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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

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Composition and substitution learning about language from algebra

Composition and Substitution:Learning about Language from Algebra

Ken Presting

University of North Carolina at Chapel Hill


Introduction
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
Role of Recursion

  • Syntax

    • Atomic symbols

    • Combination rules

    • Closure principle

  • Finiteness

    • Limited symbols, rules

    • Infinitely many expressions


Compositional semantics
CompositionalSemantics

  • The usual:

    • Choose assignments to atoms

    • Forced valuations for molecules


The two element boolean algebra
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
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
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 2ba1
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 2ba2
A Homomorphism to 2BA

…is a compositional semantics for propositional calculus


Sentence diagrams
Sentence Diagrams

  • Tree diagrams

    • Binary

    • Associativity allows n-ary nodes

      • (advanced topic: add leaves for empty expression)


Repetition
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
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
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
Comparing lattices

  • Embeddings

  • Homomorphism


Substitution for a letter
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
Substitution for Expressions

  • What do these sentences have in common?

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

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


Subalgebras
Subalgebras

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

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


Quotients
Quotients

  • Ignore specfied details

  • In our case, treat a subsentence as a letter


Sentences as functions
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 functions1
    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 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
    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
    Sentence Fractions

    • Here’s a fraction

      R

      (P & Q)

    • The numerator is R

    • The denominator is (P & Q)


    Fractions and substitution
    Fractions and Substitution

    • ‘Multiply’

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

    • by the fraction

      R

      (P & Q)

    • This will be a substitution!


    Sentence arithmetic
    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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