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Semantics Notes

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.

Contradiction 矛盾

  • 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
  • Ask questions
  • 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.

Anti-Intersective Adjective

  • aka Privative Adjective
  • 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.

Downward & Upward Entailments

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.
      • Teacher//S strikes//V (idle//ADJ kids//N)O.
    • Squad helps dog bite victim.
      • (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
          • 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.

Ambiguity e.g.

  • 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.

Mental Representation 心智表徵

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>, …}

Meaning of Expression

  • 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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