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 • 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 • (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

Composition and Substitution: Learning about Language from Algebra