Mateusz ŁełykUniversity of Warsaw | UW · Faculty of Philosophy
Mateusz Łełyk
Doctor of Philosophy
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17
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Citations since 2017
Publications
Publications (17)
G\"odel's Incompleteness Theorems suggest that no single formal system can capture the entirety of one's mathematical beliefs, while pointing at a hierarchy of systems of increasing logical strength that make progressively more explicit those \emph{implicit} assumptions. This notion of \emph{implicit commitment} motivates directly or indirectly sev...
We employ the lens provided by formal truth theory to study axiomatizations of Peano Arithmetic ${\textsf {(PA)}}$ . More specifically, let Elementary Arithmetic ${\textsf {(EA)}}$ be the fragment $\mathsf {I}\Delta _0 + \mathsf {Exp}$ of ${\textsf {PA}}$ , and let ${\textsf {CT}}^-[{\textsf {EA}}]$ be the extension of ${\textsf {EA}}$ by the commo...
Ali Enayat had asked whether two halves of Disjunctive Correctness (DC) for the compositional truth predicate are conservative over Peano Arithmetic (PA). In this paper, we show that the principle “every true disjunction has a true disjunct” is equivalent to bounded induction for the compositional truth predicate and thus it is not conservative. On...
The notion of implicit commitment has played a prominent role in recent works in logic and philosophy of mathematics. Although implicit commitment is often associated with highly technical studies, it remains an elusive notion. In particular, it is often claimed that the acceptance of a mathematical theory implicitly commits one to the acceptance o...
Ali Enayat had asked whether two halves of Disjunctive Correctness (DC) for the compositional truth predicate are conservative over Peano Arithmetic. In this article, we show that the principle "every true disjunction has a true disjunct" is equivalent to bounded induction for the compositional truth predicate and thus it is not conservative. On th...
We employ the lens provided by formal truth theory to study axiomatizations of PA (Peano Arithmetic). More specifically, let EA (Elementary Arithmetic) be the fragment I∆0 + Exp of PA, and CT − [EA] be the extension of EA by the commonly studied axioms of compositional truth CT −. We investigate both local and global properties of the family of fir...
The current paper studies the formal properties of the Global Reflection Principle, to wit the assertion "All theorems of Th are true", where Th is a theory in the language of arithmetic and the truth predicate satisfies the usual Tarskian inductive conditions for formulae in the language of arithmetic. We fix the gap in Kotlarski's proof from [15]...
We introduce a principle of local collection for compositional truth predicates and show that it is arithmetically conservative over the classically compositional theory of truth. This axiom states that upon restriction to formulae of any syntactic complexity, the resulting predicate satisfies full collection. In particular, arguments using collect...
We introduce a principle of local collection for compositional truth predicates and show that it is conservative over the classically compositional theory of truth in the arithmetical setting. This axiom states that upon restriction to formulae of any syntactic complexity, the resulting predicate satisfies full collection. In particular, arguments...
Let ${\cal T}$ be any of the three canonical truth theories CT ⁻ (compositional truth without extra induction), FS ⁻ (Friedman–Sheard truth without extra induction), or KF ⁻ (Kripke–Feferman truth without extra induction), where the base theory of ${\cal T}$ is PA (Peano arithmetic). We establish the following theorem, which implies that ${\cal T}$...
Let $\mathcal{T}$ be any of the three canonical truth theories $\textsf{CT}^-$ (Compositional truth without extra induction), $\textsf{FS}^-$ (Friedman--Sheard truth without extra induction), and $\textsf{KF}^-$ (Kripke--Feferman truth without extra induction), where the base theory of $\mathcal{T}$ is $\textsf{PA}$ (Peano arithmetic). We show that...
This paper is a follow-up to [4], in which a mistake in [6] (which spread also to [9]) was corrected. We give a strenghtening of the main result on the semantical nonconservativity of the theory of PT ⁻ with internal induction for total formulae ${(\rm{P}}{{\rm{T}}^ - } + {\rm{INT}}\left( {{\rm{tot}}} \right)$ , denoted by PT ⁻ in [9]). We show tha...
We study an extension of modal mu-calculus to sets with atoms and we study its basic properties. Model checking is decidable on orbit-finite structures, and a correspondence to parity games holds. On the other hand, satisfiability becomes undecidable. We also show expressive limitations of atom-enriched mu-calculi, and explain how their expressive...
This paper is a follow-up to "Models of PT${}^-$ with internal induction for total formulae." We give a strenghtening of the main result on the semantical non-conservativity of the theory of PT${}^-$ with internal induction for total formulae (PT${}^- +$ INT(tot)). We show that if to PT${}^-$ the axiom of internal induction for all arithmetical for...
In the following paper we propose a model-theoretical way of comparing the “strength” of various truth theories which are conservative over \( PA \). Let \({\mathfrak {Th}}\) denote the class of models of \( PA \) which admit an expansion to a model of theory \({ Th}\). We show (combining some well known results and original ideas) that $$\begin{al...
We prove that the theory of the extensional compositional truth predicate for the language of arithmetic with Δ 0 -induction scheme for the truth predicate and the full arithmetical induction scheme is not conservative over Peano Arithmetic. In addition, we show that a slightly modified theory of truth actually proves the global reflection principl...
We show that a typed compositional theory of positive truth with internal induction for total formulae (denoted by PT tot ) is not semantically conservative over Peano arithmetic. In addition, we observe that the class of models of PA expandable to models of PT tot contains every recursively saturated model of arithmetic. Our results point to a gap...
Projects
Project (1)
Contemporary conceptions of truth can be divided into two broad categories: philosophical and logical ones. Admittedly the division is not sharp, nor it should be: philosophers need formal results and the logicians often take philosophical intuitions into account in their formal constructions. The present project belongs to the realm of philosophically motivated formal investigations, devoted to the notion of truth.
The main overall goal of the project is to deepen our understanding of the notion of truth, underlying much of both our scientific and everyday thinking. In the project we will study formal theories of truth, with a clear mathematical structure, expressed in formal languages (unlike loose, intuitive conceptions, formulated in natural language). The basic assumption underlying the project is that any intuitive conception of truth has to pass the test of formalization in order to count as valid or at least promising. On the other hand, formalizations, once proposed, can be investigated with rigorous methods and confronted with intuitive conceptions which stand behind them. The results of such a confrontation may then count strongly in favour or against a given formalism. We plan to consider several directions which can lead to the realization of our overall goal and which are connected with open and important questions in this area of research. The project concentrates primarily on axiomatic theories of truth and it involves carrying out a comprehensive and systematic research on these theories. It comprises four main tasks, which are listed below. Even partial realization of these tasks can have important consequences not only for philosophy and logic, but also for theoretical and methodological foundations of formal and cognitive sciences.
Task 1: Analysis of formal properties of classical truth theories
Task 2: Truth theories in non-classical logics
Task 3: Analysis of semantic paradoxes in truth theories
Task 4: Philosophical interpretations of truth theories
