scholarly journals Sequent Calculi for the Propositional Logic of HYPE

Studia Logica ◽  
2021 ◽  
Author(s):  
Martin Fischer
Keyword(s):  
Mathematical Logic ◽  
Propositional Logic ◽  
The Other ◽  
Cut Elimination ◽  
Philosophical Logic ◽  
Admissible Rule ◽  
Sequent Calculi ◽  
Complete Proof ◽  
Elimination Procedure ◽  
The One

AbstractIn this paper we discuss sequent calculi for the propositional fragment of the logic of HYPE. The logic of HYPE was recently suggested by Leitgeb (Journal of Philosophical Logic 48:305–405, 2019) as a logic for hyperintensional contexts. On the one hand we introduce a simple $$\mathbf{G1}$$ G 1 -system employing rules of contraposition. On the other hand we present a $$\mathbf{G3}$$ G 3 -system with an admissible rule of contraposition. Both systems are equivalent as well as sound and complete proof-system of HYPE. In order to provide a cut-elimination procedure, we expand the calculus by connections as introduced in Kashima and Shimura (Mathematical Logic Quarterly 40:153–172, 1994).

Gentzen sequent calculi for some intuitionistic modal logics

2019 ◽  
Vol 27 (4) ◽  
pp. 596-623
Author(s):  
Zhe Lin ◽  
Minghui Ma
Keyword(s):  
Propositional Logic ◽  
Sequent Calculus ◽  
Modal Logics ◽  
Cut Elimination ◽  
Sequent Calculi ◽  
Intuitionistic Propositional Logic ◽  
Propositional Language ◽  
Intuitionistic Modal Logics ◽  
Subformula Property ◽  
Gentzen Sequent Calculus

Abstract Intuitionistic modal logics are extensions of intuitionistic propositional logic with modal axioms. We treat with two modal languages ${\mathscr{L}}_\Diamond $ and $\mathscr{L}_{\Diamond ,\Box }$ which extend the intuitionistic propositional language with $\Diamond $ and $\Diamond ,\Box $, respectively. Gentzen sequent calculi are established for several intuitionistic modal logics. In particular, we introduce a Gentzen sequent calculus for the well-known intuitionistic modal logic $\textsf{MIPC}$. These sequent calculi admit cut elimination and subformula property. They are decidable.

scholarly journals Relative Necessity and Propositional Quantification

2019 ◽  
Vol 49 (4) ◽  
pp. 703-726
Author(s):  
Alexander Roberts
Keyword(s):  
Mathematical Logic ◽  
Propositional Logic ◽  
Recent Article ◽  
Symbolic Logic ◽  
Philosophical Logic ◽  
Alternative Account ◽  
Standard Account ◽  
Current Article ◽  
Modal Propositional Logic ◽  
Propositional Quantification

AbstractFollowing Smiley’s (The Journal of Symbolic Logic, 28, 113–134 1963) influential proposal, it has become standard practice to characterise notions of relative necessity in terms of simple strict conditionals. However, Humberstone (Reports on Mathematical Logic, 13, 33–42 1981) and others have highlighted various flaws with Smiley’s now standard account of relative necessity. In their recent article, Hale and Leech (Journal of Philosophical Logic, 46, 1–26 2017) propose a novel account of relative necessity designed to overcome the problems facing the standard account. Nevertheless, the current article argues that Hale & Leech’s account suffers from its own defects, some of which Hale & Leech are aware of but underplay. To supplement this criticism, the article offers an alternative account of relative necessity which overcomes these defects. This alternative account is developed in a quantified modal propositional logic and is shown model-theoretically to meet several desiderata of an account of relative necessity.

scholarly journals qbf2epr: A Tool for Generating EPR Formulas from QBF

10.29007/2b5d ◽  
2018 ◽  
Author(s):  
Martina Seidl ◽  
Florian Lonsing ◽  
Armin Biere
Keyword(s):  
Propositional Logic ◽  
Decision Problem ◽  
The Other ◽  
Quantified Boolean Formulas ◽  
Other Hand ◽  
Boolean Formulas ◽  
The One

We present the tool qbf2epr which translates quantified Boolean formulas (QBF) toformulas in effectively propositional logic (EPR). The decision problem of QBF is theprototypical problem for PSPACE, whereas EPR is NEXPTIME-complete. Thus QBF isembedded in a formalism, which is potentially more succinct. The motivation for this workis twofold. On the one hand, our tool generates challenging benchmarks for EPR solvers.On the other hand, we are interested in how EPR solvers perform compared to QBF solvers and if there are techniques implemented in EPR solvers which would also be valuable in QBF solvers and vice versa.

Set theory, different systems of

2018 ◽  
Author(s):  
Michael Potter
Keyword(s):  
Twentieth Century ◽  
Mathematical Logic ◽  
Set Theory ◽  
The Other ◽  
Von Neumann ◽  
Class Theory ◽  
Iterative Conception ◽  
The One ◽  
Limitation Of Size ◽  
As If

To begin with we shall use the word ‘collection’ quite broadly to mean anything the identity of which is solely a matter of what its members are (including ‘sets’ and ‘classes’). Which collections exist? Two extreme positions are initially appealing. The first is to say that all do. Unfortunately this is inconsistent because of Russell’s paradox: the collection of all collections which are not members of themselves does not exist. The second is to say that none do, but to talk as if they did whenever such talk can be shown to be eliminable and therefore harmless. This is consistent but far too weak to be of much use. We need an intermediate theory. Various theories of collections have been proposed since the start of the twentieth century. What they share is the axiom of ‘extensionality’, which asserts that any two sets which have exactly the same elements must be identical. This is just a matter of definition: objects which do not satisfy extensionality are not collections. Beyond extensionality, theories differ. The most popular among mathematicians is Zermelo–Fraenkel set theory (ZF). A common alternative is von Neumann–Bernays–Gödel class theory (NBG), which allows for the same sets but also has proper classes, that is, collections whose members are sets but which are not themselves sets (such as the class of all sets or the class of all ordinals). Two general principles have been used to motivate the axioms of ZF and its relatives. The first is the iterative conception, according to which sets occur cumulatively in layers, each containing all the members and subsets of all previous layers. The second is the doctrine of limitation of size, according to which the ‘paradoxical sets’ (that is, the proper classes of NBG) fail to be sets because they are in some sense too big. Neither principle is altogether satisfactory as a justification for the whole of ZF: for example, the replacement schema is motivated only by limitation of size; and ‘foundation’ is motivated only by the iterative conception. Among the other systems of set theory to have been proposed, the one that has received widespread attention is Quine’s NF (from the title of an article, ‘New Foundations for Mathematical Logic’), which seeks to avoid paradox by means of a syntactic restriction but which has not been provided with an intuitive justification on the basis of any conception of set. It is known that if NF is consistent then ZF is consistent, but the converse result has still not been proved.

Multi-focused cut elimination

2017 ◽  
Vol 28 (5) ◽  
pp. 614-650
Author(s):  
TAUS BROCK-NANNESTAD ◽  
NICOLAS GUENOT
Keyword(s):  
Simple Proof ◽  
Linear Logic ◽  
Cut Elimination ◽  
Sequent Calculi ◽  
Elimination Procedure ◽  
The Impact

We investigate cut elimination in multi-focused sequent calculi and the impact on the cut elimination proof of design choices in such calculi. The particular design we advocate is illustrated by a multi-focused calculus for full linear logic using an explicitly polarised syntax and incremental focus handling, for which we provide a syntactic cut elimination procedure. We discuss the effect of cut elimination on the structure of proofs, leading to a conceptually simple proof exploiting the strong structure of multi-focused proofs.

scholarly journals Theories of truth based on four-valued infectious logics

2018 ◽  
Vol 28 (5) ◽  
pp. 712-746 ◽  
Author(s):  
Bruno Da Ré ◽  
Federico Pailos ◽  
Damian Szmuc
Keyword(s):  
Proof Theory ◽  
The Other ◽  
Sequent Calculi ◽  
Theories Of Truth ◽  
Other Hand ◽  
Truth Teller ◽  
Truth Value ◽  
The One ◽  
The Liar

AbstractInfectious logics are systems that have a truth-value that is assigned to a compound formula whenever it is assigned to one of its components. This paper studies four-valued infectious logics as the basis of transparent theories of truth. This take is motivated (i) as a way to treat different pathological sentences (like the Liar and the Truth-Teller) differently, namely, by allowing some of them to be truth-value gluts and some others to be truth-value gaps and (ii) as a way to treat the semantic pathology suffered by at least some of these sentences as infectious. This leads us to consider four distinct four-valued logics: one where truth-value gaps are infectious, but gluts are not; one where truth-value gluts are infectious, but gaps are not; and two logics where both gluts and gaps are infectious, in some sense. Additionally, we focus on the proof theory of these systems, by offering a discussion of two related topics. On the one hand, we prove some limitations regarding the possibility of providing standard Gentzen sequent calculi for these systems, by dualizing and extending some recent results for infectious logics. On the other hand, we provide sound and complete four-sided sequent calculi, arguing that the most important technical and philosophical features taken into account to usually prefer standard calculi are, indeed, enjoyed by the four-sided systems.

scholarly journals On permuting cut with contraction

2000 ◽  
Vol 10 (2) ◽  
pp. 99-136 ◽  
Author(s):  
MIRJANA BORISAVLJEVIĆ ◽  
KOSTA DOšEN ◽  
ZORAN PETRIĆ
Keyword(s):  
Propositional Logic ◽  
Cut Elimination ◽  
Intuitionistic Propositional Logic ◽  
Cut Rule ◽  
Elimination Procedure ◽  
Made In

This paper presents a cut-elimination procedure for intuitionistic propositional logic in which cut is eliminated directly, without introducing the multiple-cut rule mix, and in which pushing cut above contraction is one of the reduction steps. The presentation of this procedure is preceded by an analysis of Gentzen's mix-elimination procedure, made in the perspective of permuting cut with contraction. We also show that in the absence of implication, pushing cut above contraction does not pose problems for directly eliminating cut.

scholarly journals Qinghua School of Logic and the Origins of Taiwanese Studies in Modern Logic

Asian Studies ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 231-250
Author(s):  
Jan Vrhovski
Keyword(s):  
Mathematical Logic ◽  
Academic Discipline ◽  
The Other ◽  
Republican China ◽  
Modern Logic ◽  
Founding Fathers ◽  
Mou Zongsan ◽  
The One

The article investigates the early thought of Mou Zongsan and Yin Haiguang, two important founding fathers of Taiwanese philosophy, who contributed significantly to its formation as an academic discipline in the two decades following 1949. The article reveals how their ideas related to modern logic originated from the so-called “Qinghua School of (Mathematical) Logic”. Herewith, the article tries to provide a platform that can be used to answer the questions of continuity and succession between the studies of modern logic as conducted at the most progressive (modernised) universities in late Republican China (especially Qinghua University) on the one side, and the formation and development of studies in logic in post-1949 Taiwan, on the other.

scholarly journals Minimal Sartre: Diagonalization and Pure Reflection

2018 ◽  
Vol 1 (1) ◽  
pp. 360-379
Author(s):  
John Bova
Keyword(s):  
Mathematical Logic ◽  
Early Period ◽  
The Other ◽  
Existential Philosophy ◽  
Other Hand ◽  
Phenomenological Reflection ◽  
The One ◽  
Metaphysical Position

AbstractThese remarks take up the reflexive problematics of Being and Nothingness and related texts from a metalogical perspective. A mutually illuminating translation is posited between, on the one hand, Sartre’s theory of pure reflection, the linchpin of the works of Sartre’s early period and the site of their greatest difficulties, and, on the other hand, the quasi-formalism of diagonalization, the engine of the classical theorems of Cantor, Godel, Tarski, Turing, etc. Surprisingly, the dialectic of mathematical logic from its inception through the discovery of the diagonal theorems can be recognized as a particularly clear instance of the drama of reflection according to Sartre, especially in the positing and overcoming of its proper valueideal, viz. the synthesis of consistency and completeness. Conversely, this translation solves a number of systematic problems about pure reflection’s relations to accessory reflection, phenomenological reflection, pre-reflective self-consciousness, conversion, and value. Negative foundations, the metaphysical position emerging from this translation between existential philosophy and metalogic, concurs by different paths with Badiou’s Being and Event in rejecting both ontotheological foundationalisms and constructivist antifoundationalisms.

Medical Diagnosis: A Real Life Application of Logic

1978 ◽  
Vol 71 (6) ◽  
pp. 499-504
Author(s):  
Kenneth P. Goldberg
Keyword(s):  
Decision Making ◽  
Mathematical Logic ◽  
Medical Diagnosis ◽  
Real Life ◽  
Junior College ◽  
The Other ◽  
College Level ◽  
Human Reason ◽  
The One ◽  
Rational Reasoning

Of all the mathematical topics taught at the high school and junior college level, logic must certainly be one of the most frustrating for both students and teachers. On the one hand, the student is told that logic is the necessary basis on which all rational reasoning and decision making rests. On the other hand, since few situations involving human reason are simple enough to permit the application of mathematical logic, it is extremely difficult to discuss and illustrate the use of logic in a believable and interesting manner.

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