307 discussion 10a (2) (1)

.pptx

School

University of Illinois, Urbana Champaign *

*We aren’t endorsed by this school

Course

307

Subject

Linguistics

Date

Oct 30, 2023

Type

pptx

Pages

Uploaded by ElderSeaLion3284

LING 307: ELMNTS SEMANTICS & PRAGMATICS TOPIC 10, PART 1: GENERALIZED QUANTIFIERS DISCUSSION SECTIONS 04/14/2023, 9:00-9:50

REVIEW: Quantifiers • ∀ , are not enough to represent all English quantfers: ∃ Most dogs bark. • Mx[DOG(x) → BARK(x)] - doesn’t work; will come out true whenever there are more non-dogs than dogs • Mx[DOG(x) & BARK(x)] - doesn’t work; means that more than half of all things are dogs, and bark • Intuitvely, Most dogs bark involves comparing the size of two sets : the set of dogs that bark , and the set of dogs that don’t bark . • If the set of dogs that bark has more members, the sentence is true. • Other quantfers also can be analyzed as expressing relatons between sets ( every, some, no , etc.)

REVIEW: Set theory • Set – well-defned collecton of distnct objects • The identty of a set is determined solely by the identtes of its members. • Once you know which things are members of a given set, you know which set it is • There cannot be two different sets with the same members. • a A ∈ – object a is a member of set A • A B ⊆ – A is a subset of B [every member of a set A is also a member of a set B] • Singleton set – a set containing just one member • ∅ – the empty/null set [a set with no members]

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

REVIEW: Set theory • A ∩ B – intersection of A and B [the set of things which are members of both A and B]  If A and B have no common members, their intersecton is ∅ • A B ∪ – union of A and B [the set of things which are members of A or of B ] • A – B – complement of B relatve to A [the set of things that are members of A, but not of B] • |A| – cardinality of set A [the number of members in the set] • {x: …x…} – set of things meeting a condition : {x : HAPPY(x)} – the set of happy things {y : ADMIRE(y, m)} – the set of things that admire Mary

REVIEW: Quantifiers as relations between sets Every mongoose is hungry. • The common noun and the VP both correspond to sets: mongoose {x : MONGOOSE(x)} VP ( hungry ) {x : HUNGRY(x)} [EVERY x : MONGOOSE(x)] HUNGRY(x)  The determiner indicates that these 2 sets relate to each other in a certain way: every – 2-place predicate which takes sets as its arguments  For any sets A, B: EVERY(A,B) if A B ⊆

REVIEW: Quantifiers as relations between sets For any sets A, B: EVERY(A,B) iff A B ⊆ Every mongoose is hungry. NO(A,B) iff A ∩ B = ∅ No mongooses are hungry. TWO(A,B) iff |A ∩ B| ≥ 2 Two mongooses are hungry. Some determiners are vague : they impose conditons which can only be stated approximately [ several, many, few, a few , etc.] SEVERAL(A,B) iff |A∩B| s ⪆ (where s is the number provided by the pragmatc context for several )

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

PRACTICE: Quantifiers as relations between sets EVERY(A,B) iff A B ⊆ TWO(A,B) iff |A ∩ B| ≥ 2 NO(A,B) iff A ∩ B = ∅ SEVERAL(A,B) iff |A∩B| s (s is provided by the pragmatc context for ⪆ several ) MANY(A,B) iff |A ∩ B| m, m is provided by pragmatc context for "many" ⪆ THE(A,B) iff |A| = 1 & A B ⊆ FEW(A,B) iff |A ∩ B| f ⪅ f is provided by pragmatc context for "few"

PRACTICE: Quantifiers as relations between sets EVERY(A,B) iff A B ⊆ TWO(A,B) iff |A ∩ B| ≥ 2 NO(A,B) iff A ∩ B = ∅ SEVERAL(A,B) iff |A∩B| s (s is provided by the pragmatc context for ⪆ several ) SOME(A,B) iff |A ∩ B| ≠ ∅ MOST(A,B) iff |A ∩ B| > |A – B| BOTH(A,B) iff |A|= 2 & A B ⊆ ALL BUT ONE(A,B) iff |A - B| = 1

REVIEW: Restricted quantifier notation • Treatng quantifiers as 2-place predicates of sets is basically equivalent to treatng them as variable-binding operators , if the operators are restricted so that the variables can only take values from a specifc set • Formulas in restricted quantfer notaton: [Qx : …x…] …x… • Q – quantfer; x – variable; …x… – formula containing the variable [EVERY x : DOG(x)] BARK(x) • A formula of the form [EVERY x : p] q as true iff every constant which renders p true when substtuted for the free occurrences x in p also makes q true when substtuted for the free occurrences of x in q John likes several dogs that Mary hates. [SEVERAL x : DOG(x) & HATE(m, x)] LIKE(j, x)

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

PRACTICE: Restricted quantifier notation • Formulas in restricted quantfer notaton: [Qx : …x…] …x… Every dog barks. [EVERY x : DOG(x)] BARK(x) John likes several dogs that Mary hates. [SEVERAL x : DOG(x) & HATE(m, x)] LIKE(j, x) Some fool admires Oswald. [SOME x: FOOL(x)] ADMIRE(x,o) Most students in LING307 study. [MOST x: STUDENT(x) & IN(x,l)] STUDY(x) Mary has three dogs. [THREE x: DOG(x)] HAVE(m,x) Few vicious dogs like Mary. [FEW x: VICIOUS(x) & DOG(x)] LIKE(x,m) No student who studies fails. [NO x: STUDENT(x) & STUDY(x)] FAIL(x) John sees the desk. [THE x: DESK(x)] SEE(j,x)

REVIEW: Conservativity • All determiners represent conservative relatons, where R is conservative if for all A, B: R(A, B) if R(A, A ∩ B). • If R is conservatve, we can tell whether A and B stand in the relaton just by looking at A and A ∩ B. • We can ignore the porton of B that does not overlap with A. • We can check whether a determiner D represents a conservatve relaton by checking whether sentences of the form D A’s are A’s that are B Many dogs are vicious . ⇔ Many dogs are dogs that are vicious No dog is a stockbroker ⇔ No dog is a dog that is a stockbroker. Only dogs bark != Only dogs are dogs that bark.  Only is not a determiner.