A Characteristic Frame for Positive Intuitionistic and Relevance Logic
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A Characteristic Frame for Positive Intuitionistic and Relevance Logic
Abstract. I show that the lattice of the positive integers ordered by division is characteristic for Urquhart’s positive semilattice relevance logic; that is, a formula is valid in positive semilattice relevance logic if and only if it is valid in all models over the positive integers ordered by division. I show that the same frame is characteristic for positive intuitionistic logic, where the class of models over it is restricted to those satisfying a heredity condition. The results of this article highlight deep connections between intuitionistic and semilattice relevance logic. Keywords: Arithmetical models, Characteristic frame, Intuitionistic logic, Relevance logic, Semilattice semantics.
1.
Introduction
In a series of works in the early 1970s [12,14,15], Alasdair Urquhart presented a semilattice semantics for relevance logic. The semantics was successful in characterizing fragments of the best known relevance logics, but the basic positive semantics already fails to be complete with respect to the positive fragment of the system R. Rather, the basic semantics characterizes a distinct logic which is sometimes called (positive) semilattice relevance logic, and which I will denote by S.1 Urquhart also showed that imposing a heredity condition on the basic semilattice semantics allowed for the implicational fragment of intuitionistic logic to be characterized. Frames for S are simply join-semilattices with a bottom element. Consequently, S can be modeled in many natural mathematical structures. One particularly pleasing example consists of the positive integers ordered by division.2 This constitutes a semilattice in which join is the least common 1 Note that S is not to be confused with the “syllogistic” relevance logic (called S ) of Martin and Meyer [9]. 2 Interest in this and related structures for the purpose of modeling relevance logic extends back a long way. See, for example, Meyer [10, 11].
Presented by Heinrich Wansing; Received May 12, 2020
Studia Logica https://doi.org/10.1007/s11225-020-09921-2
c Springer Nature B.V. 2020
Y. Weiss
multiple operation and 1 is the bottom element. Weiss [16] used this structure to furnish a proof that S satisfies the variable sharing property. In this article, I show that, in fact, this frame is characteristic for S: a formula is valid in S if and only if it is valid in all models based on this frame. Since the implicational fragment of R coincides with that of S, to the extent licensed by the result just noted, this fragment may be regarded as the theory of implication of the positive integers ordered by division. Furthermore, I show that the same structure is characteristic for J+ (positive intuitionistic logic): a formula is valid in J+ if and only if it is valid in all models based on this frame satisfying a certain heredity condition. This result shows that there are deep connections between S and J+ . The plan of the article is as follows. In Section 2, I present semilattice semantics
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