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# 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

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### Composition and Substitution:Learning about Language from Algebra

Ken Presting

University of North Carolina at Chapel Hill

• 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

• 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 Truth Values

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

• Disjunction = max(a, b)

• Conjunction = min(a, b)

• Negation = 1 – a

• Boolean arithmetic

• 0  1 = 1

• 0  1 = 0

• Boolean algebra

• A  B = C

• (A  B)  ~C = C  ~C

• (A  B)  ~C = 0

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

• For example:

• f(S) = 0

• f(PQ) = 1

• etc.

• 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)

…is a compositional semantics for propositional calculus

• Tree diagrams

• Binary

• Associativity allows n-ary nodes

• Identical Subtrees

• In many sentences, certain letters appear twice or more

• P & Q  P

• Sometimes whole expressions recur

• (P & R)  (P & R)

• Identify like-labeled leaves

• Identify like-labeled nodes

• Form equivalence classes

• Redraw tree as lattice

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

• Mapping sentences to sets

• Set of letters = conjunction

• Singleton set = negation

• Associativity

• And vs. Nand

• Naturalness of negation

• Failure of associativity

• Embeddings

• Homomorphism

• Single-letter expressions

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

• Substitution for single letters is easy

• Multiple occurrences of a letter

• What do these sentences have in common?

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

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

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

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

• Ignore specfied details

• In our case, treat a subsentence as a letter

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

• 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 = 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)

• 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))

• Here’s a fraction

R

(P & Q)

• The numerator is R

• The denominator is (P & Q)

• ‘Multiply’

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

• by the fraction

R

(P & Q)

• This will be a substitution!

• (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