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Onset and Today’s Perspectives of Multilevel Syllogistic

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From Computational Logic to Computational Biology

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14070))

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Abstract

We report on the initial phases of a systematic study (undertaken over forty years ago) on decidable fragments of Set Theory, to which Alfredo Ferro contributed and which later branched out in many directions. The impact that research has had so far and will continue to have, mainly in the areas of proof-checking, program-correctness verification, declarative programming—and, more recently, reasoning within description logics—is also highlighted.

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Notes

  1. 1.

    This happening left a very profound impression on Alfredo, who afterward developed a strong religious involvement.

  2. 2.

    The case \(\textsf{at}(x) = \emptyset \) has been introduced here just to make the map \(\textsf{at}\) total on V.

  3. 3.

    The problem of producing a concrete model for any satisfiable formula (being also able to flag when none exists) is sometimes called satisfaction problem: hence, for MLS, this problem is algorithmically solvable.

  4. 4.

    Set-terms of the form \(\left\{ {s_0,\dots ,s_N}\right\} \), which can be seen as abbreviating \(\left\{ {s_0}\right\} \cup \cdots \cup \left\{ {s_N}\right\} \), enter into play for free with singleton.

  5. 5.

    See also [85, 87] and cf. footnote 14 below. Early decidability-related results concerning arb relied on the assumption that a strict well-ordering \(\lhd \) of the universe of all sets, complying with various conditions, is available and that \({\textsf {arb}}\!\left( {s}\right) \) chooses the \(\lhd \)-least element of s from each non-empty set s. One of the conditions imposed on \(\lhd \) entails that \(x\lhd y\) holds when \(x\in y\); in addition, \(\lhd \) must be anti-lexicographic over finite sets, which means: \(\left\{ {x_{m+p},\dots ,x_{m},\dots ,x_{1}}\right\} \lhd \big (\left\{ {y_{q+1},\dots ,y_{1}}\right\} \cup \left\{ {x_{m},\dots ,x_{1}}\right\} \big )\) follows from \( x_{m+p}\lhd \cdots \lhd x_m\lhd \cdots \lhd x_1\ \,{\small \& }\,\ y_{q+1}\lhd \cdots \lhd y_{1}\lhd x_m\ \,{\small \& }\,\ \big (p=0\ \,\vee \,\ (p>0\ \,{\small \& }\,\ x_{m+1}\lhd y_{1})\big )\).

  6. 6.

    I had been introduced to Alfredo by the late Filippo Chiarenza, one of his very close friends and fellow students, at a beach in the Catania shoreline, a couple of years before in the summer of 1979, when Alfredo was temporarily back to Catania to get married to his beloved wife Pina Carrà.

  7. 7.

    “Decidability issues in set theory”.

  8. 8.

    Places have been defined in Sect. 1.3.

  9. 9.

    Actually, power-set clauses impose stronger constraints than union clauses do, insofar as the implication

    figure t

    is true for all sets x and y , while the converse implication does not hold in general.

  10. 10.

    Correct modeling of also the literals of the forms \(x = y \cup z\), \(x = y \cap z\), and \(x = y \setminus z\) was ensured since the sets \(\overline{\alpha }\), with \(\alpha \in \varPi \), were maintained mutually disjoint during the stabilization phase too.

  11. 11.

    Compared with the very weak aggregate theory \(\mathsf {Z_1}\) treated by Robert L. Vaught, which is generated by the empty set axiom and the adjunction axiom alone (cf. [143, p. 21]), the systems studied by Parlamento and Policriti ensure a more transparent syntax arithmetization and a tighter connection with computability theory.

  12. 12.

    The term hyperset was first proposed in [3] to mean—in a universe modeling Aczel’s AFA—a ‘set’ \(x_0\) whose transitive closure is potentially ill-formed, in the sense that membership may form an infinite descending chain \(x_0\ni x_1\ni x_2\ni \cdots \) .

  13. 13.

    One often dubs as ‘free’ or ‘uninterpreted’ those symbols that are not subdued to any axiomatic constraint; this is not the case of \(\textsf {with}\), which must comply with the laws \(x\ \textsf {with}\ y\ \textsf {with}\ z=x\ \textsf {with}\ z\ \textsf {with}\ y\) and \(x\ \textsf {with}\ y\ \textsf {with}\ y=x\ \textsf {with}\ y\) .

  14. 14.

    In this respect, the treatment of the global choice operator is simpler-minded—but also considerably more efficient—than the one proposed in [85, 87, 104].

  15. 15.

    We mean here the problem of determining whether or not a sentence of the said form \(\exists \,x_1\cdots \exists \,x_N\,\forall \,y\,\mu \) (with \(N\geqslant 0\)) is satisfiable in the standard universe of sets. We will be more explicit on the notation \(\exists ^*\forall \) in footnote (see footnote 16).

  16. 16.

    To stay aligned with the literature cited in this section, we slightly depart from the notation used above to classify prenex formulae according to their quantificational prefix. Here and below, \(\exists ^*\) and \(\forall ^*\) denote batches (possibly empty) of alike quantifiers, while \(\exists \exists \) and \(\forall \forall \) denote two consecutive quantifiers of existential, respectively universal, type.

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Acknowledgments

We are grateful to Daniele Santamaria for contributing to Sect. 3.5.

We gratefully acknowledge partial support from project “STORAGE—Università degli Studi di Catania, Piano della Ricerca 2020/2022, Linea di intervento 2”, and from INdAM-GNCS 2019 and 2020 research funds.

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    Domenico Cantone

  2. University of Trieste, Trieste, Italy

    Eugenio G. Omodeo

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    Domenico Cantone

  2. University of Catania, Catania, Italy

    Alfredo Pulvirenti

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Cantone, D., Omodeo, E.G. (2024). Onset and Today’s Perspectives of Multilevel Syllogistic. In: Cantone, D., Pulvirenti, A. (eds) From Computational Logic to Computational Biology. Lecture Notes in Computer Science, vol 14070. Springer, Cham. https://doi.org/10.1007/978-3-031-55248-9_2

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