(Pre-)Algebras for Linguistics 5. Prelattices Carl Pollard - - PowerPoint PPT Presentation

(Pre-)Algebras for Linguistics

• 5. Prelattices

Carl Pollard

Linguistics 680: Formal Foundations

Autumn 2010

Carl Pollard (Pre-)Algebras for Linguistics

Prelattices

A prelattice is a preordered algebra P, ⊑, ⊓, ⊔ where

P, ⊑, ⊓ is a lower semilattice and

Carl Pollard (Pre-)Algebras for Linguistics

Prelattices

A prelattice is a preordered algebra P, ⊑, ⊓, ⊔ where

P, ⊑, ⊓ is a lower semilattice and P, ⊑, ⊔ is an upper semilattice.

Carl Pollard (Pre-)Algebras for Linguistics

Prelattices

A prelattice is a preordered algebra P, ⊑, ⊓, ⊔ where

P, ⊑, ⊓ is a lower semilattice and P, ⊑, ⊔ is an upper semilattice.

A bounded prelattice is a preordered algebra P, ⊑, ⊓, ⊔, ⊤, ⊥ where

P, ⊑, ⊓, ⊔ is a prelattice

Carl Pollard (Pre-)Algebras for Linguistics

Prelattices

A prelattice is a preordered algebra P, ⊑, ⊓, ⊔ where

P, ⊑, ⊓ is a lower semilattice and P, ⊑, ⊔ is an upper semilattice.

A bounded prelattice is a preordered algebra P, ⊑, ⊓, ⊔, ⊤, ⊥ where

P, ⊑, ⊓, ⊔ is a prelattice ⊤ is a top

Carl Pollard (Pre-)Algebras for Linguistics

Prelattices

A prelattice is a preordered algebra P, ⊑, ⊓, ⊔ where

P, ⊑, ⊓ is a lower semilattice and P, ⊑, ⊔ is an upper semilattice.

A bounded prelattice is a preordered algebra P, ⊑, ⊓, ⊔, ⊤, ⊥ where

P, ⊑, ⊓, ⊔ is a prelattice ⊤ is a top ⊥ is a bottom

Carl Pollard (Pre-)Algebras for Linguistics

Basic Facts about Prelattices (Review)

⊓ and ⊔ are:

monotonic in both arguments

Carl Pollard (Pre-)Algebras for Linguistics

Basic Facts about Prelattices (Review)

⊓ and ⊔ are:

monotonic in both arguments associative u.t.e.

Carl Pollard (Pre-)Algebras for Linguistics

Basic Facts about Prelattices (Review)

⊓ and ⊔ are:

monotonic in both arguments associative u.t.e. commutative u.t.e.

Carl Pollard (Pre-)Algebras for Linguistics

Basic Facts about Prelattices (Review)

⊓ and ⊔ are:

monotonic in both arguments associative u.t.e. commutative u.t.e. idempotent u.t.e.

Carl Pollard (Pre-)Algebras for Linguistics

Basic Facts about Prelattices (Review)

⊓ and ⊔ are:

monotonic in both arguments associative u.t.e. commutative u.t.e. idempotent u.t.e.

⊓ (⊔) is a glb (lub) operation

Carl Pollard (Pre-)Algebras for Linguistics

Basic Facts about Prelattices (Review)

⊓ and ⊔ are:

monotonic in both arguments associative u.t.e. commutative u.t.e. idempotent u.t.e.

⊓ (⊔) is a glb (lub) operation Interdefinability: for all p, q ∈ P, p ⊓ q ≡ p iff p ⊑ q iff p ⊔ q ≡ q

Carl Pollard (Pre-)Algebras for Linguistics

Basic Facts about Prelattices (Review)

⊓ and ⊔ are:

monotonic in both arguments associative u.t.e. commutative u.t.e. idempotent u.t.e.

⊓ (⊔) is a glb (lub) operation Interdefinability: for all p, q ∈ P, p ⊓ q ≡ p iff p ⊑ q iff p ⊔ q ≡ q Absorption u.t.e.: (p ⊔ q) ⊓ q ≡ q ≡ (p ⊓ q) ⊔ q;

Carl Pollard (Pre-)Algebras for Linguistics

More Facts about Prelattices

Semidistributivity: For all a, b ∈ P: (p ⊓ q) ⊔ (p ⊓ r) ⊑ p ⊓ (q ⊔ r)

Carl Pollard (Pre-)Algebras for Linguistics

More Facts about Prelattices

Semidistributivity: For all a, b ∈ P: (p ⊓ q) ⊔ (p ⊓ r) ⊑ p ⊓ (q ⊔ r) A prelattice is called distributive u.t.e if the inequality reverse to Semidistributivity holds: p ⊓ (q ⊔ r) ⊑ (p ⊓ q) ⊔ (p ⊓ r) so that in fact

Carl Pollard (Pre-)Algebras for Linguistics

More Facts about Prelattices

Semidistributivity: For all a, b ∈ P: (p ⊓ q) ⊔ (p ⊓ r) ⊑ p ⊓ (q ⊔ r) A prelattice is called distributive u.t.e if the inequality reverse to Semidistributivity holds: p ⊓ (q ⊔ r) ⊑ (p ⊓ q) ⊔ (p ⊓ r) so that in fact p ⊓ (q ⊔ r) ≡ (p ⊓ q) ⊔ (p ⊓ r)

Carl Pollard (Pre-)Algebras for Linguistics

More Facts about Prelattices

Semidistributivity: For all a, b ∈ P: (p ⊓ q) ⊔ (p ⊓ r) ⊑ p ⊓ (q ⊔ r) A prelattice is called distributive u.t.e if the inequality reverse to Semidistributivity holds: p ⊓ (q ⊔ r) ⊑ (p ⊓ q) ⊔ (p ⊓ r) so that in fact p ⊓ (q ⊔ r) ≡ (p ⊓ q) ⊔ (p ⊓ r) Theorem: a prelattice is distributive u.t.e. iff the following equivalence holds for all a, b, c ∈ P (obtained from the one above by interchanging ⊓ and ⊔): p ⊔ (q ⊓ r) ≡ (p ⊔ q) ⊓ (p ⊔ r)

Carl Pollard (Pre-)Algebras for Linguistics

(Pseudo-)Complement Operations

Suppose P, ⊑, ⊓, →, ⊥ is a heyting presemilattice with a bottom element ⊥. Then a unary operation ¬ on P is called a pseudocomplement operation iff, for all p ∈ P, ¬p ≡ p → ⊥

Carl Pollard (Pre-)Algebras for Linguistics

(Pseudo-)Complement Operations

Suppose P, ⊑, ⊓, →, ⊥ is a heyting presemilattice with a bottom element ⊥. Then a unary operation ¬ on P is called a pseudocomplement operation iff, for all p ∈ P, ¬p ≡ p → ⊥ Clearly a pseudocomplement operation is antitonic. It’s easy to show that ¬⊥ is a top.

Carl Pollard (Pre-)Algebras for Linguistics

(Pseudo-)Complement Operations

Suppose P, ⊑, ⊓, →, ⊥ is a heyting presemilattice with a bottom element ⊥. Then a unary operation ¬ on P is called a pseudocomplement operation iff, for all p ∈ P, ¬p ≡ p → ⊥ Clearly a pseudocomplement operation is antitonic. It’s easy to show that ¬⊥ is a top. It’s easy to show that for all p ∈ P, p ⊑ ¬(¬p).

Carl Pollard (Pre-)Algebras for Linguistics

(Pseudo-)Complement Operations

Suppose P, ⊑, ⊓, →, ⊥ is a heyting presemilattice with a bottom element ⊥. Then a unary operation ¬ on P is called a pseudocomplement operation iff, for all p ∈ P, ¬p ≡ p → ⊥ Clearly a pseudocomplement operation is antitonic. It’s easy to show that ¬⊥ is a top. It’s easy to show that for all p ∈ P, p ⊑ ¬(¬p). A pseudocomplement operation is called a complement

• peration provided, for all a ∈ P, ¬(¬a) ⊑ a, so that in

fact ¬(¬a) ≡ a.

Carl Pollard (Pre-)Algebras for Linguistics

Pre-Heyting and Pre-Boolean Algebras

A pre-heyting algebra is a preordered algebra P, ⊑, ⊓, ⊔, →, ¬, ⊤, ⊥ where:

P, ⊑, ⊓, ⊔, ⊤, ⊥ is a bounded prelattice

Carl Pollard (Pre-)Algebras for Linguistics

Pre-Heyting and Pre-Boolean Algebras

A pre-heyting algebra is a preordered algebra P, ⊑, ⊓, ⊔, →, ¬, ⊤, ⊥ where:

P, ⊑, ⊓, ⊔, ⊤, ⊥ is a bounded prelattice P, ⊑, ⊓, → is a heyting presemilattice

Carl Pollard (Pre-)Algebras for Linguistics

Pre-Heyting and Pre-Boolean Algebras

A pre-heyting algebra is a preordered algebra P, ⊑, ⊓, ⊔, →, ¬, ⊤, ⊥ where:

P, ⊑, ⊓, ⊔, ⊤, ⊥ is a bounded prelattice P, ⊑, ⊓, → is a heyting presemilattice ¬ is a pseudocomplement operation.

Carl Pollard (Pre-)Algebras for Linguistics

Pre-Heyting and Pre-Boolean Algebras

A pre-heyting algebra is a preordered algebra P, ⊑, ⊓, ⊔, →, ¬, ⊤, ⊥ where:

P, ⊑, ⊓, ⊔, ⊤, ⊥ is a bounded prelattice P, ⊑, ⊓, → is a heyting presemilattice ¬ is a pseudocomplement operation.

Theorem: pre-heyting algebras are distributive u.t.e.

Carl Pollard (Pre-)Algebras for Linguistics

Pre-Heyting and Pre-Boolean Algebras

A pre-heyting algebra is a preordered algebra P, ⊑, ⊓, ⊔, →, ¬, ⊤, ⊥ where:

P, ⊑, ⊓, ⊔, ⊤, ⊥ is a bounded prelattice P, ⊑, ⊓, → is a heyting presemilattice ¬ is a pseudocomplement operation.

Theorem: pre-heyting algebras are distributive u.t.e. Pre-heyting algebras are algebraic models of a kind of logic called intuitionistic propositional logic.

Carl Pollard (Pre-)Algebras for Linguistics

Pre-Heyting and Pre-Boolean Algebras

A pre-heyting algebra is a preordered algebra P, ⊑, ⊓, ⊔, →, ¬, ⊤, ⊥ where:

P, ⊑, ⊓, ⊔, ⊤, ⊥ is a bounded prelattice P, ⊑, ⊓, → is a heyting presemilattice ¬ is a pseudocomplement operation.

Theorem: pre-heyting algebras are distributive u.t.e. Pre-heyting algebras are algebraic models of a kind of logic called intuitionistic propositional logic. A pre-boolean algebra is a pre-heyting algebra satisfying either of the following (equivalent!) conditions:

The pseudocomplement operation ¬ is a complement

• peration, i.e. for all p ∈ P, ¬(¬p) ⊑ p.

Carl Pollard (Pre-)Algebras for Linguistics

Pre-Heyting and Pre-Boolean Algebras

A pre-heyting algebra is a preordered algebra P, ⊑, ⊓, ⊔, →, ¬, ⊤, ⊥ where:

P, ⊑, ⊓, ⊔, ⊤, ⊥ is a bounded prelattice P, ⊑, ⊓, → is a heyting presemilattice ¬ is a pseudocomplement operation.

Theorem: pre-heyting algebras are distributive u.t.e. Pre-heyting algebras are algebraic models of a kind of logic called intuitionistic propositional logic. A pre-boolean algebra is a pre-heyting algebra satisfying either of the following (equivalent!) conditions:

The pseudocomplement operation ¬ is a complement

• peration, i.e. for all p ∈ P, ¬(¬p) ⊑ p.

For all p ∈ P, p ⊔ ¬p ≡ ⊤.

Carl Pollard (Pre-)Algebras for Linguistics

Pre-Heyting and Pre-Boolean Algebras

A pre-heyting algebra is a preordered algebra P, ⊑, ⊓, ⊔, →, ¬, ⊤, ⊥ where:

P, ⊑, ⊓, ⊔, ⊤, ⊥ is a bounded prelattice P, ⊑, ⊓, → is a heyting presemilattice ¬ is a pseudocomplement operation.

Theorem: pre-heyting algebras are distributive u.t.e. Pre-heyting algebras are algebraic models of a kind of logic called intuitionistic propositional logic. A pre-boolean algebra is a pre-heyting algebra satisfying either of the following (equivalent!) conditions:

The pseudocomplement operation ¬ is a complement

• peration, i.e. for all p ∈ P, ¬(¬p) ⊑ p.

For all p ∈ P, p ⊔ ¬p ≡ ⊤.

Pre-boolean algebras are algebraic models of a kind of logic called classical propositional logic.

Carl Pollard (Pre-)Algebras for Linguistics

Heyting Algebras and Boolean Algebras

A heyting (boolean) algebra is an antisymmetric pre-heyting (pre-boolean) algebra.

Carl Pollard (Pre-)Algebras for Linguistics

Heyting Algebras and Boolean Algebras

A heyting (boolean) algebra is an antisymmetric pre-heyting (pre-boolean) algebra. An example of a heyting algebra is the set of open sets of real numbers ordered by inclusion. (Exercise: what are the

• perations?)

Carl Pollard (Pre-)Algebras for Linguistics

Heyting Algebras and Boolean Algebras

A heyting (boolean) algebra is an antisymmetric pre-heyting (pre-boolean) algebra. An example of a heyting algebra is the set of open sets of real numbers ordered by inclusion. (Exercise: what are the

• perations?)

The most familiar boolean algebras are power sets ordered by subset inclusion. (Exercise: what are the operations?)

Carl Pollard (Pre-)Algebras for Linguistics

Heyting Algebras and Boolean Algebras

A heyting (boolean) algebra is an antisymmetric pre-heyting (pre-boolean) algebra. An example of a heyting algebra is the set of open sets of real numbers ordered by inclusion. (Exercise: what are the

• perations?)

The most familiar boolean algebras are power sets ordered by subset inclusion. (Exercise: what are the operations?) Special case of preceding: 2 = ℘(1) = {0, 1}. Semanticists

• ften call this the algebra of truth values, and rename 1

and 0 to t and f respectively.

Carl Pollard (Pre-)Algebras for Linguistics

Semantic Application of Pre-Boolean Algebras

We can model the propositions (static sentence meanings) as a pre-boolean algebra where: ⊑ is entailment

Carl Pollard (Pre-)Algebras for Linguistics

Semantic Application of Pre-Boolean Algebras

We can model the propositions (static sentence meanings) as a pre-boolean algebra where: ⊑ is entailment ⊓ is the meaning of and

Carl Pollard (Pre-)Algebras for Linguistics

Semantic Application of Pre-Boolean Algebras

We can model the propositions (static sentence meanings) as a pre-boolean algebra where: ⊑ is entailment ⊓ is the meaning of and ⊔ is the meaning of or

Carl Pollard (Pre-)Algebras for Linguistics

Semantic Application of Pre-Boolean Algebras

We can model the propositions (static sentence meanings) as a pre-boolean algebra where: ⊑ is entailment ⊓ is the meaning of and ⊔ is the meaning of or → is the meaning of if . . . then

Carl Pollard (Pre-)Algebras for Linguistics

Semantic Application of Pre-Boolean Algebras

We can model the propositions (static sentence meanings) as a pre-boolean algebra where: ⊑ is entailment ⊓ is the meaning of and ⊔ is the meaning of or → is the meaning of if . . . then ¬ is the meaning of it is not the case that or no way

Carl Pollard (Pre-)Algebras for Linguistics

Semantic Application of Pre-Boolean Algebras

We can model the propositions (static sentence meanings) as a pre-boolean algebra where: ⊑ is entailment ⊓ is the meaning of and ⊔ is the meaning of or → is the meaning of if . . . then ¬ is the meaning of it is not the case that or no way ⊤ is some necessarily true proposition

Carl Pollard (Pre-)Algebras for Linguistics

Semantic Application of Pre-Boolean Algebras

We can model the propositions (static sentence meanings) as a pre-boolean algebra where: ⊑ is entailment ⊓ is the meaning of and ⊔ is the meaning of or → is the meaning of if . . . then ¬ is the meaning of it is not the case that or no way ⊤ is some necessarily true proposition ⊥ is some necessarily false proposition

Carl Pollard (Pre-)Algebras for Linguistics

Modelling Worlds (1/2)

One way to do it is Montague’s way: We take worlds to be a set W of unanalyzed primitives

Carl Pollard (Pre-)Algebras for Linguistics

Modelling Worlds (1/2)

One way to do it is Montague’s way: We take worlds to be a set W of unanalyzed primitives We model propositions as sets of worlds: P = def ℘(W)

Carl Pollard (Pre-)Algebras for Linguistics

Modelling Worlds (1/2)

One way to do it is Montague’s way: We take worlds to be a set W of unanalyzed primitives We model propositions as sets of worlds: P = def ℘(W) p@w means w ∈ p, so entailment is ⊆W

Carl Pollard (Pre-)Algebras for Linguistics

Modelling Worlds (1/2)

One way to do it is Montague’s way: We take worlds to be a set W of unanalyzed primitives We model propositions as sets of worlds: P = def ℘(W) p@w means w ∈ p, so entailment is ⊆W and′ is intersection

Carl Pollard (Pre-)Algebras for Linguistics

Modelling Worlds (1/2)

One way to do it is Montague’s way: We take worlds to be a set W of unanalyzed primitives We model propositions as sets of worlds: P = def ℘(W) p@w means w ∈ p, so entailment is ⊆W and′ is intersection

• r′ is union

Carl Pollard (Pre-)Algebras for Linguistics

Modelling Worlds (1/2)

One way to do it is Montague’s way: We take worlds to be a set W of unanalyzed primitives We model propositions as sets of worlds: P = def ℘(W) p@w means w ∈ p, so entailment is ⊆W and′ is intersection

• r′ is union

implies′ is relative complement

Carl Pollard (Pre-)Algebras for Linguistics

Modelling Worlds (1/2)

One way to do it is Montague’s way: We take worlds to be a set W of unanalyzed primitives We model propositions as sets of worlds: P = def ℘(W) p@w means w ∈ p, so entailment is ⊆W and′ is intersection

• r′ is union

implies′ is relative complement no way′ is complement

Carl Pollard (Pre-)Algebras for Linguistics

Modelling Worlds (1/2)

One way to do it is Montague’s way: We take worlds to be a set W of unanalyzed primitives We model propositions as sets of worlds: P = def ℘(W) p@w means w ∈ p, so entailment is ⊆W and′ is intersection

• r′ is union

implies′ is relative complement no way′ is complement There is only one necessary truth.

Carl Pollard (Pre-)Algebras for Linguistics

Modelling Worlds (1/2)

One way to do it is Montague’s way: We take worlds to be a set W of unanalyzed primitives We model propositions as sets of worlds: P = def ℘(W) p@w means w ∈ p, so entailment is ⊆W and′ is intersection

• r′ is union

implies′ is relative complement no way′ is complement There is only one necessary truth. There is only one necessary falsehood.

Carl Pollard (Pre-)Algebras for Linguistics

Modelling Worlds (1/2)

One way to do it is Montague’s way: We take worlds to be a set W of unanalyzed primitives We model propositions as sets of worlds: P = def ℘(W) p@w means w ∈ p, so entailment is ⊆W and′ is intersection

• r′ is union

implies′ is relative complement no way′ is complement There is only one necessary truth. There is only one necessary falsehood. Sentences with the same truth conditions have the same meaning.

Carl Pollard (Pre-)Algebras for Linguistics

Modelling Worlds (2/2)

Another way is the way of hyperintensional semantics. We take worlds to be a certain subsets of P: W ℘(P). (Which ones? We’ll come back to that.)

Carl Pollard (Pre-)Algebras for Linguistics

Modelling Worlds (2/2)

Another way is the way of hyperintensional semantics. We take worlds to be a certain subsets of P: W ℘(P). (Which ones? We’ll come back to that.) p@w means p ∈ w.

Carl Pollard (Pre-)Algebras for Linguistics

Modelling Worlds (2/2)

Another way is the way of hyperintensional semantics. We take worlds to be a certain subsets of P: W ℘(P). (Which ones? We’ll come back to that.) p@w means p ∈ w. For the preorder ⊑ in P to be a good model of entailment, it will have to be the case that for any two propositions p and q, p ⊑ q iff for every world w, if p ∈ w then q ∈ w.

Carl Pollard (Pre-)Algebras for Linguistics

Modelling Worlds (2/2)

Another way is the way of hyperintensional semantics. We take worlds to be a certain subsets of P: W ℘(P). (Which ones? We’ll come back to that.) p@w means p ∈ w. For the preorder ⊑ in P to be a good model of entailment, it will have to be the case that for any two propositions p and q, p ⊑ q iff for every world w, if p ∈ w then q ∈ w. Turning things around, for any p and q such that p ⊑ q, there must exist a w such that p ∈ w but q ∈ w.

Carl Pollard (Pre-)Algebras for Linguistics

Modelling Worlds (2/2)

Another way is the way of hyperintensional semantics. We take worlds to be a certain subsets of P: W ℘(P). (Which ones? We’ll come back to that.) p@w means p ∈ w. For the preorder ⊑ in P to be a good model of entailment, it will have to be the case that for any two propositions p and q, p ⊑ q iff for every world w, if p ∈ w then q ∈ w. Turning things around, for any p and q such that p ⊑ q, there must exist a w such that p ∈ w but q ∈ w. So, whatever worlds are, there have to be enough of them.

Carl Pollard (Pre-)Algebras for Linguistics

Modelling Worlds (2/2)

Another way is the way of hyperintensional semantics. We take worlds to be a certain subsets of P: W ℘(P). (Which ones? We’ll come back to that.) p@w means p ∈ w. For the preorder ⊑ in P to be a good model of entailment, it will have to be the case that for any two propositions p and q, p ⊑ q iff for every world w, if p ∈ w then q ∈ w. Turning things around, for any p and q such that p ⊑ q, there must exist a w such that p ∈ w but q ∈ w. So, whatever worlds are, there have to be enough of them. So which subsets of P should be in W?

Carl Pollard (Pre-)Algebras for Linguistics

Modelling Worlds (2/2)

Another way is the way of hyperintensional semantics. We take worlds to be a certain subsets of P: W ℘(P). (Which ones? We’ll come back to that.) p@w means p ∈ w. For the preorder ⊑ in P to be a good model of entailment, it will have to be the case that for any two propositions p and q, p ⊑ q iff for every world w, if p ∈ w then q ∈ w. Turning things around, for any p and q such that p ⊑ q, there must exist a w such that p ∈ w but q ∈ w. So, whatever worlds are, there have to be enough of them. So which subsets of P should be in W? To answer this, we need to know more about certain special subsets of preboolean algebras.

Carl Pollard (Pre-)Algebras for Linguistics