Where linguistic meaning meets non-linguistic cognition

Transcription

1 Where linguistic meaning meets non-linguistic cognition Tim Hunter and Paul Pietroski NASSLLI 2016

2 Friday: Putting things together (perhaps)

3 Outline 5 Experiments with kids on most 6 So: How should we think about meanings? 7 Multiple non-equivalent interpretations 8 Application to most 42 / 62

4 Outline 5 Experiments with kids on most 6 So: How should we think about meanings? 7 Multiple non-equivalent interpretations 8 Application to most 43 / 62

5 Kids and most Are most of the animals frogs, or rabbits? (Halberda et al. 2008, Odic et al. submitted) 44 / 62

6 Kids and most Are most of the animals frogs, or rabbits? Non-full-counter stage: some kids at chance some kids with above-chance but noisy accuracy (ANS) (Halberda et al. 2008, Odic et al. submitted) 44 / 62

7 Kids and most Are most of the animals frogs, or rabbits? Non-full-counter stage: some kids at chance some kids with above-chance but noisy accuracy (ANS) Full-counter stage: some kids at chance some kids with above-chance and adult-like accuracy some kids with above-chance but noisy accuracy (ANS) (Halberda et al. 2008, Odic et al. submitted) 44 / 62

8 Summary Experiment 1: adults neglect one-to-one information and persist with ANS Experiment 2: adults neglect direct number-of-non-yellows information and persist with subtraction Experiment 3: children neglect precise cardinality information and persist with ANS 45 / 62

9 Summary Most of the dots are yellow OneToOnePlus(DOT YELLOW, DOT \ YELLOW) ANS representation G(y, w 2 y 2 ) for YELLOW DOT ANS representation G(b, w 2 b 2 ) for BLUE DOT ANS representation G(d, w 2 d 2 ) for DOT Judgements with accuracy poorer than one-to-one control task Most of the dots are yellow ANS representation G(y, w 2 y 2 ) for YELLOW DOT ANS representation G(b, w 2 b 2 ) for BLUE DOT ANS representation G(d, w 2 d 2 ) for DOT Judgements with accuracy poorer than simple more task Most of the dots are yellow ANS representation G(y, w 2 y 2 ) for YELLOW DOT ANS representation G(b, w 2 b 2 ) for BLUE DOT Precise number label y for YELLOW DOT Precise number label b for BLUE DOT Judgements with accuracy dependent on ratio 46 / 62

10 Outline 5 Experiments with kids on most 6 So: How should we think about meanings? 7 Multiple non-equivalent interpretations 8 Application to most 47 / 62

11 What are meanings? Most of the dots are yellow linguistic? non-linguistic true/false processing processing 48 / 62

12 What are meanings? DOT YELLOW DOT YELLOW Most of the dots are yellow linguistic? non-linguistic true/false processing processing 48 / 62

13 What are meanings? DOT YELLOW > DOT DOT YELLOW Most of the dots are yellow linguistic? non-linguistic true/false processing processing 48 / 62

14 What are meanings? DOT YELLOW > DOT YELLOW Most of the dots are yellow linguistic? non-linguistic true/false processing processing 48 / 62

15 So: What is a meaning? We ve been guided by the idea of choosing between things like the following: DOT YELLOW DOT YELLOW DOT YELLOW > DOT YELLOW DOT YELLOW > DOT DOT YELLOW 49 / 62

16 So: What is a meaning? We ve been guided by the idea of choosing between things like the following: DOT YELLOW DOT YELLOW DOT YELLOW > DOT YELLOW DOT YELLOW > DOT DOT YELLOW But what exactly is the relation between the things we have discovered about the ways people go about making true/false judgements, and the forms of the mathematical expressions above? What linking hypotheses connect these mental representations to observations? (cf. When the state to transition to is not determined by the input, things take longer. ) 49 / 62

17 So: What is a meaning? We ve been guided by the idea of choosing between things like the following: DOT YELLOW DOT YELLOW DOT YELLOW > DOT YELLOW DOT YELLOW > DOT DOT YELLOW But what exactly is the relation between the things we have discovered about the ways people go about making true/false judgements, and the forms of the mathematical expressions above? What linking hypotheses connect these mental representations to observations? (cf. When the state to transition to is not determined by the input, things take longer. ) Idea: say something about how the symbols are interpreted by non-linguistic cognitive systems (cf. Chomsky on instructions for performance systems ) 49 / 62

18 Expectation-lowering disclaimer I will not be claiming to have derived the connection between most and subtraction-ish, non-one-to-one-ish verification procedures, from other independently justified assumptions. This remains a stipulation (for now). 50 / 62

19 Expectation-lowering disclaimer I will not be claiming to have derived the connection between most and subtraction-ish, non-one-to-one-ish verification procedures, from other independently justified assumptions. This remains a stipulation (for now). I m just trying to find a way to at least encode these stipulations into our theories. (Something like observational adequacy.) Hopefully, in the process we might get ideas about what to look for in order to get rid of some of these stipulations. 50 / 62

20 Expectation-lowering disclaimer I will not be claiming to have derived the connection between most and subtraction-ish, non-one-to-one-ish verification procedures, from other independently justified assumptions. This remains a stipulation (for now). I m just trying to find a way to at least encode these stipulations into our theories. (Something like observational adequacy.) Hopefully, in the process we might get ideas about what to look for in order to get rid of some of these stipulations. (But I don t really know whether we re getting anywhere yet... ) 50 / 62

21 Outline 5 Experiments with kids on most 6 So: How should we think about meanings? 7 Multiple non-equivalent interpretations 8 Application to most 51 / 62

22 Notation for semantic rules [ S [ NP α] [ VP β]] = [ VP β] ( [ NP α] ) [ NP α] = x [ VP β] = f [ S [ NP α] [ VP β]] = f (x) 52 / 62

23 Analogy from programming languages Program P 1 let x = 3; if ((x > 0) (f(x) > 10)) then return "yes"; else return "no"; Program P 2 let x = 3; if ((f(x) > 10) (x > 0)) then return "yes"; else return "no"; Evaluating A B (as in many programming languages) means: evaluate A if the result is true, stop; the whole expression evaluates to true otherwise, evaluate B,... A = true (A B) = true A = false B = v (A B) = v Therefore: The two programs differ in that P 1 will not evaluate (f(x) > 10), but P 2 will Nonetheless, the two programs will both return the same result 53 / 62

24 Analogy from programming languages Program P 1 let x = 3; if ((x > 0) (f(x) > 10)) then return "yes"; else return "no"; Program P 2 let x = 3; if ((f(x) > 10) (x > 0)) then return "yes"; else return "no"; What is the meaning of P 1? Of P 2? Are they the same? If we are interested only in the result returned, we can define a suitable interpretation function which maps programs to {true, false} such that P 1 = P 2 A = true (A B) = true A = false B = v (A B) = v But there are things that might make us prefer a different interpretation function 53 / 62

25 Analogy from programming languages Program P 1 let x = 3; if ((x > 0) (f(x) > 10)) then return "yes"; else return "no"; Program P 2 let x = 3; if ((f(x) > 10) (x > 0)) then return "yes"; else return "no"; Perhaps f(x) is very complicated and slow to compute. We could define an interpretation function which maps programs to ordered pairs, a boolean result and a number of steps. A = true, n 1 (A B) = true, n Here P 1 P 2. A = false, n 1 B = v, n 2 (A B) = v, n 1 + n Program P 1 (x > 0) = true, 1 (x > 0) (f(x) > 10) = true, 2 Program P 2 (f(x) > 10) = true, 100 (f(x) > 10) (x > 0) = true, / 62

26 Analogy from programming languages Program P 1 let x = 3; if ((x > 0) (f(x) > 10)) then return "yes"; else return "no"; Program P 2 let x = 3; if ((f(x) > 10) (x > 0)) then return "yes"; else return "no"; Perhaps f(x) changes the value of a global variable (say, sets y to 5) We could define s which maps programs to ordered pairs, a boolean result and a new state A s = true, s (A B) s = true, s Here P 1 s P 2 s. A s = false, s B s = v, s (A B) s = v, s Program P 1 (x > 0) {y 0} = true, {y 0} (x > 0) (f(x) > 10) {y 0} = true, {y 0} Program P 2 (f(x) > 10) {y 0} = true, {y 5} (f(x) > 10) (x > 0) {y 0} = true, {y 5} 53 / 62

27 Multiple Interpretations For many different questions we might want to ask about the interpretation of a program/expression, we can define a compositionally-determined function from programs/expressions to the appropriate answer returned value returned value and number of steps returned value and updated state 54 / 62

28 Multiple Interpretations DOT YELLOW > DOT DOT YELLOW DOT YELLOW > DOT YELLOW We can give this language a variety of interpretations, just as we gave A B multiple different interpretations, each answering a different question about its semantic properties 55 / 62

29 Multiple Interpretations DOT YELLOW > DOT DOT YELLOW DOT YELLOW > DOT YELLOW We can give this language a variety of interpretations, just as we gave A B multiple different interpretations, each answering a different question about its semantic properties A normal interpretation can answer the verification-agnostic questions about truth values A new interpretation can answer other questions: e.g. How much noise accompanies the result of a particular computation? 55 / 62

30 Outline 5 Experiments with kids on most 6 So: How should we think about meanings? 7 Multiple non-equivalent interpretations 8 Application to most 56 / 62

31 Composing a sentence meaning most of the dots are yellow SEM = D & Y > D - D & Y 57 / 62

32 Composing a sentence meaning most of the dots are yellow SEM = D & Y > D - D & Y most SEM = λaλb. a & b > a - a & b dot SEM = D yellow SEM = Y Det of the N s are Adj SEM = Det SEM ( N SEM )( Adj SEM ) 57 / 62

33 Composing a sentence meaning most of the dots are yellow SEM = D & Y > D - D & Y most SEM = λaλb. a & b > a - a & b dot SEM = D yellow SEM = Y Det of the N s are Adj SEM = Det SEM ( N SEM )( Adj SEM ) N SEM = a Adj SEM = b most of the N s are Adj SEM = a & b > a - a & b 57 / 62

34 Definition of ANS interpretation e 1 ANS = G(n 1, σ 2 1) e 2 ANS = G(n 2, σ 2 2) e 1 - e 2 ANS = G(n 1 n 2, σ σ 2 2) e 1 ANS = G(n 1, σ1) 2 e 2 ANS = G(n 2, σ2) 2 ( ) e 1 > e 2 ANS = 1 erfc n 2 n σ 2 1 +σ / 62

35 Examples Our most experiments with adults: Most of the dots are yellow SEM = D & Y > D - D & Y D ANS = G(y + b, w 2 (y + b) 2 ) D & Y ANS = G(y, w 2 y 2 ) D & Y ANS = G(y, w 2 y 2 ) D - D & Y ANS = G(b, w 2 (y + b) 2 + w 2 y 2 )) ( ) D & Y > D - D & Y ANS = 1 2 erfc b y 2 w 2 ((y+b) 2 +2y 2 ) 59 / 62

36 Examples Our most experiments with adults: Most of the dots are yellow SEM = D & Y > D - D & Y D ANS = G(y + b, w 2 (y + b) 2 ) D & Y ANS = G(y, w 2 y 2 ) D & Y ANS = G(y, w 2 y 2 ) D - D & Y ANS = G(b, w 2 (y + b) 2 + w 2 y 2 )) ( ) D & Y > D - D & Y ANS = 1 2 erfc b y 2 w 2 ((y+b) 2 +2y 2 ) Classic ANS experiment: There are more yellow dots than blue dots SEM = D & Y > D & B D & Y ANS = G(y, w 2 y 2 ) D & B ANS = G(b, w 2 b 2 ) ( ) D & Y > D & B ANS = 1 2 erfc b y 2 w 2 y 2 +w 2 b 2 59 / 62

37 Examples Our most experiments with adults: Most of the dots are yellow SEM = D & Y > D - D & Y D ANS = G(y + b, w 2 (y + b) 2 ) D & Y ANS = G(y, w 2 y 2 ) D & Y ANS = G(y, w 2 y 2 ) D - D & Y ANS = G(b, w 2 (y + b) 2 + w 2 y 2 )) ( ) D & Y > D - D & Y ANS = 1 2 erfc b y 2 w 2 ((y+b) 2 +2y 2 ) Classic ANS experiment: There are more yellow dots than blue dots SEM = D & Y > D & B D & Y ANS = G(y, w 2 y 2 ) D & B ANS = G(b, w 2 b 2 ) ( ) D & Y > D & B ANS = 1 2 erfc b y 2 w 2 y 2 +w 2 b 2 In other most situations, some reaching happens that draws a direct comparison of count-based cardinalities into answering the most question. In other most situations, some reaching happens that draws the one-to-one info into answering the most question. In other most situations, / 62

38 Subtraction vs. selection 16 yellow dots, 10 blue dots Yellow dots (primitive) Blue dots (primitive) Total dots (primitive) Blue dots (derived) With only y yellow dots and b blue dots present: Subtraction Procedure: non-yellow numerosity N (b, w 2 ((y + b) 2 + y 2 )) Selection Procedure: non-yellow numerosity N (b, w 2 b 2 ) 60 / 62

39 Definition of precise cardinality interpretation e CARD = S e CARD = S e 1 CARD = n 1 e 2 CARD = n 2 e 1 - e 2 CARD = n 1 n 2 etc. 61 / 62

40 Examples Kids who can count, but use the ANS for most : Most of the dots are yellow SEM = D & Y > D - D & Y These kids haven t yet learned to apply CARD to this expression, only ANS. So they can use D & Y ANS = G(y, w 2 y 2 ) D & B ANS = G(b, w 2 b 2 ) but they can t use D & Y CARD = y D & B CARD = b 62 / 62

41 References I Chomsky, N. (1986). Barriers. MIT Press, Cambridge, MA. Church, A. (1941). The Calculi of Lambda Conversion. Princeton University Press, Princeton. Dehaene, S. (1997). The Number Sense: How the Mind Creates Mathematics. Oxford University Press, New York. Gallistel, C. R. and Gelman, R. (2000). Non-verbal numerical cognition: from reals to integers. Trends in Cognitive Sciences, 4(2): Gallistel, C. R. and King, A. P. (2009). Memory and the computational brain. Wiley-Blackwell, Malden, MA. Halberda, J., Sires, S. F., and Feigenson, L. (2006). Multiple spatially overlapping sets can be enumerated in parallel. Psychological Science, 17: Halberda, J., Taing, L., and Lidz, J. (2008). The age of most comprehension and its potential dependence on counting ability in preschoolers. Language Learning and Development, 4: Hunter, T., Halberda, J., Lidz, J., and Pietroski, P. (2009). Beyond truth conditions: The semantics of most. In Proceedings of SALT XVIII, 2008, pages

42 References II Lidz, J., Halberda, J., Pietroski, P., and Hunter, T. (2011). Interface transparency and the psychosemantics of most. Natural Language Semantics, 19(3): Marr, D. (1982). Vision: A computational investigation into the human representation and processing of visual information. W.H. Freeman and Company, New York. Odic, D., Halberda, J., Pietroski, P., and Lidz, J. (submitted). The Language-Number Interface: Evidence from the acquisition of most. Pica, P., Lemer, C., Izard, V., and Dehaene, S. (2004). Exact and approximate arithmetic in an amazonian indigene group. Science, 306: Pietroski, P., Lidz, J., Hunter, T., and Halberda, J. (2009). The meaning of most : Semantics, numerosity and psychology. Mind and Language, 24(5): Treisman, A. and Gormican, S. (1988). Feature analysis in early vision: Evidence from search asymmetries. Psychological Review, 95: Wolfe, J. M. (1998). Visual memory: What do you know about what you saw? Current Biology, 8(9):R303 R304.