semantic typology for human i-languages paul m. pietroski, university of maryland dept. of...
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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
• 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
<t>
SADIE<e> IS-A-
HORSE(_)<e, t>
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
<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
• acquirable by normal human children given ordinary courses of experience
• pair unboundedly many “meanings”with unboundedly many pronunciations
• how many types of meanings (basic or not)?
• 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>
<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
• acquirable by normal human children given ordinary courses of experience
• pair unboundedly many “meanings”with unboundedly many pronunciations
• how many types of meanings (basic or not)?
• another answer: <M> monadic predicates
<D> dyadic predicates
possible languagesHuman
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(_)
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
Φ(_)^Ψ(_) applies to e iff Φ(_) applies to e and Ψ(_) applies to e
ON(_, _) + HORSE(_) [ON(_, _)^HORSE(_)]
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
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(_)]
Monad + Monad Monad Dyad + Monad Monad
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
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 ‘’
Monad + Monad Monad Dyad + Monad Monad
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
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)
Monad + Monad Monad Dyad + Monad Monad
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
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
possible languagesHuman
Languages
function applicationand
(type-shifting or) a rule for
adjunction
Human Languages
• acquirable by normal human children given ordinary courses of experience
• 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 _.
possible languagesHuman
Languages
ambiguous unambiguous unambiguous
(the other way)
Human Languages
• acquirable by normal human children given ordinary courses of experience
• pair unboundedly many “meanings”with unboundedly many pronunciationsin accord with language-specific constraints
hiker lost kept walking circles
possible languagesHuman
Languages
Human Languages
• acquirable by normal human children given ordinary courses of experience
• pair unboundedly many “meanings”with unboundedly many pronunciationsin accord with language-specific constraints
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)
possible languagesHuman
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)
possible languagesHuman
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)]
possible languagesHuman
Languages
Human Languages
• acquirable by children • unbounded but constrained• biologically implemented procedures that pair
“meanings” with sounds/gestures in a human way
• how many types of meanings?
• one hypothesis, via Frege on ideal languages:
<e> entity-denoters<t> truth-evaluable sentences
if <α> and <β> are types, then so is <α, β>
possible languagesHuman
Languages
Human Languages
• natural generative procedures
• how many types of meanings?
• one hypothesis, via Frege on ideal languages:
<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>
possible languagesHuman
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>
possible languagesHuman
Languages
Human Languages
• natural generative procedures
• how many types of meanings?
• one hypothesis, via Frege on ideal languages:
<e> <t> if <α> and <β>, then <α, β>
• one worry…overgeneration
<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> ???
possible languagesHuman
Languages
x[X(x) v Y(x) v Z(x)]x[X(x)] & x[Y(x) & Z(x)]
Human Languages
• natural generative procedures
• how many types of meanings?
• one hypothesis, via Frege on ideal languages:
<e> <t> if <α> and <β>, then <α, β>
• one worry…overgeneration
<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)
possible languagesHuman
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
• natural generative procedures
• how many types of meanings?
• one hypothesis, via Frege on ideal languages:
<e> <t> if <α> and <β>, then <α, β>
• THREE CONCERNS: it seems 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>
possible languagesHuman
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>
Which expressions are of type <e>?
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(_)]^…
Which expressions are of type <e>?
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
• natural generative procedures
• how many types of meanings?
• one hypothesis, via Frege on ideal languages:
<e> <t> if <α> and <β>, then <α, β>
• THREE CONCERNS: it seems 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>
possible languagesHuman
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
Which expressions are of type <t>?
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
Which expressions are of type <t>?
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
Why think TPs are of type <t> instead of <e, t> ?
Is this expression of type <<e, t>, t> or type <e, t> ?
Which expressions are of type <t>?
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
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
Which expressions are of type <t>?
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
e . T e is in the past & …
<e, t> e . T e is in the past
Which expressions are of type <t>?
?
/ \ 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
e . T e is in the past & …
<e, t> e . T e is in the past
??neg
Which expressions are of type <t>?
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
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?
Which expressions are of type <t>?
SENTENCES S = TP (or TP+)
V(P) / \ D(P) V(P) That / \
D(P) V(P) John / \
V D(P) see Mary
e . T e is (tenselessly) an event of John seeing 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…
Which expressions are of type <t>?
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
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)
Which expressions are of type <t>?
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
Human Quantification: Still Puzzling
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
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
<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>
for any assignment A of values to variables…
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>
for any assignment A of values to variables…
applies to e iff A1 ran
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>
for any assignment A of values to variables…
applies to e iff A1 ran
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>
for any assignment A of values to variables…
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
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
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
• acquirable by normal human children given ordinary courses of experience
• generatively pair meanings with gestures in accord with human constraints
possible languagesFregean Languages
with expression types: <e>, <t>, and if
<α> and <β> are types,
so is <α, β>
Human i-Languages
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
Semantic Typology for Human I-Languages
THANKS!
Human Quantification: Still Puzzling
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)
Monad + Monad Monad Dyad + Monad Monad
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
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)]
Which expressions are of type <t>?
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 & …