---
layout: linguistics
title: linguistics/semantics
---

# notes on semantics

Semantics is the study of **meaning**.

How do we know what sentences are true and which are false?<br>
What does it *mean* for a sentence to be true?<br>
What conditions must hold for a sentence to be true?

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 is [first-order/predicate logic](../mathematics/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*.


## Basic Principles

### Compositionality


### Predicate Logic & The Lambda Calculus

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?

[*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](../mathematics/logic), and [the lambda calculus](../plt/lambda-calculus).

## 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.)

### Entities and Functions

> *I am Alice.* <br>
> *Alice is bad.* <br>
> *The blue pigeon flew away.*

- 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: $⟨⟨e,t⟩,⟨e,t⟩⟩ ↝ λNoun.λx.[Adj(x) ∧ Noun(x)]$

- 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): $⟨⟨e,⟨e,t⟩⟩,⟨⟨e,⟨e,t⟩⟩,⟨e,⟨e,t⟩⟩⟩⟩ ↝ λQ.λP.λy.λx.[P(x,y) ∧ Q(x,y)]$

- 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 $λP.∀x.¬P(x))$

### Reference

### Numbers and Plurality

### Event Semantics

### Tense and Aspect

## Beyond Truth

### Necessity and Possibility

### Command, Request, Obligation

> *Alice, run!* <br>
> *Alice, please run.* <br>
> *Alice should run.*

### Questions
## Resources
- ✨ [Invitation to Formal Semantics](https://eecoppock.info/bootcamp/semantics-boot-camp.pdf)