Results for 'formulas'

177 found
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  1. Sahlqvist Formulas Unleashed in Polyadic Modal Languages.Valentin Goranko & Dimiter Vakarelov - 1998 - In Marcus Kracht, Maarten de Rijke, Heinrich Wansing & Michael Zakharyaschev (eds.), Advances in Modal Logic. CSLI Publications. pp. 221-240.
    We propose a generalization of Sahlqvist formulas to polyadic modal languages by representing such languages in a combinatorial PDL style and thus, in particular, developing what we believe to be the right syntactic approach to Sahlqvist formulas at all. The class of polyadic Sahlqvist formulas PSF defined here expands essentially the so far known one. We prove first-order definability and canonicity for the class PSF.
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  2. Formulas for Computable and Non-Computable Functions.Samuel Alexander - 2006 - Rose-Hulman Undergraduate Mathematics Journal 7 (2).
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  3. Sahlqvist Formulas Unleashed in Polyadic Modal Languages.Valentin Goranko & Dimiter Vakarelov - 1998 - In Marcus Kracht, Maarten de Rijke, Heinrich Wansing & Michael Zakharyaschev (eds.), Advances in Modal Logic. CSLI Publications. pp. 221-240.
    We propose a generalization of Sahlqvist formulae to polyadic modal languages by representing modal polyadic languages in a combinatorial style and thus, in particular, developing what we believe to be the right approach to Sahlqvist formulae at all. The class of polyadic Sahlqvist formulae PSF defined here expands essentially the so far known one. We prove first-order definability and canonicity for the class PSF.
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  4. Decidable Formulas Of Intuitionistic Primitive Recursive Arithmetic.Saeed Salehi - 2002 - Reports on Mathematical Logic 36 (1):55-61.
    By formalizing some classical facts about provably total functions of intuitionistic primitive recursive arithmetic (iPRA), we prove that the set of decidable formulas of iPRA and of iΣ1+ (intuitionistic Σ1-induction in the language of PRA) coincides with the set of its provably ∆1-formulas and coincides with the set of its provably atomic formulas. By the same methods, we shall give another proof of a theorem of Marković and De Jongh: the decidable formulas of HA are its (...)
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  5. Representing Concepts by Weighted Formulas.Daniele Porello & Claudio Masolo - 2018 - In Stefano Borgo, Pascal Hitzler & Oliver Kutz (eds.), Formal Ontology in Information Systems - Proceedings of the 10th International Conference, {FOIS} 2018, Cape Town, South Africa, 19-21 September 2018. IOS Press. pp. 55--68.
    A concept is traditionally defined via the necessary and sufficient conditions that clearly determine its extension. By contrast, cognitive views of concepts intend to account for empirical data that show that categorisation under a concept presents typicality effects and a certain degree of indeterminacy. We propose a formal language to compactly represent concepts by leveraging on weighted logical formulas. In this way, we can model the possible synergies among the qualities that are relevant for categorising an object under a (...)
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  6. Strategic Reasoning: Building Cognitive Models from Logical Formulas.Sujata Ghosh, Ben Meijering & Rineke Verbrugge - 2014 - Journal of Logic, Language and Information 23 (1):1-29.
    This paper presents an attempt to bridge the gap between logical and cognitive treatments of strategic reasoning in games. There have been extensive formal debates about the merits of the principle of backward induction among game theorists and logicians. Experimental economists and psychologists have shown that human subjects, perhaps due to their bounded resources, do not always follow the backward induction strategy, leading to unexpected outcomes. Recently, based on an eye-tracking study, it has turned out that even human subjects who (...)
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  7. A Decision Procedure for Herbrand Formulas without Skolemization.Timm Lampert - manuscript
    This paper describes a decision procedure for disjunctions of conjunctions of anti-prenex normal forms of pure first-order logic (FOLDNFs) that do not contain V within the scope of quantifiers. The disjuncts of these FOLDNFs are equivalent to prenex normal forms whose quantifier-free parts are conjunctions of atomic and negated atomic formulae (= Herbrand formulae). In contrast to the usual algorithms for Herbrand formulae, neither skolemization nor unification algorithms with function symbols are applied. Instead, a procedure is described that rests on (...)
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  8. Nietzsche e a metafísica de artista: apropriações de fórmulas kantianas, schopenhauerianas e pré-socráticas em O nascimento da tragédia.Gabriel Herkenhoff Coelho Moura - 2023 - Estudos Nietzsche 14 (1):63-93.
    In his debut book, The Birth of Tragedy, Nietzsche presents what he understands as a metaphysics of art or metaphysics of the artist. As it becomes clear throughout the argument developed in the work, his aim is to favor a justification of the world and of existence as an aesthetic phenomenon. The path to his metaphysics passes through the interaction with Kantian and, mostly, Schopenhauerian formulations, and through a deep dialogue with Greek culture in general and, indirectly, with Pre-Socratic thought. (...)
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  9. Ibn Sina’s Anticipation of Burdian and Barcan Formulas.Zia Movahed - 2006 - In Logic in Tehran. Cambridge: Cambridge University Press. pp. 248-255.
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  10. review Ibn-Sina’s Anticipation of the Formulas of Buridan and Barcan. [REVIEW]Irving H. Anellis - 2008 - The Review of Modern Logic 1: 73–86.
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  11. Elementary canonical formulae: extending Sahlqvist’s theorem.Valentin Goranko & Dimiter Vakarelov - 2006 - Annals of Pure and Applied Logic 141 (1):180-217.
    We generalize and extend the class of Sahlqvist formulae in arbitrary polyadic modal languages, to the class of so called inductive formulae. To introduce them we use a representation of modal polyadic languages in a combinatorial style and thus, in particular, develop what we believe to be a better syntactic approach to elementary canonical formulae altogether. By generalizing the method of minimal valuations à la Sahlqvist–van Benthem and the topological approach of Sambin and Vaccaro we prove that all inductive formulae (...)
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  12. Translating Metainferences Into Formulae: Satisfaction Operators and Sequent Calculi.Ariel Jonathan Roffé & Federico Pailos - 2021 - Australasian Journal of Logic 3.
    In this paper, we present a way to translate the metainferences of a mixed metainferential system into formulae of an extended-language system, called its associated σ-system. To do this, the σ-system will contain new operators (one for each standard), called the σ operators, which represent the notions of "belonging to a (given) standard". We first prove, in a model-theoretic way, that these translations preserve (in)validity. That is, that a metainference is valid in the base system if and only if its (...)
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  13. Elementary Canonical Formulae: A Survey on Syntactic, Algorithmic, and Modeltheoretic Aspects.W. Conradie, V. Goranko & D. Vakarelov - 1998 - In Marcus Kracht, Maarten de Rijke, Heinrich Wansing & Michael Zakharyaschev (eds.), Advances in Modal Logic. CSLI Publications. pp. 17-51.
    In terms of validity in Kripke frames, a modal formula expresses a universal monadic second-order condition. Those modal formulae which are equivalent to first-order conditions are called elementary. Modal formulae which have a certain persistence property which implies their validity in all canonical frames of modal logics axiomatized with them, and therefore their completeness, are called canonical. This is a survey of a recent and ongoing study of the class of elementary and canonical modal formulae. We summarize main ideas and (...)
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  14.  30
    Kant's Universal Law and Humanity Formulae.Damian Williams - forthcoming - Forthcoming.
    Kant's formulae ought to effectively produce the same result when applied to the moral validity of any particular maxim; further, no valid maxim produces contradictory results when applied against Kant's Universal Law and Humanity formulae. Where one uses all formulae in the assessment of a maxim, one gains a more complete understanding of the moral law, thereby bridging principles of reason with intuition within the agent who has undertaken to evaluate the morality of a particular action. These formulae command without (...)
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  15. Physics without Formulae.Rowan Grigg - unknown
    A scheme to help merge the ideas of quantum gravity.
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  16. Cut-conditions on sets of multiple-alternative inferences.Harold T. Hodes - 2022 - Mathematical Logic Quarterly 68 (1):95 - 106.
    I prove that the Boolean Prime Ideal Theorem is equivalent, under some weak set-theoretic assumptions, to what I will call the Cut-for-Formulas to Cut-for-Sets Theorem: for a set F and a binary relation |- on Power(F), if |- is finitary, monotonic, and satisfies cut for formulas, then it also satisfies cut for sets. I deduce the CF/CS Theorem from the Ultrafilter Theorem twice; each proof uses a different order-theoretic variant of the Tukey- Teichmüller Lemma. I then discuss relationships (...)
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  17. Supposition: A Problem for Bilateralism.Nils Kürbis - 2023 - Bulletin of the Section of Logic 53 (3):301-327.
    In bilateral logic formulas are signed by + and –, indicating the speech acts assertion and denial. I argue that making an assumption is also 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.
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  18.  63
    Molecularity in the Theory of Meaning and the Topic Neutrality of Logic.Bernhard Weiss & Nils Kürbis - 2024 - In Antonio Piccolomini D'Aragona (ed.), Perspectives on Deduction: Contemporary Studies in the Philosophy, History and Formal Theories of Deduction. Springer Verlag. pp. 187-209.
    Without directly addressing the Demarcation Problem for logic—the problem of distinguishing logical vocabulary from others—we focus on distinctive aspects of logical vocabulary in pursuit of a second goal in the philosophy of logic, namely, proposing criteria for the justification of logical rules. Our preferred approach has three components. Two of these are effectively Belnap’s, but with a twist. We agree with Belnap’s response to Prior’s challenge to inferentialist characterisations of the meanings of logical constants. Belnap argued that for a logical (...)
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  19. Polyhedral Completeness of Intermediate Logics: The Nerve Criterion.Sam Adam-day, Nick Bezhanishvili, David Gabelaia & Vincenzo Marra - 2024 - Journal of Symbolic Logic 89 (1):342-382.
    We investigate a recently devised polyhedral semantics for intermediate logics, in which formulas are interpreted in n-dimensional polyhedra. An intermediate logic is polyhedrally complete if it is complete with respect to some class of polyhedra. The first main result of this paper is a necessary and sufficient condition for the polyhedral completeness of a logic. This condition, which we call the Nerve Criterion, is expressed in terms of Alexandrov’s notion of the nerve of a poset. It affords a purely (...)
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  20. Paraconsistency.Rafael R. Testa - 2020 - In James M. Mattingly (ed.), The SAGE Encyclopedia of Theory in Science, Technology, Engineering, and Mathematics. SAGE Publications. pp. 629-632.
    Paraconsistency is the study of logical systems with a non-explosive negation such that a pair of contradictory formulas (with respect to such negation) does not necessarily imply triviality, discordant to what would be expected by contemporary logical orthodoxy. From a purely logical point of view, the significance of paraconsistency relies on the meticulous distinction between the general notions of contradictoriness and triviality of a theory—respectively, the fact that a given theory proves a proposition and its negation, and the fact (...)
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  21. A Semantics for the Impure Logic of Ground.Louis deRosset & Kit Fine - 2023 - Journal of Philosophical Logic 52 (2):415-493.
    This paper establishes a sound and complete semantics for the impure logic of ground. Fine (Review of Symbolic Logic, 5(1), 1–25, 2012a) sets out a system for the pure logic of ground, one in which the formulas between which ground-theoretic claims hold have no internal logical complexity; and it provides a sound and complete semantics for the system. Fine (2012b) [§§6-8] sets out a system for an impure logic of ground, one that extends the rules of the original pure (...)
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  22. भारतीय संस्कृति का उत्स : वैदिक वांडमय.डॉ आभा रानी - 2014 - SOCRATES 2 (1):6-11.
    This paper is a reflection of Indian culture and civilization in the lights of holy Vedas. The author in this paper considers the Holy Vedas as the origin of enriched and spiritual civilization of India. The Vedas ("knowledge") are a large body of texts originating in ancient India. Composed in Vedic Sanskrit, the texts constitute the oldest layer of Sanskrit literature and the oldest scriptures of India. The Vedas are apauruṣeya ("not of human agency").They are supposed to have been directly (...)
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  23. What is logical form?Axel Barcelo Aspeitia - manuscript
    A good metaphysical account of logical form, must make clear why logical form is logical. However, this task has proved to be very elusive. Here, I analyze different attempts to meet this challenge and defend an inferential externalism where logical form is grounded on external logical relations as our most promising option.
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  24. Existence and Modality in Kant: Lessons from Barcan.Andrew Stephenson - 2023 - Philosophical Review 132 (1):1-41.
    This essay considers Kant’s theory of modality in light of a debate in contemporary modal metaphysics and modal logic concerning the Barcan formulas. The comparison provides a new and fruitful perspective on Kant’s complex and sometimes confusing claims about possibility and necessity. Two central Kantian principles provide the starting point for the comparison: that the possible must be grounded in the actual and that existence is not a real predicate. Both are shown to be intimately connected to the Barcan (...)
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  25. 3. Planck unit quantum gravity (gravitons) for Simulation Hypothesis modeling.Malcolm J. Macleod - manuscript
    Defined are gravitational formulas in terms of Planck units and units of $\hbar c$. Mass is not assigned as a constant property but is instead treated as a discrete event defined by units of Planck mass with gravity as an interaction between these units, the gravitational orbit as the sum of these mass-mass interactions and the gravitational coupling constant as a measure of the frequency of these interactions and not the magnitude of the gravitational force itself. Each particle that (...)
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  26. The premiss-based approach to judgment aggregation.Franz Dietrich & Philippe Mongin - 2010 - Journal of Economic Theory 145 (2):562-582.
    In the framework of judgment aggregation, we assume that some formulas of the agenda are singled out as premisses, and that both Independence (formula-wise aggregation) and Unanimity Preservation hold for them. Whether premiss-based aggregation thus defined is compatible with conclusion-based aggregation, as defined by Unanimity Preservation on the non-premisses, depends on how the premisses are logically connected, both among themselves and with other formulas. We state necessary and sufficient conditions under which the combination of both approaches leads to (...)
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  27. The Genealogy of ‘∨’.Landon D. C. Elkind & Richard Zach - 2023 - Review of Symbolic Logic 16 (3):862-899.
    The use of the symbol ∨for disjunction in formal logic is ubiquitous. Where did it come from? The paper details the evolution of the symbol ∨ in its historical and logical context. Some sources say that disjunction in its use as connecting propositions or formulas was introduced by Peano; others suggest that it originated as an abbreviation of the Latin word for “or,” vel. We show that the origin of the symbol ∨ for disjunction can be traced to Whitehead (...)
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  28. A Cut-Free Sequent Calculus for Defeasible Erotetic Inferences.Jared Millson - 2019 - Studia Logica (6):1-34.
    In recent years, the e ffort to formalize erotetic inferences (i.e., inferences to and from questions) has become a central concern for those working in erotetic logic. However, few have sought to formulate a proof theory for these inferences. To fill this lacuna, we construct a calculus for (classes of) sequents that are sound and complete for two species of erotetic inferences studied by Inferential Erotetic Logic (IEL): erotetic evocation and regular erotetic implication. While an attempt has been made to (...)
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  29. What is the correct logic of necessity, actuality and apriority?Peter Fritz - 2014 - Review of Symbolic Logic 7 (3):385-414.
    This paper is concerned with a propositional modal logic with operators for necessity, actuality and apriority. The logic is characterized by a class of relational structures defined according to ideas of epistemic two-dimensional semantics, and can therefore be seen as formalizing the relations between necessity, actuality and apriority according to epistemic two-dimensional semantics. We can ask whether this logic is correct, in the sense that its theorems are all and only the informally valid formulas. This paper gives outlines of (...)
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  30. Another Side of Categorical Propositions: The Keynes–Johnson Octagon of Oppositions.Amirouche Moktefi & Fabien Schang - 2023 - History and Philosophy of Logic 44 (4):459-475.
    The aim of this paper is to make sense of the Keynes–Johnson octagon of oppositions. We will discuss Keynes' logical theory, and examine how his view is reflected on this octagon. Then we will show how this structure is to be handled by means of a semantics of partition, thus computing logical relations between matching formulas with a semantic method that combines model theory and Boolean algebra.
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  31. Polynomial ring calculus for modal logics: A new semantics and proof method for modalities: Polynomial ring calculus for modal logics.Juan C. Agudelo - 2011 - Review of Symbolic Logic 4 (1):150-170.
    A new proof style adequate for modal logics is defined from the polynomial ring calculus. The new semantics not only expresses truth conditions of modal formulas by means of polynomials, but also permits to perform deductions through polynomial handling. This paper also investigates relationships among the PRC here defined, the algebraic semantics for modal logics, equational logics, the Dijkstra???Scholten equational-proof style, and rewriting systems. The method proposed is throughly exemplified for S 5, and can be easily extended to other (...)
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  32. Proof-Theoretic Semantics and Inquisitive Logic.Will Stafford - 2021 - Journal of Philosophical Logic 50 (5):1199-1229.
    Prawitz conjectured that proof-theoretic validity offers a semantics for intuitionistic logic. This conjecture has recently been proven false by Piecha and Schroeder-Heister. This article resolves one of the questions left open by this recent result by showing the extensional alignment of proof-theoretic validity and general inquisitive logic. General inquisitive logic is a generalisation of inquisitive semantics, a uniform semantics for questions and assertions. The paper further defines a notion of quasi-proof-theoretic validity by restricting proof-theoretic validity to allow double negation elimination (...)
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  33. An enhanced model for Rosenkranz’s logic of justification.Niccolò Rossi - 2022 - Asian Journal of Philosophy 1 (1):1-9.
    Rosenkranz (2021) devised two bimodal epistemic logics: an idealized one and a realistic one. The former is shown to be sound with respect to a class of neighborhood frames called i-frames. Rosenkranz designed a specific i-frame able to invalidate a series of undesired formulas, proving that these are not theorems of the idealized logic. Nonetheless, an unwanted formula and an unwanted rule of inference are not invalidated. Invalidating the former guarantees the distinction between the two modal operators characteristic of (...)
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  34. The Greenhouse: A Welfare Assessment and Some Morals.Christoph Lumer - 2002 - Lanham, MD; New York; Oxford: University Press of America.
    In this book some options concerning the greenhouse effect are assessed from a welfarist point of view: business as usual, stabilization of greenhouse gas emissions and reduction by 25% and by 60%. Up to today only economic analyses of such options are available, which monetize welfare losses. Because this is found to be wanting from a moral point of view, the present study welfarizes (among others) monetary losses on the basis of a hedonistic utilitarianism and other, justice incorporating, welfare ethics. (...)
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  35. The Truth Table Formulation of Propositional Logic.Tristan Grøtvedt Haze - forthcoming - Teorema: International Journal of Philosophy.
    Developing a suggestion of Wittgenstein, I provide an account of truth tables as formulas of a formal language. I define the syntax and semantics of TPL (the language of Tabular Propositional Logic), and develop its proof theory. Single formulas of TPL, and finite groups of formulas with the same top row and TF matrix (depiction of possible valuations), are able to serve as their own proofs with respect to metalogical properties of interest. The situation is different, however, (...)
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  36.  55
    Proof Invariance.Blane Worley - forthcoming - Australasian Journal of Logic.
    We explore depth substitution invariance, or hyperformalism, and extend known results in this realm to justification logics extending weak relevant logics. We then examine the surprising invariance of justifications over formulas and restrict our attention to the substitution of proofs in the original relevant logic. The results of this paper indicate that depth invariance is a recalcitrant feature of the logic and that proof structures in hyperformal logics are quite inflexible.
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  37. Generality.Nathan Salmon - 2012 - Philosophical Studies 161 (3):471-481.
    A distinction is drawn among predicates, open sentences (or open formulas), and general terms, including general-term phrases. Attaching a copula, perhaps together with an article, to a general term yields a predicate. Predicates can also be obtained through lambda-abstraction on an open sentence. The issue of designation and semantic content for each type of general expression is investigated. It is argued that the designatum of a general term is a universal, e.g., a kind, whereas the designatum of a predicate (...)
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  38. The Logical Contingency of Identity.Hanoch Ben-Yami - 2018 - European Journal of Analytic Philosophy 14 (2):5-10.
    I show that intuitive and logical considerations do not justify introducing Leibniz’s Law of the Indiscernibility of Identicals in more than a limited form, as applying to atomic formulas. Once this is accepted, it follows that Leibniz’s Law generalises to all formulas of the first-order Predicate Calculus but not to modal formulas. Among other things, identity turns out to be logically contingent.
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  39. On the translation from quantified modal logic to counterpart theory.Cristina Nencha - 2022 - Synthese 200 (5):1-15.
    Lewis (1968) claims that his language of Counterpart Theory (CT) interprets modal discourse and he adverts to a translation scheme from the language of Quantifed Modal Logic (QML) to CT. However, everybody now agrees that his original translation scheme does not always work, since it does not always preserve the ‘intuitive’ meaning of the translated QML-formulas. Lewis discusses this problem with regard to the Necessitist Thesis, and I will extend his discourse to the analysis of the Converse Barcan Formula. (...)
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  40. Analytic Tableaux for all of SIXTEEN 3.Stefan Wintein & Reinhard Muskens - 2015 - Journal of Philosophical Logic 44 (5):473-487.
    In this paper we give an analytic tableau calculus P L 1 6 for a functionally complete extension of Shramko and Wansing’s logic. The calculus is based on signed formulas and a single set of tableau rules is involved in axiomatising each of the four entailment relations ⊧ t, ⊧ f, ⊧ i, and ⊧ under consideration—the differences only residing in initial assignments of signs to formulas. Proving that two sets of formulas are in one of the (...)
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  41. The Logical Burdens of Proof. Assertion and Hypothesis.Daniele Chiffi & Fabien Schang - 2017 - Logic and Logical Philosophy 26 (4):1-22.
    The paper proposes two logical analyses of (the norms of) justification. In a first, realist-minded case, truth is logically independent from justification and leads to a pragmatic logic LP including two epistemic and pragmatic operators, namely, assertion and hypothesis. In a second, antirealist-minded case, truth is not logically independent from justification and results in two logical systems of information and justification: AR4 and AR4¢, respectively, provided with a question-answer semantics. The latter proposes many more epistemic agents, each corresponding to a (...)
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  42. The logic of viewpoints.Antti Hautamäki - 1983 - Studia Logica 42 (2-3):187 - 196.
    In this paper a propositional logic of viewpoints is presented. The language of this logic consists of the usual modal operatorsL (of necessity) andM (of possibility) as well as of two new operatorsA andR. The intuitive interpretations ofA andR are from all viewpoints and from some viewpoint, respectively. Semantically the language is interpreted by using Kripke models augmented with sets of viewpoints and with a new alternativeness relation for the operatorA. Truth values of formulas are evaluated with respect to (...)
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  43. Tools for Thought: The Case of Mathematics.Valeria Giardino - 2018 - Endeavour 2 (42):172-179.
    The objective of this article is to take into account the functioning of representational cognitive tools, and in particular of notations and visualizations in mathematics. In order to explain their functioning, formulas in algebra and logic and diagrams in topology will be presented as case studies and the notion of manipulative imagination as proposed in previous work will be discussed. To better characterize the analysis, the notions of material anchor and representational affordance will be introduced.
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  44. Normalisation for Bilateral Classical Logic with some Philosophical Remarks.Nils Kürbis - 2021 - Journal of Applied Logics 2 (8):531-556.
    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 (...)
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  45. An Argument for Completely General Facts.Landon D. C. Elkind - 2021 - Journal for the History of Analytical Philosophy 9 (7).
    In his 1918 logical atomism lectures, Russell argued that there are no molecular facts. But he posed a problem for anyone wanting to avoid molecular facts: we need truth-makers for generalizations of molecular formulas, but such truth-makers seem to be both unavoidable and to have an abominably molecular character. Call this the problem of generalized molecular formulas. I clarify the problem here by distinguishing two kinds of generalized molecular formula: incompletely generalized molecular formulas and completely generalized molecular (...)
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  46. Rightness as Fairness: A Moral and Political Theory.Marcus Arvan - 2016 - New York: Palgrave MacMillan.
    This book argues that moral philosophy should be based on seven scientific principles of theory selection. It then argues that a new moral theory—Rightness as Fairness—satisfies those principles more successfully than existing theories. Chapter 1 explicates the seven principles of theory-selection, arguing that moral philosophy must conform to them to be truth-apt. Chapter 2 argues those principles jointly support founding moral philosophy in known facts of empirical moral psychology: specifically, our capacities for mental time-travel and modal imagination. Chapter 2 then (...)
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  47. A Binary Quantifier for Definite Descriptions in Intuitionist Negative Free Logic: Natural Deduction and Normalisation.Nils Kürbis - 2019 - Bulletin of the Section of Logic 48 (2):81-97.
    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.
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  48. Normalisation and subformula property for a system of intuitionistic logic with general introduction and elimination rules.Nils Kürbis - 2021 - Synthese 199 (5-6):14223-14248.
    This paper studies a formalisation of intuitionistic logic by Negri and von Plato which has general introduction and elimination rules. The philosophical 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 (...)
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  49. A small reflection principle for bounded arithmetic.Rineke Verbrugge & Albert Visser - 1994 - Journal of Symbolic Logic 59 (3):785-812.
    We investigate the theory IΔ 0 + Ω 1 and strengthen [Bu86. Theorem 8.6] to the following: if NP ≠ co-NP. then Σ-completeness for witness comparison formulas is not provable in bounded arithmetic. i.e. $I\delta_0 + \Omega_1 + \nvdash \forall b \forall c (\exists a(\operatorname{Prf}(a.c) \wedge \forall = \leq a \neg \operatorname{Prf} (z.b))\\ \rightarrow \operatorname{Prov} (\ulcorner \exists a(\operatorname{Prf}(a. \bar{c}) \wedge \forall z \leq a \neg \operatorname{Prf}(z.\bar{b})) \urcorner)).$ Next we study a "small reflection principle" in bounded arithmetic. We prove that (...)
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  50. Normalisation and subformula property for a system of classical logic with Tarski’s rule.Nils Kürbis - 2021 - Archive for Mathematical Logic 61 (1):105-129.
    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 (...)
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