Docs / Studies / Studies.Fagen2026
Fagen, L. (2026). PhD thesis, The University of Chicago
Fagen derives the exhaustive interpretation of an answer from its meaning and the question, with no theory of alternative sentences: following Schulz and van Rooij, an answer holds in the best worlds where it is true, and what varies is the order that ranks the worlds. The default order minimizes the question-property, which gives a positive answer such as Agnes its negative direction and unrestricted extent; under it exhaustive interpretation is the minimal-worlds exhaustifier over the question's propositions. A negative or mixed answer is only about its restriction, the intersection of the sets it lives on, which the positive and negative restrictions make up; the hybrid order minimizes the question-property within the first and maximizes it within the second, so that nothing outside the restriction is concluded. Positive skew, a restriction equal to the positive restriction, selects the default order. Some of the linguists but not all of the computer scientists is not positively skewed, and the hybrid order makes exactly one linguist and all but one computer scientist attend. Positive skew is exactly Spector's quasi-positivity, which Spector stated over formulae of a propositional language.
T over individuals and the question a property P : W → Set E of
worlds; exhaustive interpretation exh P T r takes the order r as a parameter, as (29) does.quantifierOf.not_transitive_ltHybrid); Fagen calls it an order and notes only that it is
antisymmetric. Exhaustive interpretation is defined from the relation itself, so his predictions
stand.not_livesOn_restriction_cofinite). The appendix assumes a finite model, where
the result holds (livesOn_restriction).The restriction of a quantifier is the intersection of the sets it lives on ((10)).
Used by notMem_restriction_iff, mem_restriction_iff, positiveRestriction_union_negativeRestriction, positiveRestriction_subset_restriction, negativeRestriction_subset_restriction, livesOn_restriction, not_livesOn_restriction_cofinite, PositivelySkewed, NegativelySkewed, ltHybrid_iff_of_inter_eq, mem_exh_ltHybrid_iff_of_inter
The positive restriction of a quantifier holds the objects whose addition turns a set the quantifier does not apply to into one it applies to ((18a)).
Used by mem_positiveRestriction_iff, positiveRestriction_union_negativeRestriction, positiveRestriction_subset_restriction, positiveRestriction_eq_empty_of_antitone, positiveRestriction_eq_empty_iff, PositivelySkewed, positivelySkewed_iff, negativelySkewed_iff, ltPos, positiveRestriction_someButNotAll, favors_pos_iff
The negative restriction of a quantifier holds the objects whose removal turns a set the quantifier does not apply to into one it applies to ((18b)).
Used by mem_negativeRestriction_iff, positiveRestriction_union_negativeRestriction, negativeRestriction_subset_restriction, negativeRestriction_eq_empty_of_monotone, negativeRestriction_eq_empty_iff, NegativelySkewed, positivelySkewed_iff, negativelySkewed_iff, ltNeg, negativeRestriction_someButNotAll, favors_neg_iff
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The positive and negative restrictions of a quantifier make up its restriction (A7).
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A quantifier is positively skewed when its restriction is its positive restriction ((2)).
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Used by positivelySkewed_iff, positivelySkewed_of_monotone, selectedOrder, not_positivelySkewed_someButNotAll, quasiPositive_iff_positivelySkewed
A quantifier is negatively skewed when its restriction is its negative restriction ((2)).
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The default order ranks worlds by inclusion of the question-property ((30)).
Used by selectedOrder, exh_ltDefault
The positive component of the hybrid order ((31a)).
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Used by leHybrid
The negative component of the hybrid order ((31b)).
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Used by leHybrid
The hybrid order keeps the pairs on which the two components do not conflict ((31d)).
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Used by selectedOrder, ltHybrid_asymm, ltHybrid_iff_of_inter_eq, mem_exh_ltHybrid_iff_of_inter, exh_ltHybrid_not, not_transitive_ltHybrid
The dual order ranks worlds by reverse inclusion of the question-property ((39)).
Used by exh_ltDual_not
Exhaustive interpretation keeps the worlds where the answer holds of the question-property
and no such world lies below by the order r ((29)).
Used by exh_ltDefault, mem_exh_ltHybrid_iff_of_inter, exh_selectedOrder_individual, exh_ltHybrid_not, exh_ltDual_not, exh_selectedOrder_someButNotAll
Order selection: a positively skewed answer takes the default order and any other answer the hybrid order ((12)).
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Used by exh_selectedOrder_individual, exh_selectedOrder_someButNotAll
Under the default order, exhaustive interpretation is the minimal-worlds exhaustifier over the propositions that the question-property holds of each individual ((83)).
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Exhaustive interpretation with the hybrid order treats alike two worlds that agree within the answer's restriction, so it concludes nothing about objects outside it (§2.2.1).
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Some of the L but not all of the C.
Used by positiveRestriction_someButNotAll, negativeRestriction_someButNotAll, not_positivelySkewed_someButNotAll, exh_selectedOrder_someButNotAll, not_transitive_ltHybrid
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Some of the L but not all of the C takes the hybrid order and says that exactly one L
and all but one C have the property ((40)).
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A proposition of Spector's question-relative language, a set of valuations of the atoms, as a quantifier over the atoms: it holds of the sets of atoms its valuations make true.
Used by favors_pos_iff, favors_neg_iff, quasiPositive_iff_positivelySkewed
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A proposition is quasi-positive in Spector's sense, strongly favoring no negative literal, just when the quantifier it determines is positively skewed (§2.1.1, after Spector's (77b)).
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