Proof theory of modal logic, edited by Heinrich Wansing, Applied logic series, vol. 2, Kluwer Academic Publishers, Dordrecht, Boston, and London, 1996, x + 317 pp.

1999 ◽  
Vol 64 (4) ◽  
pp. 1825-1828
Author(s):  
Melvin Fitting
Keyword(s):  
Modal Logic ◽  
Proof Theory ◽  
Applied Logic

Proof Theory for Modal Logic

2011 ◽  
Vol 6 (8) ◽  
pp. 523-538 ◽  
Author(s):  
Sara Negri
Keyword(s):  
Modal Logic ◽  
Proof Theory

scholarly journals A Modal Proof Theory for Polynomial Coalgebras

2021 ◽  
Author(s):  
◽  
David Friggens
Keyword(s):  
Modal Logic ◽  
Computer Science ◽  
Programming Languages ◽  
Data Structures ◽  
Proof Theory ◽  
Object Oriented ◽  
Object Oriented Programming ◽  
Algebraic Methods ◽  
Mathematical Structures ◽  
Finite Constant

<p>The abstract mathematical structures known as coalgebras are of increasing interest in computer science for their use in modelling certain types of data structures and programs. Traditional algebraic methods describe objects in terms of their construction, whilst coalgebraic methods describe objects in terms of their decomposition, or observational behaviour. The latter techniques are particularly useful for modelling infinite data structures and providing semantics for object-oriented programming languages, such as Java. There have been many different logics developed for reasoning about coalgebras of particular functors, most involving modal logic. We define a modal logic for coalgebras of polynomial functors, extending Rößiger’s logic [33], whose proof theory was limited to using finite constant sets, by adding an operator from Goldblatt [11]. From the semantics we define a canonical coalgebra that provides a natural construction of a final coalgebra for the relevant functor. We then give an infinitary axiomatization and syntactic proof relation that is sound and complete for functors constructed from countable constant sets.</p>

scholarly journals POSITIVE LOGIC WITH ADJOINT MODALITIES: PROOF THEORY, SEMANTICS, AND REASONING ABOUT INFORMATION

2010 ◽  
Vol 3 (3) ◽  
pp. 351-373 ◽  
Author(s):  
MEHRNOOSH SADRZADEH ◽  
ROY DYCKHOFF
Keyword(s):  
Modal Logic ◽  
Multiagent Systems ◽  
Epistemic Logic ◽  
Proof Theory ◽  
Sequent Calculus ◽  
Dynamic Epistemic Logic ◽  
Algebraic Semantics ◽  
Sequent Calculi ◽  
Closure Operators ◽  
Positive Logic

We consider a simple modal logic whose nonmodal part has conjunction and disjunction as connectives and whose modalities come in adjoint pairs, but are not in general closure operators. Despite absence of negation and implication, and of axioms corresponding to the characteristic axioms of (e.g.) T, S4, and S5, such logics are useful, as shown in previous work by Baltag, Coecke, and the first author, for encoding and reasoning about information and misinformation in multiagent systems. For the propositional-only fragment of such a dynamic epistemic logic, we present an algebraic semantics, using lattices with agent-indexed families of adjoint pairs of operators, and a cut-free sequent calculus. The calculus exploits operators on sequents, in the style of “nested” or “tree-sequent” calculi; cut-admissibility is shown by constructive syntactic methods. The applicability of the logic is illustrated by reasoning about the muddy children puzzle, for which the calculus is augmented with extra rules to express the facts of the muddy children scenario.

On the Proof Theory of the Modal Logic Grz

1986 ◽  
Vol 32 (10-12) ◽  
pp. 145-148 ◽  
Author(s):  
M. Borga ◽  
P. Gentilini
Keyword(s):  
Modal Logic ◽  
Proof Theory

scholarly journals A Modal Proof Theory for Polynomial Coalgebras

2021 ◽  
Author(s):  
◽  
David Friggens
Keyword(s):  
Modal Logic ◽  
Computer Science ◽  
Programming Languages ◽  
Data Structures ◽  
Proof Theory ◽  
Object Oriented ◽  
Object Oriented Programming ◽  
Algebraic Methods ◽  
Mathematical Structures ◽  
Finite Constant

<p>The abstract mathematical structures known as coalgebras are of increasing interest in computer science for their use in modelling certain types of data structures and programs. Traditional algebraic methods describe objects in terms of their construction, whilst coalgebraic methods describe objects in terms of their decomposition, or observational behaviour. The latter techniques are particularly useful for modelling infinite data structures and providing semantics for object-oriented programming languages, such as Java. There have been many different logics developed for reasoning about coalgebras of particular functors, most involving modal logic. We define a modal logic for coalgebras of polynomial functors, extending Rößiger’s logic [33], whose proof theory was limited to using finite constant sets, by adding an operator from Goldblatt [11]. From the semantics we define a canonical coalgebra that provides a natural construction of a final coalgebra for the relevant functor. We then give an infinitary axiomatization and syntactic proof relation that is sound and complete for functors constructed from countable constant sets.</p>

On the proof theory of the modal logic for arithmetic provability

1981 ◽  
Vol 46 (3) ◽  
pp. 531-538 ◽  
Author(s):  
Daniel Leivant
Keyword(s):  
Modal Logic ◽  
Proof Theory ◽  
Additional Insight ◽  
Semantical Analysis ◽  
Constructive Proofs ◽  
Definite Improvement ◽  
Original Presentation ◽  
Sequential Calculus ◽  
Insight Into

The modal logic GL has been found by Solovay [13] to formalize the provable propositional properties of the provability-predicate for Peano's Arithmetic PA (cf. §1 below). We give several sequential calculi for GL, compare their merits, and use one calculus to syntactically derive several metamathematical results about GL.Some of our results have been proved model theoretically, and similar proofs are fairly straightforward for several of the remaining ones (G. Boolos and the referee have provided such proofs for 4.1, 4.3 and 5.1 below). However, our syntactic techniques often yield more concise and obviously constructive proofs, they offer additional insight into the nature of the systems considered, and are easily adaptable to systems for which semantical analysis is problematic.I am indebted to G. Boolos and to the referee for their valuable advice. The referee has suggested the rule GL of §3 below as an axiomatization of GL; the resulting sequential calculus has allowed a definite improvement of our original presentation.

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