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Interpolation in Non-Classical Logics

Wesley Fussner

TL;DR

This chapter surveys interpolation across non-classical logics, focusing on Craig interpolation and deductive interpolation, and examines how these properties behave in broad families and their axiomatic extensions. It highlights three main analytical lenses—proof-theoretic Maehara methods, amalgamation-based algebraic techniques, and model-theoretic perspectives—while emphasizing the impact of language and structural rules on interpolation. The contribution lies in mapping the landscape for superintuitionistic, modal, fuzzy, paraconsistent, relevant, and substructural logics, summarizing key results, and outlining many open questions that guide future research. Overall, the work provides a structured big-picture view of where interpolation holds, fails, or remains unsettled across diverse non-classical logics and their extensions, with a rich bibliography for deeper study.

Abstract

This chapter surveys some of the main results on interpolation in several of the most prominent families of non-classical logics. Special attention is given to the distinction between the two most commonly studied variants of interpolation--namely, Craig interpolation and deductive interpolation. Our discussion focuses primarily on how these properties present in families of logical systems taken as a whole, particularly those comprising all axiomatic extensions of any of several notable non-classical logics. We consider a range of important examples: superintuitionistic and modal logics, fuzzy logics, paraconsistent logics, relevant logics, and substructural logics.

Interpolation in Non-Classical Logics

TL;DR

This chapter surveys interpolation across non-classical logics, focusing on Craig interpolation and deductive interpolation, and examines how these properties behave in broad families and their axiomatic extensions. It highlights three main analytical lenses—proof-theoretic Maehara methods, amalgamation-based algebraic techniques, and model-theoretic perspectives—while emphasizing the impact of language and structural rules on interpolation. The contribution lies in mapping the landscape for superintuitionistic, modal, fuzzy, paraconsistent, relevant, and substructural logics, summarizing key results, and outlining many open questions that guide future research. Overall, the work provides a structured big-picture view of where interpolation holds, fails, or remains unsettled across diverse non-classical logics and their extensions, with a rich bibliography for deeper study.

Abstract

This chapter surveys some of the main results on interpolation in several of the most prominent families of non-classical logics. Special attention is given to the distinction between the two most commonly studied variants of interpolation--namely, Craig interpolation and deductive interpolation. Our discussion focuses primarily on how these properties present in families of logical systems taken as a whole, particularly those comprising all axiomatic extensions of any of several notable non-classical logics. We consider a range of important examples: superintuitionistic and modal logics, fuzzy logics, paraconsistent logics, relevant logics, and substructural logics.

Paper Structure

This paper contains 24 sections, 23 theorems, 27 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Consequence, Interpolation, and Deduction Theorems
  3. Consequence Relations
  4. Variants of Interpolation
  5. Deduction Theorems and Relations Among Interpolation Variants
  6. Fundamental Examples
  7. Superintuitionistic Logics
  8. Modal Logics
  9. Normal Modal Logics Generally
  10. Extensions of $\mathtt{S4}$
  11. Non-Normal Modal Logics
  12. Paraconsistent Logics
  13. The Logic of Paradox
  14. Johansson's Minimal Logic
  15. Nelson's Constructive Logic With Strong Negation
  16. ...and 9 more sections

Key Result

Proposition 1

Let $\mathtt{L} = (\mathcal{L},\vdash_\mathtt{L})$ be a logic with an implication connective $\to$ in its language, and assume that: Then if $\mathtt{L}$ has the CIP, then $\mathtt{L}$ has the DIP.

Figures (1)

  • Figure 1: The basic structural rules of proof theory, as formulated in MPT23. Here each $\mathrm{\Gamma}_i$, $\Pi$, $\mathrm{\Delta}$, and so on denote finite sequences of formulas, with at most one formulas appearing to the right of ${\newline\Rightarrow{\newline}}$.

Theorems & Definitions (23)

  • Proposition 1
  • Proposition 3
  • Theorem 4: Mak77
  • Proposition 5: Mak77
  • Theorem 6: Mak91
  • Theorem 7: Mak2005LC
  • Theorem 8: Mak79GabMak2005
  • Proposition 10: Mak2012
  • Proposition 11: Mak2011
  • Theorem 14: Goranko1985
  • ...and 13 more