Intuitionism and the Sorites Paradox

2021 ◽  
pp. 393-422
Author(s):  
Crispin Wright
Keyword(s):  
Subject Matter ◽  
Classical Logic ◽  
Proof Theory ◽  
Philosophy Of Mathematics ◽  
Sorites Paradox ◽  
Major Premise ◽  
Stable Boundary

This chapter revisits and further develops all the principle themes and concepts of the preceding chapters. Epistemicism about vagueness postulates a realm of distinctions drawn by basic vague concepts that transcend our capacity to know them. Its treatment of their subject matter is thus broadly comparable to the Platonist philosophy of mathematics. An intuitionist philosophy of vagueness, as do many philosophies of the semantics and metaphysics of vague expressions, finds this idea merely superstitious and rejects it. The vagueness-intuitionist, however, credits the epistemicist with a crucial insight: that vagueness is indeed a cognitive, rather than a semantic, phenomenon—something that is not a consequence of some kind of indeterminacy, or open-endedness in the semantics of vague expressions but rather resides in our brute inability to bring, for example, yellow and orange right up against one another, so to speak, so as to mark a sharp and stable boundary. A solution to the Sorites paradox is developed that is consonant with this basic idea but, by motivating a background logic that observes (broadly) intuitionistic restrictions on the proof theory for negation, allows us to treat the paradoxical reasoning as a simple reductio of its major premise, without the unwelcome implication, sustained by classical logic, of sharp cut-offs.

On Being in a Quandary

2021 ◽  
pp. 209-260
Author(s):  
Crispin Wright
Keyword(s):  
Intuitionistic Logic ◽  
Subject Matter ◽  
Classical Logic ◽  
The Other ◽  
Sorites Paradox ◽  
Metaphysical Realism ◽  
Specific Kind ◽  
Epistemic Conception ◽  
Broad Form

This chapter addresses three problems: the problem of formulating a coherent relativism, the Sorites paradox, and a seldom noticed difficulty in the best intuitionistic case for the revision of classical logic. A response to the latter is proposed which, generalized, contributes towards the solution of the other two. The key to this response is a generalized conception of indeterminacy as a specific kind of intellectual bafflement—Quandary. Intuitionistic revisions of classical logic are merited wherever a subject matter is conceived both as liable to generate Quandary and as subject to a broad form of evidential constraint. So motivated, the distinctions enshrined in intuitionistic logic provide both for a satisfying resolution of the Sorites paradox and a coherent outlet for relativistic views about, for example, matters of taste and morals. An important corollary of the discussion is that an epistemic conception of vagueness can be prised apart from the strong metaphysical realism with which its principal supporters have associated it, and acknowledged to harbour an independent insight.

Type-theoretical checking and philosophy of Mathematics

1998 ◽  
Author(s):  
Nicolaas Govert de Bruijn
Keyword(s):  
Set Theory ◽  
Subject Matter ◽  
Mutual Influence ◽  
Mathematical Reasoning ◽  
Philosophy Of Mathematics ◽  
General Information ◽  
Point Of View ◽  
De Bruijn ◽  
Level Of Understanding ◽  
The Way

After millennia of mathematics we have reached a level of understanding that can be represented physically. Humankind has managed to disentangle the intricate mixture of language, metalanguage and interpretation, isolating a body of formal, abstract mathematics that can be completely verified by machines. Systems for computer-aided verification have philosophical aspects. The design and usage of such systems are influenced by the way we think about mathematics, but it also works the other way. A number of aspects of this mutual influence will be discussed in this paper. In particular, attention will be given to philosophical aspects of type-theoretical systems. These definitely call for new attitudes: throughout the twentieth century most mathematicians had been trained to think in terms of untyped sets. The word “philosophy” will be used lightheartedly. It does not refer to serious professional philosophy, but just to meditation about the way one does one’s job. What used to be called philosophy of mathematics in the past was for a large part subject oriented. Most people characterized mathematics by its subject matter, classifying it as the science of space and number. From the verification system’s point of view, however, subject matter is irrelevant. Verification is involved with the rules of mathematical reasoning, not with the subject. The picture may be a bit confused, however, by the fact that so many people consider set theory, in particular untyped set theory, as part of the language and foundation of mathematics, rather than as a particular subject treated by mathematics. The views expressed in this paper are quite personal, and can mainly be carried back to the author’s design of the Automath system in the late 1960s, where the way to look upon the meaning (philosophy) of mathematics is inspired by the usage of the unification system and vice versa. See de Bruijn 1994b for various philosophical items concerning Automath, and Nederpelt et al. 1994, de Bruin 1980, de Bruijn 1991a for general information about the Automath project. Some of the points of view given in this paper are matters of taste, but most of them were imposed by the task of letting a machine follow what we say, a machine without any knowledge of our mathematical culture and without any knowledge of physical laws.

Logic and Philosophical Methodology

2016 ◽  
Author(s):  
John P. Burgess
Keyword(s):  
Set Theory ◽  
Model Theory ◽  
Classical Logic ◽  
Proof Theory ◽  
Recursion Theory ◽  
Theory Model ◽  
Philosophical Methodology ◽  
Paraconsistent Logics ◽  
Philosophical Logic

This article explores the role of logic in philosophical methodology, as well as its application in philosophy. The discussion gives a roughly equal coverage to the seven branches of logic: elementary logic, set theory, model theory, recursion theory, proof theory, extraclassical logics, and anticlassical logics. Mathematical logic comprises set theory, model theory, recursion theory, and proof theory. Philosophical logic in the relevant sense is divided into the study of extensions of classical logic, such as modal or temporal or deontic or conditional logics, and the study of alternatives to classical logic, such as intuitionistic or quantum or partial or paraconsistent logics. The nonclassical consists of the extraclassical and the anticlassical, although the distinction is not clearcut.

Foreword

2020 ◽  
pp. 277-278
Author(s):  
Crispin Wright
Keyword(s):  
Intuitionistic Logic ◽  
Philosophy Of Mathematics ◽  
Sorites Paradox ◽  
Philosophy Of Logic ◽  
Metaphysical Possibility ◽  
Crispin Wright

This chapter is divided into four parts, corresponding to the partitioning of the essays in the volume. Part I, on neo-Fregeanism in the philosophy of mathematics develops replies to Demopolous, Heck, Rosen and Yablo, Boolos and Edwards; Part II, on vagueness, intuitionistic logic and the Sorites Paradox develops replies to Rumfitt and Schiffer; Part III, on revisionism in the philosophy of logic develops replies to Shieh and Tennant; and Part IV, on the epistemology of metaphysical possibility develops a reply to Hale. In each section, Crispin Wright offers an overview of the relevant area and outlines and refines his views on the relevant topics. Inter alia, he offers detailed replies to each of the ten contributed essays in the volume.

Free ordered algebraic structures towards proof theory

2001 ◽  
Vol 66 (2) ◽  
pp. 597-608
Author(s):  
Andreja Prijatelj
Keyword(s):  
Classical Logic ◽  
Proof Theory ◽  
Algebraic Structures ◽  
Ordered Algebras

AbstractIn this paper, constructions of free ordered algebras on one generator are given that correspond to some one-variable fragments of affine propositional classical logic and their extensions with n-contraction (n ≥ 2). Moreover, embeddings of the already known infinite free structures into the algebras introduced below are furnished with; thus, solving along the respective cardinality problems.

scholarly journals TINJAUAN YURIDIS TERHADAP DOKUMEN PERJANJIAN PEMBIAYAAN KONSUMEN PT BAF SURAKARTA

2020 ◽  
Vol 8 (2) ◽  
pp. 303
Author(s):  
Rima Agustina ◽  
Ambar Budhisulistyawati
Keyword(s):  
National Identity ◽  
Subject Matter ◽  
Civil Code ◽  
Major Premise ◽  
Minor Premise ◽  
Material Analysis ◽  
Analysis Technique ◽  
Identity Card ◽  
A Minor ◽  
Binding Agreement

<p>Abstract<br />This article aims to determine the suitability of the consumer financing agreement made by PT Bussan  Auto Finance (BAF) with the regulations stipulated in the Civil Code. As for analyzing these problems is done by using normative methods with approaches used through the statute approach. The legal material analysis technique used by using the syllogism method is by using the deduction method which is the opposite of submitting a major premise and then submits a minor premise and from then draws a conclusion. The results of this study indicate that the consumer financing agreement of PT Bussan  Auto Finance (BAF) is in accordance with the terms of the agreement stipulated in the Civil Code. The  conditions are as follows: (1) Their agreement is binding, agreement can be seen through the signatures of the parties in the agreement; (2) The ability to make an engagement, the parties must include a National Identity Card (KTP) to prove their skills; (3) A certain subject matter, namely regarding the financing of a motorized vehicle; (4) A reason that is not prohibited, the financing made is a reason that is lawful and does not conflict with the law. Then the agreement is valid and binding and applies as a law for the parties who make it.<br />Keywords: Agreement; Consumer Financing Agreement; Financing Company.</p><p>Abstrak<br />Artikel ini bertujuan untuk mengetahui kesesuaian antara perjanjian pembiayaan kosumen yang dibuat  oleh PT Bussan Auto Finance (BAF) dengan peraturan yang diatur dalam KUH Perdata. Adapun untuk menganalisis permasalahan tersebut dilakukan dengan menggunakan metode normatif dengan pendekatan yang digunakan melalui pendekatan undang-undang (statute approach). Teknik analisis bahan hukum yang digunakan dengan menggunakan metode silogisme yaitu dengan penggunaan metode deduksi yang bepangkal dari pengajuan premis mayor kemudian diajukan premis minor dan dari kemudian ditarik suatu kesimpulan. Hasil penelitian ini menunjukkan bahwa perjanjian pembiayaan konsumen PT Bussan Auto Finance (BAF) telah sesuai dengan syarat-syarat perjanjian yang diatur dalam KUH Perdata. Adapun syarat-syarat tersebut adalah sebagai berikut: (1) Kesepakatan mereka yang mengikatkan diri, kesepakatan dapat dilihat melalui tanda tangan para pihak dalam perjanjian; (2) Kecakapan untuk membuat perikatan, para pihak wajib mencantumkan Kartu Tanda Penduduk (KTP) untuk membuktikan kecakapannya; (3) Suatu pokok persoalan tertentu, yakni mengenai pembiayaan sebuah kendaraan bermotor; (4) Suatu sebab yang tidak terlarang, pembiayaan yang dilakukan tersebut merupakan suatu sebab yang halal dan tidak bertentangan dengan undang-undang. Maka perjanjian tersebut sah dan mengikat serta berlaku sebagai undang-undang bagi para pihak yang membuatnya. <br />Kata Kunci: Perjanjian; Perjanjian Pembiayaan Konsumen; Perusahaan Pembiayaan</p>

The finite embeddability property for noncommutative knotted extensions of RL

2015 ◽  
Vol 25 (03) ◽  
pp. 349-379 ◽  
Author(s):  
R. Cardona ◽  
N. Galatos
Keyword(s):  
Classical Logic ◽  
Proof Theory ◽  
Mathematical Linguistics ◽  
Residuated Lattices ◽  
Automata Theory ◽  
Finite Model ◽  
Algebraic Proof ◽  
Model Property ◽  
Finite Model Property ◽  
Finite Embeddability Property

The finite embeddability property (FEP) for knotted extensions of residuated lattices holds under the assumption of commutativity, but fails in the general case. We identify weaker forms of the commutativity identity which ensure that the FEP holds. The results have applications outside of order algebra to non-classical logic, establishing the strong finite model property (SFMP) and the decidability for deductions, as well as to mathematical linguistics and automata theory, providing new conditions for recognizability of languages. Our proofs make use of residuated frames, developed in the context of algebraic proof theory.

Logic, Language, and Mathematics

2020 ◽  
Keyword(s):  
Philosophy Of Mathematics ◽  
Sorites Paradox ◽  
Metaphysical Possibility ◽  
Unpublished Paper ◽  
Crispin Wright ◽  
Mathematics Section ◽  
Language And Mathematics ◽  
And Mathematics ◽  
Logical Revisionism ◽  
George Boolos

This Festschrift volume contains a series of specially commissioned papers by leading philosophers on themes from the philosophy of Crispin Wright and a previously unpublished paper by George Boolos, together with a substantial set of replies by Wright. Section I consists of five essays on Wright’s Neo-Fregean approach in the philosophy of mathematics, Section II consists of two essays on Wright’s work on vagueness, intuitionism and the Sorites Paradox, Section III contains two essays on logical revisionism, and Section IV consists of a single essay on the epistemology of metaphysical possibility. The volume also contains a full bibliography of Wright’s philosophical publications.

scholarly journals Identity of Proofs Based on Normalization and Generality

2003 ◽  
Vol 9 (4) ◽  
pp. 477-503 ◽  
Author(s):  
Kosta Došen
Keyword(s):  
Equivalence Relation ◽  
Category Theory ◽  
Classical Logic ◽  
Proof Theory ◽  
Natural Deduction ◽  
Lambda Calculus ◽  
Free Form ◽  
Low Dimensional Topology ◽  
Sequent Systems ◽  
Low Dimensional

AbstractSome thirty years ago, two proposals were made concerning criteria for identity of proofs. Prawitz proposed to analyze identity of proofs in terms of the equivalence relation based on reduction to normal formin natural deduction. Lambek worked on a normalization proposal analogous to Prawitz's, based on reduction to cut-free form in sequent systems, but he also suggested understanding identity of proofs in terms of an equivalence relation based on generality, two derivations having the same generality if after generalizing maximally the rules involved in them they yield the same premises and conclusions up to a renaming of variables. These two proposals proved to be extensionally equivalent only for limited fragments of logic.The normalization proposal stands behind very successful applications of the typed lambda calculus and of category theory in the proof theory of intuitionistic logic. In classical logic, however, it did not fare well.The generality proposal was rather neglected in logic, though related matters were much studied in pure category theory in connection with coherence problems, and there are also links to low-dimensional topology and linear algebra. This proposal seems more promising than the other one for the general proof theory of classical logic.

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