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On the Universality of Atomic and Molecular Logics via Protologics


After observing that the truth conditions of connectives of non–classical logics are generally defined in terms of formulas of first–order logic, we introduce ‘protologics’, a class of logics whose connectives are defined by arbitrary first–order formulas. Then, we introduce atomic and molecular logics, which are two subclasses of protologics that generalize our gaggle logics and which behave particularly well from a theoretical point of view. We also study and introduce a notion of equi-expressivity between two logics based on different classes of models. We prove that, according to that notion, every pure predicate logic with \(k\ge 0\) variables and constants is as expressive as a predicate atomic logic, some sort of atomic logic. Then, we prove that the class of protologics is equally expressive as the class of molecular logics. That formally supports our claim that atomic and molecular logics are somehow ‘universal’. Finally, we identify a subclass of molecular logics that we call predicate molecular logics and which constitutes its representative core: every molecular logic is as expressive as a predicate molecular logic.

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  1. I thank Peter Arndt for checking and proving that result.

  2. The definition of molecular connectives and molecular logics of [6] is less general than in the present article, in the sense that it is not possible there to have the same argument at different places in the definition of a molecular connective. This said, it is nevertheless possible to define an appropriate notion of \({\textsf {C}}\)-bisimulation for the more general definition of molecular logics of the present article.

  3. [5] differs very slightly from [4]. It essentially corrects minor mistakes and typos and proves that the rule of associativity is derivable in \(\textsf {GGL}_{{\textsf {C}}}\).


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Correspondence to Guillaume Aucher.

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This paper is the winner of the Louis Couturat Logic Prize 2021 (France)

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Aucher, G. On the Universality of Atomic and Molecular Logics via Protologics. Log. Univers. 16, 285–322 (2022).

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  • Universal logic
  • Expressivity
  • First-order logics
  • Non-classical logics

Mathematics Subject Classification

  • 03
  • 03B
  • 03C