Cariani, Santorio and Wellwood (2023): Positive gradable adjective ascriptions without positive morphemes

Cariani, Santorio and Wellwood give a gradable adjective a background ordering of states, the states of having some heat for hot, and a context-dependent threshold property, the states that count as hot. The positive form says that the subject holds a state with the threshold property; the comparative bypasses the threshold, comparing an admissible measure of the subject's states with the greatest degree of the standard's, Wellwood's comparative. The positive form therefore needs no covert pos, and neither form entails the other. The framework, with its relations to Klein's delineations and to degree thresholds, is Semantics/Degree/Background.lean; this file checks the paper's claims against it. The paper argues from two cities of equal temperature that the background is not linear, but its premises make the background total and refute only antisymmetry, the two states sharing one degree.

Main statements

• ne_and_antisymmRel_and_total: the equal-temperature cities of §4 hold distinct, tied states of a total background.
• The examples check the inference (19) on a three-state model, the split verdict of the literal reading of §4, (24) against the English fragment, and the failure of the converse of (26).

Implementation notes

The states of all holders form a type S with the background preorder, and holder : S → X is a function, each state having one holder (fn. 9), but not injective, since the than-clause maximum ranges over a holder's several states. The presupposition that a state lies in the domain of the background is membership in S. The comparative's standard degree d_b is the greatest measure of the standard's states. In (31) the comparative binds g but writes background(f).

TODO

• The crispness contrast (8) and for-phrases (33), (34), for which the paper gives no semantics.

References

• [CSW23]
• [Wel15]
• [Kle80]
• [Cre76]

The inference (19)

The equal-temperature cities (§4, fn. 10)

theorem CarianiSantorioWellwood2023.ne_and_antisymmRel_and_total {S : Type u_1} {X : Type u_2} [Preorder S] {holder : S → X} {D : Type u_3} [LinearOrder D] {μ : S → D} (hμ : ∀ (a b : S), μ a ≤ μ b → a ≤ b) {s t : S} (hne : holder s ≠ holder t) (heq : μ s = μ t) : s ≠ t ∧ AntisymmRel (fun (x1 x2 : S) => x1 ≤ x2) s t ∧ ∀ (u v : S), u ≤ v ∨ v ≤ u

In the cities of §4, states of distinct holders at equal measure are distinct and, when the measure reflects the background into a linear scale, tied, and the background is total.

Scale-mates (24)–(26)