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COUNTING TO INFINITY: GRADED MODAL LOGIC WITH AN INFINITY DIAMOND
- Part of
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- Journal:
- The Review of Symbolic Logic / Volume 17 / Issue 1 / March 2024
- Published online by Cambridge University Press:
- 08 July 2022, pp. 1-35
- Print publication:
- March 2024
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- Article
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We extend the languages of both basic and graded modal logic with the infinity diamond, a modality that expresses the existence of infinitely many successors having a certain property. In both cases we define a natural notion of bisimilarity for the resulting formalisms, that we dub
$\mathtt {ML}^{\infty }$ and 
$\mathtt {GML}^{\infty }$, respectively. We then characterise these logics as the bisimulation-invariant fragments of the naturally corresponding predicate logic, viz., the extension of first-order logic with the infinity quantifier. Furthermore, for both 
$\mathtt {ML}^{\infty }$ and 
$\mathtt {GML}^{\infty }$ we provide a sound and complete axiomatisation for the set of formulas that are valid in every Kripke frame, we prove a small model property with respect to a widened class of weighted models, and we establish decidability of the satisfiability problem.
