Sequent Calculus for HT by Mints

• Sequent calculus for HT is a structural proof system for Gödel's G3 that clarifies the boundary between intuitionistic and classical logics using three-valued semantics.
• It employs innovative interpolation techniques, including a two-stage method and the non-here operator, to facilitate efficient proof search and normalization.
• The system supports analytic, cut-free proofs and extends to first-order logic, bolstering applications in equilibrium logic and nonclassical logic programming.

The sequent calculus for the logic of here-and-there (HT) formulated by Gregory Mints provides a structural proof system for Gödel's $G_3$, a three-valued superintuitionistic logic situated strictly between intuitionistic and classical logics. Recent developments have produced significant metatheoretic results and novel interpolants in variations of Mints' system, illuminating the algebraic and proof-theoretic boundaries of HT (Wernhard, 7 Jan 2026). The system is foundational for automated reasoning in equilibrium logic, nonclassical logic programming, and modal logics with intermediate semantics (Otten et al., 7 Jan 2026).

1. Semantic Foundations of HT (Gödel’s G₃)

HT (here-and-there logic) is characterized by a three-valued (F, NF, T) semantics, assigning to each propositional atom one of:

• F (“there = F, here = F”)
• NF (“there = T, here = F”)
• T (“there = T, here = T”)

The connectives $\neg$, $\vee$, $\wedge$, and $\rightarrow$ are interpreted by the following truth tables:

$A$, $B$	$A\vee B$	$A\wedge B$	$A\rightarrow B$
F, F	F	F	T
F, NF	NF	F	T
F, T	T	F	T
NF, F	NF	F	F
NF, NF	NF	NF	T
NF, T	T	NF	T
T, F	T	F	F
T, NF	T	NF	NF
T, T	T	T	T

Negation is defined by:

• $\neg$F = T
• $\neg$NF = F
• $\neg$T = F

HT is semantically complete with respect to Gödel’s $G_3$ and strictly contains IPC (intuitionistic propositional logic) while being properly contained in classical logic. Every HT tautology is a $G_3$ tautology and vice versa.

2. Mints’ Sequent Calculus (G3–HT) and Variations

Sequents in Mints’ system take the form $\Gamma \Rightarrow \Delta$, with $\Gamma$ and $\Delta$ multisets of formulas. No explicit structural rules are listed; weakening, contraction, and exchange are admissible. The calculus comprises:

• Axiom Schemes:
  • (Ax-1): $A, \Gamma \Rightarrow A$ where $A$ is an atom or negated atom.
  • (Ax-2): $A, \neg A, \Gamma \Rightarrow$ where $A$ is an atom.
• Logical Rules:
  • $\rightarrow$-Left: $A \rightarrow B, \Gamma \Rightarrow \Delta$ follows when both $\Gamma \Rightarrow A, \Delta$ and $B, \Gamma \Rightarrow \Delta$ hold.
  • $\rightarrow$-Right: $\Gamma \Rightarrow A \rightarrow B, \Delta$ from $A, \Gamma \Rightarrow B, \Delta$.
• Double Negation and Negation-Pushing:

These auxiliary rules ensure formulas admit suitable normal forms for metatheoretic constructions, particularly interpolation.

A recent variation introduces:

• The “non-here” operator $\operatorname{nh}(A)$, interpreted as “$A$ is false here (may or may not be true there),” with truth table: | $A$ | $\operatorname{nh}(A)$ | |-----|------------------------| | F | T | | NF | T | | T | F |
• New axiom schemes for $\operatorname{nh}$ and a third right-implication ($\rightarrow^*$) rule suited to the extended interpolation technique (Wernhard, 7 Jan 2026).

3. Two-Stage Interpolation Method for HT

A Maehara-style argument enables effective Craig interpolation for HT, utilizing a provenance-annotated sequent calculus:

• Split Sequents and Interpolation Invariants:
  • (I1) $\Gamma^L \Rightarrow H \vee \Delta^L$
  • (I2) $\Gamma^R \wedge H \Rightarrow \Delta^R$
  • (I3) $$\operatorname{voc}(H) \subseteq \operatorname{voc}(\Gamma^L \cup \Delta^L) \cap \operatorname{voc}(\Gamma^R \cup \Delta^R)$$
• Stage 1 yields a preliminary interpolant $C'$ in an extended “nh-logic.”
• Stage 2 strengthens $C'$ to a genuine HT-interpolant $C$ (no $\operatorname{nh}$) by:
  1. Converting $C'$ to CNF: $C' \equiv D_1 \wedge \cdots \wedge D_k$, each $$D_i = \operatorname{nh}(E_{i1}) \vee \cdots \vee F_i$$.
  2. For each $D_i$, derive $$A \Rightarrow \neg\neg(\neg E_{i1} \vee \cdots \vee F_i)$$.
  3. Obtain the implication $$A \Rightarrow (E_{i1} \wedge \ldots \wedge E_{im} \rightarrow F_i)$$.
  4. Let $C$ be the conjunction of such implications.

This construction proves: If $A \Rightarrow B$ in HT, there is a computable HT formula $C$ with $$\operatorname{voc}(C) \subseteq \operatorname{voc}(A) \cap \operatorname{voc}(B)$$ and $A \Rightarrow C$, $C \Rightarrow B$ (Wernhard, 7 Jan 2026).

4. Metatheoretical Properties: Soundness, Completeness, Cut-Admissibility

All axioms and rules of both the original and extended Mints systems are sound for the three-valued HT semantics (with or without $\operatorname{nh}$):

• Soundness:

Verified by inspection of the HT truth tables for each rule.

• Completeness:

Demonstrated (for the original and the $\operatorname{nh}$-variation) by canonical countermodel constructions. If a sequent is unprovable, a distinguishing three-valued HT model exists.

• Cut-Admissibility:

The cut rule is admissible, with elimination by simultaneous induction on formula complexity and proof height, ensuring the analytic (subformula) property of the system.

5. Concrete Example: Interpolation Derivation

For the entailment $p \wedge q \Rightarrow p \vee r$ (shared vocabulary $\{p\}$), interpolation yields $C = p$.

• Begin with the root sequent $(p \wedge q)^L \Rightarrow^{?} (p \vee r)^R$.

• Decompose by the left-conjunction rule, then the right-disjunction rule. At each axiom leaf, assign interpolant labels: for $p^L \Rightarrow p^R$, $H=p$; for $p^L \Rightarrow r^R$, $H=\text{false}$.
• The root interpolant is $p \vee \text{false} \equiv p$, satisfying $p \wedge q \Rightarrow p$ and $p \Rightarrow p \vee r$.

This workflow generalizes: rules propagate the interpolant $H$ according to formula position; the strengthening step systematically eliminates $\operatorname{nh}$.

6. Connections to First-Order HT and Implementation

Recent work has extended the sequent calculus framework for HT to the first-order case (Otten et al., 7 Jan 2026), introducing:

• Explicit rules for quantification and variable management (using free variables and dynamic Skolemization).
• Analytic cut-free proofs and invertible rules, preserving the key properties of Mints’ propositional calculus.
• Pragmatic adjustments to the axiom schemes for efficient proof search. These developments enable effective implementation of automated HT theorem provers and broaden proof-theoretic understanding of intermediate logics.

7. Historical and Research Significance

Mints’ sequent calculus for HT and its recent interpolating and first-order variants have emerged as central tools in the study of nonclassical logics relevant for answer-set programming, modal embeddings, and constructive intermediate proof theory. The interpolation results, proof search optimizations, and modular extensions with operators such as $\operatorname{nh}$ clarify the structural position of HT, the expressive frontier between intuitionistic and classical logics, and the interface between semantic and syntactic proof procedures (Wernhard, 7 Jan 2026, Otten et al., 7 Jan 2026).