Undecidability of first-order modal and intuitionistic logics with two variables and one monadic predicate letter

We prove that the positive fragment of first-order intuitionistic logic in the language with two variables and a single monadic predicate letter, without constants and equality, is undecidable. This holds true regardless of whether we consider semantics with expanding or constant domains. We then generalise this result to intervals [QBL, QKC] and [QBL, QFL], where QKC is the logic of the weak law of the excluded middle and QBL and QFL are first-order counterparts of Visser's basic and formal logics, respectively. We also show that, for most "natural" first-order modal logics, the two-variable fragment with a single monadic predicate letter, without constants and equality, is undecidable, regardless of whether we consider semantics with expanding or constant domains. These include all sublogics of QKTB, QGL, and QGrz -- among them, QK, QT, QKB, QD, QK4, and QS4.Comment: Pre-final version of the paper published in Studia Logica,doi:10.1007/s11225-018-9815-

On Halldén Completeness of Modal Logics Determined by Homogeneous Kripke Frames

Halldén complete modal logics are defined semantically. They have a nice characterization as they are determined by homogeneous Kripke frames.Supported by the NCN, research grant DEC-2013/09/B/HS1/00701

Extensions of modal logic KTB and other topics

This thesis covers four topics. They are the extensions of the modal logic KTB, the use of normal forms in modal logic, automated reasoning in the modal logic S4 and the problem of unavoidable words. Extensions of KTB: The modal logic KTB is the logic of reflexive and symmetric frames. Dually, KTB-algebras have a unary (normal) operator f that satisfies the identities f (x){u2265}x and {u231D}x{u2264}f ({u231D}f(x)). Extensions of KTB are subvarieties of the algebra KTB. Both of these form a lattice, and we investigate the structure of the bottom of the lattice of subvarieties. The unique atom is known to correspond to the modal logic whose frame is a single reflexive point. Yutaka demonstrated that this atom has a unique cover, corresponding to the frame of the two element chain. We construct covers of this element, and so demonstrate that there are a continuum of such covers. Normal Forms in Modal Logic: Fine proposed the use of normal forms as an alternative to traditional methods of determining Kripke completeness. We expand on this paper and demonstrate the application of normal forms to a number of traditional modal logics, and define new terms needed to apply normal forms in this situation. Automated reasoning in 84: History based methods for automated reasoning are well understood and accepted. Pliu{u0161}kevi{u010D}ius & Pliu{u0161}kevi{u010D}ien{u0117} propose a new, potentially revolutionary method of applying marks and indices to sequents. We show that the method is flawed, and empirically compare a different mark/index based method to the traditional methods instead. Unavoidable words: The unavoidable words problem is concerned with repetition in strings of symbols. There are two main ways to identify a word as unavoidable, one based on generalised pattern matching and one from an algorithm. Both methods are in NP, but do not appear to be in P. We define the simple unavoidable words as a subset of the standard unavoidable words that can be identified by the algorithm in P-time. We define depth separating IX x homomorphisms as an easy way to generate a subset of the unavoidable words using the pattern matching method. We then show that the two simpler problems are equivalent to each other

Philosophical logics - a survey and a bibliography

Intensional logics attract the attention of researchers from differing academic backgrounds and various scientific interests. My aim is to sketch the philosophical background of alethic, doxastic, and deontic logics, their formal and metaphysical presumptions and their various problems and paradoxes, without attempting formal rigor. A bibliography, concise on philosophical writings, is meant to allow the reader\u27s access to the maze of literature in the field

Lewis meets Brouwer: constructive strict implication

C. I. Lewis invented modern modal logic as a theory of "strict implication". Over the classical propositional calculus one can as well work with the unary box connective. Intuitionistically, however, the strict implication has greater expressive power than the box and allows to make distinctions invisible in the ordinary syntax. In particular, the logic determined by the most popular semantics of intuitionistic K becomes a proper extension of the minimal normal logic of the binary connective. Even an extension of this minimal logic with the "strength" axiom, classically near-trivial, preserves the distinction between the binary and the unary setting. In fact, this distinction and the strong constructive strict implication itself has been also discovered by the functional programming community in their study of "arrows" as contrasted with "idioms". Our particular focus is on arithmetical interpretations of the intuitionistic strict implication in terms of preservativity in extensions of Heyting's Arithmetic.Comment: Our invited contribution to the collection "L.E.J. Brouwer, 50 years later

Deciding regular grammar logics with converse through first-order logic

We provide a simple translation of the satisfiability problem for regular grammar logics with converse into GF2, which is the intersection of the guarded fragment and the 2-variable fragment of first-order logic. This translation is theoretically interesting because it translates modal logics with certain frame conditions into first-order logic, without explicitly expressing the frame conditions. A consequence of the translation is that the general satisfiability problem for regular grammar logics with converse is in EXPTIME. This extends a previous result of the first author for grammar logics without converse. Using the same method, we show how some other modal logics can be naturally translated into GF2, including nominal tense logics and intuitionistic logic. In our view, the results in this paper show that the natural first-order fragment corresponding to regular grammar logics is simply GF2 without extra machinery such as fixed point-operators.Comment: 34 page

Substitutional Validity for Modal Logic

In the substitutional framework, validity is truth under all substitutions of the non logical vocabulary. I develop a theory where the Box is interpreted as substitutional validity. I show how to prove soundness and completeness for common modal calculi using this definition