scholarly journals From 2-Sequents and Linear Nested Sequents to Natural Deduction for Normal Modal Logics

2021 ◽  
Vol 22 (3) ◽  
pp. 1-29
Author(s):  
Simone Martini ◽  
Andrea Masini ◽  
Margherita Zorzi
Keyword(s):  
Natural Deduction ◽  
Modal Logics ◽  
Structural Rule ◽  
Elimination Rule ◽  
Accessibility Relation ◽  
Formal Systems ◽  
Explicit Reference ◽  
Deduction Rules ◽  
Nested Sequents ◽  
Normal Modal Logics

We extend to natural deduction the approach of Linear Nested Sequents and of 2-Sequents. Formulas are decorated with a spatial coordinate, which allows a formulation of formal systems in the original spirit of natural deduction: only one introduction and one elimination rule per connective, no additional (structural) rule, no explicit reference to the accessibility relation of the intended Kripke models. We give systems for the normal modal logics from K to S4. For the intuitionistic versions of the systems, we define proof reduction, and prove proof normalization, thus obtaining a syntactical proof of consistency. For logics K and K4 we use existence predicates (à la Scott) for formulating sound deduction rules.

Suppose and Tell

2020 ◽  
Author(s):  
Timothy Williamson
Keyword(s):  
Conditional Probability ◽  
Natural Deduction ◽  
Special Kind ◽  
Context Sensitivity ◽  
Modal Operator ◽  
Counterfactual Conditionals ◽  
Deduction Rules ◽  
Psychological Sense

The book argues that our use of conditionals is governed by imperfectly reliable heuristics, in the psychological sense of fast and frugal (or quick and dirty) ways of assessing them. The primary heuristic is this: to assess ‘If A, C’, suppose A and on that basis assess C; whatever attitude you take to C conditionally on A (such as acceptance, rejection, or something in between) take unconditionally to ‘If A, C’. This heuristic yields both the equation of the probability of ‘If A, C’ with the conditional probability of C on A and standard natural deduction rules for the conditional. However, these results can be shown to make the heuristic implicitly inconsistent, and so less than fully reliable. There is also a secondary heuristic: pass conditionals freely from one context to another under normal conditions for acceptance of sentences on the basis of memory and testimony. The effect of the secondary heuristic is to undermine interpretations on which ‘if’ introduces a special kind of context-sensitivity. On the interpretation which makes best sense of the two heuristics, ‘if’ is simply the truth-functional conditional. Apparent counterexamples to truth-functionality are artefacts of reliance on the primary heuristic in cases where it is unreliable. The second half of the book concerns counterfactual conditionals, as expressed with ‘if’ and ‘would’. It argues that ‘would’ is an independently meaningful modal operator for contextually restricted necessity: the meaning of counterfactuals is simply that derived compositionally from the meanings of their constituents, including ‘if’ and ‘would’, making them contextually restricted strict conditionals.

scholarly journals Fusions of Description Logics and Abstract Description Systems

2002 ◽  
Vol 16 ◽  
pp. 1-58 ◽  
Author(s):  
F. Baader ◽  
C. Lutz ◽  
H. Sturm ◽  
F. Wolter
Keyword(s):  
Large Class ◽  
Description Logics ◽  
General Setting ◽  
Modal Logics ◽  
Common Generalization ◽  
Abstract Description ◽  
Normal Modal Logics ◽  
Combining Logics

Fusions are a simple way of combining logics. For normal modal logics, fusions have been investigated in detail. In particular, it is known that, under certain conditions, decidability transfers from the component logics to their fusion. Though description logics are closely related to modal logics, they are not necessarily normal. In addition, ABox reasoning in description logics is not covered by the results from modal logics. In this paper, we extend the decidability transfer results from normal modal logics to a large class of description logics. To cover different description logics in a uniform way, we introduce abstract description systems, which can be seen as a common generalization of description and modal logics, and show the transfer results in this general setting.

Modal sequents for normal modal logics

1993 ◽  
Vol 39 (1) ◽  
pp. 231-240 ◽  
Author(s):  
Claudio Cerrato
Keyword(s):  
Modal Logics ◽  
Normal Modal Logics

Consequences of the Suppositional Rule

2020 ◽  
pp. 31-67
Author(s):  
Timothy Williamson
Keyword(s):  
Conditional Probability ◽  
Natural Deduction ◽  
David Lewis ◽  
Linguistic Practices ◽  
Deduction Rules

This chapter argues that the Suppositional Rule is a fallible heuristic, because it has inconsistent consequences. They arise in several ways: (i) it implies standard natural deduction rules for ‘if’, and analogous but incompatible rules for refutation in place of proof; (ii) it implies the equation of the probability of ‘If A, C’ with the conditional probability of C on A, which is subject to the trivialization results of David Lewis and others; (iii) its application to complex attitudes generates further inconsistencies. The Suppositional Rule is compared to inconsistent principles built into other linguistic practices: disquotation for ‘true’ and ‘false’ generate Liar-like paradoxes; tolerance principles for vague expressions generate sorites paradoxes. Their status as fallible, semantically invalid but mostly reliable heuristics is not immediately available to competent speakers.

scholarly journals Simplified Kripke-Style Semantics for Some Normal Modal Logics

Studia Logica ◽  
2019 ◽  
Vol 108 (3) ◽  
pp. 451-476 ◽  
Author(s):  
Andrzej Pietruszczak ◽  
Mateusz Klonowski ◽  
Yaroslav Petrukhin
Keyword(s):  
Modal Logics ◽  
Normal Modal Logics
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