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Two Kinds of Concept Introduction

Two Kinds of Concept Introduction. Paul M. Pietroski University of Maryland Dept. of Linguistics, Dept. of Philosophy. Sound(‘Brutus kicked Caesar’)  . Sound(‘kicked ’)  . Human Language System, tuned to “Spoken English”. Human Language System, tuned to “Spoken English”.  

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Two Kinds of Concept Introduction

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  1. Two Kinds of Concept Introduction Paul M. Pietroski University of Maryland Dept. of Linguistics, Dept. of Philosophy

  2. Sound(‘Brutus kicked Caesar’)  Sound(‘kicked’)  Human Language System, tuned to “Spoken English” Human Language System, tuned to “Spoken English”  Meaning(‘Brutus kicked Caesar’)  Meaning(‘kicked’) kicked(Brutus, Caesar) kicked(_, _) kicked(_, Caesar) y.x.kicked(x, y) Brutus kicked(_, _) Caesar

  3. Sound(‘Brutus kicked Caesar on Monday’)  Human Language System, tuned to “Spoken English”  Meaning(‘Brutus kicked Caesar on Monday’) kicked(_, Brutus, Caesar) & on(_, Monday) on(_, Monday) kicked(_, Brutus, Caesar)  kicked(_, _, Caesar) Brutus kicked(_, _, _) Caesar

  4. Sound(‘kicked’)  Human Language System, tuned to “Spoken English”  Meaning(‘kicked’) y.x.e.kicked(e, x, y) on(_, Monday) kicked(_, Brutus, Caesar)  Brutus kicked(_, _, _) Caesar

  5. Sound(‘kicked’)  Human Language System, tuned to “Spoken English”  Meaning(‘kicked’) y.x.e.kick(e, x, y) & past(e) on(_, Monday) kicked(_, Brutus, Caesar)  Brutus kicked(_, _, _) Caesar

  6. Sound(‘kicked’)  Human Language System, tuned to “Spoken English”  Meaning(‘kicked’) y.x.e.kick(e, x, y) & past(e) & agent(e, x) & patient(e, y) on(_, Monday) kicked(_, Brutus, Caesar)  Brutus kicked(_, _, _) Caesar

  7. Sound(‘kicked’)  Brutus kicked Caesar the ball Human Language System, tuned to “Spoken English” Brutus kicked Caesar (today) Brutus kicked Caesar was kicked  Meaning(‘kicked’) That kick was a good one y.x.e.kicked(e, x, y) I get no kick from champagne, but I get a kick out of you Does a child who acquires ‘kicked’ start with kicked(_, _) y.x.kicked(x, y) kick(_, _, _) y.x.e.kick(e, x, y) or some other concept(s), perhaps likee.kick(e)?

  8. Sound(‘gave’)  Brutus gave Caesar the ball (today) Human Language System, tuned to “Spoken English” Brutus gave the ball (to Caesar) Brutus gave (at the office) The painting was given/donated  Meaning(‘gave’) The rope has too much give z.y.x.e.gave(e, x, y, z) Does a child who acquires ‘gave’ start with gave(_, _, _) gave(_, _, _, _) z.y.x.e.gave(e, x, y, z) … e.give(e)

  9. Big Questions • To what degree is lexical acquisition a “cognitively passive” process in which (antecedently available) representations are simply labeled and paired with phonological forms? • To what degree is lexical acquisition a “cognitively creative” process in which (antecedently available) representations are used to introduce other representations that exhibit a new combinatorial format? • Are the concepts that kids lexicalize already as systematically combinable as words? Or do kids use these “L-concepts,” perhaps shared with other animals, to introduce related concepts that exhibit a more distinctively human format?

  10. Outline • Describe a “Fregean” mind that can use its cognitive resources to introduce mental symbols of two sorts (1) polyadic concepts that are “logically fruitful” (2) monadic concepts that are “logically boring,” but more useful than “mere abbreviations” like x[mare(x)≡dfhorse(x) & female(x) & mature(x)] • Suggest that the process of acquiring a lexicon involves concept introduction of the second sort. For example, a child might use give(_,_,_)or give(_,_,_, _)to introduce give(_). • Offer some evidence that this suggestion is correct say something about adicity

  11. Quick Reminder about Conceptual Adicity Two Common Metaphors Jigsaw Puzzles 7th Grade Chemistry -2 +1H–O–H+1

  12. Jigsaw Metaphor a Thought: Brutussang

  13. Jigsaw Metaphor one Dyadic Concept (adicity: -2) “filled by” two Saturaters (adicity +1) yields a complete Thought Sang( ) Brutus Unsaturated Saturater 2nd saturater KICK(_, _) 1st saturater oneMonadic Concept (adicity: -1) “filled by” one Saturater (adicity +1) yields a complete Thought Doubly Un- saturated Caesar Brutus

  14. 7th Grade Chemistry Metaphor a molecule of water -2 +1H(OH+1)-1 a single atom with valence -2 can combine with two atoms of valence +1 to form a stable molecule

  15. 7th Grade Chemistry Metaphor -2 +1Brutus(KickCaesar+1)-1

  16. 7th Grade Chemistry Metaphor +1BrutusSang-1 +1NaCl-1 an atom with valence -1 can combine with an atom of valence +1 to form a stable molecule

  17. Extending the Metaphor Aggie Aggie cow( ) brown( ) +1 -1 +1 -1 Aggie is brown Aggie is (a) cow brown-cow( ) Aggie -1 Aggie is (a) brown cow

  18. Extending the Metaphor Aggie Aggie cow( ) brown( ) +1 -1 +1 -1 Conjoining two monadic (-1) concepts can yield a complex monadic (-1) concept brown( ) & cow( ) Aggie -1

  19. Outline • Describe a “Fregean” mind that can use its cognitive resources to introduce mental symbols of two sorts (1) polyadic concepts that are “logically fruitful” (2) monadic concepts that are “logically boring,” but more useful than “mere abbreviations” like x[mare(x)≡dfhorse(x) & female(x) & mature(x)] • Suggest that for humans, the process of “acquiring a lexicon” involves concept introduction of the second sort. For example, a child might use give(_,_,_)to introduce give(_). • Offer some evidence that this suggestion is correct

  20. Zero is a number • every number has a successor • Zero is not a successor of any number • no two numbers have the same successor • for every property ofZero: if every number that has it is such that its successorhas it, then every number has it Frege’s Hunch: these thoughts are not as independent as axioms should be

  21. Nero is a cucumber • every cucumber has a doubter • Nero is not a doubter of any cucumber • no two cucumbers have the same doubter • for every property ofNero: if every cucumber that has it is such that its doubterhas it, then every cucumber has it Related Point: these thoughts are not as compelling as axioms should be

  22. Zero is a number () Nero is a cucumber every number has a successor x{(x)y[(y,x)]} every cucumber has a doubter Zero is not a successor of any number ~x{(, x) & (x)} Nero is not a doubter of any cucumber If we want to represent what the arithmetic thougtsimply— and what they follow from—then the formalizations on the right are better than ‘P’, ‘Q’, and ‘R’.

  23. Zero is a number () Nero is a cucumber every number has a successor x{(x)y[(y,x)]} every cucumber has a doubter Zero is not a successor of any number ~x{(, x) & (x)} Nero is not a doubter of any cucumber But we miss something if we represent the property of being a number with an atomic monadic concept, as if number( )differs from cucumber( )and carrot( ) only in terms of its specific content.

  24. Zero is a number () Nero is a cucumber every number has a successor x{(x)y[(y,x)]} every cucumber has a doubter Zero is not a successor of any number ~x{(, x) & (x)} Nero is not a doubter of any cucumber We miss something if we represent the number zero with an atomic singular concept,as if the concept Zerodiffers from concepts like Neroand Venus only in terms of its specific content.

  25. Zero is a number () Nero is a cucumber every number has a successor x{(x)y[(y,x)]} every cucumber has a doubter Zero is not a successor of any number ~x{(, x) & (x)} Nero is not a doubter of any cucumber We miss something if we represent the successor relation with an atomic relational concept,as if successor-of( , ) differs from doubter-of( , )and planet-of( , ) only in terms of its specific content.

  26. Fregean Moral • The contents of number( ), Zero, and successor-of( , ) seem to be logically related in ways that the contents of cucumber( ), Nero, and doubter-of( , ) are not • So perhaps we should try to re-present the contents of the arithmetic concepts, and reduce (re-presentations of) the arithmetic axioms to “sparer” axioms that reflect (all and only) the “nonlogical” content of arithmetic

  27. Frege’s Success (with help from Wright/Boolos/Heck) (Defs) Definitions of number(x), Zero, and successor-of(y, x) in terms of number-of(x, ) and logical notions (HP) {[x:number-of(x,)= number-of(x,)] one-to-one(,)} __________________________________________ (DP) Dedekind-Peano Axioms A mind that starts with (HP) could generate “Frege Arithmetic,” given a consistent fragment of Frege’s (2nd-order) Logic.

  28. Some Truths some initial clues to let us cotton on and “get” the concept number-of[x,] number-of[Zero,z.~(z = z)] number-of[One,z.(z = Zero)] number-of[Two,z.(z = Zero) v (z = One)] ... Zero = x:number-of[x,z.~(z = z)] One = x:number-of[x,z.(z = Zero)] Two = x:number-of[x,z.(z = Zero)v (z = One)] … x{number-of(x, )number(x)& one-to-one(,λy.precedes(y,x)]} x{number(x) (x = Zero) vprecedes(Zero,x)} x{number(x)[number-of(x,)]} We can call these definitions in the OED sense without saying that the truths are analytic.

  29. Frege’s Direction of Definition Zero =dfx:number-of[x,z.~(z = z)] x{number(x)df (x = Zero) vprecedes(Zero,x)} xy{precedes(x, y) dfANCESTRAL:predecessor(x,y)} xy{predecessor(x,y)df ∃∃{number-of(x,) & number-of(y, ) & ∃z[~z & w{w [wv (w = z)]}]} The idea is not that our concepts Zero, number( ), and predecessor(x,y)had decompositions all along. But once we have the concept number-of(x,), perhaps with help from Frege, we can imagine an ideal thinker who starts with this concept and introduces the others—either to interpret us, or to suppress many aspects of logical structure until they become relevant.

  30. A Different Direction of Analysis x{number-of(x,)df number(x)& one-to-one(, λy.precedes(y,x)]} xy{precedes(x,y)dfANCESTRAL:predecessor(x,y)} xy[predecessor(x,y)dfsuccessor(y,x)] as if primitive thinkers—who start with number(x), one-to-one(,) and successor(x,y)—were trying to interpret (or become) thinkers who use the sophisticated concept number-of(x, )and thereby come to see how the Dedekind-Peano axioms can be re-presented such thinkers don’t use number-of(x,)to introduce number(x) but they might use kick(x,y)/give(x,y,x) to introduce kick(e)/give(e)

  31. Some Potential Equivalences x{socratizes(x) (x = Socrates)} Socrates = x:socratizes(x) xy{kicked(x,y) ∃e[kicked(e,x,y)]} xye{kicked(e,x,y)past(e) &kick(e,x,y)} xye{kick(e, x,y)agent(e,x) &kick(e)&patient(e,y)} λe.kick(e)= :xye{kick(e,x,y)agent(e,x) &(e) &patient(e,y)}

  32. Some Potential Equivalences x{socratizes(x) (x = Socrates)} fixes the (one-element) extension of socratizes(_) xye{kick(e, x,y)agent(e,x) &kick(e)&patient(e,y)} λe.kick(e)= :xye{kick(e,x,y)agent(e,x) &(e) &patient(e,y)} constrains (but does not fix) the extension of kick(_), which is neither fully defined nor an unconstrained atom

  33. Two Kinds of Introduction Introduce monadic concepts in terms of nonmonadic concepts x{socratizes(x)df(x = Socrates) λe.kick(e)=df :xye{kick(e,x,y)agent(e,x) &(e) &patient(e,y)} _____________________________________________________ Introduce nonmonadic concepts in terms of monadic concepts Socrates =dfx:socratizes(x) xye{kick(e, x,y)dfagent(e,x) &kick(e)&patient(e,y)}

  34. Two Kinds of Introduction Introduce monadic concepts in terms of nonmonadic concepts x{socratizes(x)df(x = Socrates) λe.kick(e)=df :xye{kick(e,x,y)agent(e,x) &(e) &patient(e,y)} _____________________________________________________ My suggestion is not that our concepts Socratesand kick(e, x,y)have decompositions. But a child who starts with these concepts might introducekick(e)and socratizes(x).

  35. Two Kinds of Introduction Introduce monadic concepts in terms of nonmonadic concepts x{socratizes(x)df(x = Socrates) λe.kick(e)=df :xye{kick(e,x,y)agent(e,x) &(e) &patient(e,y)} _____________________________________________________ The monadic concepts—unlike number-of(x,)—won’t help much if your goal is to show how logic is related to arithmetic. But they might help if you want to specify word meanings in a way that allows for “logical forms” that involve using conjunction to build complex monadic predicates as in… ∃e{∃x[agent(e,x) & socratizes(x)] &kick(e)& past(e)}

  36. Two Conceptions of Lexicalization (1) Labelling • Acquiring words is basically a process of pairing pre-existing concepts with perceptible signals • Lexicalization is a conceptually passive operation • Word combination mirrors concept combination (2) Reformatting • Acquiring words is also a process of introducing concepts that exhibit a certain composition format • In this sense, lexicalization is cognitively creative • Word combination does not mirror combination of the pre-existing concepts that get lexicalized

  37. Bloom: How Children Learn the Meanings of Words word meanings are, at least primarily, concepts that kids have prior to lexicalization learning word meanings is, at least primarily, a process of figuring out which existing concepts are paired with which word-sized signals in this process, kids draw on many capacities—e.g., recognition of syntactic cues and speaker intentions—but no capacities specific to acquiring word meanings

  38. Lidz, Gleitman, and Gleitman “Clearly, the number of noun phrases required for the grammaticality of a verb in a sentence is a function of the number of participants logically implied by the verb meaning. It takes only one to sneeze, and therefore sneeze is intransitive, but it takes two for a kicking act (kicker and kickee), and hence kick is transitive. Of course there are quirks and provisos to these systematic form-to-meaning-correspondences…” Brutus kicked Caesar the ball Brutus kicked That kick was a good one Caesar was kicked

  39. Lidz, Gleitman, and Gleitman “Clearly, the number of noun phrases required for the grammaticality of a verb in a sentence is a function of the number of participants logically implied by the verb meaning. It takes only one to sneeze, and thereforesneeze is intransitive, but it takes two for a kicking act (kicker and kickee), and hencekick is transitive. Of course there are quirks and provisos to these systematic form-to-meaning-correspondences…”

  40. Why Not... Clearly, the number of noun phrases required for the grammaticality of a verb in a sentence isnot a function of the number of participants logically implied by the verb meaning. A paradigmatic act of kicking has exactly two participants (kicker and kickee), andyetkickneed not be transitive. Brutus kicked Caesar the ball Caesar was kicked Brutus kicked Brutus gave Caesar a swift kick Of course there are quirks and provisos. Some verbs do require a certain number of noun phrases in active voice sentences. *Brutus put the ball *Brutus put *Brutus sneezed Caesar

  41. Quirky information for lexical items like ‘kick’ Conceptof adicityn Perceptible Signal Conceptof adicityn Quirky information for lexical items like ‘put’ Conceptof adicity-1 Perceptible Signal

  42. Clearly, the number of noun phrases required for the grammaticality of a verb in a sentence is a function of the number of participants logically implied by the verb meaning. It takes only one to sneeze, and thereforesneeze is intransitive, but it takes two for a kicking act (kicker and kickee), and hence kick is transitive. Of course there are quirks and provisos to these systematic form-to-meaning-correspondences. Clearly, the number of noun phrases required for the grammaticality of a verb in a sentence isn’t a function of the number of participants logically implied by the verb meaning. It takes only one to sneeze, and usuallysneeze is intransitive. But it usually takes two to have a kicking; and yetkickcan be untransitive. Of course there are quirks and provisos. Some verbs do require a certain number of noun phrases in active voice sentences.

  43. Clearly, the number of noun phrases required for the grammaticality of a verb in a sentence is a function of the number of participants logically implied by the verb meaning. It takes only one to sneeze, and thereforesneeze is intransitive, but it takes two for a kicking act (kicker and kickee), and hence kick is transitive. Of course there are quirks and provisos to these systematic form-to-meaning-correspondences. Clearly, the number of noun phrases required for the grammaticality of a verb in a sentence isn’t a function of the number of participants logically implied by the verb meaning. It takes only one to sneeze, and sneeze is typically used intransitively; but a paradigmatic kicking has exactly two participants, and yetkickcan be used intransitivelyor ditransitively. Of course there are quirks and provisos. Some verbs do require a certain number of noun phrases in active voice sentences.

  44. Lexicalization as Concept-Introduction (not mere labeling) Concept of type T Concept of type T Perceptible Signal Concept of type T*

  45. Lexicalization as Concept-Introduction (not mere labeling) Number(_) type: <e, t> Number(_) type: <e, t> Perceptible Signal NumberOf[_, Φ(_)] type: <<e, t>, <n, t>>

  46. Lexicalization as Concept-Introduction (not mere labeling) Concept of type T Concept of type T Perceptible Signal Concept of type T*

  47. One Possible (Davidsonian) Application: Increase Adicity ARRIVE(x) ARRIVE(e, x) Concept of adicity-1 Concept of adicity-1 Perceptible Signal Concept of adicity-2

  48. One Possible (Davidsonian) Application: Increase Adicity KICK(x1, x2) KICK(e, x1, x2) Concept of adicity-2 Concept of adicity-2 Perceptible Signal Concept of adicity-3

  49. Lexicalization asConcept-Introduction: Make Monads KICK(x1, x2) KICK(e) Concept of adicityn KICK(e, x1, x2) Concept of adicityn Perceptible Signal Concept of adicity-1

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