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)

We prove that the Weihrauch lattice can be transformed into a Brouwer algebra by the consecutive application of two closure operators in the appropriate order: first completion and then parallelization. The closure operator of completion is a new closure operator that we introduce. It transforms any problem into a total problem on the completion of the respective types, where we allow any value outside of the original domain of the problem. This closure operator is of interest by itself, as it (...) generates a total version of Weihrauch reducibility that is defined like the usual version of Weihrauch reducibility, but in terms of total realizers. From a logical perspective completion can be seen as a way to make problems independent of their premises. Alongside with the completion operator and total Weihrauch reducibility we need to study precomplete representations that are required to describe these concepts. In order to show that the parallelized total Weihrauch lattice forms a Brouwer algebra, we introduce a new multiplicative version of an implication. While the parallelized total Weihrauch lattice forms a Brouwer algebra with this implication, the total Weihrauch lattice fails to be a model of intuitionistic linear logic in two different ways. In order to pinpoint the algebraic reasons for this failure, we introduce the concept of a Weihrauch algebra that allows us to formulate the failure in precise and neat terms. Finally, we show that the Medvedev Brouwer algebra can be embedded into our Brouwer algebra, which also implies that the theory of our Brouwer algebra is Jankov logic. (shrink)

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

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

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

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)

Gentzen writes in the published version of his doctoral thesis Untersuchungen über das logische Schliessen that he was able to prove the normalization theorem only for intuitionistic natural deduction, but not for classical. To cover the latter, he developed classical sequent calculus and proved a corresponding theorem, the famous cut elimination result. Its proof was organized so that a cut elimination result for an intuitionistic sequent calculus came out as a special case, namely the one in which the sequents have (...) at most one formula in the right, succedent part. Thus, there was no need for a direct proof of normalization for intuitionistic natural deduction. The only traces of such a proof in the published thesis are some convertibilities, such as when an implication introduction is followed by an implication elimination [1934–35, II.5.13]. It remained to Dag Prawitz in 1965 to work out a proof of normalization. Another, less known proof was given also in 1965 by Andres Raggio.We found in February 2005 an early handwritten version of Gentzen's thesis, with exactly the above title, but with rather different contents: Most remarkably, it contains a detailed proof of normalization for what became the standard system of natural deduction. The manuscript is located in the Paul Bernays collection at the ETH-Zurichwith the signum Hs. 974: 271. Bernays must have gotten it well before the time of his being expelled from Göttingen on the basis of the racial laws in April 1933. (shrink)

At the end of the 1980s, Tennant invented a logical system that he called “intuitionistic relevant logic”. Now he calls this same system “Core logic.” In Section 1, by reference to the rules of natural deduction for $\mathbf{IR}$, I explain why $\mathbf{IR}$ is a relevant logic in a subtle way. Sections 2, 3, and 4 give three reasons to assert that $\mathbf{IR}$ cannot be a core logic.

We show how mereotopological notions can be expressed by extending intuitionistic propositional logic with propositional quantification and a strong modal operator. We first prove completeness for the logics wrt Kripke models; then we trace the correspondence between Kripke models and topological spaces that have been enhanced with an explicit notion of expressible region. We show how some qualitative spatial notions can be expressed in topological terms. We use the semantical and topological results in order to show how in some extensions (...) of the logics it is possible to express connectedness, non-emptiness and a set of jointly exhaustive, pairwise disjoint, binary relations that play a significant role in qualitative spatial reasoning. (shrink)

We offer an alternative approach to the existing methods for intuitionistic formalization of reasoning about probability. In terms of Kripke models, each possible world is equipped with a structure of the form $$\langle H, \mu \rangle $$ that needs not be a probability space. More precisely, though H needs not be a Boolean algebra, the corresponding monotone function (we call it measure) $$\mu : H \longrightarrow [0,1]_{\mathbb {Q}}$$ satisfies the following condition: if $$\alpha $$, $$\beta $$, $$\alpha \wedge \beta (...) $$, $$\alpha \vee \beta \in H$$, then $$\mu (\alpha \vee \beta ) = \mu (\alpha ) + \mu (\beta ) - \mu (\alpha \wedge \beta )$$. Since the range of $$\mu $$ is the set $$[0,1]_{\mathbb {Q}}$$ of rational numbers from the real unit interval, our logic is not compact. In order to obtain a strong complete axiomatization, we introduce an infinitary inference rule with a countable set of premises. The main technical results are the proofs of strong completeness and decidability. (shrink)

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

This book is a specialized monograph on interpolation and definability, a notion central in pure logic and with significant meaning and applicability in all areas where logic is applied, especially computer science, artificial intelligence, logic programming, philosophy of science and natural language. Suitable for researchers and graduate students in mathematics, computer science and philosophy, this is the latest in the prestigous world-renowned Oxford Logic Guides, which contains Michael Dummet's Elements of intuitionism, J. M. Dunn and G. Hardegree's Algebraic Methods (...) in Philosophical Logic, H. Rott's Change, Choice and Inference: A Study of Belief Revision and Nonmonotonic Reasoning, P. T. Johnstone's Sketches of an Elephant: A Topos Theory Compendium: Volumes 1 and 2, and David J. Pym and Eike Ritter's Reductive Logic and Proof Search: Proof theory, semantics and control. (shrink)

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)

Two dichotomies are considered as the foundations of a scientific theory: the kind of infinity—either potential or actual-, and the kind of organization of the theory—axiomatic or problem-based. The original intuitionist program relied on the choices of potential infinity and the problem-based organization. I show that the logical theory of Kolmogorov’s 1932 paper relied on the same choices. A comparison of all other theories sharing the same foundational choices allows us to characterize their common theoretical development through a few (...) logical steps. The theory illustrated by Kolmogorov’s paper is then rationally re-constructed according to the steps of this kind of development. One obtains a new foundation of intuitionist logic, which is of a structural kind since it is based on and developed according to the structure of the above mentioned two fundamental choices. In addition, Kolmogorov’s illustration of his theory of intuitionist logic is an instance of rigorous reasoning of the intuitionist kind. (shrink)

Summary. This article develops two principal points. First, the so-called rivals of logical foundations, associated with Zermelo, Hilbert, and Brouwer, are here regarded as variants; specifically: to simplify, refine, resp. extend Frege’s scheme. Each of the variations is seen as a special case of a familiar strategy in the pursuit of knowledge. In particular, the extension provided by Brouwer’s intuitionistic logic concerns the class of propositions considered: about incompletely defined objects such as choice sequences. In contrast, Frege or, for (...) that matter, Aristotle thought that only propositions about precisely defined terms lent themselves to anything like logical theory.. This view of intuitionistic logic is fairly consistent with the mature Brouwer’s own research practice, but not at all with his rhetoric. Secondly, the article explains the silent majority’s scepticism about all variants of logical foundations, and, in note 7, sketches a genuine alternative which is almost explicit in the manifesto [#3] of Bourbaki with the aim of exploring the architecture of mathematics and our intuitive resonances to it. The alternative sees the weakness of logical foundations not in any logical defects, such as lack of precision, but in its sterility: stress on the logical elements of mathematics and mathematical reasoning draws attention away from discoveries of more significant features; more significant even for validity and reliability, though these matters are less prominent here than in logical foundations. The weakness in question is relevant beyond the present topic of foundations because this kind of weakness affects the whole analytic tradition in philosophy. Finally, for reasons explained there, an Appendix goes into some relations between the present paper and [17]. (shrink)

Meta-ethical realism faces the serious epistemological problem of how to explain our epistemic access to moral reality. In the face of this challenge many are sceptical about non-naturalist realism. Nonetheless, there is good reason to acknowledge moral objectivity: morality shows all the signs of a truth-apt discourse but doesn’t exhibit the typical relativity inducing features. This suggests a middle-ground position, a theory that embraces the virtues of realism but does avoid its vices: objectivist antirealism. In this paper, I’ll discuss, mainly (...) following Crispin Wright’s account of moral truth as superassertibility, a promising version of objectivist antirealism and show how to cope with notorious problems, notably those arising from the thought that moral disagreement might be possible in which nobody is guilty of a cognitive shortcoming, which contradicts the antirealist claim that moral truth is not beyond our epistemic reach. The solution is to deny the possibility of cognitively faultless moral disagreement by arguing that cognitively blameless thinkers either agree or stay agnostic and, therefore, never disagree about any moral proposition. Since assuming an agnostic stance on the part of such thinkers contradicts the antirealist’s conception of truth—even within the limits of intuitionistic logic—I’ll propose an alternative logical revision for the moral discourse: a three-valued logic with epistemically construed truth-values. (shrink)

. The effort in providing constructive and predicative meaning to non-constructive modes of reasoning has almost without exception been applied to theories with full classical logic [4]. In this paper we show how to combine unrestricted countable choice, induction on infinite well-founded trees and restricted classical logic in constructively given models. These models are sheaf models over a $\sigma$ -complete Boolean algebra, whose topologies are generated by finite or countable covering relations. By a judicious choice of the Boolean algebra (...) we can directly extract effective content from $\Pi_2^0$ -statements true in the model. All the arguments of the present paper can be formalised in Martin-Löf's constructive type theory with generalised inductive definitions. (shrink)

Meta-ethical realism faces the serious epistemological problem of how to explain our epistemic access to moral reality. In the face of this challenge many are sceptical about non-naturalist realism. Nonetheless, there is good reason to acknowledge moral objectivity: morality shows all the signs of a truth-apt discourse but doesn’t exhibit the typical relativity inducing features. This suggests a middle-ground position, a theory that embraces the virtues of realism but does avoid its vices: objectivist antirealism. In this paper, I’ll discuss, mainly (...) following Crispin Wright’s account of moral truth as superassertibility, a promising version of objectivist antirealism and show how to cope with notorious problems, notably those arising from the thought that moral disagreement might be possible in which nobody is guilty of a cognitive shortcoming, which contradicts the antirealist claim that moral truth is not beyond our epistemic reach. The solution is to deny the possibility of cognitively faultless moral disagreement by arguing that cognitively blameless thinkers either agree or stay agnostic and, therefore, never disagree about any moral proposition. Since assuming an agnostic stance on the part of such thinkers contradicts the antirealist’s conception of truth—even within the limits of intuitionistic logic—I’ll propose an alternative logical revision for the moral discourse: a three-valued logic with epistemically construed truth-values. (shrink)

We give a sound and complete axiomatization of a probabilistic extension of intuitionistic logic. Reasoning with probability operators is also intuitionistic (in contradistinction to other works on this topic), i.e., measure functions used for modeling probability operators are partial functions. Finally, we present a decision procedure for our logic, which is a combination of linear programming and an intuitionistic tableaux method.

We investigate in intuitionistic first-order logic various principles of preference relations alternative to the standard ones based on the transitivity and completeness of weak preference. In particular, we suggest two ways in which completeness can be formulated while remaining faithful to the spirit of constructive reasoning, and we prove that the cotransitivity of the strict preference relation is a valid intuitionistic alternative to the transitivity of weak preference. Along the way, we also show that the acyclicity axiom is not (...) finitely axiomatizable in first-order logic. (shrink)

This chapter focuses on the development of Gerhard Gentzen's structural proof theory and its connections with intuitionism. The latter is important in proof theory for several reasons. First, the methods of Hilbert's old proof theory were limited to the “finitistic” ones. These methods proved to be insufficient, and they were extended by infinitistic principles that were still intuitionistically meaningful. It is a general tendency in proof theory to try to use weak principles. A second reason for the importance of (...) intuitionism for proof theory is that the proof-theoretical study of intuitionistic logic has become a prominent part of logic, with applications in computer science. (shrink)

We discuss a propositional logic which combines classical reasoning with constructive reasoning, i.e., intuitionistic logic augmented with a class of propositional variables for which we postulate the decidability property. We call it intuitionistic logic with classical atoms. We introduce two hypersequent calculi for this logic. Our main results presented here are cut-elimination with the subformula property for the calculi. As corollaries, we show decidability, an extended form of the disjunction property, the existence of embedding into an intuitionistic modal (...) logic and a partial form of interpolation. (shrink)

This book is a specialized monograph on the development of the mathematical and computational metatheory of reductive logic and proof-search, areas of logic that are becoming important in computer science. A systematic foundational text on these emerging topics, it includes proof-theoretic, semantic/model-theoretic and algorithmic aspects. The scope ranges from the conceptual background to reductive logic, through its mathematical metatheory, to its modern applications in the computational sciences. Suitable for researchers and graduate students in mathematical, computational and philosophical logic, and in (...) theoretical computer science and artificial intelligence, this is the latest in the prestigous world-renowned Oxford Logic Guides, which contains Michael Dummet's Elements of intuitionism (2nd Edition), Dov M. Gabbay, Mark A. Reynolds, and Marcelo Finger's Temporal Logic Mathematical Foundations and Computational Aspects, J. M. Dunn and G. Hardegree's Algebraic Methods in Philosophical Logic, H. Rott's Change, Choice and Inference: A Study of Belief Revision and Nonmonotonic Reasoning, and P. T. Johnstone's Sketches of an Elephant: A Topos Theory Compendium: Volumes 1 and 2. (shrink)

The purpose of this paper is to introduce a bi-intuitionistic sequent calculus and to give proofs of admissibility for its structural rules. The calculus I will present, called SC2Int, is a sequent calculus for the bi-intuitionistic logic 2Int, which Wansing presents in [2016a]. There he also gives a natural deduction system for this logic, N2Int, to which SC2Int is equivalent in terms of what is derivable. What is important is that these calculi represent a kind of bilateralist reasoning, since (...) they do not only internalize processes of verifcation or provability but also the dual processes in terms of falsifcation or what is called dual provability. In [Wansing, 2017] a normal form theorem for N2Int is stated, here, I want to prove a cut-elimination theorem for SC2Int, i.e., if successful, this would extend the results existing so far. (shrink)

Preface -- Part I: Philosophical logic and philosophy of language -- Rules versus theorems : a new approach for mediation -- Between intuitionistic and two-valued logic -- On the relation between the partition of a whole into parts and the attribution of properties to an object -- Basic objectives of dialogic logic in historical perspective -- Pragmatic and semiotic prerequisites for predication : a dialogue model -- Pragmatics and semiotics : the peircean version of ontology and epistemology -- Intentionality and (...) its language-dependency -- Meaning postulates and rules of argumentation : remarks concerning the pragmatic tie between meaning (of terms) and truth (of propositions) -- What do language games measure? -- Features of Indian logic -- Part II: Methods in philosophy, in art, and in science -- The concept of science : some remarks on the methodological issue construction versus description in the philosophy of science -- Is and ought revisited -- Competition and cooperation : are they antagonistic or complementary? -- Another version of methodological dualism -- The pre-established harmony between the two Adams -- On the way to conceptual and perceptual knowledge -- Self and other : remarks on human nature and human culture -- On the concept of symmetry -- Procedural principles of the Eerlangen School : on the interrelation between the principles of method, of dialogue, and of reason. (shrink)

The paper deals with reconstruction of the unique reductivecounterpart of the deductive logic. The procedure results in the deductivereductive form of logic. This extension is illustrated on the base of intuitionistic logics: Heyting’s, Brouwerian and Heyting-Brouwer’s ones.

Neutrosophy has been introduced some years ago by Florentin Smarandache as a new branch of philosophy dealing with “the origin, nature and scope of neutralities, as well as their interactions with different ideational spectra”. A variety of new theories have been developed on the basic principles of neutrosophy: among them is neutrosophic logics, a family of many-valued systems that can be regarded as a generalization of fuzzy logics. In this paper we present a critical introduction to neutrosophic logics, focusing on (...) the problem of defining suitable neutrosophic propositional connectives and discussing the relationship between neutrosophic logics and other well-known frameworks for reasoning with uncertainty and vagueness, such as (intuitionistic and interval-valued) fuzzy systems and Belnap’s logic. (shrink)

This book is a specialized monograph on the development of the mathematical and computational metatheory of reductive logic and proof-search, areas of logic that are becoming important in computer science. A systematic foundational text on these emerging topics, it includes proof-theoretic, semantic/model-theoretic and algorithmic aspects. The scope ranges from the conceptual background to reductive logic, through its mathematical metatheory, to its modern applications in the computational sciences. Suitable for researchers and graduate students in mathematical, computational and philosophical logic, and in (...) theoretical computer science and artificial intelligence, this is the latest in the prestigous world-renowned Oxford Logic Guides, which contains Michael Dummet's Elements of intuitionism, Dov M. Gabbay, Mark A. Reynolds, and Marcelo Finger's Temporal Logic Mathematical Foundations and Computational Aspects, J. M. Dunn and G. Hardegree's Algebraic Methods in Philosophical Logic, H. Rott's Change, Choice and Inference: A Study of Belief Revision and Nonmonotonic Reasoning, and P. T. Johnstone's Sketches of an Elephant: A Topos Theory Compendium: Volumes 1 and 2. (shrink)

It is commonly assumed that classical logic is the embodiment of a realist ontology. In “Sets and Semantics”, however, Jonathan Lear challenged this assumption in the particular case of set theory, arguing that even if one is a set-theoretic Platonist, due attention to a special feature of set theory leads to the conclusion that the correct logic for it is intuitionistic. The feature of set theory Lear appeals to is the open-endedness of the concept of set. This article advances reasons (...) internal to Lear’s account to show that his argument for intuitionistic set theory is unsuccessful. (shrink)

The use of conventional logical connectives either in logic, in mathematics, or in both cannot determine the meanings of those connectives. This is because every model of full conventional set theory can be extended conservatively to a model of intuitionistic set plus class theory, a model in which the meanings of the connectives are decidedly intuitionistic and nonconventional. The reasoning for this conclusion is acceptable to both intuitionistic and classical mathematicians. En route, I take a detour to prove (...) that, given strictly intuitionistic principles, classical negation cannot exist. (shrink)

The most puzzling and intriguing aspect of intuitionism as a philosophy of mathematics is its claim that classical deductive reasoning in mathematics is illegitimate. The two most well-known proponents of this position are L. E. J. Brouwer and Michael Dummett. Both of their criticisms of the use of classical logic in mathematics have, by and large, been taken to depend on the thesis that the principle of bivalence does not apply to mathematical statements; and the difference between these (...) criticisms is taken to consist in their grounds for this thesis. Brouwer bases this thesis on an account of the nature of mathematics: it is an activity of mental construction essentially independent of language. Dummett, in contrast, bases the thesis on his anti-realist view of the nature of meaning. ;My first principal argument in this essay shows that the conception of anti-realism just outlined is problematic as a reading of Dummett. To begin with, Dummett explicitly rejects, not only the evidentiary restrictions of this conception, but also, more seriously, the use of specific epistemological theses in a philosophical account of meaning. Moreover, I give an alternative account of anti-realism, consisting of alternative interpretations of the manifestation requirement and of the rejection of realist truth conditions, that do not rely on an epistemology of meaning, and hence, a fortiori, not on evidential restrictions. ;My second principal argument in this essay consists of drawing a consequence of this conception of anti-realism, namely, that the failure of bivalence cannot, by itself, support a cogent criticism of classical logic. The basic reason is that, given my view of anti-realism, the claim that the legitimacy of applying classical reasoning to a class of statements depends on whether bivalence applies to that class already implies that the system of deductive reasoning applying to these statements is non-classical. Hence an argument against classical logic based on this claim is question-begging. ;Finally, I conclude by showing that Dummett's criticism of classical reasoning, as reconstructed by my third argument, fails. (shrink)

We identify a pervasive contrast between implicit and explicit stances in logical analysis and system design. Implicit systems change received meanings of logical constants and sometimes also the notion of consequence, while explicit systems conservatively extend classical systems with new vocabulary. We illustrate the contrast for intuitionistic and epistemic logic, then take it further to information dynamics, default reasoning, and other areas, to show its wide scope. This gives a working understanding of the contrast, though we stop (...) short of a formal definition, and acknowledge limitations and borderline cases. Throughout we show how awareness of the two stances suggests new logical systems and new issues about translations between implicit and explicit systems, linking up with foundational concerns about identity of logical systems. But we also show how a practical facility with these complementary working styles has philosophical consequences, as it throws doubt on strong philosophical claims made by just taking one design stance and ignoring alternative ones. We will illustrate the latter benefit for the case of logical pluralism and hyper-intensional semantics. (shrink)

Tradition is classical. Surely, nothing could be more pleonastic than that? The logical tradition, certainly, was squarely classical from Bolzano to Carnap, with, say, Frege, Moore, Russell and the Wittgenstein of the Tractatus as intermediaries. Propositions are construed as being in themselves true-or-false. Indeed, in this tradition, a declarative sentence S expresses a proposition by being true-or-false. So the meaningfulness of a sentence consists in its being true-or-false. But S is true-orfalse, or so they say, only when S is (...) true, or when S is false. On the classical account the presumption of bivalence is built into the very notion of meaningfulness: there is no difference between asserting that A is a proposition and asserting that A is true-or-false. The matter came to the fore in the foundations of set theory. In his first attempt at giving an application criterion for sets Cantor noted:Eine Mannigfaltigkeit von Elementen, die irgendwelcher Begriffssphäre angehören, nenne ich wohldefiniert, wenn auf Grund ihrer Definition und infolge des logischen Prinzips vom ausgeschlossenen Dritten es als intern bestimmt angesehen werden muss, sowohl ob irgendein derselben Begriffssphäre angehöriges Objekt zu der gedachten Mannigfaltigkeit gehört oder nicht, wie auch, ob zwei zur Mannigfaltigkeit gehörige Objekte trotz formaler Unterschiede in der Art des Gegebenseins einander gleich sind oder nicht. (shrink)

, 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. I am (...) grateful to Nathan Carter, Gary Ebbs, Janet Folina, Luise Prior McCarty, Stewart Shapiro, Neil Tennant, Christopher Tillman, Beth Tropman, Wen-fang Wang, and two anonymous referees for their comments and suggestions. CiteULike Connotea Del.icio.us What's this? (shrink)

We identify a pervasive contrast between implicit and explicit stances in logical analysis and system design. Implicit systems change received meanings of logical constants and sometimes also the notion of consequence, while explicit systems conservatively extend classical systems with new vocabulary. We illustrate the contrast for intuitionistic and epistemic logic, then take it further to information dynamics, default reasoning, and other areas, to show its wide scope. This gives a working understanding of the contrast, though we stop (...) short of a formal definition, and acknowledge limitations and borderline cases. Throughout we show how awareness of the two stances suggests new logical systems and new issues about translations between implicit and explicit systems, linking up with foundational concerns about identity of logical systems. But we also show how a practical facility with these complementary working styles has philosophical consequences, as it throws doubt on strong philosophical claims made by just taking one design stance and ignoring alternative ones. We will illustrate the latter benefit for the case of logical pluralism and hyper-intensional semantics. (shrink)

This paper is a proposal of continuation of the work of C. Rauszer. The logic of falsehood created by her may constitute the starting point for construction of logic formalising reductive reasonings. The extension of Heyting-Brouwer logic to its deductive-reductive form sheds new light upon those classical tautologies which are rejected in intuitionism. It turns out that among HBtautologies there can be found all the classical ones. Some of them are characteristic for deductive reasoning and they are accepted (...) by intuitionism. Others formulate the laws of reductive reasoning. Many of them, including the law of excluded middle has been rejected in Heyting’s intuitionism. Intuitionism only permits for those reductive tautologies, which at the same time bear deductive character. Thus, the complete HB intuitionism does not reject any of the classical tautologies. Every classical tautology appears in HB logic in its deductive or reductive part. Some are even present in both parts. At the end it is shown that the law of excluded middle α∨¬α is a law of the reductive part of the traditional, Heyting’s intuitionistic propositional calculus. (shrink)

Let $\mathcal{P}_w$ be the lattice of Muchnik degrees of nonempty $\Pi^0_1$ subsets of $2^\omega$. The lattice $\mathcal{P}$ has been studied extensively in previous publications. In this note we prove that the lattice $\mathcal{P}$ is not Brouwerian.

This book investigates the central role of reason in Islamic intellectual life. Despite widespread characterization of Islam as a system of belief based only on revelation, John Walbridge argues that rational methods, not fundamentalism, have characterized Islamic law, philosophy and education since the medieval period. His research demonstrates that this medieval Islamic rational tradition was opposed by both modernists and fundamentalists, resulting in a general collapse of traditional Islamic intellectual life and its replacement by more modern but far shallower forms (...) of thought. However, the resources of this Islamic scholarly tradition remain an integral part of the Islamic intellectual tradition and will prove vital to its revival. The future of Islam, Walbridge argues, will be marked by a return to rationalism"--Provided by publisher. (shrink)

In this work the social intuitionism defended by Jonathan Haidt is explored and compared with the dual reasoning theory (TDR), this theory belongs to a family of proposals that maintain that there is a duality in the field of the mental. On the one hand, social intuitionism has argued that it receives support from TDR, on the other hand, TDR has pointed out similarities with social intuitionism; despite the mutual references mentioned, an analysis of what the (...) precise relationship between these projects could be has not been carried out. This work is an attempt to carry out such analysis, in which it is argued that a) it is not clear that the division of social intuitionism between reasoning, intuition, and emotion is equivalent to that of the defenders of TDR; b) social intuitionism has certain differences both with the default-interventionist version and with the parallel version of the TDR; and, finally, c) it is not evident that social intuitionism is an extension of a TDR to the field of moral judgment. It is necessary to carry out more studies that delve into the different elements of social intuitionism and TDR, in order to understand how human beings make judgments and decisions, either within the scope of morality or outside it. (shrink)

In this work the social intuitionism defended by Jonathan Haidt is explored and compared with the dual reasoning theory (TDR), this theory belongs to a family of proposals that maintain that there is a duality in the field of the mental. On the one hand, social intuitionism has argued that it receives support from TDR, on the other hand, TDR has pointed out similarities with social intuitionism; despite the mutual references mentioned, an analysis of what the (...) precise relationship between these projects could be has not been carried out. This work is an attempt to carry out such analysis, in which it is argued that a) it is not clear that the division of social intuitionism between reasoning, intuition, and emotion is equivalent to that of the defenders of TDR; b) social intuitionism has certain differences both with the default-interventionist version and with the parallel version of the TDR; and, finally, c) it is not evident that social intuitionism is an extension of a TDR to the field of moral judgment. It is necessary to carry out more studies that delve into the different elements of social intuitionism and TDR, in order to understand how human beings make judgments and decisions, either within the scope of morality or outside it. (shrink)

This chapter contains sections titled: Logical and Mathematical Background Intuitive versus Abstract Knowledge Logical Intuition and Abstract Reasoning Intuitive and Abstract Knowledge of Mathematics Principium Individuationis, Physics and Idealist Metaphysics of Space and Time Schopenhauer's Philosophical Geometry Intuitive Reduction of Arithmetic to Counting in Time Notes References Further Reading.

Gentzen's three consistency proofs for elementary number theory have a common aim that originates from Hilbert's Program, namely, the aim to justify the application of classical reasoning to quantified propositions in elementary number theory. In addition to this common aim, Gentzen gave a “finitist” interpretation to every number-theoretic proposition with his 1935 and 1936 consistency proofs. In the present paper, we investigate the relationship of this interpretation with intuitionism in terms of the debate between the Hilbert School and (...) the Brouwer School over the significance of consistency proofs. First, we argue that the interpretation had the role of responding to a Brouwer-style objection against the significance of consistency proofs. Second, we propose a way of understanding Gentzen's response to this objection from an intuitionist perspective. (shrink)

The paper tries to show that when the deepest or foundational aspects of truth are at issue, both consequentially logical argument and rhetoric that aims to establish truth or justified conviction must engage with the being, or the irreplaceable particularity, of its audience’s members and also that of the arguer, what we refer to in ordinary language as who the person is. Beyond the existing discussion of existential rhetoric, the paper argues that this engagement with being is necessary to (...) establish not only truth that directly concerns or turns on the arguer’s and audience’s being, but also truth or justification about fundamental aspects of things and issues in general. Further, the address of being requires us to suspend both our own and our addressees’ familiar conceptual frameworks in order to allow being to emerge in its own terms. As a result, in contrast with our usual understanding of argumentation, the rhetorician’s initial aim and procedure will be to achieve a genuine suspension of conviction and even of the appropriate concepts under which to proceed, and so to produce a fundamental confusion. The paper then outlines some consequences for rhetoric and reasoning and also the structure of the process of working with this fundamental confusion. (shrink)

This paper introduces the special issue on the Concept of God of the Journal of Applied Logics (College Publications). The issue contains the following articles: Logic and the Concept of God, by Stanisław Krajewski and Ricardo Silvestre; Mathematical Models in Theology. A Buber-inspired Model of God and its Application to “Shema Israel”, by Stanisław Krajewski; Gödel’s God-like Essence, by Talia Leven; A Logical Solution to the Paradox of the Stone, by Héctor Hernández Ortiz and Victor Cantero; No New Solutions (...) to the Logical Problem of the Trinity, by Beau Branson; What Means ‘Tri-’ in ‘Trinity’ ? An Eastern Patristic Approach to the ‘Quasi-Ordinals’, by Basil Lourié; The Éminence Grise of Christology: Porphyry’s Logical Teaching as a Cornerstone of Argumentation in Christological Debates of the Fifth and Sixth Centruies, by Anna Zhyrkova; The Problem of Universals in Late Patristic Theology, by Dirk Krasmüller; Intuitionist Reasoning in the Tri-unitrian Theology of Nicolas of Cues, by Antonino Drago. (shrink)

This volume is a collection of papers on philosophy of mathematics which deal with a series of questions quite different from those which occupied the minds of the proponents of the three classic schools: logicism, formalism, and intuitionism. The questions of the volume are not to do with justification in the traditional sense, but with a variety of other topics. Some are concerned with discovery and the growth of mathematics. How does the semantics of mathematics change as the subject (...) develops? What heuristics are involved in mathematical discovery, and do such heuristics constitute a logic of mathematical discovery? What new problems have been introduced by the development of mathematics since the 1930s? Other questions are concerned with the applications of mathematics both to physics and to the new field of computer science. Then there is the new question of whether the axiomatic method is really so essential to mathematics as is often supposed, and the question, which goes back to Wittgenstein, of the sense in which mathematical proofs are compelling. Taking these questions together they give part of an emerging agenda which is likely to carry philosophy of mathematics forward into the twenty first century. (shrink)

The paper is a continuation of A deductive-reductive form oflogic: general theory and intuitionistic case and considers the problem of definability of modal operators on the intuitionistic base. Contrary tothe classical case, it seems that the fact whether the connective is Heyting’sor Brouwerian is essential for the intuitionistic logic. The connective of possibility has the classical interpretation, i.e. w |= ✸α iff ∃t, if it is defined on the base of the logicwith Brouwerian connective of coimplication.