Reference from Constructive View

Reference from a constructive point of view : 

7/12/01 1 Reference from a constructive point of view Pascal Boldini Université Paris IV CAMS-EHESS

The computational notions of meaning and reference : 

7/12/01 2 The computational notions of meaning and reference 2 + 3 : N 2+3 = 5 : N SS0 + SSS0 : N S(SS0 + SS0) : N SS(SS0 + S0) : N SSS(SS0 + 0) : N SSS(SS0) : N SSSSS0 : N

Σ - types : 

7/12/01 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)

DRT / Type Theory 1 : 

7/12/01 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)))

DRT / Type Theory 2 : 

7/12/01 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)))

References : 

7/12/01 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.

Reference to events : 

7/12/01 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)

Propositions as sets 1 : 

7/12/01 8 Propositions as sets 1 : Man this : Man Paul : Man : Man Judgements reference meaning meaning meaning  = this= Paul: Man

Propositions as sets 2 : 

7/12/01 9 Propositions as sets 2 : it rains  : it rains  : it rains : it rains Judgements reference meaning meaning meaning = Rain : Set  =  =  : it rains

Interface : 

7/12/01 10 Interface {Semiotic system} Judgements a : A signification (public) Reference α : A (ideal) sense (private) No entity without a type

Definite descriptions – Proper Nouns Frege : 

7/12/01 11 Definite descriptions – Proper Nouns Frege Victor Hugo the author of Les Misérables a : Person 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. »

Definite descriptions – Proper Nouns Russell : 

7/12/01 12 Definite descriptions – Proper Nouns Russell Victor Hugo the author of Les Misérables a : Person | | | | x(wrote(x,Les Misérables)  y(wrote(y,Les Misérables)  y=x)) 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!

Definite descriptions – Proper Nouns Kripke : 

7/12/01 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. »

Definite descriptions The constructive solution : 

7/12/01 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

Definite descriptions The constructive solution : 

7/12/01 15 Definite descriptions The constructive solution {Semiotic system} the biggest natural number : (Σx:N)(Πy:N)(yx)  The type is a specification for the method of access to the reference. Individuals as …. Napoléon aware of the danger concentrated his troops on the left side (Napoléon,b) with b : aware(the_danger, Napoléon) Definite descriptions without reference:

Proper nouns the constructive solution : 

7/12/01 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.

Contexts and their extensions : 

7/12/01 17 Contexts and their extensions Г=[ x1:(Σx:Donkey)owns(Paul,x), x2:beats(Paul,p(x1))] a(x1,x2) : cruel(x1,x2,Paul) Г0=[ x1:(Σx:Donkey)owns(Paul,x), x2:beats(Paul,p(x1))] Г1=[ y1:Donkey, y2: owns(Paul,y1), y3:beats(Paul,y1)] f(y1,y2,y3)=((y1,y2),y3): Г0 a(x1,x2) : cruel(x1,x2,Paul) a((y1,y2),y3) : cruel((y1,y2),y3,Paul) lifting

Time and reference : 

7/12/01 18 Time and reference John met his wife in 1968. [u : Г1948] [x0 : Г0] f a : Woman b(u) : meet(u,John,a) a : Woman b(f(x0)) : meet(f(x0),John,a) lifting 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)))

Counter-factuals : 

7/12/01 19 Counter-factuals [u : ] [x0 : 0] [v : ] f g a : Person 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) Victor Hugo might not have written Les Misérables.

Mental spaces (G. Fauconnier) : 

7/12/01 20 Mental spaces (G. Fauconnier) [u : ] [x0 : 0] [v : Luc] f g expected world In Luc’s painting, the blue eyed girl has green eyes. a : Girl 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)))

Dependant predicates : 

7/12/01 21 Dependant predicates Marlon Brando dies at the end of Apocalypse Now. [u : ] [x0 : 0] [v : Apocalypse] f g a : Person a : Person[Apocalypse, 0] 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)

Generalized modus ponens (G. Fauconnier) : 

7/12/01 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.

Generalized modus ponens : 

7/12/01 23 Generalized modus ponens [u : ] [x0 : 0] f g [v : ] [w : RWL] h a : Person a : Person[, 0] S_C(x0) : (x:Person)s_c(x0,x) p(S_C(x0)) = a: Person b(v) : spy(a)(x:Woman)love(x,a) 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)))