is influenced by religious commitments. The empirical findings suggest religious belief affects philosophical analysis and offer a challenge to theists (...) and atheists, alike: reevaluate the scope of natural theology’s conclusions or acknowledge and begin to address the influence of religious belief. (shrink)

This is a defense of John Stuart Mill’s proof of the principle of utility in the fourth chapter of his Utilitarianism. The proof is notorious as a fallacious attempt by a prominent philosopher, who ought not to have made the elementary mistakes he is supposed to have made. This book shows that he did not. The aim is not to glorify utilitarianism, in a full sweep, as the best normative ethical theory, or even to vindicate, on a more (...) specific level, Mill’s universalistic ethical hedonism as the best form of utilitarianism. The book is concerned only with Mill’s utilitarianism, and primarily with his proof of the principle of utility. The purpose is to show that Mill proceeds intelligibly and systematically in pursuing a well-defined project in the fourth chapter of Utilitarianism, and that he successfully defends what he sets out to establish in his proof of the principle of utility. (shrink)

Scientific inquiry takes onward course from the point where previous scientists had reached. But philosophical analysis initiates from scratch. Philosophy questions everything and chooses starting point for itself after having ruled out all the unsubstantiated and doubtful elements of the topic under study. Secondly, known realities must make sense. If a theory is officially 'counterintuitive', then either it is mere fiction or at the most; a distorted form of truth. This book's analysis is based on the philosophical principle (...) that knowledge is empirical and does not arise magically in absence of observational grounds. With philosophical approach, it was doubtful to accept that Georges Lemaître already knew Hubble's law in year 1927 that was yet to be found by Edwin Hubble in year 1929. Therefore this book started with denial of the claim that Lemaître already knew this law. But analysis of section I.III forced author to look the matter from original source and it came to surface that Lemaître knew this law in year 1927. But contrary to mainstream claim, Lemaître had not derived that law from general relativity (GR) equations rather had deduced from a method given by Hubble himself. Whereas whole case of the Big Bang Theory rests on misleading claim that Lemaître had derived this law solely from GR equations. The basis of this claim happened to be a manipulated translation (1931) of Lemaître's original 1927 article. People regard Big Bang Theory as truth because authoritative sources deceived them by presenting a manipulated translation in year 1931. This book is a philosophical analysis of original papers of Alexander Friedmann (1922), Georges Lemaître (1927), Edwin Hubble (1929) and Albert Einstein (1917) thus covers actual roots and origins of the Big Bang Model. In this book, only the core elements of the Big Bang Model i.e. 'Expansion of Universe' and 'CMBR' are covered. It has been sufficiently shown that 'expansion' is an illusion whereas CMBR is a proof that we live in a non-expanding infinite universe. If these two core elements of the standard Big Bang Model are precisely refuted then there is nothing crucial left with the standard model. For readers of this book at least, Big Bang Theory shall become a story of past mistakes. Author is not an authoritative source on science topics therefore readers must download all the above mentioned original papers and check all the points outlined in this book from relevant original papers. Unlike reading from an authoritative source that makes readers relaxed and careless but enables authorities to deceive them in worst way possible, this book requires readers to remain alert on all the points discussed in the book and verify everything from original sources whose links are given at the end of this description and also provided in footnotes section of the book. This book is not a judgment of the topic rather it is like a case presented by an advocate while readers are the judges. Readers are required to apply their own critical judgment to conclude the matter by themselves. After carefully reading this book, readers will also start taking 'authoritative sources' with due care and it will become difficult for the 'authorities' to deceive them again. (shrink)