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title: 邏輯語意學 Introduction to Logical Semantics (HSGF30009)

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# 邏輯語意學 Introduction to Logical Semantics (HSGF30009)
:::info
Disclaimer:
These notes are not used for commercial purposes, and are only to serve as a personal study guide. The contents of these notes are compiled and rearranged from various sources.
:::
Course Overview:
> This course aims to introduce the basic tools that semanticists use to analyze meaning: set theory, functions, compositional interpretation, predicate logic, type theory, lambda calculus, and explores the interface between syntactic and semantic representations. We will see how the tools are used in semantic treatments of some natural language phenomena, including: natural language quantifiers, plurals and kind reference, events and event structure, and tense and aspect.

Grading:
* Weekly Homework: 30% (only the top 6 will be counted)
* Mid-term: 30%
* Final: 40%

Lesson Plan:
* What meaning could be
* Compositionality and predication
* Compositionality from a formal point of view
* Type theory and lambda calculus
* Modification
* Reference: definite, proper names and pronouns
* Plurals and kind reference
* Generalized quantifier
* Events and event structure
## Lecture 1: Intro
### What this study unit is about?
* We are concerned with the question of meaning.
* Primary topics that will be focused on:
    * Predication (proposition, entity)
    * Reference
    * Quantification (scope)
    * Tense (時制), aspect (動貌), and modality (跟助動詞有關)

### Why *formal*?
> The sentence *snow is white* is true iff snow is white.

The above sentence makes a statement about a sentence. It is what we call a ==meta-sentence==. However, when we talk about meaning, we need a meta-language to avoid confusion as above. Thus, we use ==the language of logic==.

### Richard Montague
* origin of formal semantics: Russel (1905), On Denoting
* logician in the 1970s
* made use of formal tools of logic and mathematics to analyse meaning
* approach was explicitly non-psychological (in agreement with Frege; as opposed to Noah Chomsky who proposed a psychological approach)
:::spoiler Frege
Frege points out human beings require language in order to become conscious of a thought. And this, together with the fact that ordinary language – the kind of language in which human beings normally learn to think – is shaped also by other, less rational aspects of human nature, explains for Frege why actual human thought (as opposed to the bare content of pure thought, or that at which an act of thinking, as part of human consciousness, has to aim in order to be a thought at all) is prone to impurity through the influence of the language in which it is normally clad.
:::

### What is a "meaning"?
Three possible definitions:
#### Theory 1: Meaning Holism:
:::spoiler Holism: 
The theory that parts of a whole are in intimate interconnection, such that they cannot exist independently of the whole, or cannot be understood without reference to the whole
:::

##### The Classic Statement
> Our statements about the external world face the tribunal of sense experience not individually but only as a corporate body.
> —W.V.O. Quine, 1953

* We can think of language (and other linguistic practices) as a dense network of interconnected meanings.
* The meaning of an expression is determined on the basis of its position within this network.

##### A Definition
* Meaning holism is that the meaning of a linguistic expression (word, sentence) is not determined in isolation, but in relation to many other expressions (words, sentences).
    * EX. To know what *man* means, one must know what *human* and *biped* mean, and...
    * EX. To understand *Force is mass times velocity*, one must understand a whole lot of other statements.

##### A Motivating Example
> Force is mass times velocity

* This is a scientific statement which is not usually learned on its own.
    * It only makes sense within a theory (= an entire body of related statements)
* To understand *force*, we must also know the terms *mass* and *velocity* within the theory.
* Similarily, words like *tall* tend to be understood (or explained) in relations to words like *short*.

##### Problems
* If meaning is defined in terms of "relations to other linguistic units", then we run into problems: 
    
    Can we have scientific semantics?
    * Which relations count and which do not?
        (There is the possibility of no consensus.)
    * Do we really want to admit arbitrary meaning relations into our theory?
        (We might not understand the relationship.)
        
#### Theory 2: Meaning as a Social Habit (or game)
:::spoiler more
This is tied closely with pragmatics.
:::

##### The Classic Statement
> For a large class of cases—though not for all—in which we employ the word "meaning" it can be defined thus: the meaning of a word is its use in the language.
> —Wittgenstein, 1953: par. 43

##### An Example
* What does "It's raining" mean?
* Below are some possible ways it is used:
    * Statement of fact:
        *I hereby inform you that it's raining.*
    * Query/request:
        *Please give me an umbrella, shut the window,...*
    * Counter-proposal:
        *I don't want to go out.*
    * ...
* Our knowledge of the meaning of the sentence is our knowledge of how it's used or what to do in response to it, in the various situations in which it's used

##### Problems
* This theory suggests that languages are (sets of) arbitrary conventions.
    * Contemporary linguistic theory suggests that there are some universals that constrain the types of languages humans use
* It is difficult to see why some responses to some messages are appropriate, while others are not.
    * EX. Why is *It's 10 pm* appropriate as a reply for *What's the time?*
    * EX. Why don't we arbitrarily assume a convention that motivates the reply *I like sugar?*

#### Theory 3: Meaning as a Concept/Idea
##### A Characterisation
A relation depicted below holds between linguistic expressions, our minds, and the world: 
![](https://i.imgur.com/2imZCkQ.jpg)
##### The Question
* We will not dispoute the fact that we have some mental representation of the conceptual content of expressions.
* The question is: is this what we mean by "meaning"?

##### The Twin Earth Chronicles (Putnam, 1975)
* Earth and Twin Earth:
    * They are 100% identical in all aspects, except that:
        * *Water* on Earth = H₂O
        * *Water* on Twin Earth = XYZ
        * Both variations of water have the same name, look , taste, and feel the same; however, they are inherently different substances
    * When you and your Twin Earth counterpart way *water*, you are able to identify the substance on your respective planets.
##### Putnam's Conclusion
> Cut the pie any way you like, meaning just ain't in the head!
> —Putnam, 1975

* Thinking of meaning purely in mental terms might not be that useful if we want an objective theory.
* The crucial component of meaning:
    * When we use an expression, we **intend** to mean certain things in the world.
    * The crucial thing might be the things in the world that we mean (this does not apply to abstract nouns).

##### The Conclusion
* From a semantic point of view, we skip over mental representation when we refer to meaning.
    * However, we do not deny the existence of mental representation.
#### An Alternative View—*meaning* is "out there"
* Suppose we say that:
    The meaning of *dog* is whatever that describes all the things that are really dogs, out there in the world.
* Notice that we're thinking of meaning as independent of our own knowledge.
    (It doesn't metter if we don't know how to formulate the meaning itself—it exists independently of us)
##### Sentences, Worlds, and Truth Conditions
* Knowing the meaning of a sentence involves (at least) knowing the conditions under which that sentence is true
    * Just like knowing the meaning of *dog* involves knowing what things in the world are *dogs*...
* ==Take notice that this does not mean knowing the meaning of a sentence means knowing whether it is true or false.==

##### Possible Worlds and the Meaning of Sentences
* When we talk about sentence meaning, we're interested in propositions.
* A proposition can be equated with the set of worlds in which that proposition is true.
    &rarr; In other words, a sentence (proposition) describes a set of worlds in which it is true.

##### Three Arguments For Possible Worlds
Thus...
1. We can define the meaning of "logical" words like *and*, *or*, and *not*
2. We can give a precise account of sentential relationships like synonymy and entailment
3. We can deal with some aspects of human action and gency in a rational way

##### The Meaning of Logical Connectives

| $p$ | $q$ | $p$ $\land$ $q$ |
|:---:|:---:|:---------------:|
|  F  | F   | F               |
|  F  | T   | F               |
|  T  | F   | F               |
|  T  | T   | T               |

Intuitively, the meaning of p$\land$q is the intersection of the set of worlds in which $p$ is true.

##### Semantic Relations Between Propositions: Synonymy
* $p$ and $q$ are synonymous if they are true in exactly the same set of worlds.
    EX. 
    $p$: *the circle is larger than the square*
    $q$: *the square is smaller than the circle*
* It is easy to see that these propositions are true in exactly the same worlds (and false in exactly the same)

##### Semantic Relations Between Propositions: Entailment
* $p$ entails $q$ if $q$ is true whenever $p$ is true.
    EX. 
    $p$: *the circle is inside the square*
    $q$: *the circle is smaller than the square*
    &rarr; if $p$ is true, then $q$ must be true
    
##### Contrary and Contradictory
| contrary | contradictory |
|:---:|:---:|
|  both can't be true at the same time  |  both can't be true at the same time  |
|  both can be false at the same time  |  both can't be false at the same time  |

&rarr; When two statements are contradictory, one of them must be true and the other false.

#### Meaning and Human Action
* The primary purpose of language:
    * To pass on information
    * Other uses
* The reception of information impacts our beliefs about the world. We can: 
    * reformulate (or even discard) beliefs
    * form new beliefs
    * confirm old beliefs

Also, human action is founded on (rational) belief.

##### Questions
* Questions are requests for information. The form of the question hints at what sort of information ought to be given.
* Suppose we think of a question as meaning the set of propositions that are possible answers to the question.
    For example, if there is a question *What did you have for breakfast?*, it prompts the following answers:
    * I had (insert food)

##### Orders
* Recall that as a sentence allows us to categorize worlds into those where the sentence is true and those where it's false
* We could think of imperatives as categorizing worlds into those where the order is carried out and those where it isn't

### From Worlds to Models
#### Propositions as Functions