Most Meanings and Minds Paul M. Pietroski Rutgers University http://www.tinyurl.com/pietroski
function in intension function in extension (computational procedure) ( (set of input-output pairs) |x – 1| +√(x2 – 2x + 1) {…(-2, 3), (-1, 2), (0, 1), (1, 0), (2, 1), …} λx . |x – 1| ≠ λx . +√(x2 – 2x + 1) λx . |x – 1| = λx . +√(x2 – 2x + 1) Extension[λx . |x – 1|] = Extension[λx . +√(x2 – 2x + 1)] ____________________________________________________________ Are meanings more like functions in intension or functions in extension? Do meanings reflect details of certain representational formats? Or are meanings more neutral contents that abstract from such details?
Tim Hunter A W l e e l x l i w s o o d Darko Odic J e f L i d z Justin Halberda
Are most of the dots yellow? 6 blue 1 to 2 SUPER EASY How is the Yes/No question understood? What question is getting asked?
Most of the dots are yellow. MOST[DOT, YELLOW] #{DOT & YELLOW} > #{DOT}/2 More than half of the dots are yellow (9 > 15/2) #{DOT & YELLOW} > #{DOT & YELLOW} The yellow dots outnumber the nonyellow dots (9 > 6) #{DOT & YELLOW} > #{DOT} – #{DOT & YELLOW} The number of yellow dots exceeds the number of dots minus the number of yellow dots (9 > 15 – 9) (And there is at least one more option to consider.) 15 dots: 9 yellow, 6 blue
Hume’s Principle #{Triangle} = #{Heart} iff {Triangle} OneToOne {Heart} ____________________________________________ #{Triangle} > #{Heart} {Triangle} OneToOnePlus {Heart} α OneToOnePlus β iff for some α*, α* is a proper subset of α, and α* OneToOne β (and it’s not the case that β OneToOne α)
Most of the dots are yellow. MOST[DOT, YELLOW] OneToOnePlus[DOT & YELLOW, DOT & ~YELLOW] 1 to 2 SUPER EASY #{DOT & YELLOW} > #{DOT}/2 #{DOT & YELLOW} > #{DOT & YELLOW} #{DOT & YELLOW} > #{DOT} – #{DOT & YELLOW}
Most of the dots are yellow. What conditions make it easy/hard to evaluate the sentence? That might provide clues about how the sentence is understood 1 to 2 SUPER EASY (given independent accounts of the information used by the evaluators in those conditions).
‘Most of the dots are yellow’ MOST[D, Y] OneToOnePlus[{D & Y},{D & Y}] #{D & Y} > #{D & Y} #{D & Y} > #{D}/2 #{D & Y} > #{D} – #{D & Y} Number Representations These analyses are provably equivalent (for finite cases) and not crazy
function in intension function in extension (computational procedure) ( (set of input-output pairs) |x – 1| +√(x2 – 2x + 1) λx . |x – 1| ≠ λx . +√(x2 – 2x + 1) λx . |x – 1| = λx . +√(x2 – 2x + 1) ____________________________________________________________ Are meanings more like functions in intension or functions in extension? Do meanings reflect details of certain representational formats? Or are meanings more neutral contents that abstract from such details?
‘Most of the dots are yellow’ MOST[D, Y] OneToOnePlus[{D & Y},{D & Y}] #{D & Y} > #{D & Y} #{D & Y} > #{D}/2 #{D & Y} > #{D} – #{D & Y} Number Representations
‘Most of the dots are yellow’ MOST[D, Y] #{D & Y} > #{D} – #{D & Y} Number Representations ??Most of the paint is yellow??
‘Most of the dots are yellow’ MOST[D, Y] Why analyse at all? Why not take ‘Most’ to be as primitive as ‘dot’ and ‘yellow’ seem to be? Some of the yellow dogs barked Some of the dogs barked Some of the dogs barked loudly Some of the dogs barked None of the yellow dogs barked None of the dogs barked None of the dogs barked loudly None of the dogs barked
‘Most of the dots are yellow’ MOST[D, Y] Why analyse at all? Why not take ‘Most’ to be as primitive as ‘dot’ and ‘yellow’ seem to be? All of the yellow dogs barked All of the dogs barked All of the dogs barked loudly All of the dogs barked Most of the yellow dogs barked -- Most of the dogs barked Most of the dogs barked loudly Most of the dogs barked
‘Most of the dots are yellow’ MOST[D, Y] Why analyse at all? Why not take ‘Most’ to be as primitive as ‘dot’ and ‘yellow’ seem to be? Most of the dogs barked More than half of the dogs barked Most of the dogs barked More dogs barked than didn’t Most of the yellow dogs barked -- Most of the dogs barked Most of the dogs barked loudly Most of the dogs barked
Most of the dots are yellow? MOST[DOT, YELLOW] OneToOnePlus[DOT & YELLOW, DOT & ~YELLOW] 1 to 2 SUPER EASY #{DOT & YELLOW} > #{DOT}/2 #{DOT & YELLOW} > #{DOT & YELLOW} #{DOT & YELLOW} > #{DOT} – #{DOT & YELLOW}
Most of the dots are yellow? What conditions make it easy/hard to evaluate the sentence? That might provide clues about how the sentence is understood 1 to 2 SUPER EASY (given independent accounts of the information used by the evaluators in those conditions).
a model of the “Approximate Number System” (key feature: ratio-dependence of discriminability) distinguishing 8 dots from 4 (or 16 from 8) is easier than distinguishing 10 dots from 8 (or 20 from 10)
a model of the “Approximate Number System” (key feature: ratio-dependence of discriminability) correlatively, as the number of dots rises, “acuity” for estimating of cardinality decreases--but still in a ratio-dependent way, with wider “normal spreads” centered on right answers
various ways of presenting dots (8 blue, 10 yellow) scattered random scattered pairs various ways of presenting dots (8 blue, 10 yellow) column pairs mixed column pairs sorted
4:5 (blue:yellow) “scattered random” 4 to 5
1:2 (blue:yellow) “scattered random” 1 to 2 SUPER EASY
4:5 (blue:yellow) “scattered pairs”
9:10 (blue:yellow) “scattered pairs”
4:5 (blue:yellow) “column pairs mixed” 4 to 5
5:4 (blue:yellow) “column pairs mixed” 9 to 10
4:5 (blue:yellow) “column pairs sorted”
4:5 (blue:yellow) scattered random scattered pairs column pairs mixed column pairs sorted
Basic Design 12 naive adults, 360 trials for each participant 5-17 dots of each color on each trial trials varied by ratio (from 1:2 to 9:10) and type each “dot scene” displayed for 200ms target sentence: Are most of the dots yellow? answer ‘yes’ or ‘no’ by pressing buttons on a keyboard correct answer randomized controls for area (pixels) vs. number, etc.
4:5 (blue:yellow) scattered random scattered pairs THIS IS THE ONLY ODDBALL column pairs mixed column pairs sorted
better performance on easier ratios: p < .001 10 : 10 10 : 20 10 : 15
performance on Scattered Pairs and Mixed Columns was no better than on Scattered Random… looks like ANS was used to answer the question, except in Sorted Columns
ANS ANS scattered random scattered pairs Are most of the dots yellow? ANS column pairs mixed column pairs sorted
but even better performance on the components of a 1-to-1-plus task if the question is not posed with ‘most’ 10 : 20 10 : 10 10 : 15
‘Most of the dots are yellow’ OneToOnePlus[{D & Y},{D & Y}] #{D & Y} > #{D & Y} #{D & Y} > #{D} – #{D & Y} Number Representations Prima facie, posing a question with ‘most’ is not a way of posing a OneToOnePlus question
‘Most of the dots are yellow’ #{D & Y} > #{D & Y} #{D & Y} > #{D} – #{D & Y} framing the question with ‘most’ has effects that are expected if the question is understood in terms of cardinality comparison (and answered via the ANS) but unexpected if the question is understood in terms of 1-to-1 correspondence Number Representations
Most of the dots are blue. What conditions make it easy/hard to evaluate the sentence? That might provide clues about how the sentence is understood (given independent accounts of the information used by the evaluators in those conditions).
Most of the dots are blue?
Four Trial Types (2-5 colors)
‘Most of the dots are blue’ #{Dot & Blue} > #{Dot & ~Blue} in scenes with two colors, blue and red, the non-blues can be identified with the reds #{Dot & ~Blue} = #{Dot & Red} the visual system will “select” the dots, the blue dots, and the red dots; these 3 sets will be estimated for cardinality but adding colors will make it harder (and with 5 colors, impossible) to estimate the cardinality of the non-blues #{Dot & Blue} > #{Dot} − #{Dot & Blue}
‘Most of the dots are blue’ #{Dot & Blue} > #{Dot & ~Blue} verification should get harder as the number of colors increases but in the two-color case, “acuity” should be relatively good (w ≈ .15) since no estimate of the total figures in the computation #{Dot & Blue} > #{Dot} − #{Dot & Blue} predicts indifference to the number of colors “acuity” should be less good (w ≈ .3) since an estimate of the total will figure in the computation 15 dots, 9 blue 9 > 6 15 dots, 9 blue 9 > (15 − 9)
6 9 15 lower acuity #{Dot & Blue} > #{Dot & ~Blue} #{Dot} − #{Dot & Blue} 6 9 15 lower acuity
‘Most of the dots are blue’ #{Dot & Blue} > #{Dot & ~Blue} verification should get harder as the number of colors increases but in the two-color case, “acuity” should be relatively good (w ≈ .15) since no estimate of the total figures in the computation #{Dot & Blue} > #{Dot} − #{Dot & Blue} predicts indifference to the number of colors “acuity” should be less good (w ≈ .3) since an estimate of the total will figure in the computation 15 dots, 9 blue 9 > 6 15 dots, 9 blue 9 > (15 − 9)
better performance on easier ratios: p < .001
no effect of number of colors
fit to psychophysical model of ANS-driven performance
‘Most of the dots are blue’ #{D & B} > #{D & B} #{D & B} > #{D} – #{D & B} Number Representations Prima facie, posing a question with ‘most’ is not a way of posing a cardinality question formulated in terms of negation
‘Most of the dots are blue’ #{D & B} > #{D} – #{D & B} framing the question with ‘most’ has effects that are expected if the question is understood in terms of cardinality subtraction Number Representations But even if this is right for cases involving count nouns (e.g., ‘most of the dots’) what about mass nouns, as in…….Most of the paint is blue?
‘Most of the dots are blue’ #{Dot & Blue} > #{Dot} − #{Dot & Blue} mass/count flexibility Most of the dots (blobs, peas, shoes) are blue Most of the paint (blob, corn, footwear) is blue are mass nouns (somehow) disguised count nouns? #{StuffUnit & BlueUnit} > #{StuffUnit} − #{StuffUnit & BlueUnit}
Look at that goo/stuff/paint… Is most of it blue? Is most the paint blue? Was most of the blob blue? Look at those spots/blobs… Are most of them blue? Are most of the spots blue? Were most of the blobs blue?
First Study…’blob’ vs. ‘blobs’: performance is better, holding the scene constant, if the question is posed with ‘blob’. w = .20 r2 = .99 w = .29 r2 = .98 Follow-Up Studies: you can replace ‘most’ with ‘more’, or ‘blob’ with ‘goo’ It seems that performance is better, holding the scene constant, if the question is posed with a mass noun
‘Most of the dots are blue’ #{Dot & Blue} > #{Dot} − #{Dot & Blue} mass/count flexibility Most of the dots (blobs, peas, shoes) are blue Most of the paint (blob, corn, footwear) is blue are mass nouns (somehow) disguised count nouns? #{StuffUnit & BlueUnit} > #{StuffUnit} − #{StuffUnit & BlueUnit} SEEMS NOT but that raises more questions
Most of the dots are blue? #{D & B} > #{D} – #{D & B} framing the question with ‘most’ has effects that are expected if the question is understood in terms of cardinality subtraction Prima facie, this requires a representation of #{D} and a computation on this representation. Does use of ‘most’ reflect ease/difficulty of representing #{D}, the (total) number of dots?
Which side is the better example of… (1) ‘Most of the dots are blue’ left right % Of Undergrads Who Choose This Side (N= 48) “Most…” = 58% “More…” = 13% on both sides, 12 blues and 8 yellows G1: most of the dots are blue Back G2: there are more blue dots than yellow dots 2nd time: click for data Which side is the better example of… (1) ‘Most of the dots are blue’ (2) ‘More of the dots are blue’
A) More of the dots are grey. B) Most of the dots are grey. Which sentence would you choose to describe this picture? A) More of the dots are grey. B) Most of the dots are grey. 65% choose “more”
74% choose “most” A) More of the dots are grey. Which sentence would you choose to describe this picture? A) More of the dots are grey. 74% choose “most” B) Most of the dots are grey.
Not just about recognition 80 participants asked to “draw” on an iPad More/Most of the dots are blue
Not just about recognition Typical for “More of the dots are blue”
Not just about recognition Typical for “Most of the dots are blue”
Centroid distance (adults) More Most Halberda, Pietroski, Hunter, Odic, Wellwood, & Lidz. (2012). More & most: spatial vision affects word understanding on an iPad. Vision Science Society annual meeting.
4-8 yr olds (n=92) More Most
could stop here… or talk about what we’re (not) saying about how meanings are related to verification procedures why the “200 ms” data was relevant and why we don’t think there’s anything semantically special about “fast” procedures
Meanings are Verification Procedures (1a) a claim about the nature of meanings (1b) a claim about how complex meanings are related to complex VPs (2a) a bold claim about how verification is related to knowledge/justification, and how these epistemic notions are related to certain “privileged” concepts that are associated with “sensory” experience) (2b) a modes claim—about how meaning specifications are related to a semantic notion of verification—with no direct implications for how meanings are related to epistemology. (3a) a claim that applies to open-class lexical items (‘cat’, ‘dog’, ‘eat’, …) with the implication that these words have analyses in terms of privileged vocabulary (3b) a claim about how complex meanings are related to certain complex VPs (4a) a quick ride to an unhappy destination—e.g., Classical Empiricism (4b) a way of staying off the train to that destination
function in intension function in extension (computational procedure) ( (set of input-output pairs) |x – 1| +√(x2 – 2x + 1) λx . |x – 1| ≠ λx . +√(x2 – 2x + 1) λx . |x – 1| = λx . +√(x2 – 2x + 1) ____________________________________________________________ Are meanings more like functions in intension or functions in extension? Do meanings reflect details of certain representational formats? Or are meanings more neutral contents that abstract from such details?
Tim Hunter A W l e e l x l i w s o o d Darko Odic J e f L i d z Justin Halberda
Framing Matters Adam and Beth drive equal distances in a year. Adam switches from a 12-mpg to 14-mpg car. Beth switches from a 30-mpg to 40-mpg car. Who will save more gas? Adam: 10,000/12 = 833 10,000/14 = 714 saving of 119 gallons Beth: 10,000/30 = 333 10,000/40 = 250 saving of 83 gallons
Framing Matters Adam knows the axioms. The axioms imply the theorem. Adam knows the theorem. In every possible world, Hesperus is Phosphorus. ‘Hesperus’ and ‘Phosphorus’ are synonymous It’s logically necessary that … Adam knows that …
Semantic Framing: The Meaning of ‘Most’ Paul M Semantic Framing: The Meaning of ‘Most’ Paul M. Pietroski Rutgers University Dept. of Philosophy, Center for Cognitive Science
There are some yellow dots, and there are some blue dots. 1 to 2 SUPER EASY Many strategies that you might use to evaluate the conjunctive sentence in a particular situation, depending on the situation and relevant knowledge.
There are some yellow dots, and there are some blue dots. 1 to 2 SUPER EASY Many strategies that you might use to evaluate the conjunctive sentence in a particular situation. But the meaning of ‘and’ invites a “default strategy.”
α & β? Possible Background Knowledge: α & β if and only if ~(~α v ~β) α & β if and only if Brian nodded … And which question is Brian responding to: α & β, ~(~α v ~β), …
#{D & Y} > #{D} – #{D & Y} #{D & Y} > #{D} – #{D & Y}? Possible Background Knowledge: #{D & Y} > #{D} – #{D & Y} if and only if #{D & Y} > #{D & Y} #{D & Y} > #{D} – #{D & Y} if and only if Brian nodded …
Meanings are Verification Procedures Two ways of hearing this claim (1) What are meanings? (2) Can you tell us something about meanings?
Meanings are Verification Procedures Two ways of hearing this claim (1) What are meanings? Meanings are mappings from contexts to contents. Q: What are contexts? A: Mappings from indices to domain entities Q: What are contents? A: Mappings from possible worlds to truth values. Q: What are possible worlds? A: Please be quiet. I'm trying to say what meanings are.
Meanings are Verification Procedures Two ways of hearing this claim (1) What are meanings? Meanings are mappings from contexts to contents. Meanings are concepts. Q: What are concepts? A: Composable mental representations. Q: What are those? A: Go read all of Jerry Fodor's books.
Meanings are Verification Procedures Two ways of hearing this claim (1) What are meanings? Meanings are mappings from contexts to contents. Meanings are concepts. Meanings are Verification Procedures Q: What are Verification Procedures? A: Rules for connecting words with epistemically special concepts. Q: What are epistemically special concepts? A: I’ll tell you later.
Meanings are Verification Procedures Two ways of hearing this claim (1) What are meanings? Meanings are mappings from contexts to contents. Meanings are concepts. Meanings are Verification Procedures [this often goes very badly] (2) Can you tell us something about meanings? Human Languages connect meanings with pronunciations. Meanings compose. Some people think that meanings are sets of some kind. (But I don’t.) Meanings provide default strategies for how to evaluate sentences as true/false.