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Below are some philosophy papers of mine to download. None of them are about pumpkins and there aren't even any examples involving pumpkins in them, sadly.
One day I will write a paper on The Great Pumpkin objection in epistemology and the philosophy of religion. I already have a title: 'The Great Pumpkin's Objection to the Great Pumpkin Objection'. I published one of my articles in Kindle format, mostly for fun, but if inadvertently I make a fortune out of it, I won't complain. You can buy it here: "Negation: A Problem for the Proof-Theoretic Justification of Deduction". I once won the Jacobsen Essay Prize of the University of London for it. |
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Proof and Falsity
A Logical Investigation Abstract: This book argues that the meaning of negation, perhaps the most important logical constant, cannot be defined within the framework of the most comprehensive theory of proof-theoretic semantics, as formulated in the influential work of Michael Dummett and Dag Prawitz. Nils Kürbis examines three approaches that have attempted to solve the problem - defining negation in terms of metaphysical incompatibility; treating negation as an undefinable primitive; and defining negation in terms of a speech act of denial - and concludes that they cannot adequately do so. He argues that whereas proof-theoretic semantics usually only appeals to a notion of truth, it also needs to appeal to a notion of falsity, and proposes a system of natural deduction in which both are incorporated. Offering new perspectives on negation, denial and falsity, his book will be important for readers working on logic, metaphysics and the philosophy of language. A slightly more extended abstract: My book is the first systematic and detailed study of a problem proof-theoretic semantics faces in relation to what is arguably the most important logical constant: negation. I argue that the meaning of negation cannot be defined within the framework of the most comprehensive theory of proof-theoretic semantics, formulated by Michael Dummett and Dag Prawitz. I survey various possible solutions to this problem, some of them new to the literature. One is to add a primitive notion of incompatibility. This is an approach favoured by Brandom, Peacocke and Tennant. Another one is inspired by some of what Geach has to say about negation: negation itself is added as a primitive the meaning of which is not defined by proof-theoretic semantics. Finally, Price and Rumfitt propose to add a primitive speech act of denial and propose to define the meaning of negation in terms of it and assertion. All three approaches are found wanting. The book concludes with a defence of a solution that constitutes a substantial new contribution to proof-theoretic semantics. I argue for a reformulation of its basic ideas: whereas usually proof-theoretic semantics only appeals to a notion of truth, tied to notions of verification or the grounds of assertions, which is preserved by inferences, I argue that it also needs to appeal to a corresponding negative notion, namely a notion of falsity, tied to notions of falsification or the defeat of assertions. I propose a system of natural deduction with truth as well as falsity preserving rules of inference which allows the justification of classical logic on the basis of proof-theoretic semantics. Published with Cambridge University Press in April 2019 Excerpts on Google Books Reviewed by: Ivo Pezlar in Studia Logica 111 (2023): 353–356 Lavinia Picollo in Analysis 81/3 (2021): 595-599 Jaroslav Peregrin in Notre Dame Philosophical Reviews (03 November 2019) Stephen Mumford in Times Higher Education (20 June 2019, p.51) |
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Arthur Prior's Proofs of the Necessities of Identity and Difference
This paper draws attention to a proof of the necessity of identity given by Arthur Prior. In its simplicity it is comparable to a proof of Quine's, popularised by Kripke, but it is slightly different. Prior's Polish notation is transcribed into more familiar idiom. Prior's proof is followed by a proof of the necessity of difference, possibly the first such proof in the literature, which is also repeated here and transcribed. The paper concludes with a brief discussion of Prior's views on identity and difference over time. Published in History and Philosophy of Logic (2023) |
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Supposition: A Problem for Bilateralism
In bilateral logic formulas are signed by + and −, indicating the speech acts assertion and denial. I argue that making an assumption is also a speech act. Speech acts cannot be embedded within other speech acts. Hence we cannot make sense of the notion of making an assumption in bilateral logic. Attempts to solve this problem are considered and rejected. Published in Bulletin of the Section of Logic (2023), special issue Bilateralism and Proof-Theory, ed. by Sara Ayhan |
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A Cut-free, Sound and Complete Russellian Theory of Definite Descriptions
We present a sequent calculus for first-order logic with lambda terms and definite descriptions. The theory formalised by this calculus is essentially Russellian, but avoids some of its well known drawbacks and treats definite description as genuine terms. A constructive proof of the cut elimination theorem and a Henkin-style proof of completeness are the main results of this contribution. Published in R. Ramanayake and J. Urban (edd.): Tableaux 2023, Lecture Notes in Artificial Intelligence 14278 (Cham: Springer 2023), pp.112–130 |
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On a Definition of Logical Consequence
Bilateralists, who accept that there are two primitive speech acts, assertion and denial, can offer an attractive definition of consequence: X ⊢ Y if and only if it is incoherent to assert all formulas X and to deny all formulas Y. The present paper argues that this definition has consequences many will find problematic, amongst them that truth coincides with assertibility. Philosophers who reject these consequences should therefore reject this definition of consequence. Published in Thought 11/2 (2022): 64-71 |
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Comment on Mark Textor: Brentano's Positing Theory of Existence
This article is the text of a commentary on a talk delivered by Mark Textor entitled 'Brentano's Positing Theory of Existence' in December 2015. It contains ideas on implementing Textor's Neo-Brentanian theory of existence in a natural deduction proof system for negative free logic. Published online on Arxiv.org in April 2023. |
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Knowledge, Number and Reality. Encounters with the Work of Keith Hossack
Edited by Nils Kürbis, Bahram Assadian and Jonathan Nassim Throughout his career, Keith Hossack has made outstanding contributions to the theory of knowledge, metaphysics and the philosophy of mathematics. This collection of previously unpublished papers begins with a focus on Hossack's conception of the nature of knowledge, his metaphysics of facts and his account of the relations between knowledge, agents and facts. Attention moves to Hossack's philosophy of mind and the nature of consciousness, before turning to the notion of necessity and its interaction with a priori knowledge. Hossack's views on the nature of proof, logical truth, conditionals and generality are discussed in depth. In the final chapters, questions about the identity of mathematical objects and our knowledge of them take centre stage, together with questions about the necessity and generality of mathematical and logical truths. Knowledge, Number and Reality represents some of the most vibrant discussions taking place in analytic philosophy today. Table of Contents Notes on Contributors Introduction, Nils Kürbis, Bahram Assadian and Jonathan Nassim 1. A Summary of My Current Views, Keith Hossack 2. Confronting Facts: On Hossack's The Metaphysics of Knowledge, Mark Sainsbury 3. Who Knows?, MM McCabe 4. Knowledge-first Epistemology and the Input Problem, Scott Sturgeon 5. Perceiving X = Consciousness of Perceiving X. Hossack and Brentano on the Identity Thesis, Mark Textor 6. Facts, Knowledge and Knowledge of Facts, Bernhard Weiss 7. Necessity, Conditionals and A Priority, Keith Hossack 8. A Commentary on 'Necessity, Conditionals and A Priority', Dorothy Edgington 9. The Mathematicians' Use of Diagrams in Plato, Tamsin De Waal 10. Generality, Nils Kürbis 11. We Belong Together: A Plea for Modesty in Modal Plural Logic, Simon Hewitt 12. Aristotelian Aspirations, Fregean Fears: Hossack on Numbers as Magnitudes, Øystein Linnebo 13. Mathematical Structures, Universals, and Singular Terms, Bahram Assadian 14. Arithmetic in a Finite World, Peter Simons Index Published by Bloomsbury Academic in January 2022 Excerpts on Google Books The Introduction to the book as well as my contribution can be dowloaded to the left and below. |
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Generality
Hossack's The Metaphysics of Knowledge develops a theory of facts, entities in which universals are combined with universals or particulars, as the foundation of his metaphysics. While Hossack argues at length that there must be negative facts, facts in which the universal negation is combined with universals or particulars, his conclusion that there are also general facts, facts in which the universal generality is combined with universals, is reached rather more swiftly. In this paper I present Hossack with three arguments for his conclusion. They all draw, as does Hossack's theory of facts, on views Russell expressed in various writings. Two arguments are based on Russell's explanation of universals as aspects of resemblance; the third on Russell's observation that general propositions do not follow logically from exclusively particular premises. Comparison with other metaphysics of generality show them to be wanting and Russell's and Hossack's accounts superior. Published in Knowledge, Number and Reality: Encounters with the Work of Keith Hossack, edited by Nils Kürbis, Bahram Assadian and Jonathan Nassim (London: Bloomsbury Academic 2022), pp.161-176 |
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Normalisation and Subformula Property for a System of Intuitionistic Logic with General Introduction and Elimination Rules
This paper studies a formalisation of intuitionistic logic by Negri and von Plato which has general introduction and elimination rules. The philosophi- cal importance of the system is expounded. Definitions of ‘maximal formula’, ‘segment’ and ‘maximal segment’ suitable to the system are formulated and corresponding reduction procedures for maximal formulas and permutative reduction procedures for maximal segments given. Alternatives to the main method used are also considered. It is shown that deductions in the system convert into normal form and that deductions in normal form have the subformula property. Keywords: intuitionistic logic, proof theory, normalisation, general elimina- tion rules, general introduction rules, harmony, stability. Published in Synthese 199/5-6 (2021): 14223-14248 |
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A Binary Quantifier for Definite Descriptions for Cut Free Free Logics
This paper presents rules in sequent calculus for a binary quantifier I to formalise definite descriptions: Ix[F, G] means ‘The F is G’. The rules are suitable to be added to a system of positive free logic. The paper extends the proof of a cut elimination theorem for this system by Indrzejczak by proving the cases for the rules of I. There are also brief comparisons of the present approach to the more common one that formalises definite descriptions with a term forming operator. In the final section rules for I for negative free and classical logic are also mentioned. Keywords: Definite descriptions, Free logic, Sequent calculus, Cut elimination. Published in Studia Logica 110/1 (2022): 219-239 |
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Normalisation and Subformula Property for a System of Classical Logic with Tarski's Rule
This paper considers a formalisation of classical logic using general introduction rules and general elimination rules. It proposes a definition of 'maximal formula', 'segment' and 'maximal segment' suitable to the system, and gives reduction procedures for them. It is then shown that deductions in the system convert into normal form, i.e. deductions that contain neither maximal formulas nor maximal segments, and that deductions in normal form satisfy the subformula property. Tarski's Rule is treated as a general introduction rule for implication. The general introduction rule for negation has a similar form. Maximal formulas with implication or negation as main operator require reduction procedures of a more intricate kind not present in normalisation for intuitionist logic. Keywords: Proof theory, Classical logic, Normalisation, Subformula property. Mathematics Subject Classification: 03B05, 03B10, 03B20, 03F05. Published in the Archive for Mathematical Logic 61 (2022): 105–129 Correction regarding 'Normalisation and Subformula Property for a System of Classical Logic with Tarski's Rule' This note corrects an error in the above paper: Theorem 2 is mistaken, and so is a corollary drawn from it as well as a corollary that was concluded by the same mistake. Luckily this does not affect the main result of the paper. Published online on Arxiv.org. |
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Bilateral Inversion Principles
This paper formulates a bilateral account of harmony that is an alternative to one proposed by Francez. It builds on an account of harmony for unilaterallogicproposedbyKu ̈rbisandtheobservationthatreadingthe rules for the connectives of bilateral logic bottom up gives the grounds and consequences of formulas with the opposite speech act. I formulate a process I call ‘inversion’ which allows the determination of assertive elimination rules from assertive introduction rules, and rejective elimination rules from rejective introduction rules, and conversely. It corresponds to Francez’s notion of vertical harmony. I also formulate a process I call ‘conversion’, which allows the determination of rejective introduction rules from assertive elimination rules and conversely, and the determination of assertive introduction rules from rejective elimination rules and conversely. It corresponds to Francez’s notion of horizontal harmony. The account has a number of features that distinguish it from Francez’s. Published in Electronic Proceedings in Theoretical Computer Science 358 (2022): 202–215 These are the Proceedings of the 10th International Conference on Non-Classical Logics. Theory and Applications (NCL'22), Lodz, Poland, 14-18 March 2022, edited by Andrzej Indrzejczak and Michal Zawidzki. |
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Normalisation for Bilateral Classical Logic with some Philosophical Remarks
Bilateralists hold that the meanings of the connectives are determined by rules of inference for their use in deductive reasoning with asserted and denied formulas. This paper presents two bilateral connectives comparable to Prior's tonk, for which, unlike for tonk, there are reduction steps for the removal of maximal formulas arising from introducing and eliminating formulas with those connectives as main operators. Adding either of them to bilateral classical logic results in an incoherent system. One way around this problem is to count formulas as maximal that are the conclusion of reductio and major premise of an elimination rule and to require their removability from deductions. The main part of the paper consists in a proof of a normalisation theorem for bilateral logic. The closing sections address philosophical concerns whether the proof provides a satisfactory solution to the problem at hand and confronts bilateralists with the dilemma that a bilateral notion of stability sits uneasily with the core bilateral thesis. Published in the Journal of Applied Logics 8/2 (2021): 531-556 This is a special issue Assertion and Proof, edited by Massimiliano Carrara, Daniele Chiffi and Ciro de Florio. Note on 'Normalisation for Bilateral Classical Logic with some Philosophical Remarks' This brief note corrects an error in one of the reduction steps in the paper above. Published in the Journal of Applied Logics 8/7 (2021): 2259-2261 |
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Proof Theory and Semantics for a Theory of Definite Descriptions
This paper presents a sequent calculus and a dual domain semantics for a theory of definite descriptions in which these expressions are formalised in the context of complete sentences by a binary quantifier I. I forms a formula from two formulas. Ix[F, G] means 'The F is G'. This approach has the advantage of incorporating scope distinctions directly into the notation. Cut elimination is proved for a system of classical positive free logic with I and it is shown to be sound and complete for the semantics. The system has a number of novel features and is briefly compared to the usual approach of formalising 'the F' by a term forming operator. It does not coincide with Hintikka's and Lambert's preferred theories, but the divergence is well-motivated and attractive. Note. Due to an editorial mishap, the published version of this paper lacks an appendix that contains proofs of some of the theorems in the main text and references to it therein. The version uploaded here, here, here and here are the full versions. Published in Anupam Das and Sara Negri (eds.): Tableaux 2021. Lecture Notes in Artificial Intelligence, Vol. 12842 (Berlin, Heidelberg: Springer 2021). |
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The Importance of Being Erroneous
This is a commentary on MM McCabe's paper 'First Chop your logos... Socrates and the sophists on language, logic, and development' (Australasian Philosophical Review 3/2 (2019): 131-150). In her paper MM analyses Plato's Euthydemus, in which Plato tackles the problem of falsity in a way that takes into account the speaker and complements the Sophist's discussion of what is said. The dialogue looks as if it is merely a demonstration of the silly consequences of eristic combat. And so it is. But a main point of MM's paper is that there is serious philosophy in the Euthydemos, too. MM argues that to counter the sophist brothers Euthydemos and Dionysodorus, Socrates points out that that there are different aspects to the verb 'to say' that run in parallel to the different aspects of the very 'to learn'. So just as there is continuity rather than ambiguity between 'to learn' and 'to understand', so there is continuity between the different aspects of saying. Thus Socrates puts forward a teleological account of both learning and meaning. Following up on some of MM's thoughts, I argue that the sophists subscribe, despite appearance, to a theory of meaning that respects serious and widely accepted philosophical theses on meaning. I draw parallels between the sophists’ and the Socratic account of meaning that MM reconstructs from the Euthydemus and views on logic and language found in the works of classical authors of analytic philosophy. I argue that the ingredients of the sophist’s account of truth, which MM describes as ‘chopped logos’, correspond to widely held philosophical theses concerning meaning. It shares three of its four ingredients with the direct reference theory of the meanings of proper names. The sophists need a notion of meaning applicable to sayings, not names: they require a notion of truth. This is provided by the remaining ingredient, which is a version of the principle that meanings are truth conditions. The Euthydemus demonstrates dramatically that the combination of the four ingredients is unpalatable. Building on MM's point that chopped logos does not get the conditions of failure of sayings right, I conclude that, as the sophists have no notion of falsity of sayings, they have neither a notion of truth nor of meaning. Published in the Australasian Philosophical Review 3/2 (2019): 155-166 The curator of the volume is Fiona Leigh, and the committee also has Hugh Benson and Tim Clarke. The volume contains an introduction by Fiona and commentaries on MM's paper by Nicholas Denyer, myself, Russell E. Jones & Ravi Sharma, Merrick Anderson, Saloni de Souza & Daniel Vázquez, Christine J. Thomas, Matthew Duncombe, Tony Leyh, and a reply to all of them by MM. |
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Normalisation for Some Quite Interesting Many-Valued Logics (joint work with Yaroslav Petrukhin)
In this paper, we consider a set of quite interesting three- and four-valued logics and prove normalisation theorem for their natural deduction formulations. Among the logics in question are the Logic of Paradox, First Degree Entailment, Strong Kleene logic, and some of their implicative extensions, including RM3 and RM3⊃. Also, we present a detailed version of Prawitz's proof of Nelson's logic N4 and its extension by intuitionist negation. Published in Logic and Logical Philosophy 30/3 (2021): 493-534 |
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Definite Descriptions in Intuitionist Positive Free Logic
This paper presents rules of inference for a binary quantifier I for the formalisation of sentences containing definite descriptions within intuitionist positive free logic. I binds one variable and forms a formula from two formulas. Ix[F, G] means 'The F is G'. The system is shown to have desirable proof-theoretic properties: it is proved that deductions in it can be brought into normal form. The discussion is rounded up by comparisons between the approach to the formalisation of definite descriptions recommended here and the more usual approach that uses a term-forming operator ι, where ιxF means 'the F'. Published in Logic and Logical Philosophy 30/2 (2021): 327-358 |
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A Sketch of a Proof-Theoretic Semantics for Necessity
This paper considers proof-theoretic semantics for necessity within Dummett's and Prawitz's framework. Inspired by a system of Pfenning's and Davies's, the language of intuitionist logic is extended by a higher order operator which captures a notion of validity. A notion of relative necessary is defined in terms of it, which expresses a necessary connection between the assumptions and the conclusion of a deduction. Published in Advances in Modal Logic 13. Booklet of Short Papers, edited by Sara Negri, Nicola Olivetti, Rineke Verbrugge (Helsinki 2020): 37-43 |
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Two Treatments of Definite Descriptions in Intuitionist Negative Free Logic
Sentences containing definite descriptions, expressions of the form 'The F', can be formalised using a binary quantifier 𝜄 that forms a formula out of two predicates, where 𝜄x[F, G] is read as 'The F is G'. This is an innovation over the usual formalisation of definite descriptions with a term forming operator. The present paper compares the two approaches. After a brief overview of the system INF𝜄 of intuitionist negative free logic extended by such a quantifier, which was presented in , INF𝜄 is first compared to a system of Tennant's and an axiomatic treatment of a term forming 𝜄 operator within intuitionist negative free logic. Both systems are shown to be equivalent to the subsystem of INF𝜄 in which the G of INF𝜄 is restricted to identity. INF𝜄 is then compared to an intuitionist version of a system of Lambert's which in addition to the term forming operator has an operator for predicate abstraction for indicating scope distinctions. The two systems will be shown to be equivalent through a translation between their respective languages. Advantages of the present approach over the alternatives are indicated in the discussion. Published in the Bulletin of the Section of Logic 48/4 (2019): 299-317 |
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A Binary Quantifier for Definite Descriptions in Intuitionist Negative Free Logic: Natural Deduction and Normalisation
This paper presents a way of formalising definite descriptions with a binary quantifier 𝜄, where 𝜄x[F, G] is read as 'The F is G'. Introduction and elimination rules for 𝜄 in a system of intuitionist negative free logic are formulated. Procedures for removing maximal formulas of the form 𝜄x[F, G] are given, and it is shown that deductions in the system can be brought into normal form. Published in the Bulletin of the Section of Logic 48/2 (2019): 81-97 |
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An Argument for Minimal Logic
The problem of negative truth is the problem of how, if everything in the world is positive, we can speak truly about the world using negative propositions. A prominent solution is to explain negation in terms of a primitive notion of metaphysical incompatibility. I argue that if this account is correct, then minimal logic is the correct logic. The negation of a proposition A is characterised as the minimal incompatible of A composed of it and the logical constant ¬. A rule based account of the meanings of logical constants that appeals to the notion of incompatibility in the introduction rule for negation ensures the existence and uniqueness of the negation of every proposition. But it endows the negation operator with no more formal properties than those it has in minimal logic. Published in dialectica 73/1-2 (2019): 31-62 |
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Nils Kürbis: Is Fregeanism Compatible with Incompatibilism?
This paper considers whether incompatibilism, the view that negation is to be explained in terms of a primitive notion of incompatibility, and Fregeanism, the view that arithmetical truths are analytic according to Frege’s definition of that term in §3 of Foundations of Arithmetic, can be held together. Both views are attractive in their own right, in particular for a certain empiricist mind-set. They promise to account for two philosophical puzzling phenomena: the problem of negative truth and the problem of epistemic access to numbers. For an incompatibilist, proofs of numerical non-identities must appeal to primitive incompatibilities. I argue that no analytic primitive incompatibilities are forthcoming. Hence incompatibilists cannot be Fregeans. Published in the European Journal of Analytic Philosophy 14/2 (2018): 27-46 |
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Nils Kürbis: George Molnar on Truthmakers for Negative Truths
Molnar argues that the problem of truthmakers for negative truths arises because we tend to accept four metaphysical principles that entail that all negative truths have positive truthmakers. This conclusion, however, already follows from only three of Molnar's metaphysical principles. One purpose of this note is to set the record straight. I provide an alternative reading of two of Molnar's principles on which they are all needed to derive the desired conclusion. Furthermore, according to Molnar, the four principles may be inconsistent. By themselves, however, they are not. The other purpose of this note is to propose some plausible further principles that, when added to the four metaphysical theses, entail a contradiction. Published in Metaphysica 19/2 (2018): 251–257 |
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Nils Kürbis: Bilateralism: Negations, Implications and some Observations and Problems about Hypotheses
This short paper has two loosely connected parts. In the first part, I discuss the difference between classical and intuitionist logic in relation to different the role of hypotheses play in each logic. Harmony is normally understood as a relation between two ways of manipulating formulas in systems of natural deduction: their introduction and elimination. I argue, however, that there is at least a third way of manipulating formulas, namely the discharge of assumption, and that the difference between classical and intuitionist logic can be characterised as a difference of the conditions under which discharge is allowed. Harmony, as ordinarily understood, has nothing to say about discharge. This raises the question whether the notion of harmony can be suitably extended. This requires there to be a suitable fourth way of manipulating formulas that discharge can stand in harmony to. The question is whether there is such a notion: what might it be that stands to discharge of formulas as introduction stands to elimination? One that immediately comes to mind is the making of assumptions. I leave it as an open question for further research whether the notion of harmony can be fruitfully extended in the way suggested here. In the second part, I discuss bilateralism, which proposes a wholesale revision of what it is that is assumed and manipulated by rules of inference in deductions: rules apply to speech acts – assertions and denials – rather than propositions. I point out two problems for bilateralism. First, bilaterlists cannot, contrary to what they claim to be able to do, draw a distinction between the truth and assertibility of a proposition. Secondly, it is not clear what it means to assume an expression such as '+ A' that is supposed to stand for an assertion. Worse than that, it is plausible that making an assumption is a particular speech act, as argued by Dummett (Frege: Philosophy of Language, p.309ff). Bilaterlists accept that speech acts cannot be embedded in other speech acts. But then it is meaningless to assume + A or − A. Published in Beyond Logic. Proceedings of the Conference held in Cerisy-la-Salle, 22-27 May 2017, edited by Jean Fichot and Thomas Piecha |
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Nils Kürbis: Bilateralist Detours. From Intuitionist to Classical Logic and Back
There is widespread agreement that while on a Dummettian theory of meaning the justified logic is intuitionist, as its constants are governed by harmonious rules of inference, the situation is reversed on Huw Price's bilateralist account, where meanings are specified in terms of primitive speech acts assertion and denial. In bilateral logics, the rules for classical negation are in harmony. However, as it is possible to construct an intuitionist bilateral logic with harmonious rules, there is no formal argument against intuitionism from the bilateralist perspective. Price gives an informal argument for classical negation based on a pragmatic notion of belief, characterised in terms of the differences they make to speakers' actions. The main part of this paper puts Price's argument under close scrutiny by regimenting it and isolating principles Price is committed to. It is shown that Price should draw a distinction between A or ¬A making a difference. According to Price, if A makes a difference to us, we treat it as decidable. This material allows the intuitionist to block Price's argument. Abandoning classical logic also brings advantages, as within intuitionist logic there is a precise meaning to what it might mean to treat A as decidable: it is to assume A ∨ ¬A. Published in Logique et Analyse 60/239 (2017): 301-316 |
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Nils Kürbis: Some Comments on Ian Rumfitt's Bilateralism
Ian Rumfitt has proposed systems of bilateral logic for primitive speech acts of assertion and denial, with the purpose of 'exploring the possibility of specifying the classically intended senses for the connectives in terms of their deductive use'. Rumfitt formalises two systems of bilateral logic and gives two arguments for their classical nature. I assess both arguments and conclude that only one system satisfies the meaning-theoretical requirements Rumfitt imposes in his arguments. I then formalise an intuitionist system of bilateral logic which also meets those requirements. Thus Rumfitt cannot claim that only classical bilateral rules of inference succeed in imparting a coherent sense onto the connectives. My system can be extended to classical logic by adding the intuitionistically unacceptable half of a structural rule Rumfitt uses to codify the relation between assertion and denial. Thus there is a clear sense in which in the bilateral framework, the difference between classicism and intuitionism is not one of the rules of inference governing negation, but rather one of the relation between assertion and denial. Keywords: Negation, denial, proof-theoretic semantics, harmony, classical logic, intuitionist logic Published in the Journal of Philosophical Logic 45/6 (2016): 623-644 |
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Nils Kürbis: Review of Bob Hale's Necessary Beings
I summarise the main strands of Hale's Fregean approach to metaphysics and his views on fundamental modal facts. At the request of the author, I mention an error that will be corrected in the paper back edition. Hale originally thought that on his semantics, where second-order variables range only over definable subsets of the domain, compactness, completeness, and the Löwenheim-Skolem Theorems hold, as they do in the non-standard Henkin semantics, while categoricity fails. I chose a few issues for closer and critical discussion that I found particularly interesting and worth developing. I think that Hale's new amendment of his well known inferential tests for singular terms is slightly underdeveloped, as it lacks an account of immediate inferences. I suggest that there is an omission in Hale's falsity conditions for the counterfactuals and that Hale's preferred counterfactual logic should turn out to be an analogue of Lewis's VWU, based on Hale's alternative semantics in terms of possibilities rather than possible worlds. Hale defines necessity in terms of counterfactuals, but I question whether we understand counterfactuals as well as we understand necessity, possibility and contingency. I end with the suggestion that Hale's metaphysics can be used to motivate an idea put forward by Arthur Prior: to introduce a new kind of expression into the modal language, often called nominals, which name possibilities, and to adopt hybrid modal logics. The full title of Bob Hale's book is Necessary Beings: An Essay on Ontology, Modality, and the Relations Between Them, published by Oxford University Press. Published in Disputatio VII/40 (2015): 92-100 |
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Nils Kürbis: What is wrong with Classical Negation?
The focus of this paper are Dummett's meaning-theoretical arguments against classical logic based on consideration about the meaning of negation. Using Dummettian principles, I shall outline three such arguments, of increasing strength, and show that they are unsuccessful by giving responses to each argument on behalf of the classical logician. What is crucial is that in responding to these arguments a classicist need not challenge any of the basic assumptions of Dummett's outlook on the theory of meaning. In particular, I shall grant Dummett his general bias towards verificationism, encapsulated in the slogan 'meaning is use'. The second general assumption I see no need to question is Dummett's particular breed of molecularism. Some of Dummett's assumptions will have to be given up, if classical logic is to be vindicated in his meaning-theoretical framework. A major result of this paper will be that the meaning of negation cannot be defined by rules of inference in the Dummettian framework. Keywords: Proof-theoretic semantics, harmony, negation, ex falso quodlibet, compositionality, molecular theories of meaning Published in Grazer Philosophische Studien 92/1 (2015): 51-86 |
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Nils Kürbis: Proof-Theoretic Semantics, a Problem with Negation and Prospects for Modality
This paper discusses proof-theoretic semantics, the project of specifying the meanings of the logical constants in terms of rules of inference governing them. I concentrate on Michael Dummett’s and Dag Prawitz’ philosophical motivations and give precise characterisations of the crucial notions of harmony and stability, placed in the context of proving normalisation results in systems of natural deduction. I point out a problem for defining the meaning of negation in this framework and prospects for an account of the meanings of modal operators in terms of rules of inference. Keywords: Proof-Theoretic Semantics, Harmony, Stability, Negation, Modality Published in the Journal of Philosophical Logic 44/6 (2015): 713-727 |
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Nils Kürbis: How Fundamental is the Fundamental Assumption?
The fundamental assumption of Dummett’s and Prawitz’ proof-theoretic justification of deduction is that 'if we have a valid argument for a complex statement, we can construct a valid argument for it which finishes with an application of one of the introduction rules governing its principal operator'. I argue that the assumption is flawed in this general version, but should be restricted, not to apply to arguments in general, but only to proofs. I also argue that Dummett’s and Prawitz’ project of providing a logical basis for metaphysics only relies on the restricted assumption. Keywords: Proof-theoretic Semantics, Michael Dummett, Dag Prawitz, Verificationist Theories of Meaning, Realism vs. Anti-Realism. Published in Teorema XXXI/2 (2012): 5-19 |
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Nils Kürbis: What is Interpretation? A Dilemma for Davidson
The core idea of Davidson’s philosophy of language is that a theory of truth constructed as an empirical theory by a radical interpreter is a theory of meaning. I discuss an ambiguity that arises from Davidson’s notion of interpretation: it can either be understood as the hypothetical process of constructing a theory of truth for a language or as a process that actually happens when speakers communicate. I argue that each disambiguation is problematic and does not result in a theory of meaning. Published in Conceptus 40/98 (2011): 54-66 |
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Nils Kürbis: Stable Harmony
In this paper I give a formally precise definition of harmony and stability for Dummett's and Prawitz's proof-theoretic semantics. A short version of the paper below. Published in the Logica Yearbook 2008 |
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Nils Kürbis: Harmony, Normality and Stability
The paper begins with a conceptual discussion of Michael Dummett's proof-theoretic justification of deduction or proof-theoretic semantics, which is based on what we might call Gentzen's thesis: 'the introductions constitute, so to speak, the "definitions" of the symbols concerned, and the eliminations are in the end only consequences thereof, which could be expressed thus: In the elimination of a symbol, the formula in question, whose outer symbol it concerns, may only "be used as that which it means on the basis of the introduction of this symbol".' The intuitive philosophical content of Dummett's notions of harmony and stability is that harmony obtains if the grounds for asserting a proposition match the consequences of accepting it, and stability obtains if the converse also holds. Rules of inference define the meanings of a logical constant they govern if and only if they are stable. Gentzen observed that 'it should be possible to establish on the basis of certain requirements that the elimination rules are functions of the corresponding introduction rules.' One of the objectives of this paper is to specify such a function: I will specify a process by which it is possible to determine the elimination rules of logical constants from their introduction rules, and conversely, to determine the introduction rules from the elimination rules. I'll give the general forms of rules of inference and generalised reduction procedures for the normalisation of deduction. I'll give a formally precise characterisations of harmony and stability and show that deductions in logics that contain only constants governed by stable rules always normalise. Published exclusively here, by Nils Pumpkin Philosophy Publications! |
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