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Reference from a constructive point of view

Reference from a constructive point of view. Pascal Boldini Université Paris IV CAMS-EHESS. The computational notions of meaning and reference. 2 + 3 : N. SS0 + SSS0 : N. S(SS0 + SS0) : N SS(SS0 + S0) : N SSS(SS0 + 0) : N SSS(SS0) : N SSSSS0 : N. 2+3 = 5 : N. Σ - types. Introduction

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Reference from a constructive point of view

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  1. Reference from a constructive point of view Pascal Boldini Université Paris IV CAMS-EHESS

  2. The computational notions of meaning and reference 2 + 3 : N • SS0 + SSS0 : N • S(SS0 + SS0) : N • SS(SS0 + S0) : N • SSS(SS0 + 0) : N • SSS(SS0) : N • SSSSS0 : N 2+3 = 5 : N

  3. Σ - types • Introduction a : A b : B(a) ————————— (a,b) : (Σx:A)B(x) • Elimination c : (Σx:A)B(x) ——————————— p(c) : A q(c) : B(p(c)) • Equality c = (p(c),q(c)) : (Σx:A)B(x)

  4. DRT / Type Theory 1 • A man whistles. A dog follows him. • A man whistles u: (Σx:Man)whistles(x) • p(u) : Man • q(u) : whistles(p(u)) • A dog follows him • v: (Σx: Dog)follows(x,p(u)) • p(v) : Dog • q(v) : follows(p(v),p(u)) • Σ-intro • (u,v) : (Σy:(Σx:Man)whistles(x))((Σx:Dog)follows(x,p(y)))

  5. DRT / Type Theory 2 • If a farmer owns a donkey, he beats it. • If a farmer owns a donkey [u : (Σx:Farmer)(Σy:Donkey)owns(x,y)] • p(u) : Farmer • q(u) : (Σy:Donkey)owns(p(u),y) • p(q(u)) : Donkey q(q(u)) : owns(p(u),p(q(u))) • He beats it • v : beats(p(u),p(q(u))) • Σ-intro • (u,v) : (Πz: (Σx:Farmer)(Σy:Donkey)owns(x,y))beats(p(z),p(q(z)))

  6. References • G. Sundholm, ``Proof theory and meaning'', Gabbay, D. and Guenthner, F.(eds.), Handbook of Philosophical Logic, Vol. III, D. Reidel, 1986. • A.Ranta, Type-theoretical grammar, Oxford, Clarendon, 1994. • R. Ahn, Agents, Objects and Events, Technical University, Eindhoven, 2000.

  7. Reference to events • John stole the book, I saw it. u : stole(John, the_book) v : saw(I,u) (u,v) : (Σx:stole(John, the_book))saw(I,x)

  8. Propositions as sets 1 • Judgements : Man this : Man Paul : Man meaning meaning meaning  = this= Paul: Man : Man reference

  9. Propositions as sets 2 • Judgements : it rains  : it rains  : it rains meaning meaning meaning : it rains reference = Rain : Set  =  =  : it rains

  10. Interface {Semiotic system} signification (public) Judgements a : A sense (private) No entity without a type Reference α : A (ideal)

  11. Definite descriptions – Proper Nouns Frege Victor Hugo a : Person the author of Les Misérables • Well-known difficulties • some expressions do not refer: the king of France  • identity of reference is tautological: the morning star is the evening star • oblique contexts • John believes that the author of Les Misérables is Irish  • « Generally an expression denotes its reference, but within oblique contexts it denotes its meaning. »

  12. Definite descriptions – Proper Nouns Russell Victor Hugo | | the author of Les Misérables | | x(wrote(x,Les Misérables)  y(wrote(y,Les Misérables)  y=x)) a : Person Explains : definite descriptions without reference the content of referential identity Problem : The author of Les Misérables is a genious, but surely not the author of Hernani!

  13. Definite descriptions – Proper Nouns Kripke If Victor Hugo = the author of Les Misérables, the a priori (analytic) truth wrote(Victor Hugo, Les Misérables) is a necessary truth. But it makes sense to say: Victor Hugo might not have written Les Misérables. The proper name is a rigid designator « the reference of a proper named is not characterized by a definite description, nor by any bunch of properties. »

  14. Definite descriptions The constructive solution the author of Les Misérables : (Σx:Person)write(x,Les Misérables) (a,b) such that a : Person, b : write(a,Les Misérables) the author of Hernani : (Σx:Person)write(x,Hernani) (a,b’) such that a : Person, b’ : write(a,Hernani) We form without contradiction: genious(a,b)   genious(a,b’) : Prop

  15. Definite descriptions The constructive solution • Individuals as …. • Napoléon aware of the danger concentrated his troops on the left side {Semiotic system} (Napoléon,b) with b : aware(the_danger, Napoléon) • Definite descriptions without reference: the biggest natural number : (Σx:N)(Πy:N)(yx)  The type is a specification for the method of access to the reference.

  16. Proper nouns the constructive solution • Knowledge by acquaintance: Paul : Person • Knowledge by description : Victor Hugo : (Σx:Person)victor_hugo(x) Victor Hugo=(p(Victor Hugo),q(Victor Hugo)) • There is no coreference for proper nouns brave(Bonaparte)   brave(Napoléon) : Prop • There are no oblique contexts • John believes that the author of Les Misérables is Irish Assumptions required in the belief context: • p(Victor Hugo)=p(author of Les Misérables) : Person • homogeneity of the predicate to be Irish on the type Person and its sub-types.

  17. Contexts and their extensions • Г=[ x1:(Σx:Donkey)owns(Paul,x), x2:beats(Paul,p(x1))] a(x1,x2) : cruel(x1,x2,Paul) • Г1=[ y1:Donkey, y2: owns(Paul,y1), y3:beats(Paul,y1)] a((y1,y2),y3) : cruel((y1,y2),y3,Paul) f(y1,y2,y3)=((y1,y2),y3): Г0 lifting a(x1,x2) : cruel(x1,x2,Paul) Г0=[ x1:(Σx:Donkey)owns(Paul,x), x2:beats(Paul,p(x1))]

  18. Time and reference [x0 : Г0] a : Woman b(f(x0)) : meet(f(x0),John,a) • John met his wife in 1968. f his_wife(x0) : (Σx:Woman)married(x0,John,x) p(his_wife (x0))=a : Woman b(f(x0)) : meet(f(x0),John, p(his_wife(x0))) lifting a : Woman b(u) : meet(u,John,a) [u : Г1948]

  19. Victor Hugo might not have written Les Misérables. Counter-factuals a : Person Victor_Hugo(x0) : (x:Person)victor_hugo(x0,x) p(Victor_Hugo(x0)) = a : Person b(v) : write(a, Les Misérables) b(v) : write(p(Victor_Hugo(x0)), Les Misérables) v.b(v) : (v:)write(p(Victor_Hugo(x0)), Les Misérables) [v : ] [x0 : 0] g f [u : ] a : Person

  20. Mental spaces (G. Fauconnier) • In Luc’s painting, the blue eyed girl has green eyes. expected world a : Girl the_beg(x0) : (x:Girl)blue_eyed(x0,x) p(the_beg(x0)) = a : Person b(v) : green-eyed(v,a) b(v) : green-eyed(v, p(the_beg(x0))) v.b(v) : (v:Luc) green-eyed(v, p(the_beg(x0))) [v : Luc] [x0 : 0] g f [u : ] a : Girl

  21. Dependant predicates a : Person[Apocalypse, 0] • Marlon Brando dies at the end of Apocalypse Now. M_B(x0):(x:Person)m_b(x0,x) p(M_B(x0))=a:Person dies(v,a) dies(v, p(M_B(x0))) (v : Apocalypse) dies(v,p(M_B(x0))) Kurtz(v):(x:Person)kurtz(v,x) p(Kurtz(v))=a:Person dies(v,Kurtz) (v : Apocalypse) dies(v,Kurtz) [v : Apocalypse] [x0 : 0] g f [u : ] a : Person

  22. Generalized modus ponens (G. Fauconnier) • When he is a spy, all the beautiful women fall in love with Sean Connery. • In Russia With Love, Sean Connery is a spy ——————————————————— In Russia With Love, all the beautiful women fall in love with Sean Connery.

  23. Generalized modus ponens [w : RWL] a : Person[, 0] h S_C(x0) : (x:Person)s_c(x0,x) p(S_C(x0)) = a: Person b(v) : spy(a)(x:Woman)love(x,a) [v : ] [x0 : 0] g f [u : ] c(w) : spy(a) b(f(w))) : spy(a)(x:Woman)love(x,a) b(f(w))) c(w) : (x:Woman)love(x,a) b(f(w))) c(w) : (x:Woman)love(x, p(S_C(x0))) w. b(f(w))) c(w) : (w : RWL) (x:Woman)love(x, p(S_C(x0))) a : Person

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