Empty set effects in the verification of quantifiers – Evidence from reading times and picture verification Oliver Bott 1 , Fabian Schlotterbeck 2 & Udo Klein 3 1 Project CiC Xprag.de, Project B1 SFB 833 University of T¨ ubingen 2 SFB 833 University of T¨ ubingen 3 SFB 673 University of Bielefeld 17/10/2015, LCQ workshop, Budapest Bott, Klein & Schlotterbeck (LCQ Budapest 2015) 1 / 30
Introduction Introduction ◮ Aim: develop an algorithmic theory of processing quantifier scope that describes how... ◮ a verification algorithm applicable to any model is constructed during online interpretation ◮ this algorithm is executed given a specific model Bott, Klein & Schlotterbeck (LCQ Budapest 2015) 2 / 30
Introduction Introduction ◮ Aim: develop an algorithmic theory of processing quantifier scope that describes how... ◮ a verification algorithm applicable to any model is constructed during online interpretation ◮ this algorithm is executed given a specific model ◮ The automata model (e.g. van Benthem 1986, Szymanik 2009, Steinert-Threlkeld & Icard 2013) is a good candidate ◮ However, it does not account for crucial differences w.r.t. processing complexity of quantifiers (e.g. monotonicity effects) (1) a. Every A R s some B s. b. No A R s no B s. c. 00001 � 11111 � 01010 � Bott, Klein & Schlotterbeck (LCQ Budapest 2015) 2 / 30
Quantification Theory Motivation ◮ “The world is everything that is the case” (Wittgenstein, Tractatus) ◮ Not: “The world is nothing that is not the case” Bott, Klein & Schlotterbeck (LCQ Budapest 2015) 3 / 30
Quantification Theory Motivation ◮ “The world is everything that is the case” (Wittgenstein, Tractatus) ◮ Not: “The world is nothing that is not the case” Representation 1 Representation 2 ◮ Square 1: pink ◮ Square 1: pink, not blue, not red, . . . ◮ Square 2: blue ◮ Square 2: blue, not pink, not red, . . . ◮ . . . ◮ . . . ◮ Square 11: pink ◮ Square 11: pink, not blue, not red, . . . Bott, Klein & Schlotterbeck (LCQ Budapest 2015) 3 / 30
Quantification Theory Motivation ◮ “The world is everything that is the case” (Wittgenstein, Tractatus) ◮ Not: “The world is nothing that is not the case” Representation 1 Representation 2 ◮ Square 1: pink ◮ Square 1: pink, not blue, not red, . . . ◮ Square 2: blue ◮ Square 2: blue, not pink, not red, . . . ◮ . . . ◮ . . . ◮ Square 11: pink ◮ Square 11: pink, not blue, not red, . . . ◮ Representation 2 is cognitively less plausible than representation 1. We assume that - per default - humans only encode positive information Bott, Klein & Schlotterbeck (LCQ Budapest 2015) 3 / 30
Quantification Theory Interpretating multiply quantified sentences – a hard task! (2) Most boys gave exactly one girl at least two gifts. Assume: Most boys > exactly one girl > at least two gifts Bott, Klein & Schlotterbeck (LCQ Budapest 2015) 4 / 30
Quantification Theory The ‘simple’ expansion algorithm ( s-exp ) – Expanding Q 3 ◮ Aim: enlarge the verb denotation VERB by Q s starting with the Q with narrowest scope ◮ Rule: If the restrictor elements of Q participating in VERB are among the witness sets of Q , add a tuple with Q to VERB s-exp ( Q 3 , VERB ): Add � b 1 , g 1 , at least 2 � , � b 2 , g 1 , at least 2 � , � b 3 , g 2 , at least 2 � , � b 4 , g 2 , at least 2 � Bott, Klein & Schlotterbeck (LCQ Budapest 2015) 5 / 30
Quantification Theory The ‘simple’ expansion algorithm ( s-exp ) – Expanding Q 2 s-exp ( Q 3 , VERB ) : Add � b 1 , g 1 , at least 2 � , � b 2 , g 1 , at least 2 � , � b 3 , g 2 , at least 2 � , � b 4 , g 2 , at least 2 � s-exp ( Q 2 , s-exp ( Q 3 , VERB )) : Add � b 1 , exactly 1 , at least 2 � , � b 2 , exactly 1 , at least 2 � , � b 3 , exactly 1 , at least 2 � , � b 4 , exactly 1 , at least 2 � Bott, Klein & Schlotterbeck (LCQ Budapest 2015) 6 / 30
Quantification Theory The ‘simple’ expansion algorithm ( s-exp ) – Expanding Q 1 s-exp ( Q 3 , VERB ) : Add � b 1 , g 1 , at least 2 � , � b 2 , g 1 , at least 2 � , � b 3 , g 2 , at least 2 � , � b 4 , g 2 , at least 2 � s-exp ( Q 2 , s-exp ( Q 3 , VERB )) : Add � b 1 , exactly 1 , AL 2 � , � b 2 , exactly 1 , AL 2 � , � b 3 , exactly 1 , AL 2 � , � b 4 , exactly 1 , AL 2 � s-exp ( Q 1 , s-exp ( Q 2 , s-exp ( Q 3 , VERB ))) : Add � most boys , exactly 1 girl , at least 2 gifts � ⊲ Sentence is true (with linear scope) iff � Q 1 , Q 2 , Q 3 � is added by the sequence: s-exp ( Q 1 , s-exp ( Q 2 , s-exp ( Q 3 , verb ) Bott, Klein & Schlotterbeck (LCQ Budapest 2015) 7 / 30
Quantification Theory Summary s-exp ◮ Algorithm depends on a single rule ◮ s-exp of Q n : Evaluate those elements in the restrictor set of Q n that are in the scope set and check whether these belong to the set of Q n ’s witness sets ⊲ s-exp allows us to ignore ‘negative information’ Bott, Klein & Schlotterbeck (LCQ Budapest 2015) 8 / 30
Quantification Theory Empty set situations – when s-exp fails (3) Most boys gave exactly one girl at most two gifts. ◮ All boys gave at most two gifts to all of the girls ⊲ We have to consider states of affairs where boys didn’t give gifts to girls, too! Bott, Klein & Schlotterbeck (LCQ Budapest 2015) 9 / 30
Quantification Theory Introducing the complex expansion operation c-exp c-exp = s-exp + an additional rule Expansion with Q n : ◮ Add Q n if the s-exp rule succeeds ◮ Q 3 : s-exp with at most two : Add { b 1 , g 1 , at most 2 } , { b 2 , g 1 , at most 2 } , { b 3 , g 2 , at most 2 } , { b 4 , g 2 , at most 2 } ◮ Or, if Q n has the empty set among its witness sets, add Q n in empty set situations ◮ Q 3 : c-exp with at most two : Add { b 5 , g 3 , at most 2 } , { b 5 , g 4 , at most 2 } , { b 6 , g 3 , at most 2 } , { b 6 , g 4 , at most 2 } , { b 7 , g 3 , at most 2 } , { b 7 , g 4 , at most 2 } ◮ This is just for illustration purposes, in our formal system c-exp is even more complicated; we have to properly keep track of tuples in a relation as well as tuples not contained Bott, Klein & Schlotterbeck (LCQ Budapest 2015) 10 / 30
Quantification Theory Relation between s-exp and c-exp Proposition Whenever the empty set is not a witness set of any Q in the sentence, then s-exp suffices for truth evaluation irrespective of the model. However, in order to safely evaluate non-empty set quantifiers, c-exp is required. Bott, Klein & Schlotterbeck (LCQ Budapest 2015) 11 / 30
Hypotheses and predictions H1: Encoding the negative information and application of c-exp make empty set quantifiers more complex to interpret than non-empty set quantifiers ◮ Longer reading times and more difficult evaluation of sentences with empty set quantifiers. H2: Evaluation of empty set quantifiers in empty set situations is especially difficult ◮ Verification of empty set quantifiers in empty set situations leads to more errors and longer judgment times than in all other cases. Bott, Klein & Schlotterbeck (LCQ Budapest 2015) 12 / 30
Experiment 1: Establishing empty set effects Experiment 1: Establishing empty set effects A) More than five squares | are pink. (non empty set, MON ↑ ) B) Less than five squares | are pink. (empty-set, MON ↓ ) C) Exactly five squares | are pink. (non empty set, non-MON) 0-model . . . 6-model . . . 11-model . . . . . . ◮ 3 ( quantifier ) × 12 ( model ) within design ◮ 48 participants, 144 experimental items, three lists in a latin square Bott, Klein & Schlotterbeck (LCQ Budapest 2015) 13 / 30
Experiment 1: Establishing empty set effects Exp. 1 – Procedure ◮ Dependent variables: reading times RT ROI 1/2, judgment RTs and judgments Bott, Klein & Schlotterbeck (LCQ Budapest 2015) 14 / 30
Experiment 1: Establishing empty set effects Exp. 1 – Reading times ◮ Empty set, MON ↓ Q fewer than five more complex to interpret than non empty set more than five and exactly five Bott, Klein & Schlotterbeck (LCQ Budapest 2015) 15 / 30
Experiment 1: Establishing empty set effects Exp. 1 – Judgments ◮ 0-models difficult for empty set Q fewer than five (25% errors) but not for other two Qs ◮ All other conditions: > 94% correct Bott, Klein & Schlotterbeck (LCQ Budapest 2015) 16 / 30
Experiment 1: Establishing empty set effects Exp. 1 – Judgment times ◮ Comparing 0- with 1- models, we find a clear empty set effect ◮ Empty set effect not sufficient to account for this rather complex data pattern (exact counting vs approximation, general MON ↓ effect,. . . ) Bott, Klein & Schlotterbeck (LCQ Budapest 2015) 17 / 30
Experiment 1: Establishing empty set effects Exp. 1 – An alternative pragmatic explanation? ◮ It’s odd to describe an empty set situation with less than n . No would be a more informative alternative. Therefore, participants may reject such quantificational statements due to a scalar implicature. Bott, Klein & Schlotterbeck (LCQ Budapest 2015) 18 / 30
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