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Composition and Substitution: Learning about Language from Algebra

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### 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(PQ) = 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
- (advanced topic: add leaves for empty expression)

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

Comparing lattices

- 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
- Start with
- 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))

Fractions and Substitution

- ‘Multiply’

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

- by the fraction

R

(P & Q)

- This will be a substitution!

Sentence Arithmetic

Start with

- (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

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