Gajewski (2002): On Analyticity in Natural Language #

This file formalizes [Gaj02]'s principle that a sentence is ungrammatical if its logical form contains an L-analytic constituent, one that is true, or false, in virtue of its logical structure alone. Logical structure has two ingredients, and the file formalizes both halves of the paper.

The logical items are the denotations invariant under permutations of the domain, the criterion of [vB89]: a permutation of the entity domain lifts through the Montague.Ty hierarchy (permLift), and an item is logical when every lifted permutation fixes it (PermutationInvariant). Truth-functional conjunction is fixed because the lift is the identity at type t (permutationInvariant_and), expletive there, which denotes the whole domain, is fixed, and the domain and the empty set are the only invariant properties (permutationInvariant_et_iff), so a contingent predicate like new-student is not a logical item. Invariance at the determiner type is exactly the quantity invariance the Quantification substrate states on GQ (permutationInvariant_det_iff), so the logicality of every, some and no follows from the isomorphism invariance of their Lindström classes rather than being stipulated. Von Fintel's but is a logical item too (permutationInvariant_but).

The logical skeleton of a sentence replaces each maximal constituent without logical items by a distinct variable of its semantic type; a skeleton is a predicate on type-sorted assignments (Skeleton, domains by Montague.Ty.Domain), and a sentence is L-analytic when its skeleton receives one truth value under every assignment (Skeleton.IsLAnalytic). The principle puts two analyses on firmer ground. [BC81] explained the definiteness restriction on there-sentences of [Mil77] by the tautology a strong determiner produces there: the skeleton of there is every new student is an L-tautology because every set is a subset of the domain (thereSkeleton_isLTautology), while with some or two the skeleton is false under the empty assignment and true under a large enough one. [VF93] explained the restriction of but-exceptives to universal determiners by the contradiction his least-exception semantics produces under a left-upward-monotone determiner, which admits no nonempty least exception (IsExceptionSet.eq_bot_of_restrictorMonotone): the skeletons with some and three are L-contradictions, and those with every and no are contingent. Both analyses had appealed to trivial truth conditions, which war is war shows cannot be the explanation, and L-analyticity is narrower than triviality: every woman is a woman and John is smoking and John is not smoking receive skeletons with distinct variables for their repeated material, which are contingent, so the garden-variety tautologies and contradictions come out grammatical. The paper's sentences are the rows, and the principle predicts each (rows_predicted) except the most-exceptive, the gap the paper's footnote on most concedes and exceptiveSkeleton_most_not_isLAnalytic states.

Implementation notes #

The paper's type hierarchy has e, t and function types only; permLift extends it to the rest of Montague.Ty by the identity at the degree, cardinality and eventuality sorts and pointwise at intensions, with Unit as the inert index sort of the extensional fragment.
The denotation of but is the paper's schema factored compositionally, but where the printed uniqueness clause reads g ⊆ j the least-exception schema it factors requires f ⊆ j, so but is IsExceptionSet with the arguments rearranged; the exceptive skeleton carries the nonemptiness of the exception set that the paper's footnote adds to the schema.
A row's L-analyticity quantifies over every reading of its determiner and every finite nonempty domain, following the model quantification of the paper's strength and monotonicity definitions; nonemptiness reflects the remark that the some skeleton is true on every nonempty value of its variable.
The readings of every, some and most are the English fragment's; no is read from the quantification substrate because the fragment has no entry for the word, a is read as some, everyone as every, someone as some, and the numerals as at least n, the paper's own reading of one, two as left-upward-monotone determiners.
The rows cover the sentences the paper derives. Its remaining printed examples are data only: the wolf needs a partial definite determiner, John and Mary is not a determiner, many has no fixed reading, and the paper derives no verdict for exactly two or fewer than three.
The examples are Data.Examples.Gajewski2002.

TODO #

A partial definite determiner would let the the wolf row join rows_predicted.
Cardinality arguments for exactly two and fewer than three exceptives are in the later literature the paper's footnote cites, not in the paper.
The English determiner fragment lacks no; adding it would let Determiner.no read from the fragment.

Gajewski (2002): On Analyticity in Natural Language #

Implementation notes #

TODO #

References #

Logical items by permutation invariance (§3.1) #

Logical skeletons and L-analyticity (§3.2) #

The definiteness restriction (§3.3.1) #

But-exceptives (§3.3.2) #

Garden-variety tautologies and contradictions (§3.3.3) #

The paper's sentences #