scholarly journals A modal extension of intuitionist logic.

1965 ◽  
Vol 6 (2) ◽  
pp. 142-146 ◽  
Author(s):  
R. A. Bull
Keyword(s):  
Intuitionist Logic ◽  
Modal Extension

scholarly journals Tree Trimming: Four Non-Branching Rules for Priest’s Introduction to Non-Classical Logic

2017 ◽  
Vol 12 (2) ◽  
Author(s):  
Marilynn Johnson
Keyword(s):  
Modal Logic ◽  
Classical Logic ◽  
Variable Domain ◽  
Free Logic ◽  
Intuitionist Logic ◽  
Branching Rule ◽  
Branching Rules ◽  
Graham Priest

In An Introduction to Non-Classical Logic: From If to Is Graham Priest (2008) presents branching rules in Free Logic, Variable Domain Modal Logic, and Intuitionist Logic. I propose a simpler, non-branching rule to replace Priest’s rule for universal instantiation in Free Logic, a second, slightly modified version of this rule to replace Priest’s rule for universal instantiation in Variable Domain Modal Logic, and third and fourth rules, further modifying the second rule, to replace Priest’s branching universal and particular instantiation rules in Intuitionist Logic. In each of these logics the proposed rule leads to tableaux with fewer branches. In Intuitionist logic, the proposed rules allow for the resolution of a particular problem Priest grapples with throughout the chapter. In this paper, I demonstrate that the proposed rules can greatly simplify tableaux and argue that they should be used in place of the rules given by Priest.

Multiscale Variability and Uncertainty Quantification Based on a Generalized Multiscale Markov Model

2010 ◽  
Author(s):  
Yan Wang
Keyword(s):  
Probability Theory ◽  
Bayesian Inference ◽  
Markov Model ◽  
Uncertainty Quantification ◽  
System Analysis ◽  
Scale Validation ◽  
Uncertainty Propagation ◽  
Bayes Rule ◽  
Imprecise Probability ◽  
Modal Extension

Variability is inherent randomness in systems, whereas uncertainty is due to lack of knowledge. In this paper, a generalized multiscale Markov (GMM) model is proposed to quantify variability and uncertainty simultaneously in multiscale system analysis. The GMM model is based on a new imprecise probability theory that has the form of generalized interval, which is a Kaucher or modal extension of classical set-based intervals to represent uncertainties. The properties of the new definitions of independence and Bayesian inference are studied. Based on a new Bayes’ rule with generalized intervals, three cross-scale validation approaches that incorporate variability and uncertainty propagation are also developed.

A modal extension of Jaśkowski’s discussive logic $\textbf{D}_\textbf{2}$

2019 ◽  
Vol 27 (4) ◽  
pp. 451-477 ◽  
Author(s):  
Krystyna Mruczek-Nasieniewska ◽  
Marek Nasieniewski ◽  
Andrzej Pietruszczak
Keyword(s):  
Extended Version ◽  
Modal Operators ◽  
Modal Extension ◽  
Modal Systems

Abstract In Jaśkowski’s model of discussion, discussive connectives represent certain interactions that can hold between debaters. However, it is not possible within the model for participants to use explicit modal operators. In the paper we present a modal extension of the discussive logic $\textbf{D}_{\textbf{2}}$ that formally corresponds to an extended version of Jaśkowski’s model of discussion that permits such a use. This logic is denoted by $\textbf{m}\textbf{D}_{\textbf{2}}$. We present philosophical motivations for the formulation of this logic. We also give syntactic characterizations of the logic and propose a comparison with certain other modal systems. In particular, we prove that $\textbf{m}\textbf{D}_{\textbf{2}}$ is neither normal nor regular. On the basis of the axiomatization of $\textbf{D}_{\textbf{2}}$, we give an axiomatization of $\textbf{m}\textbf{D}_{\textbf{2}}$. We also give another axiomatization which is not based on the axiomatization of $\textbf{D}_{\textbf{2}}$. Furthermore, we give a natural Kripke-style semantics for $\textbf{m}\textbf{D}_{\textbf{2}}$ and prove the respective adequacy theorems.

Some results for implicational calculi

1964 ◽  
Vol 29 (1) ◽  
pp. 33-39 ◽  
Author(s):  
R. A. Bull
Keyword(s):  
Finite Models ◽  
Intuitionist Logic

I shall refer to the implicational fragment of intuitionist logic, and its extension with the further axiomCCCCpqqrCCCpqrras IIC and OIC, respectively. The purpose of this paper is to apply a result due to Garrett Birkhoff to the extensions of IIC, and to the extensions of OIC in particular. The main result obtained is that every extension of OIC is characterised by finite models.

A new “feasible” arithmetic

2002 ◽  
Vol 67 (1) ◽  
pp. 104-116 ◽  
Author(s):  
Stephen Bellantoni ◽  
Martin Hofmann
Keyword(s):  
Modal Logic ◽  
Polynomial Time ◽  
Peano Arithmetic ◽  
Quantified Modal Logic ◽  
Modal Extension ◽  
Extension Rule ◽  
Small Set ◽  
Computable Functions

AbstractA classical quantified modal logic is used to define a “feasible” arithmetic whose provably total functions are exactly the polynomial-time computable functions. Informally, one understands ⃞∝ as “∝ is feasibly demonstrable”. differs from a system that is as powerful as Peano Arithmetic only by the restriction of induction to ontic (i.e., ⃞-free) formulas. Thus, is defined without any reference to bounding terms, and admitting induction over formulas having arbitrarily many alternations of unbounded quantifiers. The system also uses only a very small set of initial functions.To obtain the characterization, one extends the Curry-Howard isomorphism to include modal operations. This leads to a realizability translation based on recent results in higher-type ramified recursion. The fact that induction formulas are not restricted in their logical complexity, allows one to use the Friedman A translation directly.The development also leads us to propose a new Frege rule, the “Modal Extension” rule: if ⊢ ∝ a then ⊢ A ↔ ∝ for new symbol A.

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