diff options
Diffstat (limited to 'ling/semantics.md')
| -rw-r--r-- | ling/semantics.md | 89 |
1 files changed, 89 insertions, 0 deletions
diff --git a/ling/semantics.md b/ling/semantics.md new file mode 100644 index 0000000..747d9c3 --- /dev/null +++ b/ling/semantics.md @@ -0,0 +1,89 @@ +--- +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](../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*. + + +## Basic Principles + +### 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 + +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](../math/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) |