📄 Presupposition Projection and Repair Strategies in Trivalent Semantics

Summary

Trivalent logic, with true, false and presupposition failure, is a standard way to model presupposition projection. The two classic Kleene systems each get half the picture right. Weak Kleene connectives never filter: for a conditional whose antecedent says Sue was once a smoker and whose consequent says she quit, they incorrectly project the presupposition carried by the consequent. Strong Kleene connectives do filter, but they produce weak conditional presuppositions with irrelevant disjuncts. This is the "proviso problem".

Winter shows that dropping truth-functionality opens up operators of intermediate strength. In these operators, the presupposition of the right operand is discarded only when the left operand actually entails it. He defines "Weak Kleene plus" (WK+) operators for left-to-right filtering and "Strong Kleene minus" (SK−) operators for conditional presuppositions, and shows that neither suffers from the proviso problem. He then generalizes these repair strategies from connectives to arbitrary binary functions. This exposes a hierarchy: SK repairs the most, WK repairs nothing, and WK+ and SK− lie strictly between them. The work matters because it reconciles filtering with non-trivial projection inside a purely trivalent semantics.

Key points

• Elementary presuppositional propositions are built with Blamey's transplication. Projection is measured by the maximal bivalent presupposition (MBP) of a sentence.
• The running examples use "Sue stopped smoking (Marlboros)". They contrast cases where filtering happens (the antecedent or first conjunct entails the presupposition) with cases where it does not (e.g., "Sue exercises a lot").
• In the generalization, the core notions are left-determination of a function's value and R-repairability vs. anti-R-repairability of a binary expression.
• The paper argues that the Tarskian entailment preorder suits natural language better than the SK lattice, which treats failure as ½. Its evidence is that "Sue has stopped shouting" entails "Sue is not shouting" even when the first sentence fails.
• The work was funded by ERC grant No 742,204.

Interesting ideas

• I1. Filtering is licensed by an actual entailment between operands rather than by truth-table shape, so a presupposition is dropped only when the earlier material really supports it.
• I2. The paper places a family of systems on a scale from "no repair" (WK) to "maximal repair" (SK) and builds controlled intermediate points instead of choosing an extreme.
• I3. Failure is treated as an exception that propagates unless specifically repaired, not as an "unknown" midpoint between true and false.
• I4. Repair is generalized to any binary function: the result can survive a failed argument only if the other argument determines the value, and only when that determination is systematically tied to the failure.
• I5. Entailment is kept separate from projection: false and failure behave the same for what a sentence entails but differ in what it projects.

For Mova

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

• I1 (frames): When the prag crate projects presuppositions of embedded clauses, it could check whether the preceding conjunct or antecedent entails the trigger's presupposition before cancelling it, and log that entailment as the explanation.
• I2 (learn): Self-induced repair rules could be graded by how permissive they are, so evaluation can compare a cautious and a liberal variant against negative controls.
• I3 (world): Unresolved facts in the world of the text (e.g., a definite with no referent) could be stored as explicit failure states, cleared only by a justified repair, rather than treated as half-true.
• I4 (answer): Arithmetic in maths word problems could return an answer despite a missing quantity only when the known operand fixes the result (e.g., multiplication by zero), with the reasoning recorded.
• I5 (answer): School-test readers could judge entailment questions on content alone, while handling presupposition-failure questions ("Did Sue stop?" when she never smoked) on a separate path.