Studia Logica

Jan Dejnožka, The Concept of Relevance and the Logic Diagram Tradition

MS-Algebras Whose e-Ideals are Kernel Ideals Abstract We consider, in the context of an MS-algebra L, the ideals I of L that are kernels of L. We characterize two kinds of de Morgan algebras: the class Boolean algebras and the absolutely indecomposable de Morgan algebras. We show that all the e-ideals I of L are kernel ideals of L if and only if the subalgebra \(L^{00}\) of L can only be these two kinds of de Morgan algebras.

Propositional Epistemic Logics with Quantification Over Agents of Knowledge (An Alternative Approach) Abstract In the previous paper with a similar title (see Shtakser in Stud Log 106(2):311–344, 2018), we presented a family of propositional epistemic logics whose languages are extended by two ingredients: (a) by quantification over modal (epistemic) operators or over agents of knowledge and (b) by predicate symbols that take modal (epistemic) operators (or agents) as arguments. We denoted this family by \({\mathcal {P}\mathcal {E}\mathcal {L}}_{(QK)}\) . The family \({\mathcal {P}\mathcal {E}\mathcal {L}}_{(QK)}\) is defined on the basis of a decidable higher-order generalization of the loosely guarded fragment (HO-LGF) of first-order logic. And since HO-LGF is decidable, we obtain the decidability of logics of \({\mathcal {P}\mathcal {E}\mathcal {L}}_{(QK)}\) . In this paper we construct an alternative family of decidable propositional epistemic logics whose languages include ingredients (a) and (b). Denote this family by \({\mathcal {P}\mathcal {E}\mathcal {L}}^{alt}_{(QK)}\) . Now we will use another decidable fragment of first-order logic: the two variable fragment of first-order logic with two equivalence relations (FO \(^2\) +2E) [the decidability of FO \(^2\) +2E was proved in Kieroński and Otto (J Symb Log 77(3):729–765, 2012)]. The families \({\mathcal {P}\mathcal {E}\mathcal {L}}^{alt}_{(QK)}\) and \({\mathcal {P}\mathcal {E}\mathcal {L}}_{(QK)}\) differ in the expressive power. In particular, we exhibit classes of epistemic sentences considered in works on first-order modal logic demonstrating this difference.

Epistemic Logic, Monotonicity, and the Halbach–Welch Rapprochement Strategy Abstract Predicate approaches to modality have been a topic of increased interest in recent intensional logic. Halbach and Welch (Mind 118(469):71–100, 2009) have proposed a new formal technique to reduce the necessity predicate to an operator, demonstrating that predicate and operator methods are ultimately compatible. This article concerns the question of whether Halbach and Welch’s approach can provide a uniform formal treatment for intensionality. I show that the monotonicity constraint in Halbach and Welch’s proof for necessity fails for almost all possible-worlds theories of knowledge. The nonmonotonicity results demonstrate that the most obvious way of emulating Halbach and Welch’s rapprochement of the predicate and operator fails in the epistemic setting.

Undecidability of First-Order Modal and Intuitionistic Logics with Two Variables and One Monadic Predicate Letter Abstract We prove that the positive fragment of first-order intuitionistic logic in the language with two individual variables and a single monadic predicate letter, without functional symbols, 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 \([\mathbf{QBL}, \mathbf{QKC}]\) and \([\mathbf{QBL}, \mathbf{QFL}]\) , 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 functional symbols, 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.

On Monadic Operators on Modal Pseudocomplemented De Morgan Algebras and Tetravalent Modal Algebras Abstract In our paper, monadic modal pseudocomplemented De Morgan algebras (or mmpM) are considered following Halmos’ studies on monadic Boolean algebras. Hence, their topological representation theory (Halmos–Priestley’s duality) is used successfully. Lattice congruences of an mmpM is characterized and the variety of mmpMs is proven semisimple via topological representation. Furthermore and among other things, the poset of principal congruences is investigated and proven to be a Boolean algebra; therefore, every principal congruence is a Boolean congruence. All these conclusions contrast sharply with known results for monadic De Morgan algebras. Finally, we show that the above results for mmpM are verified for monadic tetravalent modal algebras.

A Study in Grzegorczyk Point-Free Topology Part II: Spaces of Points Abstract In the second installment to Gruszczyński and Pietruszczak (Stud Log, 2018. https://doi.org/10.1007/s11225-018-9786-8) we carry out an analysis of spaces of points of Grzegorczyk structures. At the outset we introduce notions of a concentric and \(\omega \) -concentric topological space and we recollect some facts proven in the first part which are important for the sequel. Theorem 2.9 is a strengthening of Theorem 5.13, as we obtain stronger conclusion weakening Tychonoff separation axiom to mere regularity. This leads to a stronger version of Theorem 6.10 (in form of Corollary 2.10). Further, we show that Grzegorczyk points are maximal contracting filters in the sense of De Vries (Compact spaces and compactifications, Van Gorcum and Comp. N.V., 1962), but the converse inclusion is not necessarily true. We also compare the notions of a Grzegorczyk point and an ultrafilter, and establish several properties of topological spaces based on Grzegorczyk structures. The main results of the paper are representation and completion theorems for G-structures. We prove both set-theoretical and topological representation theorems for various classes of G-structures. We also present topological object duality theorem for the class of complete G-structures and the class of concentric spaces, both restricted to structures which satisfy countable chain condition. We conclude the paper with proving equivalence of the original Grzegorczyk axiom with the one accepted by us as axiom (G).

Sequent Calculi for Global Modal Consequence Relations Abstract The global consequence relation of a normal modal logic \(\Lambda \) is formulated as a global sequent calculus which extends the local sequent theory of \(\Lambda \) with global sequent rules. All global sequent calculi of normal modal logics admits global cut elimination. This property is utilized to show that decidability is preserved from the local to global sequent theories of any normal modal logic over \(\mathsf {K4}\) . The preservation of Craig interpolation property from local to global sequent theories of any normal modal logic is shown by proof-theoretic method.

Enrico Martino, Intuitionistic Proof Versus Classical Truth: The Role of Brouwer’s Creative Subject in Intuitionistic Mathematics, Springer, 2018

A Propositional Dynamic Logic for Instantial Neighborhood Semantics Abstract We propose a new perspective on logics of computation by combining instantial neighborhood logic \(\mathsf {INL}\) with bisimulation safe operations adapted from \(\mathsf {PDL}\) . \(\mathsf {INL}\) is a recent modal logic, based on an extended neighborhood semantics which permits quantification over individual neighborhoods plus their contents. This system has a natural interpretation as a logic of computation in open systems. Motivated by this interpretation, we show that a number of familiar program constructors can be adapted to instantial neighborhood semantics to preserve invariance for instantial neighborhood bisimulations, the appropriate bisimulation concept for \(\mathsf {INL}\) . We also prove that our extended logic \(\mathsf {IPDL}\) is a conservative extension of dual-free game logic, and its semantics generalizes the monotone neighborhood semantics of game logic. Finally, we provide a sound and complete system of axioms for \(\mathsf {IPDL}\) , and establish its finite model property and decidability.