---
title: "A new formal tool — paper by Max (1988) — summary and ideas"
description: "The paper tackles a long-standing formalization problem: an utterance has an explicit part (assertion) and an implicit part (presupposition)…"
authors: ["Ingolf Max"]
year: 1988
venue: "Coling Budapest 1988 Volume 1: International Conference on Computational Linguistics"
source: "https://aclanthology.org/C88-1083/"
licence: "CC BY-NC-SA 3.0 (ACL Anthology, before 2016)"
licence_url: "https://creativecommons.org/licenses/by-nc-sa/3.0/"
summary_licence: "Elamur Cosmic License"
canonical: "https://research.elamur.ai/free/en/papers/max-1988-new-formal-tool-functorial-variables/"
---
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# A new formal tool: Functorial variables representing assertions and presuppositions

**Authors:** Ingolf Max · **Year:** 1988 · **Venue:** [Coling Budapest 1988 Volume 1: International Conference on Computational Linguistics](https://md.research.elamur.ai/free/en/venues/coling/) ([COLING 1988](https://md.research.elamur.ai/free/en/venues/coling/1988/)) · **Licence:** [CC BY-NC-SA 3.0 (ACL Anthology, before 2016)](https://creativecommons.org/licenses/by-nc-sa/3.0/)

**Links:** [Publisher's page](https://aclanthology.org/C88-1083/) · [PDF](https://aclanthology.org/C88-1083.pdf) · [OpenAlex](https://openalex.org/W1986466188)

**Citations:** 0 citations (OpenAlex, 2026-10-14)

## Summary
The paper tackles a long-standing formalization problem: an utterance has an explicit part (assertion) and an implicit part (presupposition), and it is hard to capture both inside a single logical expression. Russell merged them by conjunction, while Strawson and later linguists such as Kiefer kept them on separate levels. Max extends classical propositional logic with a new syntactic category, functorial variables (FVs). An FV takes as its value a pair of classical two-valued functors, its first and second components, chosen from the 16 two-place truth functions. The central construction pairs an assertion variable p with a presupposition variable q, so one formula carries both. Connection rules make every formula containing FVs stand for exactly two FV-free classical formulas. A formula counts as valid only if both of these readings are valid. The resulting system is semantically equivalent to classical propositional logic. The paper argues that the approach is a purely syntactic enrichment, unlike the 4-valued or 2-dimensional treatments of Bergmann and Jung/Küstner. The gain is a compact, integrated representation that keeps classical logic's completeness and consistency.

## Key points
- The language G adds FVs of the form G²_{f,g}, where f and g index two-place functors 1–16 by their value tables. The construction G²_{4,6} p_i q_i links assertion p_i with presupposition q_i. The two variables are different sorts: p always sits on the left, and both share an index.
- Each FV formula unfolds into two closed substitutions, A^E1 and A^E2. A formula is valid iff A^E1 ∧ A^E2 is classically valid, so FVs can be eliminated and the system stays complete and consistent.
- Until one of its two readings is chosen, an FV formula receives no truth value at all. Also, A ↔ (A^E1 ∧ A^E2) often fails for formulas that are neither tautologies nor contradictions.
- In strength, the FV representation lies between conjunction and disjunction:
  - (p∧q) → G(pq) is valid.
  - G(pq) → (p∨q) is valid.
  - G(pq) → (p∧q) is not valid.
- The paper distinguishes two negations:
  - G²_{13,6} acts as presupposition-preserving negation.
  - G²_{11,11} acts as presupposition-rejecting negation.
- The paper proposes five extension steps:
  - distinct indices, so several sentences can share one presupposition;
  - multiple presuppositions joined by connectives;
  - functional dependence between assertion and presupposition;
  - more than two components;
  - combinations of these steps.

## Interesting ideas
- Store a sentence's meaning as one object with two typed slots, asserted content and presupposed content, rather than as a flat conjunction; this keeps what is claimed separate from what is taken for granted.
- Treat negation as two operators: one denies only the assertion and keeps the presupposition, the other denies the presupposition too; this explains why "X didn't stop smoking" usually still implies X smoked.
- Allow several assertions to point at one shared presupposition variable (different p indices, same q), so common background is represented once.
- Define validity of a compound representation as validity of every one of its unfolded readings; this gives a strict, checkable criterion for formulas that bundle several readings.
- Get expressive power by adding a syntactic layer that can always be compiled back to plain classical logic; this keeps the existing reasoning engine unchanged.

## For Mova

How these ideas could serve Mova, our small open language model — its files, stages and tests:

- **I1** (frames): prag frames could carry an explicit assertion/presupposition pair, so factive verbs, definite descriptions and "again/stop" triggers mark presupposed content without flattening it into the claim.
- **I2** (frames): when prag sees a negation, it could choose the preserving reading by default and switch to the rejecting reading only on explicit cues ("…because he never smoked"), recording which choice was made for explainability.
- **I3** (world): presuppositions recurring across sentences of a fable could be merged into one world-state fact that several events depend on, instead of being duplicated per sentence.
- **I4** (answer): an answer could be accepted only when it holds under both the assertion-only and the presupposition-only projection of the question, flagging cases where a question's presupposition is false in the text world.
- **I5** (world): richer frame annotations could be compiled down to ordinary knowledge-graph facts, so the existing inference over WordNet/Wikidata facts needs no new semantics.

*Our summary of the paper, in our own words — not its text. The work belongs to its authors; read it at the links above.*

## Institutions

- [Martin Luther University Halle-Wittenberg](https://md.research.elamur.ai/free/en/universities/martin-luther-university-halle-wittenberg/) — University, Halle, Germany · 3 works · 2,629 citations

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## Cite

Cite as: Ingolf Max. 1988. A new formal tool: Functorial variables representing assertions and presuppositions. Coling Budapest 1988 Volume 1: International Conference on Computational Linguistics.

The paper itself, not this page. BibTeX from the library's record of the work.

```bibtex
@inproceedings{max1988new,
  title = {{A new formal tool: Functorial variables representing assertions and presuppositions}},
  author = {Ingolf Max},
  booktitle = {Coling Budapest 1988 Volume 1: International Conference on Computational Linguistics},
  year = {1988},
  doi = {10.3115/991635.991718},
  url = {https://doi.org/10.3115/991635.991718}
}
```

- [← previous (In Amharic: Academy of Ethiopian Languages. Symposium on dictionaries organized by the section of preparation and elaboration of a dictionary), General report, Yakatit 1975 (Year of Grace February-March 1985), Addis Ababa, pp. 251.](https://md.research.elamur.ai/free/en/papers/brzuski-1988-amharic-academy-ethiopian-languages-symposium/)
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- [↑ back to papers of 1988](https://md.research.elamur.ai/free/en/papers/1988/)

*Updated 29 October 2026*