Type-shifting
from Partee (1987)
We need all of these types!
●
A man walked in. He smiled.
–
–
●
If he = a man, they need to be the same type!
Or else, we need to be able to get type e from “a man”
Mary considers John
competent in semantics and an expert in math
–
–
–
●
Adjective (phrases) are type e → t
Conjunction joins two things of the same type
So, an expert in math is type e → t here!
A prof and every student ran.
–
–
There is no possible type e or e → t meaning for “every
student”
By conjunction principle, a prof here is type (e → t) → t)
We need all of these types!
●
A man walked in. He smiled.
–
–
●
If he = a man, they need to be the same type!
Or else, we need to be able to get type e from “a man”
Mary considers John
competent in semantics and an expert in math
–
–
–
●
Adjective (phrases) are type e → t
Conjunction joins two things of the same type
So, an expert in math is type e → t here!
A prof and every student ran.
–
–
There is no possible type e or e → t meaning for “every
student”
By conjunction principle, a prof here is type (e → t) → t)
We need all of these types!
●
A man walked in. He smiled.
–
–
●
If he = a man, they need to be the same type!
Or else, we need to be able to get type e from “a man”
Mary considers John
competent in semantics and an expert in math
–
–
–
●
Adjective (phrases) are type e → t
Conjunction joins two things of the same type
So, an expert in math is type e → t here!
A prof and every student ran.
–
–
There is no possible type e or e → t meaning for “every
student”
By conjunction principle, a prof here is type (e → t) → t)
How to get from type to type
Type-shifters
●
●
Lift: j:e becomes λP [P(j)]: (e → t) → t
Lower: gets individual whose properties are in
the set: from (e → t) → t to e
–
●
lower(lift(j)) = j
Example:
Sophia and every student
λP [lift(s) (P) & Every(student)(P)]
Type-shifters (cont'd)
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A: P:e → t (a set) becomes a GQ,
λQ[∃x[P(x)&Q(x)]] :(e → t)→ t
whose element-sets have
at least one member in common with P
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Link: {a,b,...}:e → t becomes a⊕ b...: e
●
Delink: a⊕ b...: e becomes {a,b,...}:e → t
–
link(delink(P)) = P and delink(link(s)) = s
Type-shifters
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Nom: P:e → t (a set) becomes ᴖP : e (a kind)
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Pred: d:e (a kind) becomes ᴗd:e (a set)
–
●
pred(nom(P)) = P
Example:
Dodos, which are becoming a pest in Waltham,
ate my tomatoes yesterday.
Becoming-a-pest-in-Waltham(ᴖ[λx[dodo'(x)]])
&Ate-my-tomatoes(link[λx[dodo'(x)]])
Type-shifters (cont'd)
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Ident: j:e becomes λx [x = j]: (e → t)
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Iota: P:e → t (a singleton) becomes ιx[P(x)]:e
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THE: P:e → t a set turns into the GQ
λQ[∃x[∀y[P(y) ↔ x=y]&Q(x)]] :(e → t)→ t
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BE: λP.P∈ℜ:(e → t)→ t (a GQ, set of sets)
becomes λx[ (λy[y=x])∈ℜ ]:e → t (a set of entities
whose singletons are elements in the GQ)
–
BE(THE(P)) = ident(iota(P)) = P
–
John is not the prof, because the prof is a woman
Type-shifters (cont'd)
●
Example
John is not the prof, and the prof is a woman.
[~ BE(THE(prof'))(j)]
&
[BE(A(woman'))(ιx[prof'(x)])]
Type-shifting and word-meaning
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John bought Sophia's book, but he doesn't
believe it.
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Lunch was long and delicious.
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I painted a picture => new picture exists!
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I painted a wall
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I could read her face perfectly.
=/=> new wall exists