scholarly journals First-order justification logic with constant domain semantics

2020 ◽  
Vol 30 (1) ◽  
pp. 193-216
Author(s):  
Melvin Fitting ◽  
Felipe Salvatore
Keyword(s):  
Justification Logic ◽  
Propositional Justification ◽  
Logic Of Proofs ◽  
First Order ◽  
Barcan Formula ◽  
Constant Domain ◽  
New Family ◽  
Domain Semantics ◽  
Constant Domains ◽  
Completeness Theorems

Abstract Justification logic is a term used to identify a relatively new family of modal-like logics. There is an established literature about propositional justification logic, but incursions on the first-order case are scarce. In this paper we present a constant domain semantics for the first-order logic of proofs with the Barcan Formula (FOLPb); then we prove Soundness and Completeness Theorems. A monotonic semantics for a version of this logic without the Barcan Formula is already in the literature, but constant domains require substantial new machinery, which may prove useful in other contexts as well. Although we work mainly with one system, we also indicate how to generalize these results for the quantified version of JT45, the justification counterpart of the modal logic S5. We believe our methods are more generally applicable, but initially examining specific cases should make the work easier to follow.

scholarly journals On the correspondence between nested calculi and semantic systems for intuitionistic logics

2020 ◽  
Author(s):  
Tim Lyon
Keyword(s):  
Intuitionistic Logic ◽  
Structural Rules ◽  
First Order ◽  
Constant Domains ◽  
The Relationship ◽  
Intuitionistic Logics

Abstract This paper studies the relationship between labelled and nested calculi for propositional intuitionistic logic, first-order intuitionistic logic with non-constant domains and first-order intuitionistic logic with constant domains. It is shown that Fitting’s nested calculi naturally arise from their corresponding labelled calculi—for each of the aforementioned logics—via the elimination of structural rules in labelled derivations. The translational correspondence between the two types of systems is leveraged to show that the nested calculi inherit proof-theoretic properties from their associated labelled calculi, such as completeness, invertibility of rules and cut admissibility. Since labelled calculi are easily obtained via a logic’s semantics, the method presented in this paper can be seen as one whereby refined versions of labelled calculi (containing nested calculi as fragments) with favourable properties are derived directly from a logic’s semantics.

scholarly journals What the Tortoise Said to Achilles: Lewis Carroll’s Paradox in Terms of Hilbert Arithmetic

2021 ◽  
Author(s):  
Vasil Penchev
Keyword(s):  
Hidden Variables ◽  
Formal Structure ◽  
Lewis Carroll ◽  
First Order ◽  
Logical Paradox ◽  
Completeness Of Quantum Mechanics ◽  
The One ◽  
In Virtue Of ◽  
The Liar ◽  
Completeness Theorems

Lewis Carroll, both logician and writer, suggested a logical paradox containing furthermore two connotations (connotations or metaphors are inherent in literature rather than in mathematics or logics). The paradox itself refers to implication demonstrating that an intermediate implication can be always inserted in an implication therefore postponing its ultimate conclusion for the next step and those insertions can be iteratively and indefinitely added ad lib, as if ad infinitum. Both connotations clear up links due to the shared formal structure with other well-known mathematical observations: (1) the paradox of Achilles and the Turtle; (2) the transitivity of the relation of equality. Analogically to (1), one can juxtapose the paradox of the Liar (for Lewis Carroll’s paradox) and that of the arrow (for “Achilles and the Turtle”), i.e. a logical paradox, on the one hand, and an aporia of motion, on the other hand, suggesting a shared formal structure of both, which can be called “ontological”, on which basis “motion” studied by physics and “conclusion” studied by logic can be unified being able to bridge logic and physics philosophically in a Hegelian manner: even more, the bridge can be continued to mathematics in virtue of (2), which forces the equality (for its property of transitivity) of any two quantities to be postponed analogically ad lib and ad infinitum. The paper shows that Hilbert arithmetic underlies naturally Lewis Carroll’s paradox admitting at least three interpretations linked to each other by it: mathematical, physical and logical. Thus, it can be considered as both generalization and solution of his paradox therefore naturally unifying the completeness of quantum mechanics (i.e. the absence of hidden variables) and eventual completeness of mathematics as the same and isomorphic to the completeness of propositional logic in relation to set theory as a first-order logic (in the sense of Gödel (1930)’s completeness theorems).

On the strong semantical completeness of the intuitionistic predicate calculus

1968 ◽  
Vol 33 (1) ◽  
pp. 1-7 ◽  
Author(s):  
Richmond H. Thomason
Keyword(s):  
Model Theory ◽  
Completeness Theorem ◽  
Predicate Calculus ◽  
First Order ◽  
Semantic Tableaux ◽  
Weak Completeness ◽  
Skolem Theorem ◽  
Completeness Theorems

In Kripke [8] the first-order intuitionjstic predicate calculus (without identity) is proved semantically complete with respect to a certain model theory, in the sense that every formula of this calculus is shown to be provable if and only if it is valid. Metatheorems of this sort are frequently called weak completeness theorems—the object of the present paper is to extend Kripke's result to obtain a strong completeness theorem for the intuitionistic predicate calculus of first order; i.e., we will show that a formula A of this calculus can be deduced from a set Γ of formulas if and only if Γ implies A. In notes 3 and 5, below, we will indicate how to account for identity, as well. Our proof of the completeness theorem employs techniques adapted from Henkin [6], and makes no use of semantic tableaux; this proof will also yield a Löwenheim-Skolem theorem for the modeling.

New family of parasupercoherent states: entanglement, uncertainties, and statistical properties

2020 ◽  
Vol 98 (10) ◽  
pp. 953-958
Author(s):  
Amin Motamedinasab ◽  
Azam Anbaraki ◽  
Davood Afshar ◽  
Mojtaba Jafarpour
Keyword(s):  
Harmonic Oscillator ◽  
Arbitrary Order ◽  
Annihilation Operator ◽  
The Other ◽  
Mandel Parameter ◽  
First Order ◽  
New Family ◽  
Wide Range ◽  
New States ◽  
Minimum Uncertainty

The general parasupersymmetric annihilation operator of arbitrary order does not reduce to the Kornbluth–Zypman general supersymmetric annihilation operator for the first order. In this paper, we introduce an annihilation operator for a parasupersymmetric harmonic oscillator that in the first order matches with the Kornblouth–Zypman results. Then, using the latter operator, we obtain the parasupercoherent states and calculate their entanglement, uncertainties, and statistics. We observe that these states are entangled for any arbitrary order of parasupersymmetry and their entanglement goes to zero for the large values of the coherency parameter. In addition, we find that the maximum of the entanglement of parasupercoherent states is a decreasing function of the parasupersymmetry order. Moreover, these states are minimum uncertainty states for large and also small values of the coherency parameter. Furthermore, these states show squeezing in one of the quadrature operators for a wide range of the coherency parameter, while no squeezing in the other quadrature operator is observed at all. In addition, using the Mandel parameter, we find that the statistics of these new states are subPoissonian for small values of the coherency parameter.

An embedding of classical logic in S4

1970 ◽  
Vol 35 (4) ◽  
pp. 529-534 ◽  
Author(s):  
Melvin Fitting
Keyword(s):  
Intuitionistic Logic ◽  
Model Theory ◽  
Classical Logic ◽  
First Order ◽  
Barcan Formula ◽  
Complete Sequences

There are well-known embeddings of intuitionistic logic into S4 and of classical logic into S5. In this paper we give a related embedding of (first order) classical logic directly into (first order) S4, with or without the Barcan formula. If one reads the necessity operator of S4 as ‘provable’, the translation may be roughly stated as: truth may be replaced by provable consistency. A proper statement will be found below. The proof is based ultimately on the notion of complete sequences used in Cohen's technique of forcing [1], and is given in terms of Kripke's model theory [3], [4].

A NEW FAMILY OF FIRST-ORDER TIME-DELAYED CHAOTIC SYSTEMS

2011 ◽  
Vol 21 (09) ◽  
pp. 2547-2558 ◽  
Author(s):  
XIAOMING ZHANG ◽  
JUFANG CHEN ◽  
JIANHUA PENG
Keyword(s):  
Chaotic Systems ◽  
Linear Time ◽  
Period Doubling ◽  
First Order ◽  
New Family ◽  
Delayed Differential Equations ◽  
Stable Periodic ◽  
Delayed System ◽  
Bifurcation Direction ◽  
General Method

A general method for formulating first-order time-delayed chaotic systems with simple linear time-delayed term is proposed. The formulated systems are realized with electronic circuit experiments. In order to determine the unknown coefficients in a general delayed differential equations for having chaotic solutions, we follow the route of period-doubling bifurcation to chaos. Firstly, the conditions for a time-delayed system having a stable periodic solution, generating from a destablized steady state, is analyzed with Hopf bifurcation theory. Then the delay time parameter is changed according to the bifurcation direction to search the chaotic state, which is identified by the Lyapunov exponents spectra. The theoretical analysis is well confirmed by numerical simulations and circuit experiments.

scholarly journals Completeness theorems for first-order logic analysed in constructive type theory

2021 ◽  
Vol 31 (1) ◽  
pp. 112-151
Author(s):  
Yannick Forster ◽  
Dominik Kirst ◽  
Dominik Wehr
Keyword(s):  
Type Theory ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Constructive Type Theory ◽  
Constructive Type ◽  
The Standard Model ◽  
Coq Proof Assistant ◽  
Game Theoretic ◽  
Completeness Theorems

Abstract We study various formulations of the completeness of first-order logic phrased in constructive type theory and mechanised in the Coq proof assistant. Specifically, we examine the completeness of variants of classical and intuitionistic natural deduction and sequent calculi with respect to model-theoretic, algebraic, and game-theoretic semantics. As completeness with respect to the standard model-theoretic semantics à la Tarski and Kripke is not readily constructive, we analyse connections of completeness theorems to Markov’s Principle and Weak K̋nig’s Lemma and discuss non-standard semantics admitting assumption-free completeness. We contribute a reusable Coq library for first-order logic containing all results covered in this paper.

A unified completeness theorem for quantified modal logics

2002 ◽  
Vol 67 (4) ◽  
pp. 1483-1510 ◽  
Author(s):  
Giovanna Corsi
Keyword(s):  
Modal Logic ◽  
Completeness Theorem ◽  
Modal Logics ◽  
General Strategy ◽  
Quantified Modal Logic ◽  
Barcan Formula ◽  
Completeness Theorems

AbstractA general strategy for proving completeness theorems for quantified modal logics is provided. Starting from free quantified modal logic K. with or without identity, extensions obtained either by adding the principle of universal instantiation or the converse of the Barcan formula or the Barcan formula are considered and proved complete in a uniform way. Completeness theorems are also shown for systems with the extended Barcan rule as well as for some quantified extensions of the modal logic B. The incompleteness of Q°.B + BF is also proved.

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