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.
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)
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)
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)
We generalize the Kolmogorov axioms for probability calculus to obtain conditions defining, for any given logic, a class of probability functions relative to that logic, coinciding with the standard probability functions in the special case of classical logic but allowing consideration of other classes of "essentially Kolmogorovian" probability functions relative to other logics. We take a broad view of the Bayesian approach as dictating inter alia that from the perspective of a given logic, rational degrees of belief are those representable (...) by probability functions from the class appropriate to that logic. Classical Bayesianism, which fixes the logic as classical logic, is only one version of this general approach. Another, which we call Intuitionistic Bayesianism, selects intuitionistic logic as the preferred logic and the associated class of probability functions as the right class of candidate representions of epistemic states (rational allocations of degrees of belief). Various objections to classical Bayesianism are, we argue, best met by passing to intuitionistic Bayesianism—in which the probability functions are taken relative to intuitionistic logic—rather than by adopting a radically non-Kolmogorovian, for example, nonadditive, conception of (or substitute for) probability functions, in spite of the popularity of the latter response among those who have raised these objections. The interest of intuitionistic Bayesianism is further enhanced by the availability of a Dutch Book argument justifying the selection of intuitionistic probability functions as guides to rational betting behavior when due consideration is paid to the fact that bets are settled only when/if the outcome bet on becomes known. (shrink)
We reconsider the pragmatic interpretation of intuitionistic logic [21] regarded as a logic of assertions and their justi cations and its relations with classical logic. We recall an extension of this approach to a logic dealing with assertions and obligations, related by a notion of causal implication [14, 45]. We focus on the extension to co-intuitionistic logic, seen as a logic of hypotheses [8, 9, 13] and on polarized bi-intuitionistic logic as a logic of assertions and conjectures: looking at the (...) S4 modal translation, we give a de nition of a system AHL of bi-intuitionistic logic that correctly represents the duality between intuitionistic and co-intuitionistic logic, correcting a mistake in previous work [7, 10]. A computational interpretation of cointuitionism as a distributed calculus of coroutines is then used to give an operational interpretation of subtraction.Work on linear co-intuitionism is then recalled, a linear calculus of co-intuitionistic coroutines is de ned and a probabilistic interpretation of linear co-intuitionism is given as in [9]. Also we remark that by extending the language of intuitionistic logic we can express the notion of expectation, an assertion that in all situations the truth of p is possible and that in a logic of expectations the law of double negation holds. Similarly, extending co-intuitionistic logic, we can express the notion of conjecture that p, de ned as a hypothesis that in some situation the truth of p is epistemically necessary. (shrink)
We provide a direct method for proving Craig interpolation for a range of modal and intuitionistic logics, including those containing a "converse" modality. We demonstrate this method for classical tense logic, its extensions with path axioms, and for bi-intuitionistic logic. These logics do not have straightforward formalisations in the traditional Gentzen-style sequent calculus, but have all been shown to have cut-free nested sequent calculi. The proof of the interpolation theorem uses these calculi and is purely syntactic, without resorting to embeddings, (...) semantic arguments, or interpreted connectives external to the underlying logical language. A novel feature of our proof includes an orthogonality condition for defining duality between interpolants. (shrink)
This paper studies the relationship between labelled and nested calculi for propositional intuitionistic logic, first-order intuitionistic logic with non-constant domains and first-order intuitionistic logic with constant domains. It is shown that Fitting’s nested calculi naturally arise from their corresponding labelled calculi—for each of the aforementioned logics—via the elimination of structural rules in labelled derivations. The translational correspondence between the two types of systems is leveraged to show that the nested calculi inherit proof-theoretic properties from their associated labelled calculi, such as (...) completeness, invertibility of rules and cut admissibility. Since labelled calculi are easily obtained via a logic’s semantics, the method presented in this paper can be seen as one whereby refined versions of labelled calculi (containing nested calculi as fragments) with favourable properties are derived directly from a logic’s semantics. (shrink)
Intuitionism’s disagreement with classical logic is standardly based on its specific understanding of truth. But different intuitionists have actually explicated the notion of truth in fundamentally different ways. These are considered systematically and separately, and evaluated critically. It is argued that each account faces difficult problems. They all either have implausible consequences or are viciously circular.
This paper shows how to derive nested calculi from labelled calculi for propositional intuitionistic logic and first-order intuitionistic logic with constant domains, thus connecting the general results for labelled calculi with the more refined formalism of nested sequents. The extraction of nested calculi from labelled calculi obtains via considerations pertaining to the elimination of structural rules in labelled derivations. Each aspect of the extraction process is motivated and detailed, showing that each nested calculus inherits favorable proof-theoretic properties from its associated (...) labelled calculus. (shrink)
The main subject of Cusanus’ investigations was the name of God. He claimed to have achieved the best possible one, Not-Other. Since Cusanus stressed that these two words do not mean the corresponding affirmative word, i.e. the same, they represent the failure of the double negation law and therefore belong to non-classical, and above all, intuitionist logic. Some of his books implicitly applied intuitionist reasoning and the corresponding organization of a theory which is governed by intuitionist logic. A comparison of (...) two of Cusanus’ short writings shows that throughout his life he substantially improved his use of this kind of logic and ultimately was able to reason consistently within such a logic and recognize some of its basic laws. One important idea developed by him was that of a proposition composed of a triple repetition of “not-other” expressing “the Tri-unity of concordance” i.e. the “best name for the Trinity”. I complete his application of intuitionist logic to theological subjects by characterizing the inner relationships within the Trinity in such a way that there are no longer contradictions in the notion. Generally speaking, the notion of the Trinity implies a translation from intuitionist to classical logic, to which Cusanus closely approximated. Moreover, I show that the main aspects of Christian revelation, including Christ’s teachings, are represented both by this translation and by some doubly negated propositions of intuitionist logic. Hence, intuitionist logic was introduced into the history of Western theological thinking with Christian revelation, as only Cusanus partly recognized. Appendix 1 summarizes a detailed analysis of Cusanus’ second short writing. Appendix 2 shows that the Athanasian creed regarding the Christian Trinity is a consistent sequence of intuitionist propositions provided that some verbal emendations are added, showing that ancient trinitarian thinking was also close to intuitionist reasoning. (shrink)
In this article I assess Rossian Intuitionism, which is the view that the Rossian Principles of Duty are self-evident. I begin by motivating and clarifying a version of the view—Rossian Conceptual Intuitionism—that hasn’t been adequately considered by Rossians. After defending it against a series of significant objections, I show that enthusiasm for Rossian Conceptual Intuitionism should be muted. Specifically, I argue that we lack sufficient reason for thinking that the Rossian Principles are self-evident, and that insisting that (...) they are self-evident may commit Rossians to radically expanding the scope of self-evidence. (shrink)
This paper presents a way of formalising definite descriptions with a binary quantifier ι, where ιx[F, G] is read as ‘The F is G’. Introduction and elimination rules for ι in a system of intuitionist negative free logic are formulated. Procedures for removing maximal formulas of the form ιx[F, G] are given, and it is shown that deductions in the system can be brought into normal form.
Over the course of human history there appears to have been a global shift in moral values towards a broadly ‘liberal’ orientation. Huemer argues that this shift better accords with a realist than an antirealist metaethics: it is best explained by the discovery of mind-independent truths through intuition. In this article I argue, contra Huemer, that the historical data are better explained assuming the truth of moral antirealism. Realism does not fit the data as well as Huemer suggests, whereas antirealists (...) have underappreciated resources to explain the relevant historical dynamics. These resources include an appeal to socialization, to technological and economical convergences, to lessons learned from history, to changes induced by consistency reasoning and to the social function of moral norms in overcoming some of the cooperation problems that globalizing societies face. I point out that the realist’s explanans has multiple shortcomings, that the antirealist’s explanans has several explanatory virtues, and conclude that the latter provides a superior account of the historical shift towards liberal values. (shrink)
Intuitionistic Propositional Logic is proved to be an infinitely many valued logic by Gödel (Kurt Gödel collected works (Volume I) Publications 1929–1936, Oxford University Press, pp 222–225, 1932), and it is proved by Jaśkowski (Actes du Congrés International de Philosophie Scientifique, VI. Philosophie des Mathématiques, Actualités Scientifiques et Industrielles 393:58–61, 1936) to be a countably many valued logic. In this paper, we provide alternative proofs for these theorems by using models of Kripke (J Symbol Logic 24(1):1–14, 1959). Gödel’s proof gave (...) rise to an intermediate propositional logic (between intuitionistic and classical), that is known nowadays as Gödel or the Gödel-Dummett Logic, and is studied by fuzzy logicians as well. We also provide some results on the inter-definability of propositional connectives in this logic. (shrink)
Sentences containing definite descriptions, expressions of the form ‘The F’, can be formalised using a binary quantifier ι that forms a formula out of two predicates, where ιx[F, G] is read as ‘The F is G’. This is an innovation over the usual formalisation of definite descriptions with a term forming operator. The present paper compares the two approaches. After a brief overview of the system INFι of intuitionist negative free logic extended by such a quantifier, which was presented in (...) (Kürbis 2019), INFι is first compared to a system of Tennant’s and an axiomatic treatment of a term forming ι operator within intuitionist negative free logic. Both systems are shown to be equivalent to the subsystem of INFι in which the G of ιx[F, G] is restricted to identity. INFι is then compared to an intuitionist version of a system of Lambert’s which in addition to the term forming operator has an operator for predicate abstraction for indicating scope distinctions. The two systems will be shown to be equivalent through a translation between their respective languages. Advantages of the present approach over the alternatives are indicated in the discussion. (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)
This paper explores the generally overlooked relevance of an important contemporary debate in mainstream epistemology to philosophers working within ethics on questions concerning moral knowledge. It is argued that this debate, between internalists and externalists about the accessibility of epistemic justification, has the potential to be both significantly influenced by, and have a significant impact upon, the study of moral knowledge. The moral sphere provides a particular type of strong evidence in favour of externalism, and mainstream epistemologists might benefit from (...) paying attention to this fact. At the same time, the terrain of moral epistemology (approached as a sub-field of metaethics) needs to be reshaped by the realisation that externalists can steal the thunder of intuitionists when it comes to knowledge constituted by seemingly self-evident beliefs.1. (shrink)
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. (shrink)
In this book set theory INC# based on intuitionistic logic with restricted modus ponens rule is proposed. It proved that intuitionistic logic with restricted modus ponens rule can to safe Cantor naive set theory from a triviality. Similar results for paraconsistent set theories were obtained in author papers [13]-[16].
We employ a recently developed methodology -- called "structural refinement" -- to extract nested sequent systems for a sizable class of intuitionistic modal logics from their respective labelled sequent systems. This method can be seen as a means by which labelled sequent systems can be transformed into nested sequent systems through the introduction of propagation rules and the elimination of structural rules, followed by a notational translation. The nested systems we obtain incorporate propagation rules that are parameterized with formal grammars, (...) and which encode certain frame conditions expressible as first-order Horn formulae that correspond to a subclass of the Scott-Lemmon axioms. We show that our nested systems are sound, cut-free complete, and admit hp-admissibility of typical structural rules. (shrink)
This thesis sets out an argument in defence of moral objectivism. It takes Mackie as the critic of objectivism and it ends by proposing that the best defence of objectivism may be found in what I shall call Kantian intuitionism, which brings together elements of the intuitionism of Ross and a Kantian epistemology. The argument is fundamentally transcendental in form and it proceeds by first setting out what we intuitively believe, rejecting the sceptical attacks on those beliefs, and (...) by then proposing a theory that can legitimize what we already do believe. Chapter One sets out our intuitive understanding of morality: that morality is cognitive, moral beliefs can be true or false; that morality is real, we do not construct it; that morality is rational, we can learn about it by rational investigation; and that morality places us under an absolute constraint. The chapter ends by clarifying the nature of that absolute demand and by arguing that the critical idea within morality is the idea of duty. In Chapter Two Mackie’s sceptical attack on objectivism is examined. Four key arguments are identified: that moral beliefs are relative to bfferent agents; that morality is based upon on non-rational causes; that the idea of moral properties or entities is too queer to be sustainable; and that moral objectivism involves queer epistemological commitments. Essentially all of these arguments are shown to be ambiguous; however it is proposed that Mackie has an underlying epistemological and metaphysical theory, scientific empiricism, which is hostile to objectivism and a theory that many find attractive for reasons that are independent of morality. Chapter Three explores the nature of moral rationality and whether scientific empiricism can use the idea of reflective equilibrium to offer a reasonable account of moral rationality. It concludes that, while reflective equilibrium is a useful account of moral rationality, it cannot be effectively reconciled with scientific empiricism. In order to function effectively as a rational process, reflective equilibrium must be rationally constrained by our moral judgements and our moral principles. Chapter Four begins the process of exploring some alternative epistemologies and argues that the only account that remains true to objectivism and the needs of reflective equilibrium is the account of intuitionism proposed by Ross. However this account can be developed further by drawing upon number of Kantian ideas and using them to supplement Ross’s intuitionism. So Chapter Five draws upon a number of Kant's ideas, most notably some key notions from the Critique of Judgement. These ideas are: that we possess a rational will that is subject to the Moral law and determined by practical reason; that we possess a faculty of judgement which enables us to become aware of moral properties and that these two faculties together with the third faculty of thought can function to constitute the moral understanding. Using these ideas the thesis explores whether they can serve to explain how intuitions can be rational and how objectivism can be justified. (shrink)
I show that the model-theoretic meaning that can be read off the natural deduction rules for disjunction fails to have certain desirable properties. I use this result to argue against a modest form of inferentialism which uses natural deduction rules to fix model-theoretic truth-conditions for logical connectives.
Benchmarking automated theorem proving (ATP) systems using standardized problem sets is a well-established method for measuring their performance. However, the availability of such libraries for non-classical logics is very limited. In this work we propose a library for benchmarking Girard's (propositional) intuitionistic linear logic. For a quick bootstrapping of the collection of problems, and for discussing the selection of relevant problems and understanding their meaning as linear logic theorems, we use translations of the collection of Kleene's intuitionistic theorems in the (...) traditional monograph "Introduction to Metamathematics". We analyze four different translations of intuitionistic logic into linear logic and compare their proofs using a linear logic based prover with focusing. In order to enhance the set of problems in our library, we apply the three provability-preserving translations to the propositional benchmarks in the ILTP Library. Finally, we generate a comprehensive set of reachability problems for Petri nets and encode such problems as linear logic sequents, thus enlarging our collection of problems. (shrink)
Takeuti and Titani have introduced and investigated a logic they called intuitionistic fuzzy logic. This logic is characterized as the first-order Gödel logic based on the truth value set [0,1]. The logic is known to be axiomatizable, but no deduction system amenable to proof-theoretic, and hence, computational treatment, has been known. Such a system is presented here, based on previous work on hypersequent calculi for propositional Gödel logics by Avron. It is shown that the system is sound and complete, and (...) allows cut-elimination. A question by Takano regarding the eliminability of the Takeuti-Titani density rule is answered affirmatively. (shrink)
Hilbert’s choice operators τ and ε, when added to intuitionistic logic, strengthen it. In the presence of certain extensionality axioms they produce classical logic, while in the presence of weaker decidability conditions for terms they produce various superintuitionistic intermediate logics. In this thesis, I argue that there are important philosophical lessons to be learned from these results. To make the case, I begin with a historical discussion situating the development of Hilbert’s operators in relation to his evolving program in the (...) foundations of mathematics and in relation to philosophical motivations leading to the development of intuitionistic logic. This sets the stage for a brief description of the relevant part of Dummett’s program to recast debates in metaphysics, and in particular disputes about realism and anti-realism, as closely intertwined with issues in philosophical logic, with the acceptance of classical logic for a domain reflecting a commitment to realism for that domain. Then I review extant results about what is provable and what is not when one adds epsilon to intuitionistic logic, largely due to Bell and DeVidi, and I give several new proofs of intermediate logics from intuitionistic logic+ε without identity. With all this in hand, I turn to a discussion of the philosophical significance of choice operators. Among the conclusions I defend are that these results provide a finer-grained basis for Dummett’s contention that commitment to classically valid but intuitionistically invalid principles reflect metaphysical commitments by showing those principles to be derivable from certain existence assumptions; that Dummett’s framework is improved by these results as they show that questions of realism and anti-realism are not an “all or nothing” matter, but that there are plausibly metaphysical stances between the poles of anti-realism and realism, because different sorts of ontological assumptions yield intermediate rather than classical logic; and that these intermediate positions between classical and intuitionistic logic link up in interesting ways with our intuitions about issues of objectivity and reality, and do so usefully by linking to questions around intriguing everyday concepts such as “is smart,” which I suggest involve a number of distinct dimensions which might themselves be objective, but because of their multivalent structure are themselves intermediate between being objective and not. Finally, I discuss the implications of these results for ongoing debates about the status of arbitrary and ideal objects in the foundations of logic, showing among other things that much of the discussion is flawed because it does not recognize the degree to which the claims being made depend on the presumption that one is working with a very strong logic. (shrink)
This piece is a short review of a volume of papers on ethical intuitionism (Ethical Intuitionism: Re-evaluations, ed. Philip Stratton-Lake, Oxford University Press, 2002).
In his essay ‘“Wang’s Paradox”’, Crispin Wright proposes a solution to the Sorites Paradox (in particular, the form of it he calls the ‘Paradox of Sharp Boundaries’) that involves adopting intuitionistic logic when reasoning with vague predicates. He does not give a semantic theory which accounts for the validity of intuitionistic logic (and the invalidity of stronger logics) in that area. The present essay tentatively makes good the deficiency. By applying a theorem of Tarski, it shows that intuitionistic logic is (...) the strongest logic that may be applied, given certain semantic assumptions about vague predicates. The essay ends with an inconclusive discussion of whether those semantic assumptions should be accepted. (shrink)
There is widespread agreement that while on a Dummettian theory of meaning the justified logic is intuitionist, as its constants are governed by harmonious rules of inference, the situation is reversed on Huw Price's bilateralist account, where meanings are specified in terms of primitive speech acts assertion and denial. In bilateral logics, the rules for classical negation are in harmony. However, as it is possible to construct an intuitionist bilateral logic with harmonious rules, there is no formal argument against (...) class='Hi'>intuitionism from the bilateralist perspective. Price gives an informal argument for classical negation based on a pragmatic notion of belief, characterised in terms of the differences they make to speakers' actions. The main part of this paper puts Price's argument under close scrutiny by regimenting it and isolating principles Price is committed to. It is shown that Price should draw a distinction between A or ¬A making a difference. According to Price, if A makes a difference to us, we treat it as decidable. This material allows the intuitionist to block Price's argument. Abandoning classical logic also brings advantages, as within intuitionist logic there is a precise meaning to what it might mean to treat A as decidable: it is to assume A ∨ ¬A. (shrink)
This paper presents rules of inference for a binary quantifier I for the formalisation of sentences containing definite descriptions within intuitionist positive free logic. I binds one variable and forms a formula from two formulas. Ix[F, G] means ‘The F is G’. The system is shown to have desirable proof-theoretic properties: it is proved that deductions in it can be brought into normal form. The discussion is rounded up by comparisons between the approach to the formalisation of definite descriptions recommended (...) here and the more usual approach that uses a term-forming operator ι, where ιxF means ‘the F’. (shrink)
By formalizing some classical facts about provably total functions of intuitionistic primitive recursive arithmetic (iPRA), we prove that the set of decidable formulas of iPRA and of iΣ1+ (intuitionistic Σ1-induction in the language of PRA) coincides with the set of its provably ∆1-formulas and coincides with the set of its provably atomic formulas. By the same methods, we shall give another proof of a theorem of Marković and De Jongh: the decidable formulas of HA are its provably ∆1-formulas.
This paper contains five observations concerning the intended meaning of the intuitionistic logical constants: (1) if the explanations of this meaning are to be based on a non-decidable concept, that concept should not be that of 'proof'; (2) Kreisel's explanations using extra clauses can be significantly simplified; (3) the impredicativity of the definition of → can be easily and safely ameliorated; (4) the definition of → in terms of 'proofs from premises' results in a loss of the inductive character of (...) the definitions of ∨ and ∃; and (5) the same occurs with the definition of ∀ in terms of 'proofs with free variables'. (shrink)
The paper explores Hermann Weyl’s turn to intuitionism through a philosophical prism of normative framework transitions. It focuses on three central themes that occupied Weyl’s thought: the notion of the continuum, logical existence, and the necessity of intuitionism, constructivism, and formalism to adequately address the foundational crisis of mathematics. The analysis of these themes reveals Weyl’s continuous endeavor to deal with such fundamental problems and suggests a view that provides a different perspective concerning Weyl’s wavering foundational positions. Building (...) on a philosophical model of scientific framework transitions and the special role that normative indecision or ambivalence plays in the process, the paper examines Weyl’s motives for considering such a radical shift in the first place. It concludes by showing that Weyl’s shifting stances should be regarded as symptoms of a deep, convoluted intrapersonal process of self-deliberation induced by exposure to external criticism. (shrink)
In his book Intuitionism, David Kaspar is after the truth. That is to say, on his view, “philosophy is the search for the whole truth” (p. 7). Intuitionism, then, “reflects that standpoint” (p. 7). My comments are meant to reflect the same standpoint. More explicitly, my aim in these comments is to evaluate the arguments for intuitionism, as I understand them from reading Kaspar’s book. In what follows, I focus on three arguments in particular, which can be (...) found in Chapters 1, 2, and 3 of Intuitionism: an inference to the best explanation, an argument from the analogy between mathematical knowledge and moral knowledge, and an argument from the epistemic preferability of the intuitive principles. I will discuss them in this order. (shrink)
In this paper intuitionistic set theory INC# in infinitary set theoretical language is considered. External induction principle in nonstandard intuitionistic arithmetic were derived. Non trivial application in number theory is considered.
We discuss the philosophical implications of formal results showing the con- sequences of adding the epsilon operator to intuitionistic predicate logic. These results are related to Diaconescu’s theorem, a result originating in topos theory that, translated to constructive set theory, says that the axiom of choice (an “existence principle”) implies the law of excluded middle (which purports to be a logical principle). As a logical choice principle, epsilon allows us to translate that result to a logical setting, where one can (...) get an analogue of Diaconescu’s result, but also can disentangle the roles of certain other assumptions that are hidden in mathematical presentations. It is our view that these results have not received the attention they deserve: logicians are unlikely to read a discussion because the results considered are “already well known,” while the results are simultaneously unknown to philosophers who do not specialize in what most philosophers will regard as esoteric logics. This is a problem, since these results have important implications for and promise signif i cant illumination of contem- porary debates in metaphysics. The point of this paper is to make the nature of the results clear in a way accessible to philosophers who do not specialize in logic, and in a way that makes clear their implications for contemporary philo- sophical discussions. To make the latter point, we will focus on Dummettian discussions of realism and anti-realism. Keywords: epsilon, axiom of choice, metaphysics, intuitionistic logic, Dummett, realism, antirealism. (shrink)
În acest studiu, îmi propun să arăt că modelul social intuiţionist al judecăţii morale propus de Haidt este la rândul său prea restrictiv faţă de influenţa raţionării morale, poate tot aşa cum modelul raţionalist subestima influenţa emoţiilor morale. Mai întâi, voi prezenta modelul raţionalist despre natura judecăţii morale şi voi evidenţia rezultatele empirice care au contribuit la erodarea sa. Apoi, voi prezenta şi critica modelul social intuiţionist revigorat de revoluţia „afectivă” din psihologia morală, argumentând că rezultatele din psihologia experimentală, neuroştiinţă (...) şi psihologia evoluţionistă acordă raţionării morale o influenţă cauzală mai mare decât admite Haidt. (shrink)
In this dissertation I discuss the epistemology of ethical intuitionism, in particular the claim that mature moral agents possess self-evident moral knowledge. Traditional intuitionists such as W.D. Ross have claimed that by reflection, we can acquire knowledge of our basic moral duties such as the duty of veracity or benevolence. Recent defenders of intuitionism such as Robert Audi have further developed this theory and argued that adequate understanding can be sufficient for moral knowledge. I criticize this view and (...) argue that such accounts fail to make a convincing case for a foundationalist moral epistemology. Instead, I propose to separate the question of how we acquire moral knowledge from an account that justifies moral beliefs. In response to the first issue, I draw an analogy between our moral intuitions and chosmkian linguistics; in both areas, I argue, human beings possess a universal, unconscious and (partly) inaccessible system of rules that explains how we come to learn language and to make moral judgments. In regards to the justificatory issue, I address recent evolutionary debunking arguments designed to undermine the claim that our moral judgments track stance-independent truths. I try to show that this conclusion only follows under the assumption of an instrumentalist interpretation of moral reasoning which the intuitionist is not forced to accept. (shrink)
In this book set theory INC# based on intuitionistic logic with restricted modus ponens rule is proposed. It proved that intuitionistic logic with restricted modus ponens rule can to safe Cantor naive set theory from a triviality.
I discuss Whewell’s philosophy of morality, as opposed to systematic morality, not unlike Kant’s distinction between a pure and an empirical moral philosophy. Whewell worked out a systematization of traditional normative ethics as a first step before its rational justification; he believed that the point in the philosophy of morality is justifying a few rational truths about the structure of morality such as to rule hedonism, eudemonism, and consequentialism; yet a system of positive morality cannot be derived solely from such (...) rational truths but requires consideration of the ongoing dialectics between idea and fact in historically given moralities. Whewell’s intuitionism turns out to be both more similar to Kantian ethics and more different from Sidgwick’s idea of intuitionism. (shrink)
A natural problem from elementary arithmetic which is so strongly undecidable that it is not even Trial and Error decidable (in other words, not decidable in the limit) is presented. As a corollary, a natural, elementary arithmetical property which makes a diﬀerence between intuitionistic and classical theories is isolated.
Moral intuitionism, which claims that some moral seemings are justification-conferring, has become an increasingly popular account in moral epistemology. Defenses of the position have largely focused on the standard account, according to which the justification-conferring power of a moral seeming is determined by its phenomenal credentials alone. Unfortunately, the standard account is a less plausible version of moral intuitionism because it does not take etiology seriously. In this paper, I provide an outline and defense of a non-standard account (...) of moral intuitionism that I dub the “Prudent Conscience View.” According to this view, phenomenal credentials only partially determine the justification-conferring power of a moral seeming, for a seeming’s justification-conferring power is also determined by its etiology. In brief, a moral seeming is justification-conferring to the degree that the conscience that gave rise to it is functioning properly, and a person's conscience functions properly to the degree that the person is prudent. (shrink)
Various studies have reported that moral intuitions about the permissibility of acts are subject to framing effects. This paper reports the results of a series of experiments which further examine the susceptibility of moral intuitions to framing effects. The main aim was to test recent speculation that intuitions about the moral relevance of certain properties of cases might be relatively resistent to framing effects. If correct, this would provide a certain type of moral intuitionist with the resources to resist challenges (...) to the reliability of moral intuitions based on such framing effects. And, fortunately for such intuitionists, although the results can’t be used to mount a strident defence of intuitionism, the results do serve to shift the burden of proof onto those who would claim that intuitions about moral relevance are problematically sensitive to framing effects. (shrink)
Critical notice of Robert Audi's The Good in the Right in which doubts are raised about the epistemological and ethical doctrines it defends. It doubts that an appeal to Kant is a profitable way to defend Rossian normative intuitionism.
This paper deals with, prepositional calculi with strong negation (N-logics) in which the Craig interpolation theorem holds. N-logics are defined to be axiomatic strengthenings of the intuitionistic calculus enriched with a unary connective called strong negation. There exists continuum of N-logics, but the Craig interpolation theorem holds only in 14 of them.
This paper considers logics which are formally dual to intuitionistic logic in order to investigate a co-constructive logic for proofs and refutations. This is philosophically motivated by a set of problems regarding the nature of constructive truth, and its relation to falsity. It is well known both that intuitionism can not deal constructively with negative information, and that defining falsity by means of intuitionistic negation leads, under widely-held assumptions, to a justification of bivalence. For example, we do not want (...) to equate falsity with the non-existence of a proof since this would render a statement such as “pi is transcendental” false prior to 1882. In addition, the intuitionist account of negation as shorthand for the derivation of absurdity is inadequate, particularly outside of purely mathematical contexts. To deal with these issues, I investigate the dual of intuitionistic logic, co-intuitionistic logic, as a logic of refutation, alongside intuitionistic logic of proofs. Direct proof and refutation are dual to each other, and are constructive, whilst there also exist syntactic, weak, negations within both logics. In this respect, the logic of refutation is weakly paraconsistent in the sense that it allows for statements for which, neither they, nor their negation, are refuted. I provide a proof theory for the co-constructive logic, a formal dualizing map between the logics, and a Kripke-style semantics. This is given an intuitive philosophical rendering in a re-interpretation of Kolmogorov’s logic of problems. (shrink)
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