This presentation is the property of its rightful owner.
1 / 25

# Composition and Substitution: Learning about Language from Algebra PowerPoint PPT Presentation

Composition and Substitution: Learning about Language from Algebra. Ken Presting University of North Carolina at Chapel Hill. Introduction. Intensional contexts are defined by substitution failure Johnny heard that Venus is the Morning Star Johnny heard that Venus is Venus

Composition and Substitution: Learning about Language from Algebra

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

## Composition and Substitution:Learning about Language from Algebra

Ken Presting

University of North Carolina at Chapel Hill

### Introduction

• Intensional contexts are defined by substitution failure

• Johnny heard that Venus is the Morning Star

• Johnny heard that Venus is Venus

• Composition accounts for indefinite application of finite knowledge

• ‘p and q’ is a sentence

• ‘p and q and r’ is a sentence

### Role of Recursion

• Syntax

• Atomic symbols

• Combination rules

• Closure principle

• Finiteness

• Limited symbols, rules

• Infinitely many expressions

### CompositionalSemantics

• The usual:

• Choose assignments to atoms

• Forced valuations for molecules

### The Two-Element Boolean Algebra

• The Truth Values

• Just two atomic objects: 2BA = {0, 1}

• Disjunction = max(a, b)

• Conjunction = min(a, b)

• Negation = 1 – a

### It’s almost familiar

• Boolean arithmetic

• 0  1 = 1

• 0  1 = 0

• Boolean algebra

• A  B = C

• (A  B)  ~C = C  ~C

• (A  B)  ~C = 0

### A Homomorphism to 2BA

• Take any old function that labels sentences with 0 or 1.

• For example:

• f(S) = 0

• f(PQ) = 1

• etc.

### A Homomorphism to 2BA

• Ask: Does this function have the ‘distributive’

• a(b + c) = ab + ac

• f(S  P) = f(S)  f(P)

• and ‘commutative’ properties?

• ac = ca

• f(~S) = ~f(S)

### A Homomorphism to 2BA

…is a compositional semantics for propositional calculus

### Sentence Diagrams

• Tree diagrams

• Binary

• Associativity allows n-ary nodes

### Repetition

• Identical Subtrees

• In many sentences, certain letters appear twice or more

• P & Q  P

• Sometimes whole expressions recur

• (P & R)  (P & R)

### Reducing the diagram

• Identify like-labeled leaves

• Identify like-labeled nodes

• Form equivalence classes

• Redraw tree as lattice

• (advanced topics: empty expression as zero; quotient)

### Set Membership Model

• Mapping sentences to sets

• Set of letters = conjunction

• Singleton set = negation

• Associativity

• And vs. Nand

• Naturalness of negation

• Failure of associativity

• Embeddings

• Homomorphism

### Substitution for a Letter

• Single-letter expressions

• Every sentence is a substitution-instance of ‘P’

• Substitution for single letters is easy

• Multiple occurrences of a letter

### Substitution for Expressions

• What do these sentences have in common?

(P & Q) v ~(P & Q)

(T & S) v ~(T & S)

### Subalgebras

• A subalgebra is a subset which follows the same rules as its container

• In our case, that means ‘is also a sentence’

### Quotients

• Ignore specfied details

• In our case, treat a subsentence as a letter

### Sentences as Functions

In Algebra, formulas map numbers to each other

• F(x) = mx + b

• Sentences map the language to itself

• (P v ~P)(Q) = Q v ~Q

• ### Sentences as Functions

• Mapping the language to itself

• Atomic Sentence letters map L to itself

• No other sentence does

• Complex sentences map the language to a subset of itself

### Image of a Sentence

• Image = all the substitution-instances

Image of ‘P v ~P’ is:

Q v ~Q

R v ~R

(Q & R) v ~(Q & R)

(P & Q) v ~(P & Q)

### Composition of mappings

• Substitute into a substitution-instance

• P v ~P

• Substitute for P

• (Q v R) v ~(Q v R)

• Substitute for R

• (Q v (S & T)) v ~(Q v (S & T))

### Sentence Fractions

• Here’s a fraction

R

(P & Q)

• The numerator is R

• The denominator is (P & Q)

### Fractions and Substitution

• ‘Multiply’

(P & Q) v ~(P & Q)

• by the fraction

R

(P & Q)

• This will be a substitution!

### Sentence Arithmetic

• (P & Q) v ~(P & Q)

Dividing by (P & Q), gives a lattice with a missing label:

• ‘x’ v ~ ‘x’

But R replaces ‘x’ (this step is by fiat)

• R v ~R