From 8f7194de7767f8cede241682a368d01bcda66abc Mon Sep 17 00:00:00 2001
From: JJ
Date: Sun, 5 May 2024 17:02:30 -0700
Subject: meow

---
 linguistics/semantics.md | 11 ++++++++---
 1 file changed, 8 insertions(+), 3 deletions(-)

diff --git a/linguistics/semantics.md b/linguistics/semantics.md
index a590de3..96638aa 100644
--- a/linguistics/semantics.md
+++ b/linguistics/semantics.md
@@ -20,6 +20,11 @@ This framework is [first-order/predicate logic](../mathematics/logic) and the [s
 
 ### 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).
+
+### 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.
 
 ### Predicate Logic & The Lambda Calculus
 
@@ -37,8 +42,8 @@ With basic logic and the lambda calculus under our belt, we may simply get strai
 > *Alice is bad.* <br>
 > *The blue pigeon flew away.*
 
-- Noun: $⟨e,t⟩ ↝ λx.Noun(x)$
-- Verb (intransitive): $⟨e,t⟩ ↝ λx.Verb(x)$
+- Noun $↝ λx.Noun(x)$ $: ⟨e,t⟩$
+- Verb (intransitive) ↝ $λx.Verb(x)$: $⟨e,t⟩$
 - Verb (transitive): $⟨e,⟨e,t⟩⟩ ↝ λy.λx.Verb(x, y)$
 - Verb (meaningless): $⟨⟨e,t⟩,⟨e,t⟩⟩ ↝ λP.λx.P(x)$
 - Adj: $⟨⟨e,t⟩,⟨e,t⟩⟩ ↝ λNoun.λx.[Adj(x) ∧ Noun(x)]$
@@ -59,7 +64,7 @@ With basic logic and the lambda calculus under our belt, we may simply get strai
 - 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 $λP.∀x.¬P(x))$
+  - nothing: $⟨⟨e,t⟩,t⟩ ↝ λP.¬∃x.P(x)$ (or $λP.∀x.¬P(x)$)
 
 ### Reference
 
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