meow

1 files changed, 120 insertions, 31 deletions

diff --git a/ling/semantics.md b/ling/semantics.md
index 656cf80..23d873d 100644
--- a/ling/semantics.md
+++ b/ling/semantics.md
@@ -20,14 +20,16 @@ What *conditions* must hold for a sentence to be true?
   - [Compositionality](#compositionality)
   - [Substitution](#substitution)
   - [Higher-Order Logic & The Lambda Calculus](#higher-order-logic-the-lambda-calculus)
-  - Models and Denotation
+  - [Types](#types)
+  - Models
 - Denotational Semantics
   - Entities and Functions
   - Quantification
+  - Negation
+  - Coordination
   - Reference
   - Numbers and Plurality
-  - Event Semantics
-  - Situation Semantics
+  - Event and Situation Semantics
 - Possible Worlds
   - Necessity and Possibility
   - Knowledge and Belief
@@ -79,9 +81,9 @@ Comfortability with the following concepts will be assumed:
 - higher-order logic
 - intuitionistic logic
 
-If this is not the case, there are a variety of wonderful resources for learning such topics. I am partial to *An Introduction to Non-Standard Logics* myself, and think it gives a good, operational, syntactic motivation for possible worlds and accessibility relations. I have heard praise for *Boxes and Diamonds* (which is free and open!) but have yet to read it myself. Wikipedia is also a wonderful reference. Best of all, however, is finding yourself a friend who is a nerd about logic! (thanks alex)
+If this is not the case, there are a variety of wonderful resources for learning such topics. I am partial to [*An Introduction to Non-Standard Logics*](https://annas-archive.org/md5/21cdfde7ee1a125c1bfe6d03d4541970) myself, and think it gives a good, operational, syntactic motivation for possible worlds and accessibility relations. I have heard much praise for [*Boxes and Diamonds*](https://bd.openlogicproject.org/) (which is free and open!) but have yet to read it myself. Wikipedia is also a wonderful reference. Best of all, however, is finding yourself a friend who is a nerd about logic!
 
-These notes chart a standard course through undergraduate/graduate semantics that is taken by the canonical texts: *Semantics in Generative Grammar* and *Invitation to Formal Semantics* at the undergraduate level, and *Intensional Semantics* and *Logic, Language, and Meaning* at the graduate level. I strongly recommend *Invitation to Formal Semantics* for those experienced in and new to logic alike. The beginning chapters give a comfortable introduction to predicate logic and the lambda calculus, which can be skipped easily by those already well versed in formalism.
+These notes chart a standard course through undergraduate/graduate semantics that is taken by the canonical texts: [*Semantics in Generative Grammar*](https://annas-archive.org/md5/2d9c2174690df454700fedcd4a9b237c) and [*Invitation to Formal Semantics*](https://eecoppock.info/bootcamp/semantics-boot-camp.pdf) at the undergraduate level, and [*Intensional Semantics*](https://web.mit.edu/fintel/fintel-heim-intensional.pdf) ([source](https://github.com/fintelkai/fintel-heim-intensional-notes)) and [*Logic, Language, and Meaning*](https://annas-archive.org/md5/359c3ff1e391376cb2ac110c4e8a8d71) at the graduate level. I strongly recommend *Invitation to Formal Semantics* for those experienced in and new to logic alike. The beginning chapters give a comfortable introduction to predicate logic and the lambda calculus, which can be skipped easily by those already well versed in formalism.
 
 $$∧ ∨ + × ⊕ ↑ ↓ ∼ ¬ ⇁ → ⇒ ⊃ ⊐ ⥽ > ⊢ ⊨$$
 
@@ -97,6 +99,8 @@ This is important, and not entirely obvious - given how much *context* is wrappe
 
 The *Principle of Substitution* states that substituting one part of an expression with something else of the same meaning *preserves* the meaning of the expression as a whole. This may typically be thought of as a given, but semantics has its roots in philosophy, and philosophers care very much about enumerating their givens. And there are cases in which it is not only not a given, but does not hold entirely! We shall highlight such cases as we come to them.
 
+...
+
 ### Higher-Order Logic & The Lambda Calculus
 
 Formal semantics attempts to answer those questions by providing a *framework* for determining what conditions must hold for a sentence to be true. This framework can be thought of as a meta-language (c.f. meta/object language distinction) for all natural language: a language in which to *describe meaning*.
@@ -107,7 +111,7 @@ Our logic needs to be higher-order as natural language has the need to quantify
 
 [*An Invitation to Formal Semantics*](https://eecoppock.info/bootcamp/semantics-boot-camp.pdf) covers basic logic and the lambda calculus well in its first six chapters. Otherwise, for a worse introduction, see [my notes on logic](../math/logic) and [the lambda calculus](../plt/lambda-calculus).
 
-$$λ\ ∀\ ∃\ ∂\ ☐\ ◇$$
+$$λ\ ∀\ ∃\ ι\ ∂\ ☐\ ◇$$
 
 ### Types
 
@@ -124,58 +128,143 @@ The notion of *entities* is not necessarily straightforward... We shall consider
 
 We also, on occasion, consider the types of *situations* and *events*...
 
+$$e\ t\ ⟨σ,τ⟩\ ⟨s,σ⟩$$
+
+### Models
+
 ## Denotational Semantics
 
-With basic logic and the lambda calculus under our belt, we may simply get straight to assigning *meaning* to language. We consider two *basic types* to start: the type of entities, $e$, and the type of truth values, $t$. Our function types we denote by ordered pairs: that is, a function from $e$ to $t$ is of type $⟨e,t⟩$. This is perhaps clunkier notation than the type-theoretic $e→t$, but it is what it is. (And does avoid issues of precedence.)
+With basic logic and $λ→$ under our belt, we may get straight to assigning *meaning* to language.
 
 ### Entities and Functions
 
 > *I am Alice.* <br>
 > *Alice is pretty.* <br>
-> *The blue pigeon flew away.*
+> *The blue bird flew away.*
+
+How do we even begin to represent basic sentences?
+
+Let's start with something a little simpler.
 
-- Noun: $⟨e,t⟩ ↝ λx.Noun(x)$
-- Verb (intransitive): $⟨e,t⟩ ↝ λx.Verb(x)$
-- Verb (transitive): $⟨e,⟨e,t⟩⟩ ↝ λy.λx.Verb(x, y)$
-- Verb (meaningless): $⟨⟨e,t⟩,⟨e,t⟩⟩ ↝ λP.λx.P(x)$
-- Adj: $⟨⟨e,t⟩,⟨e,t⟩⟩ ↝ λNoun.λx.[Adj(x) ∧ Noun(x)]$
+> *Alice.*
 
-- or (clausal): $⟨t,⟨t,t⟩⟩ ↝ λq.λp.[p ∨ q]$
-- and (clausal): $⟨t,⟨t,t⟩⟩ ↝ λq.λp.[p ∧ q]$
-- or (verbal): $⟨⟨e,t⟩,⟨⟨e,t⟩,⟨e,t⟩⟩⟩ ↝ λQ.λP.λx.[P(x) ∨ Q(x)]$
-- and (verbal): $⟨⟨e,t⟩,⟨⟨e,t⟩,⟨e,t⟩⟩⟩ ↝ λQ.λP.λx.[P(x) ∧ Q(x)]$
-- or (quantifiers): $⟨⟨e,⟨e,t⟩⟩,⟨⟨e,⟨e,t⟩⟩,⟨e,⟨e,t⟩⟩⟩⟩ ↝ λQ.λP.λy.λx.[P(x,y) ∨ Q(x,y)]$
-- and (quantifiers): $⟨⟨e,⟨e,t⟩⟩,⟨⟨e,⟨e,t⟩⟩,⟨e,⟨e,t⟩⟩⟩⟩ ↝ λQ.λP.λy.λx.[P(x,y) ∧ Q(x,y)]$
+We consider *Alice* to be an *entity*.
+- $⟦\text{Alice}⟧^M ↝ Alice_e$
+
+- $⟦\text{Alice is pretty}⟧^M$
+  - $⟦\text{Alice}⟧^M = \text{Alice}_e$
+  - $⟦\text{pretty}⟧^M = \text{pretty} ↝ λx_e \text{pretty}(x)$
+- $⟦\text{Alice is pretty}⟧^M ↝ \text{pretty}(\text{Alice}_e)$
 
-- not: $⟨⟨e,t⟩,⟨e,t⟩⟩ ↝ λP.λx.¬P(x)$
 
 ### Quantification
 
-- every: $⟨⟨e,t⟩,⟨⟨e,t⟩,t⟩⟩ ↝ λQ.λP.∀x.[P(x) → Q(x)]$
-  - everything: $⟨⟨e,t⟩,t⟩ ↝ λP.∀x.P(x)$
-- some: $⟨⟨e,t⟩,⟨⟨e,t⟩,t⟩⟩ ↝ λQ.λP.∃x.[P(x) ∧ Q(x)]$
-  - something: $⟨⟨e,t⟩,t⟩ ↝ λP.∃x.P(x)$
-- no: $⟨⟨e,t⟩,⟨⟨e,t⟩,t⟩⟩ ↝ λQ.λP.∀x.[P(x) → ¬Q(x)] (or λQ.λP.¬∃x.[P(x) ∧ Q(x)])$
-  - nothing: $⟨⟨e,t⟩,t⟩ ↝ λP.¬∃x.P(x)$ (or $λP.∀x.¬P(x)$)
+> *Every bug saw Alice.* \
+> *No bug saw Alice.* \
+
+How might we represent sentences that involve *quantification*?
+
+As it turns out, we must extend our formal system.
+
+- $\text{every} ↝ λQ_{⟨e,t⟩} λP_{⟨e,t⟩} ∀x [P(x) → Q(x)]$
+  - $\text{everything} ↝ λP_{⟨e,t⟩} ∀x P(x)$
+- $\text{some} ↝ λQ_{⟨e,t⟩} λP_{⟨e,t⟩} ∃x [P(x) ∧ Q(x)]$
+  - $\text{something} ↝ λP_{⟨e,t⟩} ∃x P(x)$
+- $\text{no} ↝ λQ_{⟨e,t⟩} λP_{⟨e,t⟩} ∀x [P(x) → ¬Q(x)]$
+  - $↝ λQ_{⟨e,t⟩} λP_{⟨e,t⟩} ∀x [¬P(x) ∨ ¬Q(x)]$
+  - $↝ λQ_{⟨e,t⟩} λP_{⟨e,t⟩} ¬∃x [P(x) ∧ Q(x)]$
+- $\text{nothing} ↝ λP_{⟨e,t⟩} ¬∃x P(x) ↝ λP_{⟨e,t⟩} ∀x ¬P(x)$
+- "except": ↝
+- "many": ↝
+- "three": ↝
+- "most": ↝
+- "few": ↝
+### Negation
+- not: ↝ $λP_{⟨e,t⟩} λx_e.¬P(x)$
+
+### Coordination
+
+talk abt generics
+
+clausal coordination:
+- $\text{or}_C ↝ λq_t λp_t [p ∨ q]$
+- $\text{and}_C ↝ λq_t λp_t [p ∧ q]$
+
+verbal coordination:
+- $\text{or}_V ↝ λQ_{⟨e,t⟩} λP_{⟨e,t⟩} λx_e [P(x) ∨ Q(x)]$
+- $\text{and}_V ↝ λQ_{⟨e,t⟩} λP_{⟨e,t⟩} λx_e [P(x) ∧ Q(x)]$
+
+quantifier coordination:
+- $\text{or}_Q ↝ λQ_{⟨e,⟨e,t⟩⟩} λP_{⟨e,⟨e,t⟩⟩} λy_e λx_e [P(x,y) ∨ Q(x,y)]$
+- $\text{and}_Q ↝ λQ_{⟨e,⟨e,t⟩⟩} λP_{⟨e,⟨e,t⟩⟩} λy_e λx_e [P(x,y) ∧ Q(x,y)]$
 
 ### Reference
 
+co-indexation
+
 ### Numbers and Plurality
 
-### Event Semantics
+having a quantity function
 
-### Tense and Aspect
+### Event and Situation Semantics
+
+what are these??
+
+## Possible Worlds
 
-## Beyond Truth
 
 ### Necessity and Possibility
 
+> *Alice may run.*  \
+> *Alice must run.* \
+> *Alice should run.* \
+> *Alice could run.*
+
+### Knowledge and Belief
+
 ### Command, Request, Obligation
 
 > *Alice, run!* <br>
 > *Alice, please run.* <br>
 > *Alice should run.*
 
+### Strength and Flavour
+
+### Drawing Distinctions
+
+### Accessibility Relations
+
+- $ρ$ reflexivity
+  - $∀x : x∼x$
+- $σ$ symmetry
+  - $∀x,y : x∼y ⇒ y∼x$
+- $τ$ transitivity
+  - $∀x,y,z : x∼y ∧ y∼z ⇒ x∼z$
+- $η$ extensionality
+  - $∀x, ∃y : x∼y$
+
+when do you have these accessibility relations?
+
+### Propositions and Worlds
+
+To quote Kratzer:
+- A proposition $p$ is *true* in a world $w ∈ W$ iff $w ∈ p$.
+- A proposition $p$ *follows* from a set of propositions $P$ iff $p ⊆ ⋂P$.
+- A set of propositions $P$ is *consistent* iff $⋂P ≠ ∅$.
+- A proposition $p$ is *compatible with* a set of propositions $P$ iff $P∪\{p\}$ is consistent.
+
+### Conversational Backgrounds
+
+## Beyond Truth
+
+So far, we have only dealt with sentences that can be, for some notion of truth, considered *true* or *false*.
+
 ### Questions
-## Resources
-- ✨ [Invitation to Formal Semantics](https://eecoppock.info/bootcamp/semantics-boot-camp.pdf)
+
+> *Did Alice run?*
+
+### Tense and Aspect
+
+> *Alice ran.* \
+> *Alice runs.* \
+> *Alice will run.* \