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

Compositional Semantics

• 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))

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

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