Geurts (2010): Quantity Implicatures

This file formalizes the textbook's case against conventionalism in section 7.3 of [Geu10], which rests on [GP09]'s verification experiment. Conventionalist theories of scalar inference agree that some is read as some but not all under an upward-entailing quantifier such as all or more than one, and the stronger versions predict the same under a non-monotone quantifier such as exactly two, while none predicts it under a downward-entailing one; the experiment found no such readings. The three quantifiers of (62) are the earlier study's Quant, and the weak and strong predictions are its PredictsLocalSIWeak and PredictsLocalSI (conventionalist_predictions). Chapter 3's observation that a downward-entailing environment reverses a scale, so that not all blocks the local inference, is the negated case (downward_entailing_blocks).

References

• [Geu10]
• [GP09]

theorem Geurts2010.exactlyTwo_not_scopeUp : ¬GeurtsPouscoulous2009.Quant.exactlyTwo.sem.ScopeMonotone

Exactly two is not upward entailing in its scope: two of three squares connected with some circle, all three with some circle or other.

theorem Geurts2010.notAll_not_scopeUp : ¬GeurtsPouscoulous2009.Quant.notAll.sem.ScopeMonotone

Not all is not upward entailing in its scope.

theorem Geurts2010.conventionalist_predictions : GeurtsPouscoulous2009.PredictsLocalSIWeak GeurtsPouscoulous2009.Quant.all.sem ∧ GeurtsPouscoulous2009.PredictsLocalSIWeak GeurtsPouscoulous2009.Quant.moreThanOne.sem ∧ ¬GeurtsPouscoulous2009.PredictsLocalSIWeak GeurtsPouscoulous2009.Quant.exactlyTwo.sem ∧ GeurtsPouscoulous2009.PredictsLocalSI GeurtsPouscoulous2009.Quant.exactlyTwo.sem

Section 7.3, (62): every conventionalist theory predicts the local reading under all and more than one; only the stronger versions predict it under exactly two, which is neither upward nor downward entailing in its scope.

theorem Geurts2010.downward_entailing_blocks : ¬GeurtsPouscoulous2009.PredictsLocalSI GeurtsPouscoulous2009.Quant.notAll.sem ∧ ¬GeurtsPouscoulous2009.PredictsLocalSIWeak GeurtsPouscoulous2009.Quant.notAll.sem

Section 3.2: the downward-entailing scope of not all supports no local inference on either version of the prediction.