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$0F = T
• $\neg$1NF = F
• $\neg$2T = F

HT is semantically complete with respect to Gödel’s $\neg$3 and strictly contains IPC (intuitionistic propositional logic) while being properly contained in classical logic. Every HT tautology is a $\neg$4 tautology and vice versa.

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

Sequents in Mints’ system take the form $\neg$5, with $\neg$6 and $\neg$7 multisets of formulas. No explicit structural rules are listed; weakening, contraction, and exchange are admissible. The calculus comprises:

• Axiom Schemes:
  • (Ax-1): $\neg$8 where $\neg$9 is an atom or negated atom.
  • (Ax-2): $\vee$0 where $\vee$1 is an atom.
• Logical Rules:
  • $\vee$2-Left: $\vee$3 follows when both $\vee$4 and $\vee$5 hold.
  • $\vee$6-Right: $\vee$7 from $\vee$8.
• 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 $\vee$9, interpreted as “$\wedge$0 is false here (may or may not be true there),” with truth table: | $\wedge$1 | $\wedge$2 | |-----|------------------------| | F | T | | NF | T | | T | F |
• New axiom schemes for $\wedge$3 and a third right-implication ($\wedge$4) 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) $\rightarrow$0
  • (I2) $\rightarrow$1
  • (I3) $\rightarrow$2
• Stage 1 yields a preliminary interpolant $\rightarrow$3 in an extended “nh-logic.”
• Stage 2 strengthens $\rightarrow$4 to a genuine HT-interpolant $\rightarrow$5 (no $\rightarrow$6) by:
  1. Converting $\rightarrow$7 to CNF: $\rightarrow$8, each $\rightarrow$9.
  2. For each $A$0, derive $A$1.
  3. Obtain the implication $A$2.
  4. Let $A$3 be the conjunction of such implications.

This construction proves: If $A$4 in HT, there is a computable HT formula $A$5 with $A$6 and $A$7, $A$8 (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 $A$9):

• Soundness:

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

• Completeness:

Demonstrated (for the original and the $B$0-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 $B$1 (shared vocabulary $B$2), interpolation yields $B$3.

• Begin with the root sequent $B$4.

• Decompose by the left-conjunction rule, then the right-disjunction rule. At each axiom leaf, assign interpolant labels: for $B$5, $B$6; for $B$7, $B$8.
• The root interpolant is $B$9, satisfying $A\vee B$0 and $A\vee B$1.

This workflow generalizes: rules propagate the interpolant $A\vee B$2 according to formula position; the strengthening step systematically eliminates $A\vee B$3.

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 $A\vee B$4 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).