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1 files changed, 38 insertions, 12 deletions

diff --git a/ling/semantics.md b/ling/semantics.md
index 7a4fa4a..656cf80 100644
--- a/ling/semantics.md
+++ b/ling/semantics.md
@@ -17,9 +17,10 @@ What *conditions* must hold for a sentence to be true?
 - History
 - Prerequisites
 - Basic Principles
-  - Compositionality
-  - Substitution
-  - Predicate Logic & The Lambda Calculus
+  - [Compositionality](#compositionality)
+  - [Substitution](#substitution)
+  - [Higher-Order Logic & The Lambda Calculus](#higher-order-logic-the-lambda-calculus)
+  - Models and Denotation
 - Denotational Semantics
   - Entities and Functions
   - Quantification
@@ -67,18 +68,20 @@ Modern approaches to semantics largely fell out of historical work in logic...
 Formal semantics builds atop a bevy of concepts in formal logic.
 Comfortability with the following concepts will be assumed:
 - object languages and meta languages
-- zeroth-order/propositional logic
-- first-order/predicate logic
+- propositional / zeroth-order logic
+- predicate / first-order logic
 - the lambda calculus
 - simple types
 - logical models
 - modal logic
   - possible worlds
   - accessibility relations
-- second-order/higher-order logic
+- 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, 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 look at 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* 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)
+
+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.
 
 $$∧ ∨ + × ⊕ ↑ ↓ ∼ ¬ ⇁ → ⇒ ⊃ ⊐ ⥽ > ⊢ ⊨$$
 
@@ -86,17 +89,40 @@ $$∧ ∨ + × ⊕ ↑ ↓ ∼ ¬ ⇁ → ⇒ ⊃ ⊐ ⥽ > ⊢ ⊨$$
 
 ### Compositionality
 
-The *Principle of Compositionality* states that the meaning of a *constituent* is determined entirely by its *components*. This is *the* fundamental underlying principle behind formal logic and subsequently semantics. It holds for not just sentence composition (syntax), but also *word formation* (morphology), and what's of interest to us here - meaning (semantics).
+The *Principle of Compositionality* states that the meaning of a *constituent* is determined **entirely** by its *components*. This is *the* fundamental underlying principle behind formal logic and subsequently semantics. It holds for not just sentence composition (syntax), but also *word formation* (morphology), and what is of interest to us here - meaning (semantics).
+
+This is important, and not entirely obvious - given how much *context* is wrapped up in everyday speech, it might seem like a bit of a stretch to claim that ex. the meaning of a sentence is contained entirely within its parts. And, in many regards, it is. But we must start somewhere. We will address the problem of context when we come to it, but as a sneak peek: we typically consider sentences (and expressions as a whole) with respect to a *model*, which can contain broader context (among other things). In other (more [PLT](/plt)-y) words, we *parametrize* our sentences across domains.
 
 ### Substitution
 
-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 might be thought of as a given, but semantics has its roots in philosophy, and philosophers care very much about enumerating their givens.
+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*.
+
+This language is [higher-order logic](../math/logic) and the [simply-typed lambda calculus](../plt/lambda-calculus). On top of this, we often build set theory, relying on *characteristic functions* of the lambda calculus as denotations of *set membership*. While we could in principle use natural language as our framework - and historically, this has been the case - formal semantics begets a formal system for such semantics. And a formal system has a number of advantages - an unambiguous presentation, cross-linguistic accessibility, computational interpretations... semantics is the study of meaning, and what is logic but a system for expressing meaning?
+
+Our logic needs to be higher-order as natural language has the need to quantify over quite a lot of constructions. Propositional (zeroth-order) logic is too weak, as we may not quantify over any entities. Predicate (first-order logic) is still too weak as while we may quantify over entities, we have no way to quantify over propositions. Thus we need at least second-order logic - which, interestingly enough, is equivalent to any higher nth-order logic (and so is typically just called higher-order logic).
+
+[*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).
+
+$$λ\ ∀\ ∃\ ∂\ ☐\ ◇$$
+
+### Types
+
+Formal semantics has a rather minimal notion of types. To quote *Intensional Semantics*:
+
+> 1. $e$ and $t$ (entities and truth values) are semantic types.
+> 2. If $σ$ and $τ$ are semantic types, then $⟨σ,τ⟩$ is a semantic type.
+> 3. If $σ$ is a semantic type, then $⟨s,σ⟩$ is a semantic type.
+> 4. Nothing else is a semantic type.
 
-### Predicate Logic & The Lambda Calculus
+The semantic product type $⟨σ,τ⟩$ is more similar in usage to the function type $σ → τ$ from programming language theory. Nonetheless, $⟨σ,τ⟩$ is convention. It is fairly common to write such basic types as $στ$ for compactness, though this will be avoided in these notes for explicitness.
 
-Formal semantics begets a formal system for such semantics, and *first-order logic* and *the lambda calculus* are a natural fit. Semantics is the study of meaning - and what is logic but a system for expressing meaning? As discussed above, language functions by composition - and what are functions but their property of composition?
+The notion of *entities* is not necessarily straightforward... We shall consider our truth values to be taken from the classical set $\{0,1\}$ unless otherwise specified.
 
-[*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 [logic](../math/logic), and [the lambda calculus](../plt/lambda-calculus).
+We also, on occasion, consider the types of *situations* and *events*...
 
 ## Denotational Semantics