TY - CHAP

T1 - Lewisian Fixed Points I

T2 - Two Incomparable Constructions

AU - Litak, Tadeusz

AU - Visser, Albert

N1 - Publisher Copyright:
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024.

PY - 2024/7/26

Y1 - 2024/7/26

N2 - Our paper studies what one might call “the reverse mathematics of explicit fixed points”. We discuss two methods of constructing such fixed points for formulas whose principal connective is the intuitionistic Lewis arrow⇝. Our main motivation comes from metatheory of constructive arithmetic, but the systems in question allow several natural kinds of semantics. The first of these methods, inspired by de Jongh and Visser, turns out to yield a modal system La♭, extending the “gathering” axiom 4a with the standard (“box”) version of the Löb axiom. The second one, inspired by de Jongh and Sambin, seemingly simpler, leads to a modal theory JS♭, which proves harder to axiomatize in an elegant way. Apart from showing that both theories are mutually incomparable, we axiomatize their join and investigate several subtheories, whose axioms are obtained as fixed points of simple formulas. We also show that both La♭ and JS♭ are extension stable, that is, their validity in the corresponding preservativity logic of a given arithmetical theory transfers to its finite extensions.

AB - Our paper studies what one might call “the reverse mathematics of explicit fixed points”. We discuss two methods of constructing such fixed points for formulas whose principal connective is the intuitionistic Lewis arrow⇝. Our main motivation comes from metatheory of constructive arithmetic, but the systems in question allow several natural kinds of semantics. The first of these methods, inspired by de Jongh and Visser, turns out to yield a modal system La♭, extending the “gathering” axiom 4a with the standard (“box”) version of the Löb axiom. The second one, inspired by de Jongh and Sambin, seemingly simpler, leads to a modal theory JS♭, which proves harder to axiomatize in an elegant way. Apart from showing that both theories are mutually incomparable, we axiomatize their join and investigate several subtheories, whose axioms are obtained as fixed points of simple formulas. We also show that both La♭ and JS♭ are extension stable, that is, their validity in the corresponding preservativity logic of a given arithmetical theory transfers to its finite extensions.

KW - Constructivism

KW - Fixed points

KW - Lewis arrow

KW - Provability logic

UR - http://www.scopus.com/inward/record.url?scp=85202037367&partnerID=8YFLogxK

U2 - 10.1007/978-3-031-47921-2_2

DO - 10.1007/978-3-031-47921-2_2

M3 - Chapter

AN - SCOPUS:85202037367

SN - 978-3-031-47920-5

T3 - Outstanding Contributions to Logic

SP - 33

EP - 73

BT - Dick de Jongh on Intuitionistic and Provability Logics

PB - Springer

ER -