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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.* \ |