The guise of the good doctrine is the view that whatever we desire, we desire it under the guise of the good, i.e. it appears good to us in some way. In this paper I first clarify the role that the doctrine of the guise of the good plays in the first step of J. S. Mill’s proof of the principle of utility (in which he shows that one’s happiness is desirable as an end). Then (...) I provide textual evidence in favour of ascribing the doctrine to Mill, arguing that he commits to it to the extent that he equates finding something pleasant and thinking it desirable. Finally I counter two potential sources of evidence against ascribing the guise of the good to Mill: apparent desires based on ‘fixed ideas’, and those habitual desires which are no longer associated with finding their objects pleasant. I argue that ‘fixed ideas’ do not feed actual desires, and that the habitual desires which seem to escape the guise of the good, even if not uncommon, have a secondary status as desires. (shrink)

This paper explains a way of understanding Kant's proof of God's existence in the Critique of Practical Reason that has hitherto gone unnoticed and argues that this interpretation possesses several advantages over its rivals. By first looking at examples where Kant indicates the role that faith plays in moral life and then reconstructing the proof of the second Critique with this in view, I argue that, for Kant, we must adopt a certain conception of the highest good, (...) and so also must choose to believe in the kind of God that can make it possible, because this is essentially a way of actively striving for virtue. One advantage of this interpretation, I argue, is that it is able to make sense of the strong link Kant draws between morality and religion. (shrink)

The opening chapters of Anselm's Monologion contain a ‘proof’ of a perfect being, which has received far less attention than the more famous Proslogion proof, and the ontological arguments derived from it. I wish to rectify this by developing an argument in defence of a crucial premise of the Monologion proof. This premise states that ‘the Good’, i.e. that in virtue of which numerically distinct things may all be good, must itself be a supremely (...) class='Hi'>good thing. I motivate the argument before considering objections to both premises, as well as putative ‘parodies’ of my argument. Part of the motivation of my argument will involve the claim that the Good, if it is good at all, must be a paradigm good thing. I conclude that theists have a second kind of ontological argument at their disposal. (shrink)

Bruineberg and colleagues argue that the patellar reflex cannot be modeled sufficiently with a Friston blanket due to counterintuitive sensorimotor boundaries. Although I agree with their theoretical discussion, their model of the patellar reflex is insufficiently based on clinical knowledge. Consequently, this example should not be applied to challenge Friston blankets. I will provide an alternative example.

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)

The traditional view of evidence in mathematics is that evidence is just proof and proof is just derivation. There are good reasons for thinking that this view should be rejected: it misrepresents both historical and current mathematical practice. Nonetheless, evidence, proof, and derivation are closely intertwined. This paper seeks to tease these concepts apart. It emphasizes the role of argumentation as a context shared by evidence, proofs, and derivations. The utility of argumentation theory, in general, and (...) argumentation schemes, in particular, as a methodology for the study of mathematical practice is thereby demonstrated. Argumentation schemes represent an almost untapped resource for mathematics education. Notably, they provide a consistent treatment of rigorous and non-rigorous argumentation, thereby working to exhibit the continuity of reasoning in mathematics with reasoning in other areas. Moreover, since argumentation schemes are a comparatively mature methodology, there is a substantial body of existing work to draw upon, including some increasingly sophisticated software tools. Such tools have significant potential for the analysis and evaluation of mathematical argumentation. The first four sections of the paper address the relationships of evidence to proof, proof to derivation, argument to proof, and argument to evidence, respectively. The final section directly addresses some of the educational implications of an argumentation scheme account of mathematical reasoning. (shrink)

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)

This paper studies proof systems for the logics of super-strict implication ST2–ST5, which correspond to C.I. Lewis’ systems S2–S5 freed of paradoxes of strict implication. First, Hilbert-style axiomatic systems are introduced and shown to be sound and complete by simulating STn in Sn and backsimulating Sn in STn, respectively(for n=2,...,5). Next, G3-style labelled sequent calculi are investigated. It is shown that these calculi have the good structural properties that are distinctive of G3-style calculi, that they are sound and (...) complete, and it is shown that the proof search for G3. ST2 is terminating and therefore the logic is decidable. (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)

This book provides a detailed, updated exposition and defense of five of the historically most important (but in recent years largely neglected) philosophical proofs of God’s existence: the Aristotelian, the Neo-Platonic, the Augustinian, the Thomistic, and the Rationalist. It also offers a thorough treatment of each of the key divine attributes—unity, simplicity, eternity, omnipotence, omniscience, perfect goodness, and so forth—showing that they must be possessed by the God whose existence is demonstrated by the proofs. Finally, it answers at length all (...) of the objections that have been leveled against these proofs. This work provides as ambitious and complete a defense of traditional natural theology as is currently in print. Its aim is to vindicate the view of the greatest philosophers of the past— thinkers like Aristotle, Plotinus, Augustine, Aquinas, Leibniz, and many others— that the existence of God can be established with certainty by way of purely rational arguments. It thereby serves as a refutation both of atheism and of the fideism that gives aid and comfort to atheism. (shrink)

This paper provides a proof-theoretic study of quantified non-normal modal logics. It introduces labelled sequent calculi based on neighbourhood semantics for the first-order extension, with both varying and constant domains, of monotone NNML, and studies the role of the Barcan formulas in these calculi. It will be shown that the calculi introduced have good structural properties: invertibility of the rules, height-preserving admissibility of weakening and contraction and syntactic cut elimination. It will also be shown that each of the (...) calculi introduced is sound and complete with respect to the appropriate class of neighbourhood frames. In particular, the completeness proof constructs a formal derivation for derivable sequents and a countermodel for non-derivable ones, and gives a semantic proof of the admissibility of cut. (shrink)

A sequent calculus methodology for systems of agency based on branching-time frames with agents and choices is proposed, starting with a complete and cut-free system for multi-agent deliberative STIT; the methodology allows a transparent justification of the rules, good structural properties, analyticity, direct completeness and decidability proofs.

This paper studies proof systems for the logics of super-strict implication \(\textsf{ST2}\) – \(\textsf{ST5}\), which correspond to C.I. Lewis’ systems \(\textsf{S2}\) – \(\textsf{S5}\) freed of paradoxes of strict implication. First, Hilbert-style axiomatic systems are introduced and shown to be sound and complete by simulating \(\textsf{STn}\) in \(\textsf{Sn}\) and backsimulating \(\textsf{Sn}\) in \(\textsf{STn}\), respectively (for \({\textsf{n}} =2, \ldots, 5\) ). Next, \(\textsf{G3}\) -style labelled sequent calculi are investigated. It is shown that these calculi have the good structural properties that (...) are distinctive of \(\textsf{G3}\) -style calculi, that they are sound and complete, and it is shown that the proof search for \(\mathsf {G3.ST2}\) is terminating and therefore the logic is decidable. (shrink)

Mathematicians only use deductive proofs to establish that mathematical claims are true. They never use inductive evidence, such as probabilistic proofs, for this task. Don Fallis (1997 and 2002) has argued that mathematicians do not have good epistemic grounds for this complete rejection of probabilistic proofs. But Kenny Easwaran (2009) points out that there is a gap in this argument. Fallis only considered how mathematical proofs serve the epistemic goals of individual mathematicians. Easwaran suggests that deductive proofs might be (...) epistemically superior to probabilistic proofs because they are transferable. That is, one mathematician can give such a proof to another mathematician who can then verify for herself that the mathematical claim in question is true without having to rely at all on the testimony of the first mathematician. In this paper, I argue that collective epistemic goals are critical to understanding the methodological choices of mathematicians. But I argue that the collective epistemic goals promoted by transferability do not explain the complete rejection of probabilistic proofs. (shrink)

In principle, a proof can be any sequence of logical deductions from axioms and previously-proved statements that concludes with the proposition in question. This freedom in constructing a proof can seem overwhelming at first. How do you even start a proof? Here’s the good news: many proofs follow one of a handful of standard templates. Proofs all differ in the details, of course, but these templates at least provide you with an outline to fill in. We’ll (...) go through several of these standard patterns, pointing out the basic idea and common pitfalls and giving some examples. Many of these templates fit together; one may give you a top-level outline while others help you at the next level of detail. And we’ll show you other, more sophisticated proof techniques later on. The recipes below are very specific at times, telling you exactly which words to write down on your piece of paper. You’re certainly free to say things your own way instead; we’re just giving you something you could say so that you’re never at a complete loss. (shrink)

This Element looks at the contemporary debate on the nature of mathematical rigour and informal proofs as found in mathematical practice. The central argument is for rigour pluralism: that multiple different models of informal proof are good at accounting for different features and functions of the concept of rigour. To illustrate this pluralism, the Element surveys some of the main options in the literature: the 'standard view' that rigour is just formal, logical rigour; the models of proofs as (...) arguments and dialogues; the recipe model of proofs as guiding actions and activities; and the idea of mathematical rigour as an intellectual virtue. The strengths and weaknesses of each are assessed, thereby providing an accessible and empirically-informed introduction to the key issues and ideas found in the current discussion. (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)

The specific characteristics of mathematical argumentation all depend on the centrality that writing has in the practice of mathematics, but blindness to this fact is near universal. What follows concerns just one of those characteristics, justification by proof. There is a prevalent view that long proofs pose a problem for the thesis that mathematical knowledge is justified by proof. I argue that there is no such problem: in fact, virtually all the justifications of mathematical knowledge are ‘long proofs’, (...) but because these real justifications are distributed in the written archive of mathematics, proofs remain surveyable, hence good. (shrink)

Beer’s (2003) paper is a tour de force of detailed comments on the more general notion of “situated- dynamical modeling, and provides a concrete sample ness”, Beer suggests that “on this view, situated action of the kinds of understanding dynamicists may realis- is the fundamental concern and cognition is … one tically hope to achieve. The analysis is thus, as Beer resource among many that can be brought to bear as an states, a “tool for building intuition”, and in this (...) it suc- agent encounters its world”. In the worked example of ceeds brilliantly. But it is also an attempt to show that an agent that approaches circles and avoids diamonds, dynamical approaches can get a foothold in the expla- we see direct evidence of this in the claim that a certain nation of “minimally cognitive behaviors”; that is to three-dimensional projection provides a potent analytic say, behaviors that seem, on the surface at least, good tool. For this projection happens to be one that involves contenders for more traditional forms of problem de- one environmental state variable (vertical object posi- composition and analysis. In these brief comments, I tion), one body state variable (horizontal position rela- want to focus on one important question that I think tive to the object) and one neuronal state variable remains unanswered, and that bears rather directly on (output of interneuron 9). By dispensing with talk of this enterprise of “scaling up”. representations and their contents, and restricting the The question concerns the notion of an integrated depiction of the inner realm to a depiction of inner state dynamical explanation itself. The point about such alone (section 9.3), the dynamicist makes it easy to.. (shrink)

Lighten up about mathematics! Have fun. If you read this book, you will have to endure bad math puns and jokes and out-of-date pop culture references. You'll learn some really cool mathematics to boot. In the process, you will immerse yourself in living, thinking, and breathing logical reasoning. We like to call this proofs, which to some is a bogey word, but to us it is a boogie word. You will learn how to solve problems, real and imagined. After all, (...) math is a game where, although the rules are pretty much set, we are left to our imaginations to create. Think of this book as blueprints, but you are the architect of what structures you want to build. Make sure you lay a good foundation, for otherwise your buildings might fall down. To help you through this, we guide you to think and plan carefully. Our playground consists of basic math, with a loving emphasis on number theory. We will encounter the known and the unknown. Ancient and modern inquirers left us with elementary-sounding mathematical puzzles that are unsolved to this day. You will learn induction, logic, set theory, arithmetic, and algebra, and you may one day solve one of these puzzles. (shrink)

: This paper offers a new interpretation of the first chapter of Aristotle’s Rhetoric and of Aristotle’s understanding of rhetoric throughout the treatise. I defend the view that, for Aristotle, rhetoric was a skill in offering the listener ‘proofs’, that is, proper grounds for conviction. His arguments in the opening chapters of the treatise state and defend this controversial, epistemically normative view against the rival views of Gorgias, Thrasymachus and the rhetorical handbook writers, on the one hand, and against those (...) of Plato, on the other. Aristotle defends his view on the basis that rhetoric is a skill in discharging an important role in the state – the role of helping citizens to good publicly-deliberated judgements. (shrink)

Two claims have been explored, the first, that fool-proof proofs of the sort that there could be if there were a God like the God of Abraham, Isaac and Jacob are not to be expected, on good religious grounds (a claim I found wanting); and second, that there cannot be philosophical proofs of God which work beyond reasonable doubt.The argument that there cannot be philosophical proofs beyond a reasonable doubt is supported by an examination of some of the (...) fundamental issues in the traditional discussions of proofs for God's existence, and by claims about the relativity of methodological rules to world-views which, I maintain, the traditional discussions indicate. I do not claim to have proved that relativity, only to have illustrated the claim that it is there.It is my further opinion, but I do not claim really to have proved it, that the failure of religious excuses for the lack of public demon strations constitutes a good reason for concluding that there is no God of the sort described as the God of Abraham, Isaac and Jacob; hence that if there is a God, it must be the God of the philosophers. However I admit that there might be sufficient hidden reasons which would offer persuasive excuses for the God of the ordinary believer.Lastly, I have made some comments about what I think is the more valuable way to view the “proofs for God.” Such an interpretation does justice to the otherwise baffling and continual philosophical disagreements better than rival theories. It is time we take these disagreements with utmost seriousness, and one can hardly do that while treating basic metaphysical arguments as fool-proof proofs. (shrink)

According to common interpretations of mill's 'proof' of utility (chapter 4 utilitarianism), The conclusion, "the general happiness is desirable", Is said to be a simple maximizing conception and to ignore the competing desirability or deontic claims of justice. I offer a construction of the 'proof' such that the term "general happiness" in the conclusion is seen to include equitable distribution of happiness among persons as a rational condition of goodness. This construction turns crucially on the idea that each (...) person has an equal rational claim for as much happiness as possible; accordingly, As a consequence of resolving the conflict of rational claims, The general happiness is conceived as that state of affairs in which each person gains as much happiness as possible compatible with a like result for all others. (shrink)

This article explores the “fair trial” as a good practice for the construction of public proof. If proof signifies closure on matter at hand, and publicness is taken to signify both “access to” and “participation in” the construction of proof by the publics concerned, the authors contend that the “fair trial” is a good example of building public proof and that its backbone constraints can be of great interest to the defenders and advocates of (...) participative Technology Assessment, especially citizens' juries. (shrink)

This paper is a continuation of dag-like proof compression research initiated in [9]. We investigate proof compression phenomenon in a particular, most transparent case of propositional DNF Logic. We define and analyze a very efficient semi-analytic sequent calculus SEQ*0 for propositional DNF. The efficiency is achieved by adding two special rules CQ and CS; the latter rule is a variant of the weakened substitution rule WS from [9], while the former one being specially designed for DNF sequents. We (...) show that dag-like SEQ*0 has good deterministic proof search strategy. Furthermore, we expose six examples showing that dag-like deducibility in SEQ*0 admits exponential-size proof compression, as compared to familiar proof systems like cutfree sequent calculi, resolution and cutting plane proof systems. (shrink)

This paper puts forward a new account of rigorous mathematical proof and its epistemology. One novel feature is a focus on how the skill of reading and writing valid proofs is learnt, as a way of understanding what validity itself amounts to. The account is used to address two current questions in the literature: that of how mathematicians are so good at resolving disputes about validity, and that of whether rigorous proofs are necessarily formalizable.

John Stuart Mill's version of utilitarianism is that there is something that is a value as an end of action and that all actions, rules for action, laws, policies, and so on, are to evaluated by their promotion of that value or reduction of the negative of that value. The value judgment, that promotion of happiness and reduction of unhappiness are the normative ends of action is the “principle of utility,” and the “proof” is designed to argue for that (...) principle. The proof is an argument from introspective psychology, on what people desire, and are averse to, as ends. The claim is that when properly analyzed each person desires his or her own happiness, and this is evidence that happiness is the kind of thing that is valuable as an end, whether it exists in the agent's or another's consciousness. It is a “comprehensive formula, including all things which are in themselves good, and that whatever else is good, is not so as an end, but as a means…”. (shrink)

Does the past rationally bear on the future? David Hume argued that we lack good reason to think that it does. He insisted in particular that we lack — and forever will lack — anything like a demonstrative proof of such a rational bearing. A surprising mathematical result can be read as an invitation to reconsider Hume's confidence.

Numbers. Natural numbers -- Integers -- Real numbers -- Irrational and transcendental numbers -- Funny numbers. Zero -- e : the unnatural natural number -- [Phi] : the golden ratio -- i : the imaginary number -- Writing numbers. Roman numerals -- Egyptian fractions -- Continued fractions -- Logic. Mr. Spock is not logical -- Proofs, truth, and trees : oh my! -- Programming with logic -- Temporal reasoning -- Sets. Cantor's diagonalization : infinity isn't just infinity -- Axiomatic set (...) theory : keep the good, dump the bad -- Models : using sets as the LEGOs of the math world -- Transfinite numbers : counting and ordering infinite sets -- Group theory : finding symmetries with sets -- Mechanical math. Finite state machines : simplicity goes far -- The turing machine -- Pathology and the heart of computing -- Calculus : no, not that calculus- [lambda] calculus -- Numbers, booleans, and recursion -- Types, types, types : modeling [lambda] calculus -- The halting problem. (shrink)

§ 1. Introduction. Perhaps the most surprising recent development in complexity theory is the discovery that the class NP can be characterized using a form of randomized proof checker that only examines a constant number of bits of the “proof” that a string is in a language [6, 5, 31, 3, 4]. More specifically, writing ∣x∣ for the length of a string x, a language L in the class NP of languages recognizable in Nondeterministic polynomial time is traditionally (...) given by a polynomial p and a polynomial-time predicate P such that a string x is in L iff there is some string y satisfying P, where ∣y∣ ≤ p. Intuitively, we can think of a string y as a possible proof that x ϵ L, with the predicate P some kind of proof checker that distinguishes good proofs from bad ones. A string x is therefore in a language L ϵ NP if there is a short proof that x ϵ L, and not in L otherwise. The surprising discovery is that the deterministic proof checker that reads the entire input and proof can be replaced by a probabilistic verifier that on input x and possible proof y, where y is presented in a certain way, flips at most O coins and reads at most a constant number of bits of x and y. Obviously, a probabilistic verifier that does not read the whole proof will sometimes make mistakes. However, the surprising and essentially non-intuitive mathematical fact is that for each language L in NP, it is possible to find a polynomial q and verifier V flipping a logarithmic number of coins and reading a constant number of bits such that, for any x ϵ L, there exists y with ∣y∣ ≤ q and with V accepting with probability 1 and, for x ∉ L, there is no y with ∣y∣ ≤ q and with V accepting with probability ≥ 1/4. This idea canalsobeextended to PSPACE [10, 9], using a game that alternates between two players instead of a proof presented by a “single player.”. (shrink)

Free logics are a family of first-order logics which came about as a result of examining the existence assumptions of classical logic (Hintikka _The Journal of Philosophy_, _56_, 125–137 1959 ; Lambert _Notre Dame Journal of Formal Logic_, _8_, 133–144 1967, 1997, 2001 ). What those assumptions are varies, but the central ones are that (i) the domain of interpretation is not empty, (ii) every name denotes exactly one object in the domain and (iii) the quantifiers have existential import. Free (...) logics reject the claim that names need to denote in (ii). _Positive_ free logic concedes that some atomic formulas containing non-denoting names (including self-identity) are true, _negative_ free logic treats them as uniformly false, and _neutral_ free logic as taking a third value. There has been a renewed interest in analyzing proof theory of free logic in recent years, based on intuitionistic logic in Maffezioli and Orlandelli (_Bulletin of the Section of Logic_, _48_(2), 137–158 2019 ) as well as classical logic in Pavlović and Gratzl (_Journal of Philosophical Logic_, _50_, 117–148 2021 ), there for the positive and negative variants. While the latter streamlines the presentation of free logics and offers a more unified approach to the variants under consideration, it does not cover neutral free logic, since there is some lack of both clear formal intuitions on the semantic status of formulas with empty names, as well as a satisfying account of the conditional in this context. We discuss extending the results to this third major variant of free logics. We present a series of G3 sequent calculi adapted from Fjellstad (_Studia Logica_, _105_(1), 93–119 2017, _Journal of Applied Non-Classical Logics_, _30_(3), 272–289 2020 ), which possess all the desired structural properties of a good proof system, including admissibility of contraction and all versions of the cut rule. At the same time, we maintain the unified approach to free logics and moreover argue that greater clarity of intuitions is achieved once neutral free logic is conceptualized as consisting of two sub-varieties. (shrink)

This chapter contains sections titled: Kant's Anti‐Cartesianism Kant on the Highest Good and the Practical Necessity of Belief in God's Existence The Moral Proof at the Tübinger Stift and Its Fate Self‐Positing and the “Only True and Thinkable Creation out of Nothing” The Way to Absolute Knowing in Hegel's Phenomenology.

Roughly, a proof of a theorem, is “pure” if it draws only on what is “close” or “intrinsic” to that theorem. Mathematicians employ a variety of terms to identify pure proofs, saying that a pure proof is one that avoids what is “extrinsic,” “extraneous,” “distant,” “remote,” “alien,” or “foreign” to the problem or theorem under investigation. In the background of these attributions is the view that there is a distance measure (or a variety of such measures) between mathematical (...) statements and proofs. Mathematicians have paid little attention to specifying such distance measures precisely because in practice certain methods of proof have seemed self- evidently impure by design: think for instance of analytic geometry and analytic number theory. By contrast, mathematicians have paid considerable attention to whether such impurities are a good thing or to be avoided, and some have claimed that they are valuable because generally impure proofs are simpler than pure proofs. This article is an investigation of this claim, formulated more precisely by proof- theoretic means. After assembling evidence from proof theory that may be thought to support this claim, we will argue that on the contrary this evidence does not support the claim. (shrink)

Chapter 1 constitutes an introduction to Gentzen calculi from two perspectives, logical and philosophical. It introduces the notion of generalisations of Gentzen sequent calculus and the discussion on properties that characterize good inferential systems. Among the variety of Gentzen-style sequent calculi, I divide them in two groups: syntactic and semantic generalisations. In the context of such a discussion, the inferentialist philosophy of the meaning of logical constants is introduced, and some potential objections – mainly concerning the choice of working (...) with semantic generalizations – are addressed. Finally, I’ll introduce the case studies that I’ll be dealing with in part II. Chapter 2 is concerned with the origins and development of Jaśkowski’s discussive logic. The main idea of this chapter is to systematize the various stages of the development of discussive logic related researches from two different angles, i.e., its connections to modal logics and its proof theory, by highlighting virtues and vices. Chapter 3 focuses on the Gentzen-style proof theory of discussive logic, by providing a characterization of it in terms of labelled sequent calculi. Chapter 4 deals with the Gentzen-style proof theory of relevant logics, again by employing the methodology of labelled sequent calculi. This time, instead of working with a single logic, I’ll deal with a whole family of them. More precisely, I’ll study in terms of proof systems those relevant logics that can be characterised, at the semantic level, by reduced Routley-Meyer models, i.e., relational structures with a ternary relation between states and a unique base element. Chapter 5 investigates the proof theory of a modal expansion of intuitionistic propositional logic obtained by adding an ‘actuality’ operator to the connectives. This logic was introduced also using Gentzen sequents. Unfortunately, the original proof system is not cut-free. This chapter shows how to solve this problem by moving to hypersequents. Chapter 6 concludes the investigations and discusses the future of the research presented throughout the dissertation. (shrink)

Since the beginning of human history, there has been no society that did not have any religion. Man meets his need to believe, encoded in his nature by turning to God. God has not left humans alone in their journey on earth, and from time to time, He has intervened in the world through his prophets. The prophethood, which constitutes one of the main subjects of theology, is an important institution in God-human communication. The messengers chosen by God convey to (...) people the information that they received through revelation and offer teachings that will bring them to salvation. The idea of salvation by God when suffering or feeling helpless has driven people to the expectation of a savior. Nearly all religions except Shintoism have the awaited savior belief in the world. Accordingly, the savior will emerge in an environment of chaos where disasters are experienced, and a moral collapse is observed. The expected savior will drive people away from this turmoil, bring them happiness, well-being, and lead them to salvation. Although there is a savior belief in Judaism and Christianity, Muslims claim that the awaited savior in these religions is Muhammad. Muslim thinkers who believe that Muhammad was heralded in previous scripts, tried to find evidence on the subject from the Bible. One of the Islamic scholars who did researches for this purpose is Shamsuddin al-Samarqandi. Shamsuddin Mohammed b. Ashraf al-Husayni al-Samarqandi is an important Turkish-Islamic scholar who lived in Turkestan at the end of the 7th century and the first quarter of the 8th century. Samarqandi is a versatile scholar who produced important works in various fields such as philosophy, logic, mathematics, discussion, debate and astronomy. Shamsuddin al-Samarqandi is one of the eminent scholars who have written his name in the history of science and philosophy with his works. He adopted a critical and analytical style in almost every field, putting the science of logic at the center of his system. His constructive criticisms had an important function in understanding and interpreting the ideas of numerous thinkers, from Greek philosophers to Islamic philosophers and representatives of philosophical theology. In addition to all these, Samarqandi did not hesitate to criticize the Christian and Jewish belief systems, religions such as Sanawiyyah and Zoroastrianism, and the teachings of the naturalist philosophers regarding the relationship between God and the world in the light of rational and scriptural evidence. Samarqandi, who distinguished himself in rational sciences, also showed himself with his background in theology. Due to his exceptional intellectual interest, he analyzes some key concepts and divine codes related to the Bible, making original inferences about the loyalty of Muhammad, the prophet of Islam. Samarqandi interprets some passages of the Torah and the New Testament, which he considers as signs of Muhammad's prophethood, in the axis of language, logic and the rules of shariah, and concludes that Muhammad is the last prophet to be announced. The conceptual framework that he mostly focuses on in this regard contains Mount Paran, Paraklite, comparison with Moses, and the good news given to the Ishmaelites. Samarqandi makes inferences about the environment in which Muhammad appeared, his qualities, and his lineage, based on the aforementioned concepts. Samarqandi also responds to the criticisms raised by the Ahl al-Kitab about the principles, such as the abolition of other religious rules constituting the basis of Muhammad's legitimacy and establishment of Islam instead, and him being a universal prophet. An important privilege of Samarqandi is that he not only grounds his claims on a theoretical scale but also engages in debates with the leaders of the Ahl al-Kitab. Although Samarqandi exhibits a satisfactory analytical attitude in most of his claims, he displays a sophisticated approach based on the method of coercion (ilzām) in some of his claims. Although he adheres to the tradition of beshair al-nubuwwah in general, he gives a different perspective and strong reasoning to the same evidence within the tradition with the effect of his identity as a verifier (muhaqqik). (shrink)

In Chapter 4 of his essay Utilitarianism, “Of what sort of Proof the Principle of Utility is susceptible,” J. S. Mill undertakes to prove, in some sense of that term, the principle of utility. It has very commonly been argued that in the course of this “proof” Mill commits two very obvious fallacies. The first is the naturalistic fallacy which he is held to commit when he argues that since “the only proof capable of being given that (...) an object is visible, is that people actually see it. The only proof that a sound is audible is that people hear it: and so of the other sources of our experience. In like manner … the sole evidence it is possible to produce that anything is desirable, is that people do actually desire it.”1 Here Mill appears to hold that “X is desirable ”—a value judgment—follows deductively from “People desire x”—a factual statement. And the second is the fallacy of composition which seems to be involved in Mill'zs argument that since “each person's happiness is a good to that person … the general happiness, therefore, a good to the aggregate of all persons.”2. (shrink)

How many serious mistakes can a brilliant philosopher make in a single paragraph? Many think that Mill answers this question by example—in the third paragraph of Chapter IV of Utilitarianism. Here is the notorious paragraph: The only proof capable of being given that an object is visible, is that people actually see it. The only proof that a sound is audible, is that people hear it: and so of the other sources of our experience. In like manner, I (...) apprehend, the sole evidence it is possible to produce that anything is desirable, is that people do actually desire it. If the end which the utilitarian doctrine proposes to itself were not, in theory and in practice, acknowledged to be an end, nothing could ever convince any person that it was so. No reason can be given why the general happiness is desirable, except that each person, so far as he believes it to be attainable, desires his own happiness. This, however, being a fact, we have not only all the proof which the case admits of, but all which it is possible to require, that happiness is a good: that each person's happiness is a good to that person, and the general happiness, therefore, a good to the aggregate of all persons. Happiness has made out its title as one of the ends of conduct, and consequently one of the criteria of morality. (shrink)

The goals of reduction andreductionism in the natural sciences are mainly explanatoryin character, while those inmathematics are primarily foundational.In contrast to global reductionistprograms which aim to reduce all ofmathematics to one supposedly ``universal'' system or foundational scheme, reductive proof theory pursues local reductions of one formal system to another which is more justified in some sense. In this direction, two specific rationales have been proposed as aims for reductive proof theory, the constructive consistency-proof rationale and the foundational (...) reduction rationale. However, recent advances in proof theory force one to consider the viability of these rationales. Despite the genuine problems of foundational significance raised by that work, the paper concludes with a defense of reductive proof theory at a minimum as one of the principal means to lay out what rests on what in mathematics. In an extensive appendix to the paper,various reduction relations betweensystems are explained and compared, and arguments against proof-theoretic reduction as a ``good'' reducibilityrelation are taken up and rebutted. (shrink)

The true key to all the perplexities of the human condition, Rousseau boldly claims, is the “natural goodness of man.” It is also the key to his own notoriously contradictory writings, which, he insists, are actually the disassembled parts of a rigorous philosophical system rooted in that fundamental principle. What if this problematic claim—so often repeated, but as often dismissed—were resolutely followed and explored? Arthur M. Melzer adopts this approach in The Natural Goodness of Man. The first two parts of (...) the book restore the original, revolutionary significance of this now time-worn principle and examine the arguments Rousseau offers in proof of it. The final section unfolds and explains Rousseau’s programmatic thought, especially the Social Contract, as a precise solution to the human problem as redefined by the principle of natural goodness. The result is a systematic reconstruction of Rousseau’s philosophy that discloses with unparalleled clarity both the complex weave of his argument and the majestic unity of his vision. Melzer persuasively resolves one after another of the famous Rousseauian paradoxes–enlarging, in the process, our understanding of modern philosophy and politics. Engagingly and lucidly written, The Natural Goodness of Man will be of interest to general as well as scholarly readers. (shrink)

In 1969, De Jongh proved the “maximality” of a fragment of intuitionistic predicate calculus forHA. Leivant strengthened the theorem in 1975, using proof-theoretical tools (normalisation of infinitary sequent calculi). By a refinement of De Jongh's original method (using Beth models instead of Kripke models and sheafs of partial combinatory algebras), a semantical proof is given of a result that is almost as good as Leivant's. Furthermore, it is shown thatHA can be extended to Higher Order Heyting Arithmetic+all (...) trueΠ 2 0 -sentences + transfinite induction over primitive recursive well-orderings. As a corollary of the proof, maximality of intuitionistic predicate calculus is established wrt. an abstract realisability notion defined over a suitable expansion ofHA. (shrink)

In Chapter 4 of his essay Utilitarianism, “Of what sort of Proof the Principle of Utility is susceptible,” J. S. Mill undertakes to prove, in some sense of that term, the principle of utility. It has very commonly been argued that in the course of this “proof” Mill commits two very obvious fallacies. The first is the naturalistic fallacy which he is held to commit when he argues that since “the only proof capable of being given that (...) an object is visible, is that people actually see it. The only proof that a sound is audible is that people hear it: and so of the other sources of our experience. In like manner … the sole evidence it is possible to produce that anything is desirable, is that people do actually desire it.”1 Here Mill appears to hold that “X is desirable ”—a value judgment—follows deductively from “People desire x”—a factual statement. And the second is the fallacy of composition which seems to be involved in Mill'zs argument that since “each person's happiness is a good to that person … the general happiness, therefore, a good to the aggregate of all persons.”2. (shrink)

This chapter contains sections titled: Generic Argument for Traditional Utilitarianism Mill's Proof of Utilitarianism (Straightforward Interpretation) Mill's Proof of Utilitarianism (One Alternative Interpretation) Mill's Proof of Utilitarianism (Another Alternative Interpretation).

In this paper, I develop a quasi-transcendental argument to justify Kant’s infamous claim “man is evil by nature.” The cornerstone of my reconstruction lies in drawing a systematic distinction between the seemingly identical concepts of “evil disposition” (böseGesinnung) and “propensity to evil” (Hang zumBösen). The former, I argue, Kant reserves to describe the fundamental moral outlook of a single individual; the latter, the moral orientation of the whole species. Moreover, the appellative “evil” ranges over two different types of moral failure: (...) while an “evil disposition” is a failure to realize the good (i.e., to adopt the motive of duty as limiting condition for all one’s desires), an “evil propensity” is a failure to realize the highest good (i.e., to engage in the collective project of transforming the legal order into an ethical community). This correlation between units of moral analysis and types of obligation suggests a way to offer a deduction of the universal propensity on behalf of Kant. It consists in tracing the source of radical evil to the same subjective necessity that gives rise to the doctrine of the highest good. For, at the basis of Kant’s two doctrines lies the same natural dialectic between happiness and morality. While the highest good brings about the critically acceptable resolution of this dialectic, the propensity to evil perpetuates and aggravates it. Instead of connecting happiness and morality in an objective relation, the human will subordinatesmorality to the pursuit of happiness according to the subjective order of association. If this reading is correct, it would explain why prior attempts at a transcendental deduction have failed: interpreters have looked for the key to the deduction in the body of Kant’s text, where it is not to be found, for it is tucked, instead, in the Preface to the first edition. (shrink)

In the introductory chapter of his essay on Utilitarianism, John Stuart Mill says his aim is to contribute towards the understanding of utilitarianism and towards ‘such proof as it is susceptible of’. He immediately adds that ‘this cannot be proof in the ordinary and popular meaning of the term’ because ‘ultimate ends are not amenable to direct proof’. A proof that something is good has to show that it is ‘a means to something admitted to (...) be good without proof’. But, he goes on, this does not imply that a formula of ultimate ends can only be accepted on ‘blind impulse, or arbitrary choice’. It can be rationally discussed and subjected to proof in a wider sense of that word. ‘Considerations may be presented capable of determining the intellect either to give or withhold its assent to the doctrine; and this is equivalent to proof.’. (shrink)