📄 A new formal tool: Functorial variables representing assertions and presuppositions

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

• I1. 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.
• I2. 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.
• I3. Allow several assertions to point at one shared presupposition variable (different p indices, same q), so common background is represented once.
• I4. 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.
• I5. 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.