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Held by Ryan Ruoshui Yan.

Here is the world of a nerdy guy.

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Disclaimer: The notes below are fully/partially NOT created by myself. They are from slides and/or wikipedia and/or textbook. The purpose of this post is simply to learn and review for the course. If you think something is inappropriate, please contact me at “ryan_yrs [at] hotmail [dot] com” immediately and I will remove it as soon as possible.

Truth-Condition 真值條件

• To know meaning of a declarative sentence is to know conditions in which it is true.

Entailment 蘊涵

• p entails q iff every situation in which p is true is a situation in which q is true.
• p => q
• Note: Even if p entails q, there may be situations in which q is true and p is false.

• When 2 sentences cannot be true in same situation, they contradict each other.
• p entails q iff p & ¬q are contradictory.

#

It indicates following sentence is semantically unacceptable.

Literal & Non-Literal Meaning （非）字面意思

• 2 forms of non-literal meaning of a sentence are:
• Implicature
• Metaphor
• Not same as truth-conditions, which can have both literal and non-literal meaning.

Implicature 話裡有話

• Non-literal meaning.
• A type of inference.
• Conclusion reached by reasoning about literal meaning of a sentence.
• Can be denied without creating a contradiction.

Metaphor 隱喻

• Literal meaning of a phrase is used as a symbol for something else.

Presupposition 預設

• Sometimes, when speakers utter certain sentences, they make presuppositions.
• A piece of info that is presented as taken for granted when a speaker utters a sentence.
• It behaves differently in NON-VERIDICAL env.
• When putting a clause in a non-veridical env, its entailments are trapped in there. They are not entailments of whole sentence.
• Projected over scope of negation.
• Presuppositions of a sentence are also entailments. But it may NOT.

Negation

• Non-veridical
• If negating a sentence, it loses its entailments, but not PRESUPPOSITION, which is projected over scope of negation.
• e.g. - Did he stop smoking? - He used to smoke.

Speech Act

• Assertion
• Present sentence as being a true and relevant piece of info.
• Give orders
• Make promises

Set: Basic ({})

• A collection of objects
• {}
• Using curly braces
• List notation for sets
• List notation is called an extensional definition
• Heterogeneous
• ORDER does not matter.
• {a, b} = {b, a}
• REPETITION does not matter.
• {a, b, b} = {a ,b}

Set: Membership (∈, ∉)

• A relation between a set and something that is a member of this set.
• Symbol: ∈ (“is a member of”)
• e.g. 3 ∈ {1, 2, 3}
• Symbol: ∉ (“is not a member of”)
• e.g. 4 ∉ {1, 2, 3}

Set: Predicate (:)

• {x: x is a natural number}
• Set of all natural numbers
• Set of every x such that x is a natural number
• {x: 3 is a natural number}
• Set of everything
• {x: pi is a natural number}
• Set has no members.
• {x is a natural number}
• Only member of set is formula “x is a natural number”.

Set: Identity (=)

• 2 sets are identical iff they have same members
• e.g.
• {1, 2} = {2, 1}
• {1, 2} = {1, 2}

Set: Inclusion (⊆, ⊈, ⊂)

• Let S_1 & S_2 be 2 sets
• S_1 is included in S_2 iff EVERY member of S_1 is a member of S_2.
• If S_1 is included in S_2, then S_1 is a subset of S_2.
• Note: if A = B, then A ⊆ B.
• Note: A = B iff (A ⊆ B and B ⊆ A)
• If A is included in B but A is not identical to B, then A is PROPERLY INCLUDED (⊂) in B.
• If A ⊆ B, then
• A is a subset of B.
• B is a superset of A.
• If A ⊂ B, then
• A is a PROPER subset of B.
• B is a PROPER superset of A.

Set: Union (⋃)

• Union of 2 sets A and B is set that contains every element that is either in A OR B, or both.
• A⋃B = {x: x∈A or x∈B}
• e.g.
• A = {1,2,3}
• B = {3,4,5}
• then A⋃B = {1,2,3,4,5}

Set: Intersection (⋂)

• Intersection fo 2 sets A and B is set that contains every element that is both in A AND B.
• A⋂B = {x: x∈A and x∈B}
• e.g.
• A = {1,2,3}
• B = {3,4,5}
• A⋂B = {3}

Set: Difference (-)

• Diff btw a set A and B is set that contains ANY element in A that is NOT in B.
• A - B = {X: x∈A and x∉B}
• e.g.
• A = {1,2,3}
• B = {3,4,5}
• A - B = {1,2}

Set: Empty (∅)

• Empty set is set that hes no member
• IMPORTANT: empty is NOT “nothing”, it is a set.
• ∅ = {}

Meaning of Expression (⟦⟧)

• Meaning of common nouns
• ⟦dog⟧ = {x: x is a dog}
• ⟦dog⟧ = set of all dogs
• Meaning of modifiers
• ⟦cute⟧ = {x: x is cute}
• ⟦who smokes⟧ = {x: x smokes}
• e.g.
• ⟦cute dog⟧
• = ⟦cute⟧ ⋂ ⟦dog⟧
• = {x: x is cute} ⋂ {x: x is a dog}
• = {x: x is cute and x is a dog}
• ⟦linguist who smokes⟧
• = ⟦linguist⟧ ⋂ ⟦who smokes⟧
• = {x: x is a linguist} ⋂ {x: x smokes}
• = {x: x is a linguist and x smokes}
• IMPORTANT: It does not work for ANTI-INTERSECTIVE (PRIVATIVE) adjectives.

• If A is an anti-intersective adjective and N is a noun
• then ⟦A N⟧ ⋂ ⟦N⟧ = ∅
• e.g.
• ⟦fake diamond⟧ ⋂ ⟦diamond⟧ = ∅
• ⟦counterfeit Rembrandt⟧ ⋂ ⟦Rembrandt⟧ = ∅
• Adjs like “fake” and “counterfeit” are anti-intersective.

Subset & Superset Inferences

• e.g.
1. If Mary bought chocolate, it is in the freezer.
2. If Mary bought milk chocolate, it is in the freezer.
3. 1 entails 2, but not opposite (To subset). Antecedent.
• e.g.
• If Mary went shopping, she bought chocolate.
• If Mary went shopping, she bought milk chocolate.
1. 2 entails 1, but not opposite (To superset). Consequent.

DE (Downward Entailing)

• Antecedent of a conditional is a DE env.
• An env X_Y is DE iff:
• If A ⊆ B, then XBY entails XAY. (To subset)

UE (Upward Entailing)

• Consequent of a conditional is an UE env.
• An env X_Y is UE iff:
• If A ⊆ B, then XAY entails XBY. (To superset)

Word Analysis: “Any”

• Grammatical in:
• Negative clause
• There isn’t anyone to talk to.
• Antecedents of conditionals
• If we talk to anyone, they will discover our secret.
• Complements of “doubt”
• I doubt that there is anyone to talk to.
• Ungrammatical in:
• Unembedded positive clauses
• *There is anyone to talk to.
• Consequents of conditional
• * If we go home, there is anyone to talk to.
• Complements of “believe”
• I believe that there is anyone to talk to.

DE-ness & UE-ness

• Distribution of “any” if sensitive to DE-ness of its env.
• DE-ness is a logical property of envs.
• Grammar is sensitive to logical properties of sentences.
• This supports our use of truth-conditions and logical concepts like entailments in our study of natural lnaguage semantics.
• “Exactly 1” is neither UE or DE.

Ambiguity

• Lexical
• Police begin campaign to run down jaywalker.
• run down
• knock down
• chase
• Safety experts say school bus passengers should be belted.
• belted
• secured with a belt
• beaten with a belt
• Bank Drive-in Window Blocked by Board.
• board
• committee
• piece of wood
• Bar trying to help alcoholic laywers.
• bar
• tribunal
• drinking establishment
• Structural
• Teacher strikes idle kids.
• (Teacher strikes)//S idle//V kids//O.
• Squad helps dog bite victim.
• (Squad helps dog) bite victim.
• Bill hit the man with the hammer.
• Bill (hit the man) with the hammer.
• Bill hit (the man with the hammer).
• Syntactic (Beyond modification)
• John told the girl that Bill liked the story.
• John told (the girl that Bill liked) the story.
• John told the girl that (Bill liked the story).
• Every student read a book.
• For every student, there is a book that this student read.
• There is a book that all of the students read.

Principle of Modification

• If an expression X (constituent or word) modifies an expression Y, then X must c-command Y.

C-Command (Constituent Command)

• Node A c-commands node B if every node dominating A also dominates B, and neither A nor B dominates the other.

Test of Ambiguity

• Type
• Not ambiguous but context dependent
• I am hungry.
• Diff contexts, diff speakers.
• Not ambiguous but (lexical) under-specification
• I am going to visit my aunt.
• My aunt: My father’s sister, or my mother’s sister.
• Not ambiguous but vague
• John is bald.
• How much hair loss counts as baldness?
• Test
• Truth-Conditions
• If a sentence can both be true and false in same situation, it is AMBIGUOUS.
• John hit the man with the hammer.
• Situation: hammer was used as an instrument.
• Sentence is true in 1 reading, false in another.
• Thus AMBIGUOUS.
• I am hungry
• No matter who speaker is, I refers to speaker.
• Thus NOT AMBIGUOUS.
• Ellipsis (?)
• An elided expression must hae same meaning as its antecedent.
• If an elided expression and its antecedent can have diff meanings, diff in meaning is not a case of ambiguity.

SDCH (Scope/Domain Correspondence Hypothesis)

• Semantic scope of a linguistic operator in a sentence corresponds to its c-command domain in some syntactic representation of sentence.

Scope Ambiguity

• John didn’t leave because he was sick.
• Meaning
• John didn’t leave, because he was sick.
• John didn’t (leave because he was sick).
• Expression
• Negation
• Subordinating conjunction “because”
• We need single men or women.
• Meaning
• We need (single men) or women.
• We need single (men or women).

DetQ (Determiner Quantifier)

• Determiner quantifiers are expressions like
• Type
• every
• ⟦Every cat is cute⟧
• T iff ⟦cat⟧ ⊆ ⟦cute⟧
• some
• ⟦Some cats are cute⟧
• T iff ⟦cat⟧ ⋂ ⟦cute⟧ ≠ ∅
• most
• all
• e.g.
• Every child read some book.
• Meaning
1. For every child, there is some book that this child read. (Surface Scope)
2. There is some book such that every child read this book. (Inverse Scope)
• Situations
1. is true in abc
1. is true in bc only

LF (Logical Form)

• In order to keep a strict correspondence between c-command and scope, we assume that there is a level of syntactic representation where quantifiers can be moved according to their scope.
• This level of syntactic representation is called Logical Form.

Logical Form Scope Principle

• At level of LF, an element α has scope over β iff β is in domain of α.
• “Domain” means c-command domain.

Entailments e.g.

• DE
• Every DOG is a pet.
• At most 5 STUDENTS watched the video.
• At most 5 students WATCHED THE VIDEO.
• Every cat who ate BANANAS got sick.
• No CAT got sick.
• No cat GOT SICK.
• UE
• Every dog is A PET.
• Some cat who ate BANANAS got sick.
• Neither
• Exactly one DOG is a pet.
• The CAT ate the canary.

• Lexical
• Syntactical
• Both

Principle of Compositionality 複合性原理

• Meaning of a composite expression is a function of meaning of its immediate constituents and way these constituents are put together.

Reference

• The meaning of the expression X as a relation btw this expression and the real object called X.
• The name X refers to (denotes) the object called X.

Referent (Denotation)

• The object that is called X is the referent (or denotation) of the name X.

Definite Description

• Formed by combining a definite determiner with an NP.
• Possessive phrases are another type of definite descriptions.
• It describes its denotation, while proper names do not.

CN (Common Noun)

• Singular common nouns refer to SETS.
• e.g.
• CAT refers to set of cats
• ⟦X is cute⟧ = 1 iff ⟦X⟧ \in {x: x is a cute INDIVIDUAL}

Word Analysis: “Every”

• ⟦Every boy is happy⟧ = 1 iff ⟦boy⟧ ⊆ ⟦is happy⟧
• ⟦Every NP VP⟧ = 1 iff ⟦NP⟧ ⊆ ⟦VP⟧

Word Analysis: “Some”

• ⟦Some NP VP⟧ = 1 iff ⟦NP⟧ ⋂ ⟦VP⟧ ≠ ∅

Word Analysis: “A(n)”

• ⟦Some NP VP⟧ = 1 iff ⟦NP⟧ ⋂ ⟦VP⟧ ≠ ∅

Relational Noun

• A class of nouns that do not refer to sets of individuals
• e.g.
• mother
• friend
• side
• owner
• birthplace

Functional Noun

• Relations that satisfy the following condition are functions:
• A 2 place relation R is a function iff \forall x,u,v: if both <x,u> \in R and <x,v> \in R, then u=v.

Transitivity

• 0: Intransitive
• 1: Transitive
• 2: Ditransitive

Intransitive Verb

• Like CNs, they denote sets of individuals
• ⟦smokes⟧ = the set of individuals who smoke
• ⟦is sleeping⟧ = the set of individuals who are sleeping
• ⟦was sleeping⟧ = the set of individuals who were sleeping
• ⟦slept⟧ = the set of individuals who slept
• Predication as set membership
• ⟦A is sleeping⟧ = 1 iff ⟦A⟧ \in ⟦is sleeping⟧
• ⟦A is sleeping⟧ = 1 iff ⟦A⟧ \in {x: x is sleeping}

Transitive Verb

• Similar to that of relational nouns.
• A likes B.
• ⟦likes⟧ = {<A, B>, <A, C>, …}
• Order: <SUBJECT, OBJECT>
• ⟦A likes B⟧ = 1 iff <A, B> \in ⟦likes⟧
• Not symmetric
• A likes B but B might not like A.

Ditransitive Verb

• Have triples in its extensions.
• ⟦A gave B to C⟧ 1 iff <A, B, C> \in ⟦gave⟧
• ⟦gave⟧ = {<A, B, C>, <D, E, F>, …}

• Extension
• Intension

Extension

• Denotation of these (Proper Name, CN, etc.) expressions are called their extensions.
• Extension of an expression is its denotation.
• e.g.
• Extension of sentence “A likes B” is the truth value ‘true’.
• Extension of sentence “A does not like B” is the truth value ‘false’.

Review for Midterm

- Truth Condition
- Entailment
- p entails q iff sentence "p and not q" IS contradictory.
- Situation:
- Test
1. p && q
2. p && !q
- If 1 && !2 => p entails q
- If 1 && 2 => p does not entail q
- If !1 => p and q are contradictory
- Implicature
- Cancellation Test (.... Indeed, ...)
- I saw two elks. I didn't see three elks. => I saw two elks. Indeed, I saw three of them. (No contradiction: Implicature)
- I saw two elks. I saw more than one elk. => \*I saw two elks. Indeed, I didn't see more than one elk. (Contradiction: Entailment)
- Presupposition
- Projection Test
- Non-Verdical Environment
- Antecedent of Conditionals
- Negation
- Complement of "Doubt"
- Question
- Inclusion
- ⊆: Inclusion
- ⊂: Proper Inclusion
- Union
- A∪B = {x: x∈A or x∈B}
- Intersection
- A∩B = {x: x∈A and x∈B}
- Difference
- A-B = {x: x∈A and x∉B}
- Ambiguity
- Test
- Truth-Conditions
- Ellipsis
- Constituent Command
- Extension (Set of ...)
- Intension (Concept of ...)
- Set Theory
- \{\{1,2,3,6},{4,5,8\}\} - {4,5,8} = \{\{1,2,3,6},{4,5,8\}\}
- \{\{1,2,3,6},{4,5,8\}\} - \{\{4,5,8\}\} = \{\{1,2,3,6\}\}
- {1,2,3,4,5,6} - {4,5,8} = {1,2,3,6}
- \{\{\{2,6\}\}\}
- \{\{8},\{\{2,6\}\}\} - {2,6,{8\}\} = \{\{\{2,6\}\}\}
- \{\{8},\{\{2,6\}\}\} - \{\{2,6},{8\}\} = \{\{\{2,6\}\}\}
- \{\{8},\{\{2,6\}\}\} - \{\{\{2,6\}\},{8\}\} = empty


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