semantic typology for human i-languages paul m. pietroski, university of maryland dept. of...

Semantic Typology for Human i-Languages

Paul M. Pietroski, University of MarylandDept. of Linguistics, Dept. of Philosophy

Human Languages

• acquirable by normal human children given ordinary courses of experience

• pair unboundedly many “meanings”with unboundedly many pronunciations

• how many types of meanings?

• one answer, via Frege on ideal languages:<e> entity-denoters<t> truth-evaluable sentences

possible languagesHuman

Languages

if <α> and <β> are types, then so is <α, β>

Human Languages




• one answer, via Frege on ideal languages:<e> entity-denoters <t> truth-evaluable sentences


Languages

<t>

SADIE<e> IS-A-

HORSE(_)<e, t>

Human Languages




• one answer, via Frege on ideal languages:<e> entity-denoters <t> truth-evaluable sentences


Languages

<e, t>

SADIE<e> SAW(_, _)<e,

<e, t>>

<t>

SOPHIE<e>

Barbara Partee (2006), “Do We Need Two Basic Types?”

“…Carstairs-McCarthy argues that the apparently universal distinction in human languages between sentences and noun phrases cannot be assumed to be inevitable….His work suggests…that there is also no conceptual necessity for the distinction between basic types e and t….

If I am asked why we take e and t as the two basic semantic types, I am ready to acknowledge that it is in part because of tradition, and in part because doing so has worked well.…”

Some of Partee’s Suggested Ingredients for an Alternative

• eventish semantics for VPs: barked Bark(e, e’) & Past(e)

chase Chase(e, e’, e’’) • Heim/Kamp for indefinite NPs: a dog Dog(e)• entity/event neutrality, and

maybe predicate/sentence neutrality (cp. Tarski)

Human Languages



• how many types of meanings (basic or not)?

• one answer, via Frege on ideal languages:<e> entity-denoters<t> truth-evaluable sentences


Languages


<e> <t>

if <α> and <β>, then <α, β>

0. <e> <t>(2)

1. <e, e> <e, t> <t, e> <t, t>(4) of <0, 0>

2. eight of <0, 1> eight of <1, 0> (32), includingsixteen of <1, 1>

<e, et> and <et, t>

3. 64 of <0, 2> 64 of <2, 0>(1408),

128 of <1, 2> 128 of <2, 1> including <e, <e, et>> 1024 of <2, 2>

and <et, <et, t>>

4. 2816 of <0, 3> 2816 of <3, 0>5632 of <1, 3> 5632 of <1, 3> (2,089,472), including45,056 of <2, 3> 45,056 of <3, 2> <e, <e, <e, <et>>1,982,464 of <3, 3>

that’s a lot of types

possible languagesFregean Languages

with expression types: <e>, <t>, and if

<α> and <β> are types,

so is <α, β>

Human i-Languages

Level-n Fregean Languages with expression types:

<e>, <t>, and the nonbasic types up to Level-n

Pseudo-Fregean Languages with expression types:

<e>, <t>, and a few of the nonbasic types

Human Languages



• how many types of meanings (basic or not)?

• another answer: <M> monadic predicates

<D> dyadic predicates


Languages

Horse(_)

On(_, _)

<e, t><e, <e, t>>

We can imagine/invent a language that has…

(1) finitely many atomic monadic predicates: M1(_) … Mk(_)

(2) finitely many atomic dyadic predicates: D1(_, _) … Dj(_, _)

(3) boundlessly many complex monadic predicates

Monad + Monad Monad Dyad + Monad Monad

BROWN(_) + HORSE(_) BROWN(_)^HORSE(_)

FAST(_) + BROWN(_)^HORSE(_) FAST(_)^BROWN(_)^HORSE(_)






Φ(_)^Ψ(_) applies to e iff Φ(_) applies to e and Ψ(_) applies to e

ON(_, _) + HORSE(_) [ON(_, _)^HORSE(_)]






Φ(_)^Ψ(_) applies to e iff Δ(_, _)^Φ(_) applies to e iff

Φ(_) applies to e and e bears Δ(_, _) to something

Ψ(_) applies to e that Φ(_) applies to

BROWN(_)^HORSE(_) [ON(_, _)^HORSE(_)]





FAST(_)^BROWN(_)^HORSE(_) [ON(_, _)^HORSE(_)]

FAST(e) & BROWN(e) & HORSE(e) e’[ON(e, e’) & HORSE(e’)]

v[BETWEEN(x, y, z) & SOLD(z, w, v, x)]

Triad & Tetrad Pentad

v[Pentad(…v…)] Tetrad

but ‘&’ and ‘v[…v…]’ permit a lot more than ‘^’ and ‘’





FAST(_)^BROWN(_)^HORSE(_) [ON(_, _)^HORSE(_)]

FAST(e) & BROWN(e) & HORSE(e) e’[ON(e, e’) & HORSE(e’)]

[AGENT(_, _)^HORSE(_)]^FAST(_)^RUN(_) e’[AGENT(e, e’) & HORSE(e’)] & FAST(e) &

RUN(e)

[AGENT(_, _)^HORSE(_)^FAST(_)]^RUN(_)

e’[AGENT(e, e’) & HORSE(e’) & FAST(e’)] & RUN(e)





PAST(_)^SEE(_)^[THEME(_, _)^HORSE(_)] PAST(e) & SEE(e) & e’[THEME(e, e’) & HORSE(e’)]

PAST(_)^SEE(_)^[THEME(_, _)^RUN(_)^[AGENT(_,

_)^HORSE(_)]]

PAST(e) & SEE(e) & e’[THEME(e, e’) &

RUN(e’) & e’’[AGENT(e’, e’’)^HORSE(e’’)]]

/ \saw / \

a horse

/ \saw / \

/ \ run

a horse

How many types of meanings in human languages?

How do the meanings combine?

<t> <β> <M> <M>

/ \ / \ / \ / \

<e> <e, t> <α, β> <α> <M> <M> <D> <M>

<e, <e, t>> <e> <<e, t>, t> <e, t>

… … BROWN(_)^HORSE(_)

[ON(_, _)^HORSE(_)] <e, t>

/ \ <e, t> <e, t>

brown horse


Languages

function applicationand

(type-shifting or) a rule for

adjunction

Human Languages


• pair unboundedly many “meanings”with unboundedly many pronunciationsin accord with language-specific constraints

Bingley was ready _ to please _. Georgiana was eager _ to please _. Darcy was easy _ to please _.


Languages

ambiguous unambiguous unambiguous

(the other way)

Human Languages



hiker lost kept walking circles


Languages

Human Languages



The hiker who was lost kept walking in circles. The hiker who lost was kept walking in circles.

Was the hiker who lost kept walking in circles? (meaning 2)


Languages

Human Languages

• acquirable by children • unbounded but constrained and so presumably

• i-Languages in Chomsky’s (Church-inspired) sense

function-in-intension vs. function-in-extension

--a procedure that pairs inputs with outputs in a certain way

--a set of ordered pairs (with no <x,y> and <x, z> where y ≠ z)


Languages

Human Languages

• acquirable by children • unbounded but constrained and so presumably

• i-Languages in Chomsky’s (Church-inspired) sense

function-in-intension vs. function-in-extension

|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)]


Languages

Human Languages

• acquirable by children • unbounded but constrained• biologically implemented procedures that pair

“meanings” with sounds/gestures in a human way


• one hypothesis, via Frege on ideal languages:

<e> entity-denoters<t> truth-evaluable sentences



Languages

Human Languages

• natural generative procedures



<e> <t> if <α> and <β>, then <α, β>

• THREE CONCERNS: available evidence suggests that…

the proposed generative principle overgenerates (massively)

Human Languages don’t generate expressions of type <e>

Human Languages don’t generate expressions of type <t>


Languages

<e> <t>

if <α> and <β>, then <α, β>

• one worry…overgeneration

0. <e> <t>(2)

1. <e, e> <e, t> <t, e> <t, t>(4)

2. eight of <0, 1> eight of <1, 0>sixteen of <1, 1>

(32)

3. 64 of <0, 2> 64 of <2, 0> 128 of <1, 2> 128 of <2, 1>

1024 of <2, 2> (1408)

4. 2816 of <0, 3> 2816 of <3, 0>5632 of <1, 3> 5632 of <1, 3>45,056 of <2, 3> 45,056 of <3, 2> (2,089,471)1,982,464 of <3, 3>


Languages

Human Languages






<e, <e, <e, <e, <e, t>>>>> λv. λw. λz . λy. λx . GRONK(x, y, z, w, v)

<<<e, t>, <<e, t> , <e, t>>> λZ . λY. λX . GLONK(X, Y, Z)

<<e, t>, <t, e>>

<<e, t>, e> ??? <e, e> ???


Languages

x[X(x) v Y(x) v Z(x)]x[X(x)] & x[Y(x) & Z(x)]

Human Languages






<e, <e, <e, <e, <e, t>>>>> λw. λz . λy. λx . λe. SELL(e, x, y, z, w)

<e, <e, <e, <e, t>>>> λz . λy. λx . λe. GIVE(e, x, y, z)

<e, <e, <e, t>>> λy. λx . λe. KICK(e, x, y)


Languages

claim: verbs don’t provide evidence for “supradyadic” types

a linguist sold a car to a friend for a dollar (x) (y) (z) (w)

a linguist sold a friend a car for a dollar (x) (z) (y) (w)

a linguist sold a friend a car a dollar

‘sold’ λz. λy . λw. λx . x sold y to z for w

(x) (z) (y) (w)

Why not just…

λz. λy . λw. λx . λe . e was a selling by x of y to z for w

a triple-object construction

she her this that

(x) (y) (z)

a thief jimmied a lock with a knife

(x) (y) (z)

a thief jimmied a lock a knife

Why not instead…

‘jimmied’ λz. λy . λx . x jimmied y with z

(x) (y) (z)

a rock betweens a lock a knife

Why not…

‘betweens’ λz. λy . λx . x is between y and z

(x) kicked (y)gave

a linguist tossed a coin

(x) kicked (y) (z) gave

a linguist tossed a coin to a friend

a linguist tossed a friend a coin

‘tossed’ λz . λy. λx . x tossed y to z‘kicked’ λz . λy. λx . x kicked y to z

(x) kicked (z) (y)

Why not…

λz . λy. λx . λe . e was a kicking by x of y to z

(y) kicked (x)

a coin was tossed (by a linguist)

‘tossed’ λy. λe . e was a tossing of y‘kicked’ λy. λe . e was a kicking of y

if Kratzer andothers are

on the right track…

representing an Agent is as optional as representing a Recipient

(x) kicked (y)

a linguist tossed a coin

if Kratzer andothers are

on the right track…

‘kicked’ λy. λe . e was a kicking of y

[AGENT(_, _)^LINGUIST(_)]^PAST(_)^[KICK(_, _)^COIN(_)]

or … ^PAST(_)^KICK(_)^[PATIENT(_, _)^COIN(_)]

Chris hailed from Boston

Why …

and not…

Chris frommed Bostonλy . λx . λe .

e was a hailing by x from y

Chris hailed from Boston Chris was taller than Sam

Why …

and not…

Chris frommed Boston Chris talled

Samoutheighted

Human Languages





• THREE CONCERNS: it seems that…

✔ the proposed generative principle overgenerates (massively)

Human Languages don’t generate expressions of type <e>



Languages

Which expressions are of type <e>?

NAMES ® ©

--|-- --|--

‖Robin<e>‖= / \ ‖Cruso<e>‖= / \

‖Robin<<e, t>, t>‖= P . P(®) ‖Cruso<<e, t>, t>‖= P . P(©)

‖Robin<e, t>‖= x . x = ® ‖Cruso<e, t>‖= x . x = ©

x . Robin(x) x . Cruso(x)

‖[D1<e, t> Robin<e, t>]<e, t>‖ = x . Indexes(1, x) & Called(x, ‘Robin’)

Proper Nouns

• even English tells against the idea that lexical proper nouns are i-language expressions of type <e>

• Every Tyler I saw was a philosopherEvery philosopher I saw was a Tyler That Tyler stayed late, and so did this oneThere were three Tylers at the party, and Tylers are cleverThe Tylers are coming to dinner

(That nice) Professor Tyler Burge was sitting next to John Jacob Jingleheimer

Schmidt

• proper nouns seem to be of type <M>, even if they are related to singular concepts of type <e>

D(P) D(P) N(P)

/ \ / \ / \

D N D N N C(P)

every tiger most tigers tiger(s) that I saw

the Tyler some Tylers Tyler(s)

this those

?/ \

Tyler Burge<e> <e>

/ \ that <e, t>

/ \ nice ?

/ \Professor Burge

<e>

<e, t> / \

Prof. <e, t> <e, t> / \

Tyler Burge

<e, t> <e, t>


V(P) / \

D(P) V(P) / \ / \ D N V D(P)that woman tossed / \

D N

this coin

if the nouns are of type <e, t>then presumably, the indexed determiners are not of type <e>;

if ‘that’ and ‘this’ are of type <e, t>then presumably, the indices are not of type <e>;

1

2

[AGENT(_, _)^THAT(_)^1(_)^WOMAN(_)]^…


V(P) / \

/ V(P) / / \ D(P) V D(P) she tossed \

it

if ‘the pronouns ‘she’ and ‘it’ are of type <e, t>then presumably, the indices are not of type <e>

‖she<e, t>‖= x . x is female ‖1<e, t>‖= x . Indexes(1, x)

12

[AGENT(_, _)^FEMALE(_)^1(_)]^…

Human Languages





• THREE CONCERNS: it seems that…

✔ the proposed generative principle overgenerates (massively)

✔ Human Languages don’t generate expressions of type <e>



Languages

Which expressions are of type <t>?

SENTENCES S NP aux VP NP aux VP

D + N D(P) V + D(P) V(P) every tree / \ saw / \ / \

D N D N V D(P) every tree every tree saw / \

D N every tree

? + ?? S / \ ? ??

S


SENTENCES S = TP

T(P) / \

T V(P) past / \

D(P) V(P) John / \

V D(P) see Mary

e . T e is (tenselessly) an event of John seeing Mary

y . x . e . T e is (tenselessly) an event of x seeing y


SENTENCES S = TP

T(P) / \

T V(P) past / \

D(P) V(P) John / \

V D(P) see Mary


Why think TPs are of type <t> instead of <e, t> ?

Is this expression of type <<e, t>, t> or type <e, t> ?


SENTENCES S = TP

T(P) / \

T V(P) past / \

D(P) V(P) John / \

V D(P) see Mary


Why think TPs are of type <t> instead of <e, t> ?

<<e, t>, t> E . T e[e is in the past & E(e) = T] <e, t>

e . T e is in the past


SENTENCES S = TP+ ?

/ \ T(P)

/ \ T V(P) past / \

D(P) V(P) John / \

V D(P) see Mary


e . T e is in the past & …

<e, t> e . T e is in the past


?

/ \ T(P)

/ \ T V(P) past / \

D(P) V(P) John / \

V D(P) see Mary


e . T e is in the past & …

<e, t> e . T e is in the past

??neg


SENTENCES S = TP

T(P) / \

T V(P) past / \

D(P) V(P) John / \

V D(P) see Mary


T e[e is in the past & …]

<<e, t>, t> E . T e[e is in the past & E(e) = T]

vv

is T a quantificational argument of V and a conjunctive adjunct?


SENTENCES S = TP (or TP+)

V(P) / \ D(P) V(P) That / \

D(P) V(P) John / \

V D(P) see Mary


T That is (tenselessly) an event of J seeing M

Tense may be needed (in matrix clauses). But does it do two semantic jobs: adding time information via the ‘e’-variable, like the adjunct ‘yesterday’; and closing the ‘e’ variable, as if tense is the 3rd argument of a verb that

can’t take a 3rd argument?

if T is (semantically) the verb’s 3rd argument, then why not…


SENTENCES S = TP

T(P) / \

T V(P) past / \

D(P) V(P) John / \

V D(P) see Mary


T e[PAST(e) & …]

<<e, t>, t> E . T e[PAST(e) & E(e) = T]

“God likes Fregean Semantics” theory of tense

(e < RefTime) & (RefTime = SpeechTime)


Maybe None:

T(P) / \

T V(P) past / \

D(P) V(P) John / \

V D(P) see Mary

PastSeeingOfMaryByJohn(_)

a monadic predicate M can be “polarized” into a predicate +M that applies to e iff M applies to somethingor a predicate -M that applies to e iff M applies to nothing

+Polarized |

_ is such that [PastSeeingOfMaryByJohn(_)]

But what about Quantification?

<t> / \

<et, t> <et> / \ ran

<et, <et, t>> <et> every cow

Not at all clear that the “external argument” of ‘every cow’ is—or even can be—an expression of type <et>

<?>

/ \ Fido <?>

/ \ chased <et, t>

/ \

every cow

<et> / \

<et> <et> / \ today <e> <e, et> Bessie ran

<?>/ \

<et> today

<et> / \

<?> <et> / \ today <et, t> <e, et>

/ \ ran every cow

<et> / \

<?> <et> / \ today <et, t> <e, et>

/ \ ran every cow

<et> / \

<et> <et> / \ today

<e> <e, et> (i)t1 ran

every cow

<t>

<t> <et>

1

<et, t>

<t> / \

<et, t> <et> / \ / \

every cow which ran

<et> / \

<et> <et> / \ today

<e> <e, et> (i)t1 ran

every cow

<t>

<t> <et>

1

<et, t>

Why not…

very syncategorematic

<?>/ \

<et> today

<?>

/ \ Fido <?>

/ \ chased <et, t>

/ \

every cow

<et>/ \

<et> today

<et>

/ \ Fido <e, et>

/ \

chased <et, t> / \

every cow

<<et, t>, <<e, et>>

We can discuss the difficulties for this kind of view in Q&A.But my point is not that an <et, t> analysis of quantifierscannot be preserved. My point is that there is no argument here for the standard typology, especially given doubts about <e>.

<<1, 0>, <<0, 1>><2, 2>

<3>

Human Quantification: Still Puzzling

But maybe…

• ‘every’ is a plural Dyad:EVERY(_, _) applies to <the Xs, the Ys> iff the Xs include the Ys

• ‘every cow’ is a plural Monad:[EVERY(_, _)^THE-COWS(_)]applies to the Xs iff the Xs include the cows

• ‘every cow ran’ is a plural Monad:[EVERY(_, _)^THE-COWS(_)]^RAN(_)applies to the Xs iff the Xs include the cows &

the Xs ran


But maybe…

• ‘most’ is a plural Dyad:MOST(_, _) applies to <the Xs, the Ys> iff the Ys that are Xs outnumber the Ys that are

not Xs

• ‘most cows’ is a plural Monad:[MOST (_, _)^THE-COWS(_)] applies to the Xs

iff the cows that are Xs outnumber the cows that

are not Xs

• ‘most cows ran’ is a plural Monad:[MOST(_, _)^THE-COWS(_)]^RAN(_) applies to

the Xs iff the cows that are Xs outnumber the other cows


But maybe…




the Xs ran

<M> / \

t1 ran every cow

<M>

<M>

polarity<M>

for any assignment A of values to variables…

applies to e iff there was an event of A1 running

applies to e iff e was an event of A1 running

<M> / \

t1 ran every cow

<M>

<M>

polarity<M>


applies to e iff A1 ran

so if e is the value of 1 (but A is otherwise the same),then the “polarized” predicate applies to e iff e ran

<M> / \

t1 ran every cow

<M>

<M>

polarity<M>



and we can define a “Tarski Relation” such that TARSKI(e, Polarity[t1 ran], 1) iff e ran

<M> / \

t1 ran every cow

<M>

<M>

polarity<M>



TARSKI(e, Polarity[t1 ran], 1)A*:A*≈1A{Satisfies(A*, Polarity[t1 ran]) & (e = A*[1])}

<M> / \

t1 ran every cow

<M>

<M>

polarity<M>


applies to e iff e ran

we can define a “Tarski Relation” such that TARSKI(e, Polarity[t1 ran], 1) iff e ran

syncategorematic,but honestly so


But maybe…




the Xs ran

Lots of Further Issues

• Quantification Homework (including conservativity)

• Mary saw John, and John didn’t see Mary

• lexicalization of singular and relational concepts

• lexical inflexibilities

*Chris sneezed the baby

*Chris put the letter

• Gleitman-esque acquisition of verbs

• Your Objection Here

Human Languages


• generatively pair meanings with gestures in accord with human constraints




so is <α, β>

Human i-Languages




so is <α, β>

Human i-Languages

Level-n Fregean Languages with expression types:

<e>, <t>, and the nonbasic types up to Level-n

Pseudo-Fregean Languages with expression types:

<e>, <t>, and a few of the nonbasic types

Semantic Typology for Human I-Languages

THANKS!


But maybe...

• DET(_, _) applies to <the Xs, the Ys> only if the Xs are among the Ys

• EVERY(_, _) applies to <the Xs, the Ys> only if the Xs are the Ys

• MOST(_, _) applies to <the Xs, the Ys> only if #(X) > #(Y) − #(X)





FAST(_)^BROWN(_)^HORSE(_) ON(_, _)^HORSE(_)

FAST(e) & BROWN(e) & HORSE(e) e[ON(e’, e)^HORSE(e)]

HORSE(e)s[EXEMPLIFIES(e, s) & HORSEY(s)] & i[AT(s, i) & NOW(i)]


Maybe None

<t> / \

<t> <t, t>Mary saw John / \

<t, <t, t>> <t> and John saw Mary

<M> / \

<M> <M>Mary saw John / \

<D> <M> before John saw Mary

[Before(_, _)^JohnSeeMary(_)]

Past(_)^MarySeeJohn(_) & …

<M> / \

<M> <M>Mary saw John / \

<e, M> <M> before John saw Mary

and

f[Before(e, f) & f is an event of John seeing Mary]

e was an event of Mary seeing John & …

semantic typology for human i-languages paul m. pietroski, university of maryland dept. of...

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