This book defends a form of ethical intuitionism, according to which (i) there are objective moral truths; (ii) we know some of these truths through a kind of immediate, intellectual awareness, or "intuition"; and (iii) our knowledge of moral truths gives us reasons for action independent of our desires. The author rebuts all the major objections to this theory and shows that the alternative theories about the nature of ethics all face grave difficulties.

According to moral intuitionism, at least some moral seeming states are justification-conferring. The primary defense of this view currently comes from advocates of the standard account, who take the justification-conferring power of a moral seeming to be determined by its phenomenological credentials alone. However, the standard account is vulnerable to a problem. In brief, the standard account implies that moral knowledge is seriously undermined by those commonplace moral disagreements in which both agents have equally good phenomenological credentials supporting their (...) disputed moral beliefs. However, it is implausible to think that commonplace disagreement seriously undermines moral knowledge, and thus it is implausible to think that the standard account of moral intuitionism is true. (shrink)

I argue that, given evidence of the factors that tend to distort our intuitions, ethical intuitionists should disown a wide range of common moral intuitions, and that they should typically give preference to abstract, formal intuitions over more substantive ethical intuitions. In place of the common sense morality with which intuitionism has traditionally allied, the suggested approach may lead to a highly revisionary normative ethics.

Ethical Intuitionism was the dominant moral theory in Britain for much of the 18th, 19th and the first third of the twentieth century. However, during the middle decades of the twentieth century ethical intuitionism came to be regarded as utterly untenable. It was thought to be either empty, or metaphysically and epistemologically extravagant, or both. This hostility led to a neglect of the central intuitionist texts, and encouraged the growth of a caricature of intuitionism that could easily (...) be rejected before moving on to 'more serious' philosophical theories. More recently, however, this hostility towards ethical intuitionism has subsided. A wide range of moral philosophers, from Aristotelians, to rule-consequentialists, to expressivists, Kantians and deontologists, are beginning to look to the ethical intuitionists's work as a positive resource. It is, therefore, a good time to get clear on what it was that intuitionists said, and re-evaluate their contribution to our understanding of morality. This volume is the first serious engagement with ethical intuitionism in the light of contemporary developments in ethical theory. It contains essays by eminent moral philosophers working in very different traditions whose aim is to clarify and assess ethical intuitionism. Issues addressed include whether the plurality of basic principles intuitionists adhere to can be grounded in some more fundamental principle; the autonomy of ethics and self-evidence; moral realism and internalism; and the open question argument and naturalism. (shrink)

We review the state of the art in moral psychology to answer 6 questions: 1) Where do moral beliefs and motivations come from? 2) How does moral judgment work? 3) What is the evidence for the social intuitionist model? 4) What exactly are the moral intuitions? 5) How does morality develop? And 6) Why do people vary in their morality? We describe the intuitionist approach to moral psychology. The mind makes rapid affective evaluations of everything it encounters, and these evaluations (...) (intuitions) shape and push subsequent moral reasoning. This approach to moral judgment has a variety of implications for moral philosophy and for the law in that it questions common assumptions about the reliability and causal efficacy of private, conscious reasoning. (shrink)

The aims of this paper are twofold: firstly, to say something about that philosophy of mathematics known as 'intuitionism' and, secondly, to fit these remarks into a more general message for the philosophy of mathematics as a whole. What I have to say on the first score can, without too much inaccuracy, be compressed into two theses. The first is that the intuitionistic critique of classical mathematics can be seen as based primarily on epistemological rather than on meaning-theoretic considerations. (...) The second is that the intuitionist's chief objection to the classical mathematician's use of logic does not center on the use of particular logical principles (in particular, the law of excluded middle and its ilk). Rather on the role the classical mathematician assigns (or at least extends) generally (i.e. regardless of the particular principles used) to the use of logic in the production mathematical proofs. Thus, the intuitionist critique of logic that we shall be presenting is far more radical than that which has commonly been presented. -/- Concerning the second, more general, theme, my claim is this: some restriction of the role of logical inference in mathematical proof such as that mentioned above is necessary if one is to account for the seeming difference in the epistemic conditions of provers whose reasoning is based on genuine insight into the subject-matter being investigated, and would-be provers whose reasoning is based not on such insight, but rather on principles of inference which hold of every subject-matter indifferently. (shrink)

Intuitionistic logic provides an elegant solution to the Sorites Paradox. Its acceptance has been hampered by two factors. First, the lack of an accepted semantics for languages containing vague terms has led even philosophers sympathetic to intuitionism to complain that no explanation has been given of why intuitionistic logic is the correct logic for such languages. Second, switching from classical to intuitionistic logic, while it may help with the Sorites, does not appear to offer any advantages when dealing with (...) the so-called paradoxes of higher-order vagueness. We offer a proposal that makes strides on both issues. We argue that the intuitionist’s characteristic rejection of any third alethic value alongside true and false is best elaborated by taking the normal modal system S4M to be the sentential logic of the operator ‘it is clearly the case that’. S4M opens the way to an account of higher-order vagueness which avoids the paradoxes that have been thought to infect the notion. S4M is one of the modal counterparts of the intuitionistic sentential calculus and we use this fact to explain why IPC is the correct sentential logic to use when reasoning with vague statements. We also show that our key results go through in an intuitionistic version of S4M. Finally, we deploy our analysis to reply to Timothy Williamson’s objections to intuitionistic treatments of vagueness. (shrink)

Thinking about morality -- Story of contemporary intuitionism -- Moral knowledge -- New challenges to intuitionism -- Grounds of morality -- Right and the good reconsidered -- Intuitionism's rivals -- Being moral: how and why.

Conditional logics have traditionally been intended to formalize various intuitively correct modes of reasoning involving conditional expressions in natural language. Although conditional logics have by now been thoroughly studied in a classical context, they have yet to be systematically examined in an intuitionistic context, despite compelling philosophical and technical reasons to do so. This paper addresses this gap by thoroughly examining the basic intuitionistic conditional logic ICK, the intuitionistic counterpart of Chellas’ important classical system CK. I give ICK both worlds (...) semantics and algebraic semantics, and prove that these are equivalent. I give a Gödel-type embedding of ICK into CK and a Glivenko-type embedding of CK into ICK. I axiomatize ICK and prove soundness, completeness, and decidability results. Finally, I discuss extending ICK. (shrink)

It is a central tenet of ethical intuitionism as defended by W. D. Ross and others that moral theory should reflect the convictions of mature moral agents. Hence, intuitionism is plausible to the extent that it corresponds to our well-considered moral judgments. After arguing for this claim, I discuss whether intuitionists offer an empirically adequate account of our moral obligations. I do this by applying recent empirical research by John Mikhail that is based on the idea of a (...) universal moral grammar to a number of claims implicit in W. D. Ross’s normative theory. I argue that the results at least partly vindicate intuitionism. (shrink)

This paper is a reaction to the following remark by grzegorczyk: "the compound sentences are not a product of experiment. they arise from reasoning. this concerns also negations; we see that the lemon is yellow, we do not see that it is not blue." generally, in science the truth is ascertained as indirectly as falsehood. an example: a litmus-paper is used to verify the sentence "the solution is acid." this approach gives rise to a (very intuitionistic indeed) conservative extension of (...) the heyting logic satisfying natural duality laws. (shrink)

The present work is motivated by two questions. (1) What should an intuitionistic epistemic logic look like? (2) How should one interpret the knowledge operator in a Kripke-model for it? In what follows we outline an answer to (2) and give a model-theoretic definition of the operator K. This will shed some light also on (1), since it turns out that K, defined as we do, fulfills the properties of a necessity operator for a normal modal logic. The interest of (...) our construction also lies in a better insight into the intuitionistic solution to Fitch's paradox, which is discussed in the third section. In particular we examine, in the light of our definition, De Vidi and Solomon's proposal of formulating the verification thesis as Φ → ¬¬KΦ. We show, as our main result, that this definition excapes the paradox, though it is validated only under restrictive conditions on the models. (shrink)

The aim of this paper is to establish a phenomenological mathematical intuitionism that is based on fundamental phenomenological-epistemological principles. According to this intuitionism, mathematical intuitions are sui generis mental states, namely experiences that exhibit a distinctive phenomenal character. The focus is on two questions: what does it mean to undergo a mathematical intuition and what role do mathematical intuitions play in mathematical reasoning? While I crucially draw on Husserlian principles and adopt ideas we find in phenomenologically minded mathematicians (...) such as Hermann Weyl and Kurt Gödel, the overall objective is systematic in nature: to offer a plausible approach towards mathematics. (shrink)

In this paper we give a brief treatment of a theory of proofs for a system of Full Intuitionistic Linear Logic. This system is distinct from Classical Linear Logic, but unlike the standard Intuitionistic Linear Logic of Girard and Lafont includes the multiplicative disjunction par. This connective does have an entirely natural interpretation in a variety of categorical models of Intuitionistic Linear Logic. The main proof-theoretic problem arises from the observation of Schellinx that cut elimination fails outright for an intuitive (...) formulation of Full Intuitionistic Linear Logic; the nub of the problem is the interaction between par and linear implication. We present here a term assignment system which gives an interpretation of proofs as some kind of non-deterministic function. In this way we find a system which does enjoy cut elimination. The system is a direct result of an analysis of the categorical semantics, though we make an effort to present the system as if it were purely a proof-theoretic construction. (shrink)

In this paper, I defend a non-skeptical intuitionist approach to moral epistemology from recent criticisms. Starting with Sinnott-Armstrong's skeptical attacks, I argue that a familiar sort of skeptical argument rests on a problematic conception of the evidential grounds of our moral judgments. The success of his argument turns on whether we conceive of the evidential grounds of our moral judgments as consisting entirely of non-normative considerations. While we cannot avoid skepticism if we accept this conception of our evidential grounds, that's (...) because accepting this conception of our evidential grounds is tantamount to accepting the skeptic's conclusion. We have nothing to fear from arguments for skepticism from skepticism. (shrink)

This is a long-awaited new edition of one of the best known Oxford Logic Guides. The book gives an introduction to intuitionistic mathematics, leading the reader gently through the fundamental mathematical and philosophical concepts. The treatment of various topics, for example Brouwer's proof of the Bar Theorem, valuation systems, and the completeness of intuitionistic first-order logic, have been completely revised.

Intuitionistic propositional logic with Galois negations ( $$\mathsf {IGN}$$ ) is introduced. Heyting algebras with Galois negations are obtained from Heyting algebras by adding the Galois pair $$(\lnot,{\sim })$$ and dual Galois pair $$(\dot{\lnot },\dot{\sim })$$ of negations. Discrete duality between GN-frames and algebras as well as the relational semantics for $$\mathsf {IGN}$$ are developed. A Hilbert-style axiomatic system $$\mathsf {HN}$$ is given for $$\mathsf {IGN}$$, and Galois negation logics are defined as extensions of $$\mathsf {IGN}$$. We give the bi-tense (...) logic $$\mathsf {S4N}_t$$ which is obtained from the minimal tense extension of the modal logic $$\mathsf {S4}$$ by adding tense operators. We give a new extended Gödel translation $$\tau $$ and prove that $$\mathsf {IGN}$$ is embedded into $$\mathsf {S4N}_t$$ by $$\tau $$. Moreover, every Kripke-complete Galois negation logic L is embedded into its tense companion $$\tau (L)$$. (shrink)

We show that, if a suitable intuitionistic metatheory proves that consistency implies satisfiability for subfinite sets of propositional formulas relative either to standard structures or to Kripke models, then that metatheory also proves every negative instance of every classical propositional tautology. Since reasonable intuitionistic set theories such as HAS or IZF do not demonstrate all such negative instances, these theories cannot prove completeness for intuitionistic propositional logic in the present sense.

Hybrid logics are a principled generalization of both modal logics and description logics, a standard formalism for knowledge representation. In this paper we give the first constructive version of hybrid logic, thereby showing that it is possible to hybridize constructive modal logics. Alternative systems are discussed, but we fix on a reasonable and well-motivated version of intuitionistic hybrid logic and prove essential proof-theoretical results for a natural deduction formulation of it. Our natural deduction system is also extended with additional inference (...) rules corresponding to conditions on the accessibility relations expressed by so-called geometric theories. Thus, we give natural deduction systems in a uniform way for a wide class of constructive hybrid logics. This shows that constructive hybrid logics are a viable enterprise and opens up the way for future applications. (shrink)

Walter Sinnott-Armstrong has developed and progressively refined an argument against moral intuitionism—the view on which some moral beliefs enjoy non-inferential justification. He has stated his argument in a few different forms, but the basic idea is straightforward. To start with, Sinnott-Armstrong highlights facts relevant to the truth of moral beliefs: such beliefs are sometimes biased, influenced by various irrelevant factors, and often subject to disagreement. Given these facts, Sinnott-Armstrong infers that many moral beliefs are false. What then shall we (...) think of our own moral beliefs? Either we have reason to think some of our moral beliefs are reliably formed or we have no such reason. If the latter, our moral beliefs are unjustified. If we have reason to think some moral beliefs are reliably formed, then those beliefs are not non-inferentially justified, because then we’ll have reason to accept something—namely, that they are reliably formed—that entails or supports those beliefs. But then, either way, our moral beliefs are not non-inferentially justified, and so moral intuitionism is false. This paper takes issue with Sinnott-Armstrong’s interesting and widely discussed argument, which we here call the Empirical Defeat Argument (EDA). According to us, the EDA does not defeat moral intuitionism. In section 1, we will set out the argument, briefly reviewing the rationale Sinnott-Armstrong offers for the premises. Then, in section 2, we identify a critical but dubious epistemological assumption concerning the nature of defeat that undergirds the argument. Finally, in section 3, we will defend our challenge to the EDA by answering two objections. (shrink)

Inquisitive logic is a research program seeking to expand the purview of logic beyond declarative sentences to include the logic of questions. To this end, inquisitive propositional logic extends classical propositional logic for declarative sentences with principles governing a new binary connective of inquisitive disjunction, which allows the formation of questions. Recently inquisitive logicians have considered what happens if the logic of declarative sentences is assumed to be intuitionistic rather than classical. In short, what should inquisitive logic be on an (...) intuitionistic base? In this paper, we provide an answer to this question from the perspective of nuclear semantics, an approach to classical and intuitionistic semantics pursued in our previous work. In particular, we show how Beth semantics for intuitionistic logic naturally extends to a semantics for inquisitive intuitionistic logic. In addition, we show how an explicit view of inquisitive intuitionistic logic comes via a translation into propositional lax logic, whose completeness we prove with respect to Beth semantics. (shrink)

The purpose of this paper is to define a new logic $${\mathcal {SI}}$$ called semi-intuitionistic logic such that the semi-Heyting algebras introduced in [ 4 ] by Sankappanavar are the semantics for $${\mathcal {SI}}$$ . Besides, the intuitionistic logic will be an axiomatic extension of $${\mathcal {SI}}$$.

I define ethical intuitionism as the view that it is appropriate to appeal to inferentially unsupported moral beliefs in the course of moral reasoning. I mention four common objections to this view, including the view that all such appeals to intuition make ethical theory politically and noetically conservative. I defend intuitionism from versions of this criticism expressed by R.B. Brandt, R.M. Hare and Richard Miller.

In the recent metaethical literature there has been significant interest in the prospects for what I am denoting ‘Perceptual Intuitionism’: the view that normal ethical agents can and do have non-inferential justification for first-order ethical beliefs by having ethical perceptual experiences, e.g., Cullison 2010, McBrayer 2010, Vayrynen 2008. If true, it promises to constitute an independent a posteriori intuitionist epistemology, providing an alternative to intuitionist accounts which posit a priori intuition and/or emotion as sources of non-inferentially justified ethical beliefs. (...) As it is formulated, it is plausible that a necessary con- dition for the view is the truth of Ethical Perception: normal ethical agents can and do have perceptual experiences as of the instantiation of ethical properties. In this paper a sophisticated and promising account of Ethical Perception is offered. Extant objections are shown to fail. However, it will be argued that it is far from obvious that the account of Perceptual Intuitionism which emerges constitutes an independent alternative to other intuitionist accounts. This is because we have reason to think that ethical perceptual experience may be epistemically dependent on other epistemic sources, e.g. a priori intuition or emotion. (shrink)

There are several open problems in the study of the calculi which result from adding either of Hilbert's ϵ- or τ-operators to the first order intuitionistic predicate calculus. This paper provides answers to several of them. In particular, the first complete and sound semantics for these calculi are presented, in both a “quasi-extensional” version which uses choice functions in a straightforward way to interpret the ϵ- or τ-terms, and in a form which does not require extensionality assumptions. Unlike the classical (...) case, the addition of either operator to intuitionistic logic is non-conservative. Several interesting consequences of the addition of each operator are proved. Finally, the independence of several other schemes in either calculus are also proved, making use of the semantics supplied earlier in the paper. (shrink)

Let T be a first-order theory. A T-normal Kripke structure is one in which every world is a classical model of T. This paper gives a characterization of the intuitionistic theory T of sentences intuitionistically valid in all T-normal Kripke structures and proves the corresponding soundness and completeness theorems. For Peano arithmetic , the theory PA is a proper subtheory of Heyting arithmetic , so HA is complete but not sound for PA-normal Kripke structures.

Berkeley’s ‘master argument’ for idealism has been the subject of extensive criticism. Two of his strongest critics, A.N. Prior and J.L. Mackie, argue that due to various logical confusions on the part of Berkeley, the master argument fails to establish his idealist conclusion. Prior argues that Berkeley’s argument ‘proves too little’ in its conclusion, while Mackie contends that Berkeley confuses two different kinds of self-refutation in his argument. This paper proposes a defence of the master argument based on intuitionistic argument. (...) It begins by giving a brief exposition of the master argument and Prior's and Mackie's criticism. The following section explains why we might read the master argument along intuitionistic lines. The final section demonstrates that, according to intuitionistic logic, Berkeley's argument withstands the criticisms of Prior and Mackie. (shrink)

Rationalism about the psychology of moral judgment holds, among other things, that the justifying moral reasons we have for our judgments are also the causally effective reasons for why we make those judgments. This can be called the ‘effectiveness’-thesis regarding moral reasoning. The theory that best exemplifies the thesis is the traditional conscious reasoning-paradigm. Current empirical moral psychology, however, poses a serious challenge to this thesis: it argues that in fact, emotional reactions are necessary and sufficient to account for moral (...) judgment, and that typically, moral reasoning is a matter of mere confabulation. In this survey, the empirical challenge to this thesis made by the ‘social intuitionist’ model of moral judgment and reasoning is discussed. The model claims that moral reasoning is essentially ineffective and, psychologically speaking, a matter of mere post hoc-rationalizations of cognitively impenetratable gut reactions. Several interpretations of this evidence are discussed and it is shown that there is room for a psychology of moral reasoning that can account for the available empirical evidence and yet does not have to give up the most central elements of a normative picture of moral reasoning. (shrink)

In this paper it is shown that the intuitionistic .xed point theory equation image for α times iterated fixed points of strictly positive operator forms is conservative for negative arithmetic and equation image sentences over the theory equation image for α times iterated arithmetic comprehension without set parameters.This generalizes results previously due to Buchholz [5] and Arai [2].

Ethical intuitionists regard moral knowledge as deriving from moral intuition, moral observation, moral emotion and inference. However, moral intuitions, observations and emotions are cultural artefacts which often differ starkly between cultures. Intuitionists attribute uncongenial moral intuitions, observations or emotions to bias or to intellectual or moral failings; but that leads to sectarian ad hominen attacks. Intuitionists try to avoid that by restricting epistemically genuine intuitions, observations or emotions to those which are widely agreed. That does not avoid the problem. It (...) also limits epistemically genuine intuitions, observations or emotions to those with meagre content, and the intuitionists offer no plausible explanation for how inference from such insubstantial propositions can engender substantial moral knowledge. Instead of moral knowledge, intuitionism offers the prospect of mutual name-calling between intellectually stagnant groups. I criticise and reject the principle of phenomenal conservatism, to which intuitionists sometimes appeal. (shrink)

We consider a logic which is semantically dual (in some precise sense of the term) to intuitionistic. This logic can be labeled as “falsification logic”: it embodies the Popperian methodology of scientific discovery. Whereas intuitionistic logic deals with constructive truth and non-constructive falsity, and Nelson's logic takes both truth and falsity as constructive notions, in the falsification logic truth is essentially non-constructive as opposed to falsity that is conceived constructively. We also briefly clarify the relationships of our falsification logic to (...) some other logical systems. (shrink)

A decade ago, Isham and Butterfield proposed a topos-theoretic approach to quantum mechanics, which meanwhile has been extended by Döring and Isham so as to provide a new mathematical foundation for all of physics. Last year, three of the present authors redeveloped and refined these ideas by combining the C*-algebraic approach to quantum theory with the so-called internal language of topos theory (Heunen et al. in arXiv:0709.4364). The goal of the present paper is to illustrate our abstract setup through the (...) concrete example of the C*-algebra M n (ℂ) of complex n×n matrices. This leads to an explicit expression for the pointfree quantum phase space Σ n and the associated logical structure and Gelfand transform of an n-level system. We also determine the pertinent non-probabilisitic state-proposition pairing (or valuation) and give a very natural topos-theoretic reformulation of the Kochen–Specker Theorem.In our approach, the nondistributive lattice ℘(M n (ℂ)) of projections in M n (ℂ) (which forms the basis of the traditional quantum logic of Birkhoff and von Neumann) is replaced by a specific distributive lattice $\mathcal{O}(\Sigma_{n})$ of functions from the poset $\mathcal{C}(M_{n}(\mathbb{C}))$ of all unital commutative C*-subalgebras C of M n (ℂ) to ℘(M n (ℂ)). The lattice $\mathcal{O}(\Sigma_{n})$ is essentially the (pointfree) topology of the quantum phase space Σ n , and as such defines a Heyting algebra. Each element of $\mathcal{O}(\Sigma_{n})$ corresponds to a “Bohrified” proposition, in the sense that to each classical context $C\in\mathcal{C}(M_{n}(\mathbb{C}))$ it associates a yes-no question (i.e. an element of the Boolean lattice ℘(C) of projections in C), rather than being a single projection as in standard quantum logic. Distributivity is recovered at the expense of the law of the excluded middle (Tertium Non Datur), whose demise is in our opinion to be welcomed, not just in intuitionistic logic in the spirit of Brouwer, but also in quantum logic in the spirit of von Neumann. (shrink)

In 1926, Ernst Mally proposed a number of deontic postulates. He added them as axioms to classical propositional logic. The resulting system was unsatisfactory because it had the consequence that A is the case if and only if it is obligatory that A. We present an intuitionistic reformulation of Mally’s deontic logic. We show that this system does not provide the just-mentioned objectionable theorem while most of the theorems that Mally considered acceptable are still derivable. The resulting system is unacceptable (...) as a deontic logic, but it does make sense as a lax logic in the modern sense of the word. (shrink)

In the pioneering article and two papers, written jointly with McKinsey, Tarski developed the so-called algebraic and topological frameworks for the Intuitionistic Logic and the Lewis modal system. In this paper, we present an outline of modern systems with a topological tinge. We consider topological interpretation of basic systems GL and G of the provability logic in terms of the Cantor derivative and the Hausdorff residue.

Perennial philosophers' hopes are unlikely victims of swift, natural deduction. Yet anti-realism has been thought one. Not hoping for anti-realism myself I here show it, lest it be underestimated, to survive the following argument, adapted from W. D.Hart pp. 156, 164-5; he credits first publication to Fitch).

The sequent system LDJ is formulated using the same connectives as Gentzen's intuitionistic sequent system LJ, but is dual in the following sense: (i) whereas LJ is singular in the consequent, LDJ is singular in the antecedent; (ii) whereas LJ has the same sentential counter-theorems as classical LK but not the same theorems, LDJ has the same sentential theorems as LK but not the same counter-theorems. In particular, LDJ does not reject all contradictions and is accordingly paraconsistent. To obtain a (...) more precise mapping, both LJ and LDJ are extended by adding a "pseudo-difference" operator which is the dual of intuitionistic implication. Cut-elimination and decidability are proved for the extended systems and , and a simply consistent but -inconsistent Set Theory with Unrestricted Comprehension Schema based on LDJ is sketched. (shrink)

According to pluralistic intuitionist theories, some of our moral beliefs are non-inferentially justified, and these beliefs come in both an a priori and an a posteriori variety. In this paper I present new support for this pluralistic form of intuitionism by examining the deeply social nature of moral inquiry. This is something that intuitionists have tended to neglect. It does play an important role in an intuitionist theory offered by Bengson, Cuneo, and Shafer-Landau (forth), but whilst they invoke the (...) social nature of moral inquiry in order to argue that ordinary moral intuitions are trustworthy, my argument focuses on what I will call the ‘frontiers’ of moral inquiry. I will show that inclusive and cooperative dialogue is necessary at moral inquiry’s frontiers, and that intuitionists can expect such dialogue to result in both a priori and a posteriori moral beliefs. (shrink)

The geometric system of deduction called N-Graphs was introduced by de Oliveira in 2001. The proofs in this system are represented by means of digraphs and, while its derivations are mostly based on Gentzen's sequent calculus, the system gets its inspiration from geometrically based systems, such as the Kneales' tables of development, Statman's proofs-as-graphs, Buss' logical flow graphs, and Girard's proof-nets. Given that all these geometric systems appeal to the classical symmetry between premises and conclusions, providing an intuitionistic version of (...) any of these is an interesting exercise in extending the range of applicability of the geometric system in question. In this article we produce an intuitionistic version of N-Graphs, based on Maehara's LJ' system, as described by Takeuti. Recall that LJ' has multiple conclusions in all but the essential intuitionistic rules, e.g., implication right and negation right. We show soundness and completeness of our intuitionistic N-Graphs with respect to LJ'. We also discuss how we expect to extend this work to a version of N-Graphs corresponding to the intuitionistic logic system FIL (Full Intuitionistic Logic) of de Paiva and Pereira and sketch future developments. (shrink)

Dijkstra and Scholten have proposed a formalization of classical predicate logic on a novel deductive system as an alternative to Hilbert's style of proof and Gentzen's deductive systems. In this context we call it CED (Calculus of Equational Deduction). This deductive method promotes logical equivalence over implication and shows that there are easy ways to prove predicate formulas without the introduction of hypotheses or metamathematical tools such as the deduction theorem. Moreover, syntactic considerations (in Dijkstra's words, "letting the symbols do (...) the work") have led to the "calculational style," an impressive array of techniques for elegant proof constructions. In this paper, we formalize intuitionistic predicate logic according to CED with similar success. In this system (I-CED), we prove Leibniz's principle for intuitionistic logic and also prove that any (intuitionistic) valid formula of predicate logic can be proved in I-CED. (shrink)

We consider notions of boundedness of subsets of the natural numbers ℕ that occur when doing mathematics in the context of intuitionistic logic. We obtain a new characterization of the notion of a pseudobounded subset and we formulate the closely related notion of a detachably finite subset. We establish metric equivalents for a subset of ℕ to be detachably finite and to satisfy the ascending chain condition. Following Ishihara, we spell out the relationship between detachable finiteness and sequential continuity. Most (...) of the results do not require countable choice. (shrink)

I define ethical intuitionism as the view that it is appropriate to appeal to inferentially unsupported moral beliefs in the course of moral reasoning. I mention four common objections to this view, including the view that all such appeals to intuitionism collapse into “subjectivism”, i.e., that they make truth in ethical theory depend on what people believe. I defend intuitionism from versions of this criticism expressed by R.M. Hare and Peter Singer.

In this paper I give a critical overview of the views of the main Rational Intuitionists from 18th to 20th century.