Effects of Domain Complexity on Verifica7on Procedures for Most and More Than Half Hadas Kotek, Yasutada Sudo, Edwin Howard, Micha Y. Breakstone and Mar9n Hackl MIT Summary Accuracy Rates [1] and [3] propose two compe9ng views of how statements of the form Most As are Bs are verified. This study shows that the complexity of the complement set (A–B) affects the verifica9on strategies for most and more than half in different ways. We argue that these results accord with the superla9ve seman9cs for most proposed by [1], but are not captured by the alterna9ve put forward by [3]. Most 2C 3C 47.9 55.6 86.8 66.7 More Than Half 100 80 60 40 20 Background Most vs. More than Half [1] proposes a superla9ve meaning for most and relates differences in verifica9on procedures between most and more than half as revealed in Self-­‐Paced Coun9ng (SPC) experiments to differences in their seman9cs. [[ most ]](A)(B) = 1 iff ∃X [ A(X) ∧ B(X) ∧ ∀Y∈C [Y≠X→|X|>|Y|] ] where C = A (closed under i-­‐sum forma9on) [[ more than half ]](A)(B) = 1 iff |A∩B| > ½|A| 0 Alterna7ve Account [3] conducted a flash presenta9on experiment where par9cipants judged the truth of Most of the dots are blue against pictures of dots differing in the number of colors, and report no effect of the number of colors on the accuracy rate. Based on these results, [3] suggests a different representa9on for most: [[ most ]](A)(B) = 1
iff |A∩B| > |A|–|A∩B| (i)
(ii)
The Self-­‐Paced Coun9ng (SPC) method of [1]: Par9cipants answered ques9ons such as Are most of the dots blue? rela9ve to dot arrays of varying sizes (14-­‐18). The dots are uncovered in groups of 2 or 3 as the spacebar is pressed. In our study, the number of colors varied between 2 and 3. A third color appears in the 4th frame and on. •  Each par9cipant was randomly assigned most or more than half •  24 target items and 96 filler items •  Half of the target items have the target-­‐to-­‐non-­‐target ra9os 9:8 or 8:7; the other half 8:9 or 7:8 •  4 ra9os are used in 3C: 9:4+4, 8:3+4, 8:5+4, 7:4+4 •  51 par9cipants (24 for most, 27 for more than half; all accuracy r ate >75%) References: [1] Hackl, M. (2009) On the grammar and processing of propor9onal quan9fiers: Most versus more than half. Natural Language Seman4cs. [2] Huang, Y., Spelke, E. & Snedeker, J. “What exactly do numbers mean?” Ms., Harvard University. [3] Lidz, J., P. Pietroski, T. Hunter & J. Halberda (in press) Interface Transparency and the Psychoseman9cs of ‘Most’. Natural Language Seman4cs. Corresponding author: Hadas Kotek ( hkotek@mit.edu ) Support through NSF Grant 0642478: “Experimental Inves9ga9ons of Verifica9on Procedures for Quan9fied Statements” Yes 50 0 98.6 96.5 92.4 92.4 >n (>1/2 Subj.) 100 50 0 No 95.7 94.4 93.2 87.7 Yes No model with random subject and item effects; p<0.001 for both determiners) An interac9on of the correct answer type and the number of colors for most (Contrast coded mixed logit model with random subject and item effects; p<0.001 for most, p=0.884 for more than half) We propose: (a)
(b)
Observa9on (i) is due to a ‘significantly-­‐more’ bias for both determiners Observa9on (ii) is due to a superla7ve reading of most in subject posi9on Superla7ve Reading Most of the dots are blue has two readings: Propor7onal reading:
|blue dots| > |non-­‐blue dots| Superla7ve reading:
For all non-­‐blue colors C, |blue dots| > |C dots| These readings disagree in 3 Color condi9ons such as 7:4+4 (Prop. → “No”; Sup. → “Yes”) A covered box experiment (cf. [2]) on Amazon Mechanical Turk confirmed that the superla9ve reading exists for most in subject posi9on, but not for more than half. More Than Half (n=19) Most (n=17) Sup 32% Procedure No 100 More correct answers for ‘No’ for both most and more than half (Contrast coded mixed logit SPC Experiment 91.4 90.7 Yes Observa9ons: Predic7on [1]’s account predicts that the verifica9on of Most of the As are Bs is sensi9ve to the complexity of A, because all non-­‐overlapping subsets of A are compared with A∩B. 35.2 33.3 >n (Most Subj.) Superla9ve Picture Box 68% Box 100% Most of the dots are blue was judged true 32% of the 9me for superla9ve pictures (9 subjects judged them true at least once) False Picture Covered Box Reac7on Times Analysis for SPC Experiment Mean Reac7on Times (All Trials) 2.5 Sup 2C Sup 3C Prop 2C Prop 3C >1/2 2C •  Par9cipants for most were divided into “superla9ve” and “propor9onal” speakers (n=12 for each; if accuracy of 2C-­‐no < accuracy of 3C-­‐no, then “superla9ve”) •  We observe a significant difference in RTs (Contrast coded >1/2 3C 2 1.5 1 0.5 1 2 3 4 5 6 mixed effects model with random subject and item effects; p<0.001 for Frames 1-­‐5) Discussions •  The superla9ve reading of most is straighxorwardly accounted for by [1]’s seman9cs: C=A ⇒ the propor9onal reading C={D1, D2, D3} (C is a salient cover of A) ⇒ the superla9ve reading •  Why are superla9ve speakers slower than propor9onal speakers? The superla9ve reading is hard to verify in 3C; A more accurate, slower strategy is chosen (and generalized to 2C)