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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.

One of Mill’s primary targets, throughout his work, is intuitionism. In this paper, I distinguish two strands of intuitionism, against which Mill offers separate arguments. The first strand, ‘a priorism’ makes an epistemic claim about how we come to know norms. The second strand, ‘first principle pluralism’, makes a structural claim about how many fundamental norms there are. In this paper, I suggest that one natural reading of Mill’s argument against first principle pluralism is incompatible with the naturalism that drives his argument against a priorism. It must, therefore, be discarded. Such a reading, however, covertly attributes Mill realist commitments about the normative. These commitments are unnecessary. To the extent that Mill’s argument against first principle pluralism is taken seriously, I suggest, it is an argument that points towards Mill as having an antirealist approach to the normative.

Marshall’s recipe for mathematical economics can be described as generous doses of the imagery (‘language’) of mathematics combined with as little space as possible for its demonstrative methods (‘reasonings’). In other words, use the semantics of mathematics, but be wary of its syntax. The article traces the origins of this recipe in the broadly defined ‘intuitionist’ philosophy of mathematics which marked the reformed Cambridge Mathematical Tripos during the mid-nineteenth century. Marshall’s early philosophical papers reveal that he meditated and reconstructed this philosophy in terms of his own. His later approach to mathematical economics can be accounted for as an adaptation of this intuitionist background to the needs of economic theory as he perceived them.

Several theories, including disposition-based theories, depend on moral judgments about narrative characters' actions to describe how audiences understand characters in media narratives. However, more needs to be known about how moral judgments about characters' actions are made and affective dispositions toward characters are formed and change as a story unfolds. There is evidence that the intentionality underlying a character's actions is an important judgment that audiences use to understand a character's actions. We propose two sequences: a rationalist sequence in which perceived intentionality of a character's actions informs moral judgments and affective dispositions, and an intuitionist sequence in which affective dispositions are determined first and moral and intentionality judgments are made congruent with that judgment. Implications for narrative theories are discussed.

INTRODUCTIONEarly on in The Origin of Our Knowledge of Right and Wrong, Brentano presents his metaethical stance succinctly:Is there such a thing as a moral truth taught by nature itself and independent of ecclesiastical, political, and every other kind of social authority? Is there a moral law that is natural in the sense of being universally and incontestably valid-valid for men at all places and all times, indeed valid for any being that thinks and feels-and are we capable of knowing that there is such a law? . . . My own answer is emphatically affirmative. (Brentano 1969, 6)Brentano's metaethical stance thus appears thoroughly realist and objectivist. However, his theory has several features that modern readers do not expect to find in a theory that holds that moral truths are objective and universal. The two most important unorthodox features of Brentano's realism concern moral knowledge and the nature of moral judgment. According to Brentano, emotions are a presupposition for both moral knowledge and moral judgment. Brentano denies that there is a separate realm of values and norms, independent of the emotive attitudes of love and hate, and he also denies that evaluative properties and facts are among the content of acts of valuing.These unorthodox features of Brentanos view all tie in with notorious difficulties for moral realism, chief among which are accounting for (i) the nature of moral truths and how we come to know them, and (ii) the connection between moral judgments and attitudes that motivate action. In order to illuminate Brentanos moral realism, and how it purports to deal with such difficulties it is useful to compare it to another metaethical view with which it shares many commitments and which is also sometimes said to resist ready categorization, namely Thomas Reid's. Brentano and Reid share a commitment to moral realism and they oppose nonobjectivist views like Hume's on similar grounds. These shared commitments and similarities are explored in §1.I shall also argue that it is apt to classify both Brentano and Reid as intuitionists concerning moral knowledge and the nature of moral judgment. However, their respective versions of intuitionism are importantly different, in ways that connect in interesting ways to questions concerning (i) and (ii). Reid's brand of intuitionism resembles more closely than does Brentano's, the intuitionist views of G.E. Moore and W.D. Ross, which became highly influential in twentieth century metaethics.1 In this respect, Brentanian intuitionism is an underexplored alternative.2 As we shall see in §11, the differences between Brentanian and Reidian ethical intuitionism are reflective of more general differences between Brentano's and Reid's epistemological views.In §§III-IV, I focus more exclusively on Brentano's view, in particular on the notion of correct emotions and its relevance for our capacity to acquire moral knowledge. Brentano explains valuing in terms of the emotive attitudes of love, hate, and preference; and he explains value in terms correct love, hate, and preference. On Brentano's view, the good is ethically prior to the right, and the correct is conceptually prior to the good, in ways I shall explain in §§III and IV. As I shall explain in §IV, the priority of the correct over the good has the virtue of making Brentano's account immune to the 'wrong kind of reason' problem, which has been much discussed in recent metaethical debate. In §V, finally, I sum up and return briefly to the comparison between Brentano and Reid.I. BRENTANO AND REID: SHARED METAETHICAL COMMITMENTSWe have already seen that Brentano endorsed a metaethical view that appears thoroughly realist. Reid at several places expresses a similar commitment to moral realism. For example, he states that "the principles of morals stand upon the immutable foundation of truth, and can undergo no change by any difference of fabric, or structure of those who judge of them. …

Although they might not make an obvious generational pair, I am not the first to suggest that we should read Philip Roth and Colson Whitehead together. Michele Elam, for instance, has noted that Roth and Whitehead both produced millennial novels about racial passing: Roth’s The Human Stain (2000) and Whitehead’s The Intuitionist (1999) are surely two of the most important American novels to have emerged right around the turn of the century. And while I am interested in both novels, my focus here is on what we might usefully glean from the fact that Roth and Whitehead followed—Whitehead immediately, Roth, after returning to his Kepesh novels in The Dying Animal (2001)—these millennial fictions about the sometimes invisible histories of race with novels that make stamps central. Whitehead’s John Henry Days (2001) is about the release of a commemorative John Henry stamp at a special postal service event in Talcott, West Virginia, and Roth’s The Plot Against America (2004), his hugely successful counterfactual historical novel about the years 1940–42, stresses postal culture: Charles Lindbergh gets his start as an airmail pilot; the young Philip Roth in the novel is an avid stamp collector; and as “the country’s foremost philatelist” (1), much is made of Franklin Roosevelt’s interest in them. In the novel, stamps represent and symbolize a shared (if flawed) national history, and thus play a crucial role. To turn from racial passing to stamps may sound deflationary or even trivial, but I argue that these authors’ common interest in a (coming) post-postal age is part of a larger discussion about the nature of fiction’s efficacy for historical critique and allows us to begin thinking through how the American fiction of the twenty-first century distinguishes itself

It is sometimes held that rules of inference determine the meaning of the logical constants: the meaning of, say, conjunction is fully determined by either its introduction or its elimination rules, or both; similarly for the other connectives. In a recent paper, Panu Raatikainen argues that this view call it logical inferentialism is undermined by some Very little known' considerations by Carnap (1943) to the effect that 'in a definite sense, it is not true that the standard rules of inference' themselves suffice to 'determine the meanings of [the] logical constants' (Raatikainen 2000: 283). In a nutshell, Carnap showed that the rules allow for non-normal interpretations of negation and disjunction. Raatikainen concludes that 'no ordinary formalization of logic ... is sufficient to "fully formalize" all the essential properties of the logical constants' (2000: 283). We suggest that this is a mistake. Pace Raatikainen, intuitionists like Dummett and Prawitz need not worry about Carnap's problem. And although bilateral solutions for classical inferentialists as proposed by Timothy Smiley and Ian Rumfitt seem inadequate, it is not excluded that the classical inferentialists may be in a position to address the problem too.

The core doctrine of ethical intuitionism is that some of our ethical knowledge is non-inferential. Against this, Sturgeon has recently objected that if ethical intuitionists accept a certain plausible rationale for the autonomy of ethics, then their foundationalism commits them to an implausible epistemology outside ethics. I show that irrespective of whether ethical intuitionists take non-inferential ethical knowledge to be a priori or a posteriori, their commitment to the autonomy of ethics and foundationalism does not entail any implausible non-inferential knowledge in areas outside ethics (such as the past, the future, or the unobservable). However, each form of intuitionism does require a controversial stand on certain unresolved issues outside ethics.

In Logical Syntax of Language, Rudolf Carnap became a chief proponent of the doctrine that the statements of intuitionism carry nonstandard intuitionistic meanings. This doctrine is linked to Carnap's ‘Principle of Tolerance’ and claims he made on behalf of his notion of pure syntax. From premises independent of intuitionism, we argue that the doctrine, the Principle, and the attendant claims are mistaken, especially Carnap's repeated insistence that, in defining languages, logicians are free of commitment to mathematical statements intuitionists would reject.

From the point of view of proof-theoretic semantics, we examine the logical background invoked by Neil Tennant's abstractionist realist account of mathematical existence. To prepare the way, we must first look closely at the rule of existential elimination familiar from classical and intuitionist logics and at rules governing identity. We then examine how well free logics meet the harmony and uniqueness constraints familiar from the proof-theoretic semantics project. Tennant assigns a special role to atomic formulas containing singular terms. This, we find, secures harmony and uniqueness but militates against the putative realism.

International Relations (IR) theorists have traditionally viewed rationality and consciousness as defining features of human behavior. But recently this approach has come under fire from several angles. Not only do many psychologists now consider rationality to be dependent on emotions, recent developments in the logics of action debate in IR theory explicitly argue for de-emphasizing the role of consciousness and rationality in theorizing agency. In an effort to put rationality back into its proper place, we critique two recent contributions to the logics of action debate: practice theory and the logic of habit. Both logics of action are useful in many respects, but we argue that they rest on a view of individual agency that is too structural and insufficiently cognitive to fully understand how individuals make decisions in international politics. We do not doubt that unconscious reflexes often control individual decisions, but we maintain that cognitive control and deliberation play a much larger role in constructing these reflexes than practice theory and the logic of habit recognize. We sketch a rational intuitionist logic of action for IR theory. We argue that intuitions constitute the most useful way to theorize the unconscious determinants of individual action and that although intuitions control how people behave in particular circumstances, they are subject to rational recalibration through internal and intersubjective reasoning.

In its bearing upon mathematical intuitionism, antirealism has been and still is a project and not, as too readily assumed, an accomplishment. The project of antirealism, as set out in [2], [3], and [4], was to engage philosophers and logicians in creating a semantics that took a notion of justified assertibility as key and proved sufficient, at least, for intuitionistic formal predicate logic. There were three constraints on such assertibility semantics. First, the intended semantics was to afford a firm basis on which the general law of the excluded third, namely

Godel regarded the Dialectica interpretation as giving constructive content to intuitionism, which otherwise failed to meet reasonable conditions of constructivity. He founded his theory of primitive recursive functions, in which the interpretation is given, on the concept of computable function of finite type. I will 1) criticize this foundation, 2) propose a quite different one, and 3) note that essentially the latter foundation also underlies the Curry-Howard type theory and hence Heyting’s intuitionistic conception of logic. Thus the Dialectica interpretation (in so far as its aim was to give constructive content to intuitionism) is superfluous. In the set of notes [1938a] for an informal lecture, Godel refers to a hierarchy of constructive or finitary systems, the lowest level of which he called finitary number theory and is, in fact, primitive recursive arithmetic (PRA). He mentions three constructive extensions of this system: the intuitionism of Brouwer and Heyting (the “modal-logical route”), concerning which he is quite critical, the use of transfinite induction, inspired by Gentzen’s consistency proof for PA, and the use of functions of finite or even transfinite type over the type N of natural numbers. The latter extension seems to be mentioned here for the first time in connection with constructive extensions of finitary number theory. He ranks it the most satisfactory of these extensions ∗Versions of this paper were presented at a joint workshop on Hilbert at CarnegieMellon University and the University of Pittsburgh in June 1998, at a workshop on foundations of mathematics at UCLA in May 2005 and at the IMLA ’05 workshop [Intuitionistic Modal Logic and Applications, of course] in Chicago. I received helpful comments on all of these occasions. As for the history of the idea of a hierarchy of types, I know only of these examples: The hierarchy of transfinite types in which the natural numbers constitute the lowest type

The expression ‘platonism in mathematics’ or ‘mathematical platonism’ is familiar in the philosophy of mathematics at least since the use Paul Bernays made of it in his paper of 1934, ‘Sur le Platonisme dans les Mathematiques’. But he was not the first to point out the similarities between the conception of the defenders of mathematical realism and the ideas of Plato. Poincare had already stressed the ‘platonistic’ orientation of the mathematicians he called‘Cantorian’, as opposed to those who (like himself) were ‘pragmatist’ ones. I examine in this paper some very perplexing aspects of the use which is made at that time of a number of concepts, particularly ‘idealism’ (which generally designates what we would call ‘mathematical realism’) and ‘empiricism’ (which can designate almost any form of antirealism, even if, like for example intuitionism, it is not empiricist at all). There are, of course, historical reasons that may explain why it was for a time so easy and natural to use the words and the concepts in a way that may seem now very strange and to treat as if they were equivalent the two oppositions: realism/antirealism and idealism/empiricism.

I am not a deflationist. I believe that truth and falsity are substantial. The truth of a proposition consists in its having a constructive proof, or truthmaker. The falsity of a proposition consists in its having a constructive disproof, or falsitymaker. Such proofs and disproofs will need to be given modulo acceptable premisses. The choice of these premisses will depend on the discourse in question. In the case of an arithmetical proposition, a truthmaker would be an intuitionistic proof from acceptable first principles, or axioms; and a falsitymaker would be an intuitionistic disproof from the same. For many an arithmetical proposition (p, one is not at present in possession of either a proof or a disproof of (p-indeed, one is not possessed of any effective method that one knows will eventually produce either a proof or a disproof of cp.1 That, says the intuitionist, is why one should refrain from asserting (p V -ip. In my paper 'Deflationism and the Godel Phenomena', I was not wearing the hat of an anti-realist committed to a substantial notion of truth. Rather, I was playing devil's advocate. The devil-an unwitting client, for whom I was appearing pro bono was the deflationist. I was seeking to show that a certain argument appealing to the orthodox treatment of Godel sentences in arithmetic did not count against deflationism about truth. Other writers, such as Shapiro and Ketland, had claimed that deflationism about truth could be discredited by consideration of the special status of independent Godel sentences. Ketland, in particular, had written (1999, p. 88)

It is difficult for prioritarians to explain the degree to which justice requires redress for misfortune in a way that avoids imposing unreasonably high costs on more advantaged individuals whilst also economising on intuitionist appeals to judgment. An appeal to hypothetical insurance may be able to solve the problems of cost and judgment more successfully, and can also be defended from critics who claim that resource egalitarianism is best understood to favour the ex post elimination of envy over individual endowments.

PART I: ONTOLOGY, MODELS, AND INDETERMINACY PART II: MATHEMATICS, SCIENCE, AND METHOD PART III: FINITISM AND INTUITIONISM PART IV: FREGE AND THE FOUNDATIONS OF ARITHMETIC PART V: SETS, STRUCTURE, AND ABSTRACTION

After presenting the current version of principlism, in the process repudiating a widespread deductivist misinterpretation, a fundamental metaethical disagreement is developed by outlining the deductivistic critique of principlism. Once the grounds for this critique have been understood, the dispute between casuistry, deductivism and principlism can be restructured, and the model of "application" proven to be the central difference. In the concluding section it is argued that principlism is the most attractive position, if the perceptual model of weak intuitionism is made more explicit.

ed from its qualitative aspects. It is useful to think of the views of Weyl and Brouwer in connection with the phenomenology of the consciousness of internal time because Husserl probably invested more effort in describing the phenomenon in question than any other philosopher [13]. What should be of interest to those who know phenomenology is that there is a mathematics associated with the phenomenon of internal time. What should be of interest to students of Brouwerian intuitionism is that there is a phenomenology of the intuitive continuum. Focusing on internal time as a model of the intuitive continuum should also help to give logicians, philosophers and mathematicians a firmer grasp of what it is that is being analyzed by people like Brouwer and Weyl, and it should help to allay some of the skepticism that has been expressed about whether Brouwer’s description of the move of time has any real content (e.g., in [21]). We open the paper with some informal descriptions of the intuitive continuum. The mathematics of the intuitive continuum depends on formalizing aspects of the informal descriptions. In their subsequent mathematical and logical analyses of the intuitive continuum Brouwer and Weyl do not agree on all details. Both Brouwer and Weyl use choice sequences to characterize real numbers and they describe modifications of classical logic that are required for this purpose, but there are some important differences between their views. One central difference is that Weyl evidently thinks it is possible to take full advantage of the concept of choice sequence without being ontologically committed to sequences other than those that are lawlike. In his account, existential quantification over non-lawlike sequences is not permitted. Since this view is quite different from Brouwer’s, and since it may have been motivated in part by his reading of Husserl, we will consider whether this is a workable alternative to Brouwer’s development of intuitionistic real analysis. We should clear up a matter of terminology right from the beginning. For Brouwer, ‘choice sequence’ is the all-inclusive concept, with lawless sequences and lawlike sequences as the two limiting cases. Even as, according to Brouwer, all sequences are generated by the subject, the subject may limit its own freedom in generating a particular sequence. In particular, it may freely decide to let a rule or law prescribe its next choice. (In a Kantian vein, following a law may be a perfect expression of freedom.) In Weyl’s usage, on the other hand, ‘choice sequence’ or ‘free choice sequence’ never refers to a lawlike sequence. We will adhere to Brouwer’s terminology. 1 Internal Time and the Intuitive Continuum Brouwer and Weyl both distinguish ‘internal’, intuitive time from ‘external’ time. They follow the idealist tradition in holding that time is the basic form of the stream of consciousness. Brouwer makes a number of comments about the intuitive continuum that can be readily understood in connection with Husserl’s phenomenological descriptions of internal time. First, Brouwer describes the primordial intuition of mathematics as the substratum, divested of all quality

Kant famously said that one could not do morality a worse disservice than to derive it from examples, and this pronouncement, taken together with his formulations and explanations of the categorical imperative, has led some critics to regard him as too abstract. Ross, by contrast, has been widely viewed as taking individual cases of duty to have a kind of epistemic priority over principles of duty, and some of his critics have thus considered him insufficiently systematic, or even dogmatically limited to deliverances of intuition. This paper arises from the conviction that understanding of the categorical imperative may be enhanced by reflection on Rossian principles, and conversely. Kant and other systematic philosophers who have done moral philosophy in the grand style have had too little faith in intuitive singular moral judgement; Ross and other intuitionists have had too little faith in comprehensive moral theory. Drawing in part on an independent account of self-evidence and its relation to intuition, the paper shows how a Rossian view can be integrated with a Kantian moral theory in a way that yields the major benefits of both positions: the moral unification possible through the categorical imperative and other notions prominent in Kantian ethics, and the relative closeness to moral practice of Rossian principles.

Ethical intuitionists are often criticized on the ground that their view makes it possible for an agent to believe that she ought to Φ whilst lacking any motive to Φ - that is, on the ground that it involves, or implies a form of externalism. I begin by distinguishing this form of externalism (what I call belief externalism) from two other forms of ethical externalism - moral externalism and reasons externalism. I then consider various reasons why one might think that ethical intuitionism is defective in so far as it involves, or implies belief externalism, and argue that these objections are unpersuasive

The first part of the paper introduces the varieties of modern constructive mathematics, concentrating on Bishop’s constructive mathematics (BISH). It gives a sketch of both Myhill’s axiomatic system for BISH and a constructive axiomatic development of the real line R. The second part of the paper focusses on the relation between constructive mathematics and programming, with emphasis on Martin-Lof’s theory of types as a formal system for BISH. 1 What is Constructive Mathematics? The story of modern constructive mathematics begins with the publication, in 1907, of L.E.J. Brouwer’s doctoral dissertation Over de Grondslagen der Wiskunde [18], in which he gave the first exposition of his philosophy of intuitionism (a general philosophy, not merely one for mathematics). According to Brouwer, mathematics is a creation of the human mind, and precedes logic: the logic we use in mathematics grows from mathematical practice, and is not some a priori given before mathematical activity can be undertaken. It is not difficult to see how, with this view of mathematics as a strictly creative activity, Brouwer came to the view that the phrase “there exists” should be interpreted strictly and uniquely as “there can be constructed” or, in more modern parlance, “we can compute”. In turn, this interpretation of existence led Brouwer to reject the unbridled use of the Law of Excluded Middle (LEM), P ∨¬P, in mathematical arguments. For example, consider the following statement, the Limited Principle of Omniscience (LPO): ∀a ∈ {0, 1} (a = 0 ∨ a 6= 0) . (1) Here, N = {0, 1, 2, . . .} is the set of natural numbers, {0, 1} is the set of all

Rejetant l'argument formel de G. Evans concluant a l'impossibilite des objets vagues, l'A. defend l'existence d'une propriete semantique faible dans les affirmations d'identite qui garantit le caractere vague de l'objet dont il est question. Soulignant la necessite de distinguer les axiomes des regles d'inference, ainsi que l'insatisfaisabilite d'une formule de la validite de sa negation, dans le cadre de la logique non-classique et modele-theorique du vague, l'A. definit les objets vague a la lumiere de la notion de deduction et a la lumiere de la loi de Leibniz, d'une part, et defend la these anti-realiste de l'indefinitisme contre Evans, d'autre part

Peter Milne (this issue, [1]) has produced a neat counterexample to my conjecture in [2] that every sequent provable in the system IR of intuitionistic relevant logic is a substitution instance of an intuitionistically valid sequent that has no intuitionistically valid proper subsequent. While the Popperian in me rejoices, the relevantist/intuitionist in me might have to resign itself with disappointment to the prospect that the conjecture might not hold for any system worthy of the title of intuitionistic relevant logic. Now while I do not think the truth of the conjecture is essential to IR, it would still be welcome if it were so. What, then, can be done to save the situation in the light of Milne's counterexample? The sequent that Milne provides as his counterexample is A v B, A, B : A & B and its proof in IR is as follows:1 (1) -(1) (0) A B A B AvB A&B A&B A&B Let us recall the guiding idea behind the treatment of relevance embodied in the systems CR and IR. The main aim was to avoid thinning, or dilution structural inferences (in a sequent system) that become sources of irrelevant connections between sets of premisses and conclusions drawn from them. The slogan here could be

I call myself an anti-philosopher wrote F.R.Leavis: but was he accurate or is this a case of a great reader misreading his own texts?

Whether one should accept a principle like DDE cannot be settled independent of one's more general moral theory. In this, I take it, I agree with Professor Boyle, though I do not think he has shown that DDE has a role only in his particular form of absolutism. Still, since his theory does require DDE, an important question is what the alternatives are--whether we must choose between this absolutism and either utilitarianism or intuitionism. A form of contractualism, the requirements of which derive to a large extent from institutionally or conventionally established rights, is sketched here as an attractive alternative. It does not lead, so far as I can see, to DDE.

L'avantage de la logique intuitionniste sur la logique classique et la theorie des degres de verite comme logique du vague, reside dans le fait qu'elle n'a pas besoin d'imposer des criteres de demarcation des termes vagues. Discussion de G. Rea, " Degrees of Truth versus Intuitionism " (ibid, 1989, vol. 49, pp. 31-32)

In 'What Dummett Says About Truth and Linguistic Competence', Richard Kirkham distinguishes a number of interpretations of Dummett's views on truth and meaning. After providing evidence that various interpreters have misunderstood Dummett, Kirkham lays out one of Dummett's arguments for the conclusion that truth conditional theories of meaning are without content, which supports the thesis that truth does not play a fundamental role in the theory of meaning. Finally, Kirkham attempts to refute Dummett's argument by offering a counter-example to the crucial premiss that the behaviour of competent speakers can be explained by appeal to implicit knowledge of justification conditions, without also attributing to them implicit knowledge of truth conditions distinct form justification conditions. I shall not address the issue of accuracy of interpretation or representation of Dummett's views. What I shall argue is that Kirkham's proposed counter-example does not succeed. Drawing on an example offered by Crispin Wright, Kirkham considers a speaker suffering from a congenital lack of a sense of taste. For such a person propositions like 'this is sweet' and 'this is salty' are undecidable, since there are circumstances in which she cannot ascertain their truth-values. Still, she knows justification conditions for such propositions, conditions largely in terms of the behaviour of the unafflicted when dealing with stuff that is sweet or salty. Wright also allows her the ability to discern relations of logical consequence between various attributions of taste to a substance (such as Kirkham's example, that 'this is salty or sweet' and 'this is not salty' entails 'this is sweet'), an ability which he thinks can be accounted for on the basis of her knowledge of justification conditions. Kirkham admits that this ability can be explained on a justification conditional model of meaning and lingusitic competence by use of an intuitionist/constructivist redefinition of 'logical consequence' in terms of justification conditions: X is a logical consequence of Y if and only if it is impossible for X not to be justified if Y iS (p. 221). (Hereinafter I adopt Kirham's convention of using double quotations to indicate this use of the expression, to distinguish it from the classically defined sense, for which no quotation marks are used.) However, although our tasteless person might assert

From Plato' on there have been many attempts to 'define' negation. The motivation has often been a rooted aversion to the idea of negativeness as such. The usual strategy has been to try to explain negation by some suitable positive thought or action. For example, Locke2 says that assertion is the joining of two ideas, and negation is their separation. He saw separation as a positive act on the part of the thinker, thus rescuing negation from mere negativeness. The trouble with attempts of this sort to define negation is that they fail to get the laws of logic right. Thus Locke's theory loses its appeal when one reflects that double negation is affirmation, whereas on his account a separation followed by a further separation ought to be emphatic negation if anything. Similarly the law of Excluded Middle appears to fail on Locke's account, since he does not exclude the possibility that the things corresponding to our ideas may in reality be neither joined nor separated. The correct conclusion from the failure of attempts to define negation is that it is a primitive idea. The fact that we cannot define it does not mean there is a problem about how we can learn its meaning. For, on the account normally given by a classical logician, if someone understands a proposition, they know the circumstances under which it would be true. That in turn involves knowing the difference between these circumstances

I think without doubt that the thorniest problem for one who wishes to transfer something resembling the intuitionist account of the meanings of mathematical statements to the whole of discourse is what account he can give of tensed statements. (Dummett (1978), p. 368) Let atomic sentences be sentences not containing any of the usual operators of first-order predicate calculi, and let compound sentences be sentences formed from such atomic sentences by applying one or more of the usual operators of first-order predicate calculi. So far, we cannot classify tensed sentences like

If it is an unknown truth that p, it is an unknowable truth that it is an unknown truth that p (cf F. B. Fitch, 'A logical analysis of some value concepts', The Journal of Symbolic Logic 28, 1963, pp. 135-42, at p. 138). It follows, by classical logic, that if all truths are knowable then all truths are known. This hardish fact makes life difficult for the verificationist who wishes to assert that all truths are knowable, but to deny that all truths are known. He might try rejecting classical logic (cf. my 'Intuitionism disproved?', ANALYSIS 42.4, October, 1982, pp. 203-7). Dorothy Edgington has recently suggested a different way out (cf. 'The paradox of knowability', Mind 94, 1985, pp. 557-68). She admits that there are unknowable truths in the sense of Fitch's argument, but interestingly revises the verificationist principle, to give it this form: if, in a situation s, it is a truth that p, then there is a possible situation s' in which it is known that, in s, p. Although the truth (for some values of '/?') that, in s, it is an unknown truth that p cannot be known in s, it might be known in some possible situation other than s. I have argued elsewhere that this account faces difficulties of principle in explaining how the knower in s' can pick out the situations in thought with adequate specificity (cf. 'On the paradox of knowability', Mind forthcoming). I waive those objections here, to examine Edgington's application of her principle to a particular case of knowledge in one situation that something is an unknown truth in another. I shall argue that her account does not generalize to the full range of cases which it needs to cover.

Though my title speaks of Kant’s mathematical realism, I want in this essay to explore Kant’s relation to a famous mathematical anti-realist. Specifically, I want to discuss Kant’s influence on L. E. J. Brouwer, the 20th-century Dutch mathematician who built a contemporary philosophy of mathematics on constructivist themes which were quite explicitly Kantian.1 Brouwer’s theory (called intuitionism) is perhaps most notable for its belief that constructivism (whatever that means) requires us to abandon the traditional (classical) logic of mathematical reasoning in favor of a different canon of reasoning, called intuitionistic logic. Brouwer thought that classical logic is intrinsically bound up with a nonconstructive (or “realistic”) view of mathematics. This means that, according to Brouwer, when we do mathematics we must give up bivalence (the principle that a given sentence either is true or is determinately false), we must no longer use such familiar logical laws as excluded middle, and we must sometimes forebear from the classic method of reductio ad absurdum. All of these are intuitionistically invalid classical principles.

William K. Frankena has himself authoritatively and engagingly nar rated the itinerarium of his mind from youthful cognitivism in ethics, as a beginner 'of Calvinistic background and Hegelian sympathies' who con trived to combine 'naturalism about "good" with intuitionism about "ought" ', to his mature noncognitivist rationalism as a major philosopher of sophisticated analytic technique and Calvinist sympathies.1 A number of his characteristic earlier opinions were elaborated in response to the writings of G. E. Moore; and this body of work as a young man contains the seeds of his later development. Yet the past thirty years have radically altered the perspective from which Moore and his influence are now viewed. What changes does our altered perspective on Moore make to our understanding of Frankena?

Some things are absolutely wrong. They are not to be done, no matter what the consequences or circumstances. To put it another way, certain descriptions of acts are such that if a proposed course of action satisfies any of them it is thereby ruled out, regardless of whatever other descriptions it may satisfy. For some things are morally impossible; and if a thing is impossible for any reason then it is an error even to think of doing such a thing, and still more of an error to try it, whatever the good that might be attained or evil averted if one were, per impossibile, to do that thing.' Declaring that some things are morally impossible exposes one to charges of incoherence from both intuitionists and consequentialists. It is not that either need doubt that the things declared to be morally impossible are indeed evil. When an absolutist declares that murder for example is absolutely wrong both consequentialist and intuitionist can agree with him that there is something bad about murder. Their point will be rather the formal one that whatever content you give it an absolutist ethic is structurally absurd. In this paper I want to look at the consequentialist and intuitionist objections to absolutism and at their alternative conceptions of the structure that a moral theory should have. I will show by an analogy with chess that the objections are not sound, and that a moral theory can coherently be given an absolutist structure. I will concede nevertheless that there is some truth in both consequentialism and intuitionism, and will use again the chess analogy to show how we may build upon an absolutist ethic to do justice to those truths about the correct structure for moral theory. Let me start with the consequentialist objection to absolutist ethics. To focus discussion let us assume that our absolutism holds murder to be absolutely wrong. Now circumstances might