A basic result in intuitionism is Π02-conservativity. Take any proof p in classical arithmetic of some Π02-statement , with P decidable). Then we may effectively turn p in some intuitionistic proof of the same statement. In a previous paper [1], we generalized this result: any classical proof p of an arithmetical statement ∀x.∃y.P, with P of degree k, may be effectively turned into some proof of the same statement, using Excluded Middle only over degree k formulas. When k = (...) 0, we get the original conservativity result as particular case. This result was a by-product of a semantical construction. J. Avigad of Carnegie Mellon University, found a short, direct syntactical derivation of the same result, using H. Friedman's A-translation. His proof is included here with his permission. (shrink)

A Plithogenic Logical proposition P is a proposition that is characterized by many degrees of truth-values with respect to many corresponding attribute-values (or random variables) that characterize P. Each degree of truth-value may be classical, fuzzy, intuitionistic fuzzy, neutrosophic, or other fuzzy extension type logic. At the end, a cumulative truth of P is computed.

We give an equivalent formulation of topological algebras, interpreting S4, as boolean algebras equipped with intuitionistic negation. The intuitionistic substructure—Heyting algebra—of such an algebra can be then seen as an “epistemic subuniverse”, and modalities arise from the interaction between the intuitionistic and classical negations or, we might perhaps say, between the epistemic and the ontological aspects: they are not relations between arbitrary alternatives but between intuitionistic substructures and one common world governed by the classical (propositional) logic. As an example of (...) the generality of the obtained view, we apply it also to S5. We give a sound, complete and decidable sequent calculus, extending a classical system with the rules for handling the intuitionistic negation, in which one can prove all classical, intuitionistic and S4 valid sequents. (shrink)

We define a family of intuitionistic non-normal modal logics; they can be seen as intuitionistic counterparts of classical ones. We first consider monomodal logics, which contain only Necessity or Possibility. We then consider the more important case of bimodal logics, which contain both modal operators. In this case we define several interactions between Necessity and Possibility of increasing strength, although weaker than duality. We thereby obtain a lattice of 24 distinct bimodal logics. For all logics we provide both a Hilbert (...) axiomatisation and a cut-free sequent calculus, on its basis we also prove their decidability. We then define a semantic characterisation of our logics in terms of neighbourhood models containing two distinct neighbourhood functions corresponding to the two modalities. Our semantic framework captures modularly not only our systems but also already known intuitionistic non-normal modal logics such as Constructive K and the propositional fragment of Wijesekera’s Constructive Concurrent Dynamic Logic. (shrink)

The standard Pawlak approach to rough set theory, as an approximation space consisting of a universe U and an equivalence (indiscernibility) relation R U x U, can be equivalently described by the induced preclusivity ("discernibility") relation U x U \ R, which is irreflexive and symmetric.We generalize the notion of approximation space as a pair consisting of a universe U and a discernibility or preclusivity (irreflexive and symmetric) relation, not necessarily induced from an equivalence relation. In this case the "elementary" (...) sets are not mutually disjoint, but all the theory of generalized rough sets can be developed in analogy with the standard Pawlak approach. On the power set of the universe, the algebraic structure of the quasi fuzzy-intuitionistic "classical" (BZ) lattice is introduced and the sets of all "closed" and of all "open" definable sets with the associated complete (in general nondistributive) ortholattice structures are singled out. (shrink)

Intuitionism, the second type of descriptivism, is the theory that the truth conditions of moral statements depend on irreducible moral properties, which must be defined in moral terms. The intuitionist claims that we have knowledge of moral truths derived from moral intuition. However, because it is a subjective experience, one person's intuition may differ from another's, and the theory offers no way to decide between them. Intuitionism, Hare argues, is really a kind of Subjectivist Naturalism, or Subjectivism; and, (...) as with Naturalism, it also leads to relativism. Therefore, because it is exhausted by these two theories, descriptivism in general must be abandoned if we are to attain the sort of objectivity that will allow us to avoid relativism. (shrink)

It has long been known that (classical) Peano arithmetic is, in some strong sense, “equivalent” to the variant of (classical) Zermelo–Fraenkel set theory (including choice) in which the axiom of infinity is replaced by its negation. The intended model of the latter is the set of hereditarily finite sets. The connection between the theories is so tight that they may be taken as notational variants of each other. Our purpose here is to develop and establish a constructive version of this. (...) We present an intuitionistic theory of the hereditarily finite sets, and show that it is definitionally equivalent to Heyting Arithmetic, in a sense to be made precise. Our main target theory, the intuitionistic small set theory is remarkably simple, and intuitive. It has just one non-logical primitive, for membership, and three straightforward axioms plus one axiom scheme. We locate our theory within intuitionistic mathematics generally. (shrink)

IntroductionEthical intuitionists have never known quite what to make of the emotions. Generally speaking, these philosophers fall into two camps: rational intuitionists and moral sense theorists. And by my lights, neither camp has been able to tell a convincing story about the exact role and significance of emotion in moral judgment. Rational intuitionists are for the most part too dismissive of the emotions, either regarding emotions as little more than distractions to moral judgment,Samuel Clarke, for instance, after naming our “faculties (...) of reason and will, whereby [we] are enabled to distinguish good from evil,” laments that these faculties are sometimes “imposed upon and deceived in matters of good and evil, right and wrong… by absurd passions and corrupt or partial affections” (A Discourse Concerning the Being and Attributes of God, the Obligations of Natural Religion, and the Truth and Certainty of the Christian Revelation (Glasgow: Richard Griffin and Co., 1823), pp. 166-167). (shrink)

Semi-intuitionistic logic is the logic counterpart to semi-Heyting algebras, which were defined by H. P. Sankappanavar as a generalization of Heyting algebras. We present a new, more streamlined set of axioms for semi-intuitionistic logic, which we prove translationally equivalent to the original one. We then study some formulas that define a semi-Heyting implication, and specialize this study to the case in which the formulas use only the lattice operators and the intuitionistic implication. We prove then that all the logics (...) thus obtained are equivalent to intuitionistic logic, and give their Kripke semantics. (shrink)

Hume bequeathed to rational intuitionists a problem concerning moral judgment and the will – a problem of sufficient severity that it is still cited as one of the major reasons why intuitionism is untenable.1 Stated in general terms, the problem concerns how an intuitionist moral theory can account for the intimate connection between moral judgment and moral motivation. One reason that this is still considered to be a problem for intuitionists is that it is widely assumed that the early (...) intuitionists made little progress towards solving it. In this essay, I wish to challenge this assumption by examining one of the more subtle intuitionist responses to Hume, viz., that offered by Thomas Reid. For reasons that remain unclear to me, Reid's response to Hume on this issue has been almost entirely neglected. I shall argue that it is nonetheless one that merits our attention, for at least two reasons. In the first place, Reid's response to Hume's challenge to rational intuitionism bears a close affinity to the type of response that he offers to Hume's broadly skeptical challenge to realist views regarding our perception of the external world. Since Reid's strategy in the latter case is widely regarded as exhibiting significant promise, it is natural to wonder whether, when applied to the moral domain, this type of strategy displays similar promise.2 I will suggest that it does. That is, I will suggest that since Reid's broadly nativist position in perception is one well worth considering, then so also is his broadly nativist account of moral motivation. Second, Reid's position regarding moral motivation represents an intriguing attempt to blend a broadly intuitionist view with important insights from the sentimentalist tradition. In this respect, Reid's view is a genuine hybrid position unlike that offered by other intuitionists such as Richard Price. The synthetic character of Reid's position, I claim, gives it a unique type of theoretical richness, since it incorporates some very attractive features of both rational intuitionism and sentimentalism. (shrink)

Recent research on algebraic models of _quasi-Nelson logic_ has brought new attention to a number of classes of algebras which result from enriching (subreducts of) Heyting algebras with a special modal operator, known in the literature as a _nucleus_. Among these various algebraic structures, for which we employ the umbrella term _intuitionistic modal algebras_, some have been studied since at least the 1970s, usually within the framework of topology and sheaf theory. Others may seem more exotic, for their primitive operations (...) arise from algebraic terms of the intuitionistic modal language which have not been previously considered. We shall for instance investigate the variety of _weak implicative semilattices_, whose members are (non-necessarily distributive) meet semilattices endowed with a nucleus and an implication operation which is not a relative pseudo-complement but satisfies the postulates of Celani and Jansana’s strict implication. For each of these new classes of algebras we establish a representation and a topological duality which generalize the known ones for Heyting algebras enriched with a nucleus. (shrink)

This paper provides a novel critique of Walter Sinnott-Armstrong’s influential argument against epistemological moral intuitionism, the view that some people are non-inferentially justified in believing some moral propositions. At the beginning of the twenty-first century, this view experienced a revival, which coincided with an increasing interest in empirical research on intuitions. The results of that research are seen by some as casting serious doubt on the reliability of our moral intuitions. According to Sinnott-Armstrong, empirical evidence shows that our moral (...) beliefs have a high error rate in general, which creates a need for inferential confirmation for every single moral belief. His argument involves the problematic assumption that it is reasonable for informed moral believers to ascribe a high probability of error to every particular moral belief unless the believer has some special evidence that this particular moral belief belongs to a class that has a lower probability of error than the class of moral beliefs. Focussing on the non-moral example that Sinnott-Armstrong uses in the latest reformulation of his argument, the “Californian wine example”, I argue that apart from exceptional circumstances, the description of moral agents as ascribing correctness probabilities to their moral beliefs is odd, ERROR reveals an awkward picture of how agents relate to their moral beliefs and ERROR is problematic from the perspective of moral competence. This critique goes deeper than the worries raised by other critics to earlier versions of the argument, and part of it applies to moral intuitionism as well. (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)

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)

Multisource information systems and multigranulation intuitionistic fuzzy rough sets are important extended types of Pawlak’s classical rough set model. Multigranulation intuitionistic fuzzy rough sets have been investigated in depth in recent years. However, few studies have considered this combination of multisource information systems and intuitionistic fuzzy rough sets. In this paper, we give the uncertainty measure for multisource intuitionistic fuzzy information system. Against the background of multisource intuitionistic fuzzy information system, each information source is regarded as a granularity level. Considering (...) the different importance of information sources, we assign different weights to them. Firstly, the paper proposes an optimal source selection method. Secondly, we study the weighted generalized, weighted optimistic, and weighted pessimistic multigranularity intuitionistic fuzzy rough set models and uncertainty measurement methods in the multisource intuitionistic fuzzy information system, and we further study the relationship between the three models and related properties. Finally, an example is given to verify the validity of the models and methods. (shrink)

We introduce a general recipe for presenting non-classical logics in a modular and uniform way as labelled deduction systems. Our recipe is based on a labelling mechanism where labels are general entities that are present, in one way or another, in all logics, namely truth-values. More specifically, the main idea underlying our approach is the use of algebras of truth-values, whose operators reflect the semantics we have in mind, as the labelling algebras of our labelled deduction systems. The “truth-values as (...) labels” approach allows us to give generalized systems for multiple-valued logics within the same formalism: since we can take multiple-valued logics as meaning not only finitely or infinitely many-valued logics but also power-set logics, i.e. logics for which the denotation of a formula can be seen as a set of worlds, our recipe allows us to capture also logics such as modal, intuitionistic and relevance logics, thus providing a first step towards the fibring of these logics with many-valued ones. (shrink)

Sensibility theorists such as John McDowell have argued that once we appreciate certain similarities between moral values and secondary qualities, a new meta-ethical position might emerge, one that avoids the alleged difficulties with moral intuitionism and non-cognitivism. The aim of this paper is to examine the meta-ethical prospects of this secondary-quality analogy. Of particular concern will be the extent to which McDowell’s comparison of values to secondary qualities supports a viewpoint unique from that of the moral intuitionist. Once we (...) disentangle the various metaphysical and epistemological strands of McDowell’s analogy, McDowell’s position might appear closer to moral intuitionism than initially supposed. This discussion will also help clarify the intended meaning of the secondary-quality analogy, as well as its significance for ethics more generally. (shrink)

Intuitionistic dependence logic was introduced by Abramsky and Väänänen [1] as a variant of dependence logic under a general construction of Hodges’ (trump) team semantics. It was proven that there is a translation from intuitionistic dependence logic sentences into second order logic sentences. In this paper, we prove that the other direction is also true, therefore intuitionistic dependence logic is equivalent to second order logic on the level of sentences.

We generalize intuitionistic tense logics to the multi-modal case by placing grammar logics on an intuitionistic footing. We provide axiomatizations for a class of base intuitionistic grammar logics as well as provide axiomatizations for extensions with combinations of seriality axioms and what we call "intuitionistic path axioms". We show that each axiomatization is sound and complete with completeness being shown via a typical canonical model construction.

We present $\in_I$-Logic (Epsilon-I-Logic), a non-Fregean intuitionistic logic with a truth predicate and a falsity predicate as intuitionistic negation. $\in_I$ is an extension and intuitionistic generalization of the classical logic $\in_T$ (without quantifiers) designed by Sträter as a theory of truth with propositional self-reference. The intensional semantics of $\in_T$ offers a new solution to semantic paradoxes. In the present paper we introduce an intuitionistic semantics and study some semantic notions in this broader context. Also we enrich the quantifier-free language (...) by the new connective < that expresses reference between statements and yields a finer characterization of intensional models. Our results in the intuitionistic setting lead to a clear distinction between the notion of denotation of a sentence and the here-proposed notion of extension of a sentence (both concepts are equivalent in the classical context). We generalize the Fregean Axiom to an intuitionistic version not valid in $\in_I$. A main result of the paper is the development of several model constructions. We construct intensional models and present a method for the construction of standard models which contain specific (self-)referential propositions. (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)

Heyting is considered to be the first individual to place the previously informal logic of the Intuitionist movement on a rigorous formal foundation; he is probably the most likely candidate one might select for a book about Intuitionism. The first edition appeared in 1956, and the revisions have been brief. Only the seventh of eight sections deals with the Intuitionistic formulation of sentential and predicate logics; the first chapter is in the form of a dialogue among an Intuitonist [[sic]], (...) Logicist, Formalist, and several supporting characters; this discussion is predominantly philosophical and deals with various views concerning the foundations of mathematics. The five succeeding chapters consider first arithmetic, spreads and species, algebra, plane pointspecies, measure and integration. The last chapter considers two controversial subjects—infinitely proceeding sequences and negationless mathematics. There is a bibliography containing references of recent as well as classical articles, an index and glossary of symbols. The reader with a classical mathematical intuition will have sticky going in certain sections and should take care not to conflate his own notions with those of the author; Intuitionism and its attendant mathematics are not merely formal systems or philosophies of mathematics, they are a way of thinking. (shrink)

Intuitionism has for many years been a poor relation among the various metaethical theories, commonly thought both parochial and irrational. Most recent writers who attempt a survey of ethical theory mention it briefly in an embarrassed sort of way, and then dismiss it in a paragraph or two. John Rawls, however, does not share this common attitude. In his recent book he represents his own theory as being an alternative both to intuitionism and to utilitarianism, and it is (...) apparent from his discussion that he takes both of these latter theories seriously, as having substantial attraction to reasonable men. Those sympathetic to intuitionism have no doubt been cheered by Rawls’ respectful treatment; but they are likely also to have been dismayed by the fact that, because he focuses upon a somewhat unusual understanding of it, he does not make clear how his theory stands in relation to intuitionism in its most general form, nor whether there are varieties of intuitionism consistent with the whole of his theory or at least with important parts of it. I shall try to answer these questions here. (shrink)

We introduce an extended intuitionistic linear logic with strong negation and modality. The logic presented is a modal extension of Wansing's extended linear logic with strong negation. First, we propose three types of cut-free sequent calculi for this new logic. The first one is named a subformula calculus, which yields the subformula property. The second one is termed a dual calculus, which has positive and negative sequents. The third one is called a triple-context calculus, which is regarded as a natural (...) extension or generalization of Hodas and Miller's dual-context calculus appearing in a linear logic programming language. Second, we present a concurrent process calculus based on the logic. This calculus is an extension of Okada's process calculus. Third, we introduce a Kripke type semantics for a fragment of the logic, and show the completeness theorems with respect to the semantics. Finally, we mention a logic programming language based on the triple-context calculus. (shrink)

We present a general notion of realizability encompassing both standard Kleene style realizability over partial combinatory algebras and Kleene style realizability over more general structures, including all partial cartesian closed categories. We shown how the general notion of realizability can be used to get models of dependent predicate logic, thus obtaining as a corollary (the known result) that the category Equ of equilogical spaces models dependent predicate logic. Moreover, we characterize when the general notion of realizability gives rise to a (...) topos, i.e., a model of impredicative intuitionistic higher-order logic. (shrink)

According to moral intuitionism, moral properties are objective, but our cognitions of them are not always based on premises. In this paper, I develop a novel version of moral intuitionism and argue that this new intuitionism is worthy of closer attention. The intuitionistic theory I propose, while inspired by the early twentieth-century intuitionism of W. D. Ross, avoids the alleged errors of his view. Furthermore, unlike Robert Audi's contemporary formulation of intuitionism, my theory has the (...) resources to account for the noninferential character of particular, as opposed to merely general, moral beliefs. I achieve this result by avoiding the appeal to self-evidence to explain the possibility of noninferential moral knowledge. (shrink)

In this paper, we introduce the plithogenic set (as generalization of crisp, fuzzy, intuitionistic fuzzy, and neutrosophic sets), which is a set whose elements are characterized by many attributes (parameters)’ values. An attribute value v has a corresponding (fuzzy, intuitionistic fuzzy, or neutrosophic) degree of appurtenance d(x,v) of the element x, to the set P, with respect to some given criteria. In order to obtain a better accuracy for the plithogenic aggregation operators in the plithogenic set, and for a (...) more exact inclusion (partial order), a (fuzzy, intuitionistic fuzzy, or neutrosophic) contradiction (dissimilarity) degree is defined be-tween each attribute value and the dominant (most important) attribute value. The plithogenic intersection and union are linear combinations of the fuzzy operators tnorm and tconorm, while the plithogenic complement, inclusion (inequality), equality are influenced by the attribute values contradiction (dissimilarity) degrees. This article offers some examples and applications of these new concepts in our everyday life. (shrink)

The interplay of introduction and elimination rules for propositional connectives is often seen as suggesting a distinguished role for intuitionistic logic. We prove three formal results concerning intuitionistic propositional logic that bear on that perspective, and discuss their significance. First, for a range of connectives including both negation and the falsum, there are no classically or intuitionistically correct introduction rules. Second, irrespective of the choice of negation or the falsum as a primitive connective, classical and intuitionistic consequence satisfy exactly the (...) same structural, introduction, and elimination (briefly, elementary) rules. Third, for falsum as primitive only, intuitionistic consequence is the least consequence relation that satisfies all classically correct elementary rules. (shrink)

We call a logic regular for a semantics when the satisfaction predicate for at least one of its nontheorems is closed under double negation. Such intuitionistic theories as second-order Heyting arithmetic HAS and the intuitionistic set theory IZF prove completeness for no regular logics, no matter how simple or complicated. Any extensions of those theories proving completeness for regular logics are classical, i.e., they derive the tertium non datur. When an intuitionistic metatheory features anticlassical principles or recognizes that a logic (...) regular for a semantics is nonclassical, it proves explicitly that the logic is incomplete with respect to that semantics. Logics regular relative to Tarski. Beth and Kripke semantics form a large collection that includes propositional and predicate intuitionistic, intermediate and classical logics. These results are corollaries of a single theorem. A variant of its proof yields a generalization of the Gödel-Kreisel Theorem linking weak completeness for intuitionistic predicate logic to Markov's Principle. (shrink)

A remarkable philosophical affinity may be observed between the intuitionistic conception of mathematics and the transformational generative approach to the study of language: both disciplines profess a mentalistic ontology, both posit an idealized subject, and both insist on their autonomy with respect to other disciplines. This philosophical parallel is formalized in terms of a generalization of the intuitionistic notion of creative subject; resulting are the foundations of a unified theory of mental acts based on intuitionistic logic — capturing, inter (...) alia, similarities between proof acts and speech acts. As an application of the theory, it is then shown how the notion of mental act may provide for an insightful formalization of various hypotheses pertaining to the linguistic dependence or relativity of mathematics. (shrink)

This paper introduces a generalized version of inquisitive semantics, denoted as GIS, and concentrates especially on the role of disjunction in this general framework. Two alternative semantic conditions for disjunction are compared: the first one corresponds to the so-called tensor operator of dependence logic, and the second one is the standard condition for inquisitive disjunction. It is shown that GIS is intimately related to intuitionistic logic and its Kripke semantics. Using this framework, it is shown that the main results concerning (...) inquisitive semantics, especially the axiomatization of inquisitive logic, can be viewed as particular cases of more general phenomena. In this connection, a class of non-standard superintuitionistic logics is introduced and studied. These logics share many interesting features with inquisitive logic, which is the strongest logic of this class. (shrink)

We take the well-known intuitionistic modal logic of Fischer Servi with semantics in bi-relational Kripke frames, and give the natural extension to topological Kripke frames. Fischer Servi’s two interaction conditions relating the intuitionistic pre-order with the modal accessibility relation generalize to the requirement that the relation and its inverse be lower semi-continuous with respect to the topology. We then investigate the notion of topological bisimulation relations between topological Kripke frames, as introduced by Aiello and van Benthem, and show that their (...) topology-preserving conditions are equivalent to the properties that the inverse relation and the relation are lower semi-continuous with respect to the topologies on the two models. The first main result is that this notion of topological bisimulation yields semantic preservation w.r.t. topological Kripke models for both intuitionistic tense logics, and for their classical companion multi-modal logics in the setting of the Gödel translation. After giving canonical topological Kripke models for the Hilbert-style axiomatizations of the Fischer Servi logic and its classical companion logic, we use the canonical model in a second main result to characterize a Hennessy–Milner class of topological models between any pair of which there is a maximal topological bisimulation that preserve the intuitionistic semantics. (shrink)

In Radical Skepticism & Epistemic Intuition Michael Bergmann attempts to overcome the threat of radical skepticism as it arises in several different forms. The key to Bergmann’s response to skepticism is his method of intuitionist particularism wherein we give our intuitions about particular beliefs being justified more weight than we do intuitions about the premises of arguments for skepticism. There are two general problems for Bergmann’s response to skepticism. First, he fails to accurately portray the key principle of the skeptical (...) argument. As a result, much of the apparent motivation for looking to intuitionist particularism as opposed to other responses to skepticism is merely apparent. Second, intuitionist particularism faces significant problems when it comes to the Problem of the Criterion and the resolution of conflicting intuitions. However, a related intuitionist method, one incorporating explanationism, may be able to deliver what intuitionist particularism promises while avoiding its problems. (shrink)

In this thesis I investigate the implications, for one's account of mathematics, of holding an anti-realist view. The primary aim is to appraise the scope of revision imposed by anti-realism on classical inferential practice in mathematics. That appraisal has consequences both for our understanding of the nature of mathematics and for our attitude towards anti-realism itself. If an anti-realist position seems inevitably to be absurdly revisionary then we have grounds for suspecting the coherence of arguments canvassed in favour of anti-realism. (...) I attempt to defend the anti-realist position by arguing, i) that it is not internally incoherent for anti-realism to be a potentially revisionary position, and ii) that an anti-realist position can, plausibly, be seen to result in a stable intuitionistic position with regard to the logic it condones. The use of impredicative methods in classical mathematics is a site of traditional intuitionistic attacks. I undertake an examination of what the anti-realist attitude towards such methods should be. This question is of interest both because such methods are deeply implicated in classical mathematical theory of analysis and because intuitionistic semantic theories make use of impredicative methods. I attempt to construct the outlines of a set theory which is anti-realistically acceptable but which, although having no antecedent repugnance for impredicative methods as such, appears to be too weak to offer an anti-realistic vindication of impredicative methods in general. I attempt to exonerate intuitionistic semantic theories in their use of impredicative methods by showing that a partial order relying on the nature of our grasp of the intuitionistic meaning stipulations for the logical constants precludes a possible circularity. (shrink)

This paper proposes a generalization of the Kripke semantics of intuitionistic logic IL appropriate for intuitionistic Łukasiewicz logic IŁL — a logic in the intersection between IL and (classical) Łukasiewicz logic. This generalised Kripke semantics is based on the poset sum construction, used in Bova and Montagna (Theoret Comput Sci 410(12):1143–1158, 2009) to show the decidability (and PSPACE completeness) of the quasiequational theory of commutative, integral and bounded GBL algebras. The main idea is that w \Vdash \sigma—which for IL (...) is a relation between worlds w and formulas \sigma, and can be seen as a function taking values in the booleans w \Vdash \sigma \in {B}—becomes a function taking values in the unit interval w \Vdash \sigma \in [0,1]. An appropriate monotonicity restriction (which we call sloping functions) needs to be put on such functions in order to ensure soundness and completeness of the semantics. (shrink)

Issues about information spring up wherever one scratches the surface of logic. Here is a case that raises delicate issues of 'factual' versus 'procedural' information, or 'statics' versus 'dynamics'. What does intuitionistic logic, perhaps the earliest source of informational and procedural thinking in contemporary logic, really tell us about information? How does its view relate to its 'cousin' epistemic logic? We discuss connections between intuitionistic models and recent protocol models for dynamic-epistemic logic, as well as more general issues that emerge.

This paper investigates two intuitionistic mereological systems based on Tarski’s axiomatisation of general mereology. These systems use two intuitionistically non-equivalent formalisations of the notion of fusion. I study extensionality and supplementation properties as well as some variants of these systems, and defend parthood as a suitable primitive notion for intuitionistic mereology if working with Tarski’s axiomatisation. Furthermore, I arrive at an equi-interpretability result for one of the atomistic variants with intuitionistic plural logic. I discuss to what extent these results support (...) the philosophical pertinence of the mereological systems under investigation as intuitionistic theories of parthood, thereby reacting to a conceptual challenge that we are confronted with when engaging in intuitionistic mereology. (shrink)

We construct an extension P of the standard language of classical propositional logic by adjoining to the alphabet of a new category of logical-pragmatic signs. The well formed formulas of are calledradical formulas (rfs) of P;rfs preceded by theassertion sign constituteelementary assertive formulas of P, which can be connected together by means of thepragmatic connectives N, K, A, C, E, so as to obtain the set of all theassertive formulas (afs). Everyrf of P is endowed with atruth value defined classically, (...) and everyaf is endowed with ajustification value, defined in terms of the intuitive notion of proof and depending on the truth values of its radical subformulas. In this framework, we define the notion ofpragmatic validity in P and yield a list of criteria of pragmatic validity which hold under the assumption that only classical metalinguistic procedures of proof be accepted. We translate the classical propositional calculus (CPC) and the intuitionistic propositional calculus (IPC) into the assertive part of P and show that this translation allows us to interpret Intuitionistic Logic as an axiomatic theory of the constructive proof concept rather than an alternative to Classical Logic. Finally, we show that our framework provides a suitable background for discussing classical problems in the philosophy of logic. (shrink)

Dutch Mathematician Luitzen Egbertus Jan Brouwer (1881-1966) was a rebel. His doctoral thesis... was the manifesto of an angry young man taking on the mathematical establishment on all fronts. In a short time he established a world-wide reputation for himself; his genius and originality were acknowledged by the great mathematicians of his time... The Intuitionist-Formalist debate became a personal feud between the mathematical giants Brouwer and Hilbert, and ended in 1928 with the expulsion of Brouwer from the editorial board of (...) the Mathematische Annalen by dictat of Hilbert. Forsaken, humiliated and disillusioned Brouwer abandoned his Intuitionist Programme and withdrew into silence just about the time when the Formalist Programme appeared to be fundamentally flawed and major opposition collapsed... This book attempts to follow the `genetic' development of Brouwer's ideas, linking the man Brouwer, his Weltanschauung, his philosophy of mathematics and his reconstruction of mathematics. Brouwer's own writings, his publications as well as his unpublished papers, are its immediate and main source of reference. It is the second volume in the new series Studies in the History and Philosophy of Mathematics, and is written for the specialist as well as for the general reader interested in mathematics and the interpretation of its status and function. (shrink)

Brouwer's demonstration of his Bar Theorem gives rise to provocative questions regarding the proper explanation of the logical connectives within intuitionistic and constructivist frameworks, respectively, and, more generally, regarding the role of logic within intuitionism. It is the purpose of the present note to discuss a number of these issues, both from an historical, as well as a systematic point of view.

We present the interval-valued intuitionistic fuzzy ordered weighted cosine similarity measure in this paper, which combines the interval-valued intuitionistic fuzzy cosine similarity measure with the generalized ordered weighted averaging operator. The main advantage of the IVIFOWCS measure provides a parameterized family of similarity measures, and the decision maker can use the IVIFOWCS measure to consider a lot of possibilities and select the aggregation operator in accordance with his interests. We have studied some of its main properties and particular cases such (...) as the interval-valued intuitionistic fuzzy ordered weighted arithmetic cosine similarity measure and the interval-valued intuitionistic fuzzy maximum cosine similarity measure. The IVIFOWCS measure not only is a generalization of some similarity measure, but also it can deal with the correlation of different decision matrices for interval-valued intuitionistic fuzzy values. Furthermore, we present an application of IVIFOWCS measure to the group decision-making problem. Finally the existing similarity measures are compared with the IVIFOWCS measure by an illustrative example. (shrink)

The aim of the paper is to present two natural deduction systems for Intuitionistic Sentential Calculus with Identity ( ISCI ); a syntactically motivated \(\mathsf {ND}^1_{\mathsf {ISCI}}\) and a semantically motivated \(\mathsf {ND}^2_{\mathsf {ISCI}}\). The formulation of \(\mathsf {ND}^1_{\mathsf {ISCI}}\) is based on the axiomatic formulation of ISCI. Its rules cannot be straightforwardly classified as introduction or elimination rules; ISCI -specific rules are based on axioms characterizing the identity connective. The system does not enjoy the standard subformula property, but due (...) to the normalization procedure non-subformulas can label only leaves of proofs. In \(\mathsf {ND}^2_{\mathsf {ISCI}}\), we propose only two general identity-related rules, in reference to the treatment of the identity connective in First-Order Logic. (shrink)

In this paper we consider an intuitionistic variant of the modal logic S4 (which we call IS4). The novelty of this paper is that we place particular importance on the natural deduction formulation of IS4— our formulation has several important metatheoretic properties. In addition, we study models of IS4— not in the framework of Kirpke semantics, but in the more general framework of category theory. This allows not only a more abstract definition of a whole class of models but also (...) a means of modelling proofs as well as provability. (shrink)

Working in constructive set theory we formulate notions of constructive topological space and set-generated locale so as to get a good constructive general version of the classical Galois adjunction between topological spaces and locales. Our notion of constructive topological space allows for the space to have a class of points that need not be a set. Also our notion of locale allows the locale to have a class of elements that need not be a set. Class sized mathematical structures need (...) to be allowed for in constructive set theory because the powerset axiom and the full separation scheme are necessarily missing from constructive set theory. We also consider the notion of a formal topology, usually treated in Intuitionistic type theory, and show that the category of set-generated locales is equivalent to the category of formal topologies. We exploit ideas of Palmgren and Curi to obtain versions of their results about when the class of formal points of a set-presentable formal topology form a set. (shrink)

We propose a model of physics that blends Rovelli’s relational quantum mechanics (RQM) interpretation with the language of finite information quantities (FIQs), defined by Gisin and Del Santo in the spirit of intuitionistic mathematics. We discuss deficiencies of using real numbers to model physical systems in general, and particularly under the RQM interpretation. With this motivation for an alternative mathematical language, we propose the use of FIQs to model the world under the RQM interpretation, wherein we view the propensities that (...) make up a FIQ as quantifications of potential interaction. Under this model, the stable facts, relative facts, and shifting perspectives that make up the relational interpretation correspond to shifting digits and propensities of the FIQs. The model’s predictions agree with those of both classical and quantum physics, and it is indeterministic. We also propose explanations, with examples, for how the propensities of a FIQ are distributed, and how its digits become actualized. This is equivalent to the notion of the measurement problem, and the question of what causes wave function collapse. In short, by stepping through the “new door” opened by the language of FIQs, we attempt to describe the world under the relational interpretation. (shrink)

L.E.J. Brouwer famously took the subject’s intuition of time to be foundational and from there ventured to build up mathematics. Despite being largely critical of formal methods, Brouwer valued axiomatic systems for their use in both communication and memory. Through the Dutch Mathematical Society, Gerrit Mannoury posed a challenge in 1927 to provide an axiomatization of intuitionistic arithmetic. Arend Heyting’s 1928 axiomatization was chosen as the winner and has since enjoyed the status of being the _de facto_ formalization of intuitionistic (...) arithmetic. We argue that axiomatizations of intuitionistic arithmetic ought to make explicit the role of the subject’s activity in the intuitionistic arithmetical process. While Heyting Arithmetic is useful when we want to contrast constructed objects with platonistic ones, Heyting Arithmetic omits the contribution of the subject and thus falls short as a response to Mannoury’s challenge. We offer our own solution, Doxastic Heyting Arithmetic, or \({\textsf{DHA}}\), which we contend axiomatizes Brouwerian intuitionistic arithmetic. (shrink)

Truth-maker analyses construe truth as existence of proof, a well-known example being that offered by Wittgenstein in theTractatus. The paper subsumes the intuitionistic view of truth as existence of proof under the general truth-maker scheme. Two generic constraints on truth-maker analysis are noted and positioned with respect to the writings of Michael Dummett and theTractatus. Examination of the writings of Brouwer, Heyting and Weyl indicates the specific notions of truth-maker and existence that are at issue in the intuitionistic truth-maker analysis, (...) namely that of proof in the sense of proof-object (Brouwer, Heyting) and existence in the nonpropositional sense of a judgement abstract (Weyl). Furthermore, possible anticipations in the writings of Schlick and Pfänder are noted. (shrink)

Recently, research on uncertainty modeling is progressing rapidly and many essential and breakthrough stud ies have already been done. There are various ways such as fuzzy, intuitionistic and neutrosophic sets to handle these uncertainties. Although these concepts can handle incomplete information in various real-world issues, they cannot address all types of uncertainty such as indeterminate and inconsistent information. Also, plithogenic sets as a generalization of crisp, fuzzy, intuitionistic fuzzy, and neutrosophic sets, which is a set whose elements are characterized (...) by many attributes’ values. In this paper, our aim is to demonstrate and review the history of fuzzy, intuitionistic and neutrosophic sets. For this purpose, we divided the paper as: section 1. History of Fuzzy Sets, section 2. History of Intuitionistic Fuzzy Sets and section 3. History of Neutrosophic Theories and Applications, section 4. History of Plithogenic Sets. (shrink)