Museum of Outstanding Figures in Ukrainian Culture

This paper deals with a famous problem of epistemic logic – logical omniscience. Logical omniscience occurs in the logical systems where the axiomatics is complete and consequently an agent using inference rules knows everything about the system. Logical omniscience is a major problem due to complexity problems and the inability for adequate human reasoning modeling. It is studied both informal logic and philosophy of psychology (bounded rationality). It is important for bounded rationality because it reflects the problem of formal characterization of purely psychological mechanisms. Paper proposes to solve it using the ideas from the philosophical bounded rationality and intuitionistic logic. Special regions of deductible formulas developed according to psychologistic criterion should guide the deductive model. The method is compared to other ones presented in the literature on logical omniscience such as Hintikka’s and Vinkov and Fominuh. Views from different perspectives such as computer science and artificial intelligence are also provided.

The Presheaf of Conditional Expectations, a Bridge Between Probability and Intuitionistic Truth Value

Research for a theory of quantum gravity has recently led to the use of presheaf topos. Quantum uncertainty is linked to the truth values of intuitionistic logic. This paper proposes transposing this model into a classic probability context, that of conditional mathematical expectations. A simulation of Brownian motion is offered for illustrative purposes.

Schema Complexity in Propositional-Based Logics

The essential structure of derivations is used as a tool for measuring the complexity of schema consequences in propositional-based logics. Our schema derivations allow the use of schema lemmas and this is reflected on the schema complexity. In particular, the number of times a schema lemma is used in a derivation is not relevant. We also address the application of metatheorems and compare the complexity of a schema derivation after eliminating the metatheorem and before doing so. As illustrations, we consider a propositional modal logic presented by a Hilbert calculus and an intuitionist propositional logic presented by a Gentzen calculus. For the former, we discuss the use of the metatheorem of deduction and its elimination, and for the latter, we analyze the cut and its elimination. Furthermore, we capitalize on the result for the cut elimination for intuitionistic logic, to obtain a similar result for Nelson’s logic via a language translation.

Normalisation and subformula property for a system of intuitionistic logic with general introduction and elimination rules

This paper studies a formalisation of intuitionistic logic by Negri and von Plato which has general introduction and elimination rules. The philosophical importance of the system is expounded. Definitions of ‘maximal formula’, ‘segment’ and ‘maximal segment’ suitable to the system are formulated and corresponding reduction procedures for maximal formulas and permutative reduction procedures for maximal segments given. Alternatives to the main method used are also considered. It is shown that deductions in the system convert into normal form and that deductions in normal form have the subformula property.

Set Theory INC# ∞# Based on Innitary Intuitionistic Logic with Restricted Modus Ponens Rule. Hyper Inductive Denitions. Application in Transcendental Number Theory. Generalized Lindemann-Weierstrass Theorem

In this paper intuitionistic set theory INC#∞# in infinitary set theoretical language is considered. External induction principle in nonstandard intuitionistic arithmetic were derived. Non trivial application in number theory is considered.The Goldbach-Euler theorem is obtained without any references to Catalan conjecture. Main results are: (i) number ee is transcendental; (ii) the both numbers e + π and e − π are irrational.

Maximality of bi-intuitionistic propositional logic

In the style of Lindström’s theorem for classical first-order logic, this article characterizes propositional bi-intuitionistic logic as the maximal (with respect to expressive power) abstract logic satisfying a certain form of compactness, the Tarski union property and preservation under bi-asimulations. Since bi-intuitionistic logic introduces new complexities in the intuitionistic setting by adding the analogue of a backwards looking modality, the present paper constitutes a non-trivial modification of the previous work done by the authors for intuitionistic logic (Badia and Olkhovikov, 2020, Notre Dame Journal of Formal Logic, 61, 11–30).

Note on the Intuitionistic Logic of False Belief

In this paper we analyse logic of false belief in the intuitionistic setting. This logic, studied in its classical version by Steinsvold, Fan, Gilbert and Venturi, describes the following situation: a formula $\varphi$ is not satisfied in a given world, but we still believe in it (or we think that it should be accepted). Another interpretations are also possible: e.g. that we do not accept $\varphi$ but it is imposed on us by a kind of council or advisory board. From the mathematical point of view, the idea is expressed by an adequate form of modal operator $\mathsf{W}$ which is interpreted in relational frames with neighborhoods. We discuss monotonicity of forcing, soundness, completeness and several other issues. We present also some simple systems in which confirmation of previously accepted formula is modelled.

On co-Filters in Semigroups with Apartness

The logical environment of this research is the Intuitionistic Logic and principled-philosophical orientation of the Bishop’s Constructive Mathematics. In this paper, basing our consideration on the sets with the apartness relation, we analyze the lattices of all co-filters of an ordered semigroup under a co-quasiorder as a continuation of our article [?]. We prove a number of results related to co-filters in a semigroup with apartness and the lattice of all co-filters of such semigroups.

Hairier than Putnam Thought

This chapter is a short note, co-written with Stephen Read, reacting to Hilary Putnam’s observation in his ‘Vagueness and Alternative Logic’ that intuitionistic logic would block the transition from the negation of the usual universally quantified conditional form of major premise for a Sorites to the assertion of a sharp boundary to the target predicate in the series concerned, and would thus allow the paradox to be reconceived as a straightforward reductio of its major premise. It is pointed out that a Sorites need not employ that form of major premise but can instead proceed, in intuitionistic logic, from the negation of the existential claim that the series in question contains a sharp boundary and that, while an intuitionistically suspect double negation elimination step would still be needed to enforce the unpalatable conclusion that the predicate in question indeed has a sharp boundary, nothing like the semantic motivation that the Intuitionists have favoured in mathematics for a restriction on double negation elimination can be operative in this context.