unicode-fol-kit

A Python toolkit for first-order logic with Unicode operators — parse, transform, and reason about formulas — with a reasoning layer that reaches well beyond classical FOL into modal, temporal, hybrid, many-valued, fuzzy, intuitionistic, relevant, second-order, description, dependence/IF, substructural, and a range of further non-classical logics.

from unicode_fol_kit import MSFLParser, is_valid

phi = MSFLParser().parse(
    "∀x (Human(x) → Mortal(x)) ∧ Human(socrates) → Mortal(socrates)")
print(is_valid(phi))   # True

One parser class, MSFLParser, has nine modes (classical FOL, many-sorted FOL, many-sorted and single-sorted Łukasiewicz fuzzy logic, modal/temporal/epistemic/deontic/ hybrid, second-order, team-semantic dependence/IF logic, intuitionistic linear logic, and the Lambek calculus) selected by constructor flags, with natural Unicode surface syntax (∀ ∃ ∧ ∨ ¬ → ↔ ⊕ ⊗ □ ◇ @ ⊸ 𝟙 …) and no ASCII fallbacks.

On top of the AST sits a full reasoning stack — four proof methods (a built-in resolution prover, Fitch natural deduction with checker and searcher, the Gentzen sequent calculi LK/LJ — the latter backed by Dyckhoff's G4ip, a genuine terminating decision procedure for propositional intuitionistic logic — and analytic tableaux), a finite model finder, SMT (Z3) and external-prover (Prover9 / Vampire) backends, truth tables, and dedicated semantics for every logic. Formulas import/export to TPTP, Prover9, SMT-LIB, LaTeX, and JSON.

📖 Documentation

Full guide and API reference: https://unicode-fol-kit.readthedocs.io/

The documentation walks through every logic with runnable examples — start with the Quickstart and Choosing a tool.

Installation

pip install unicode-fol-kit

Requires Python 3.10+. Z3 ships with the package; Prover9, Vampire, and Isabelle are optional external tools you install separately to unlock the corresponding backends.

Logics at a glance

Logic	Enable / entry point	Decide / reason with
Classical FOL / MSFOL	`MSFLParser()` / `many_sorted=True`	resolution, Z3, Prover9/Vampire, tableaux, Fitch, LK, finite model finder
Fuzzy Łukasiewicz / Gödel / product	`MSFLParser(fuzzy=True)`	`fuzzy_evaluate`, `fuzzy_is_valid(…, tnorm=…)` (Z3 reals, quantifier grounding); classical routes (`is_valid`, normal forms, resolution) refuse fuzzy input rather than silently collapsing it — opt in explicitly with `to_fol(node)`
Modal / temporal / epistemic / deontic	`MSFLParser(modal=True)`	`satisfies_modal`, `standard_translation`, native `is_modal_valid` / `modal_decide` (K…S5, B, KD45)
Quantified modal	`KripkeModel(domains=…)`	`qml_is_valid` per domain regime + frame; THF / Isabelle export
Many-valued K3 / LP / Belnap FDE	`truth_table`, `semantics.matrix`	`matrix_is_valid` / `matrix_entails` over any finite `TruthMatrix`, incl. THF/Isabelle export
Intuitionistic	`int_valid` / `int_countermodel`	propositional decision procedure (`int_prove`/`int_decide`, G4ip) + bounded first-order Kripke search; LJ checker
Second-order	`MSFLParser(second_order=True)`	`satisfies_so`, bounded `so_is_valid_finite` / `so_find_countermodel`
Description logic ALC	`unicode_fol_kit.dl`	`concept_satisfiable` / `subsumes` / `abox_consistent` (tableau, TBox + ABox); `parse_concept`/`parse_gci` plus `concept_to_fol`/`tbox_to_fol`/`abox_to_fol` reuse the FOL provers and Isabelle/THF exports
Free · public-announcement · counterfactual · circumscription	`semantics.free_logic` / `dynamic_epistemic` / `conditional` / `nonmonotonic`	`free_is_valid`/`free_entails` (bounded search); `[φ!]ψ`/`⟨φ!⟩ψ` parse in modal mode and decide via `reduce_announcements`; `cf_valid`; `minimal_entails` and the unbounded `circumscription_entails_so`
Hybrid H(@) (nominals, `@i φ`)	`MSFLParser(modal=True)`	`KripkeModel(nominals=…)`, `hybrid_is_valid` per frame (standard translation + Z3)
Relevant logic B	classical syntax + `semantics.relevant`	`rel_valid` / `rel_countermodel` (Routley–Meyer, bounded exhaustive search); `isabelle_decide_relevant` certifies both directions
Dependence / IF (team semantics)	`MSFLParser(dependence=True)`	`team_satisfies` / `team_models` over finite structures; `dependence_to_eso` translates the guarded/slashed sentence fragment to ESO for `satisfies_so` / Isabelle export
Linear logic (ILL, incl. `⊤`/`𝟘`) · Lambek calculus	`MSFLParser(linear=True)` / `lambek=True`	`ill_prove` (cut-free; complete for !-free) · `lambek_derivable` (decision procedure); either derivation replays as a machine-checked Isabelle lemma

With a local Isabelle installed, the hol subpackage's shallow embeddings become proofs: isabelle_decide_modal / isabelle_decide_fol / isabelle_decide_relevant actually run the prover, and hol.isabelle_substructural replays a Python-found ILL/Lambek derivation as a machine-checked lemma. The hol.deepshallow subpackage goes further, emitting — for propositional modal, intuitionistic, Lewis-conditional and relevant logic — the deep, maximal-shallow and minimal-shallow embeddings side by side with machine-checked faithfulness proofs between them (Benzmüller, arXiv:2502.19311), verified end to end by Isabelle. See the higher-order guide.

Command line

python -m unicode_fol_kit "∀x P(x)" --to latex

--mode selects the parser dialect — fol (default), msfol, msfl, fl, modal, second_order, dependence, linear, lambek — and --to the rendering: tree, unicode, latex, tptp, prover9, json, dot.

Building the documentation locally

pip install -e ".[docs]"
sphinx-build -b html docs docs/_build/html

Citation

If you use this toolkit in academic work, please cite the accompanying preprint:

@misc{vossel2025advancingnaturallanguageformalization,
      title={Advancing Natural Language Formalization to First Order Logic with Fine-tuned LLMs},
      author={Felix Vossel and Till Mossakowski and Björn Gehrke},
      year={2025},
      eprint={2509.22338},
      archivePrefix={arXiv},
      primaryClass={cs.CL},
      url={https://arxiv.org/abs/2509.22338},
}

Vossel, F., Mossakowski, T., & Gehrke, B. (2025). Advancing Natural Language Formalization to First Order Logic with Fine-tuned LLMs. arXiv preprint arXiv:2509.22338.

License

MIT

Metadata

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