A hierarchy of dependencies - Helsinki

A hierarchy of dependencies

Vít Puncochár

Institute of PhilosophyCzech Academy of Sciences

Czech Republic

The aims of this talk

I to use propositional dependence logic to illustrate twological phenomena that I call semantic relativity andsyntactic sensitivity .

I to develop a semantics that model these phenomenaI to look at propositional dependence logic through the

prism of this semantics









A uniform account of information

I In logic a piece of information is often understood as aclassifier of some semantic objecs (possible worlds inclassical logic or some more abstract states in variousnon-classical logics).

I In most logical systems it is implicitly assumed that there isonly one type of information and all indicative sentencesconvey information of this type.

I All pieces of information classify semantic objects of thesame type.









Propositional dependence logic

I Atomic formulas classify primarily possible worlds.I Dependence statements (the truth value of q is functionally

dependent on the truth value of p) classify primarily sets ofpossible worlds (teams).

Propositional dependence logic

I Atomic formulas classify primarily possible worlds.I Dependence statements (the truth value of q is functionally

dependent on the truth value of p) classify primarily sets ofpossible worlds (teams).

Principle of semantic relativity

Different kinds of sentences may classify differentkinds of semantic objects.

Principle of syntactic sensitivity

The behaviour of logical operators is sensitive to thesyntactic features of the statements to which theoperators are applied.

A simple game

I We have a deck of cards each of which has one of thethree values: 1, 2, 3.

I Two cards will be drawn randomly.

A simple game

I We have a deck of cards each of which has one of thethree values: 1, 2, 3.

I Two cards will be drawn randomly.

Propositions

I Pn . . . The resulting sum is exactly nI P<n . . . The resulting sum is less than nI P>n . . . The resulting sum is greater than n

Fix a possible world: w = 〈2,3〉Context of degree 0

Examples of statements that have a truth-value in w :I P5,I P>4,I ¬P2,I P>3 ∧ P<6,I P1 ∨ P>3,I ¬(P2 ∨ P3).

(Statements of degree 0)

















Examples of statements that have no truth-value in w :I dep(P>3,P3),I ¬dep(P>4,P4),I ♦P4,I P>4 → P5

I ¬(P>3 → P4)



I ¬(P>3 → P4)



I ¬(P>3 → P4)



I ¬(P>3 → P4)



I ¬(P>3 → P4)

Fix a team: X = {〈2,1〉, 〈2,2〉, 〈2,3〉}Context of degree 1

Examples of statements that have a truth-value in X :I P>2,I ¬P6,I P5 ∨ P<5,I dep(P>3,P3),I ¬dep(P>4,P4),I dep(P>3,P3) ∨ dep(P>4,P4),I ♦P4,I ¬♦P6,I P>4 → P5,I ¬(P>3 → P4),I P>3 → (¬P4 → P5).




































Examples of statements that have no truth-value in X :I dep(♦P6,♦P3)

I ♦dep(P>4,P4)

I ♦P2 → ¬♦P5

I ¬(♦P3 → ¬♦P5)

I (P<5 → P4)→ ♦P6

Fix a set of teams: M = {{〈1,1〉, 〈1,2〉, 〈1,3〉},{〈2,1〉, 〈2,2〉, 〈2,3〉},{〈3,1〉, 〈3,2〉, 〈3,3〉}, }

Context of degree 2

Examples of statements that have a truth-value in M:I ♦P4

I dep(♦P6,♦P3)

I ♦dep(P>4,P4)

I ♦P2 → ¬♦P5

I ¬(♦P3 → ¬♦P5)

I (P<5 → P4)→ ♦P6



Context of degree 2


I dep(♦P6,♦P3)

I ♦dep(P>4,P4)

I ♦P2 → ¬♦P5

I ¬(♦P3 → ¬♦P5)

I (P<5 → P4)→ ♦P6



Context of degree 2


I dep(♦P6,♦P3)

I ♦dep(P>4,P4)

I ♦P2 → ¬♦P5

I ¬(♦P3 → ¬♦P5)

I (P<5 → P4)→ ♦P6



Context of degree 2


I dep(♦P6,♦P3)

I ♦dep(P>4,P4)

I ♦P2 → ¬♦P5

I ¬(♦P3 → ¬♦P5)

I (P<5 → P4)→ ♦P6



Context of degree 2


I dep(♦P6,♦P3)

I ♦dep(P>4,P4)

I ♦P2 → ¬♦P5

I ¬(♦P3 → ¬♦P5)

I (P<5 → P4)→ ♦P6



Context of degree 2


I dep(♦P6,♦P3)

I ♦dep(P>4,P4)

I ♦P2 → ¬♦P5

I ¬(♦P3 → ¬♦P5)

I (P<5 → P4)→ ♦P6


Context and its degree

DefinitionI A possible world is a function that assigns to every atomic

formula a unique truth value (either T , or F ).I Every possible world will be called a context of degree 0.I A context of degree n + 1 is defined as a nonempty set of

contexts of degree n.I The empty set is called a context of infinite degree.

The degree of a context C will be denoted as d(C).

Language

I ϕ ::= p | ¬ϕ | ϕ ∧ ϕ | ϕ ∨ ϕ | ϕ→ ϕ

The degree of a formula

For every L-formula ϕ we define the degree of ϕ, denoted asd(ϕ), in the following way:I d(p) = 0, for every atomic formula p,I d(¬ϕ) = d(ϕ),I d(ϕ ∧ ψ) = d(ϕ ∨ ψ) = max{d(ϕ),d(ψ)},I d(ϕ→ ψ) = max{d(ϕ) + 1,d(ψ)},

Truth in a context

Now we define a relation of truth between contexts andformulas. However, we impose the following restriction:

C ϕ is defined if and only if d(ϕ) ≤ d(C).

Truth in a context

if d(ϕ) < d(C), then C ϕ iff for all D ∈ C, D ϕ.

Truth in a context

Assume that the degree of the formula on the right is equal tothe degree of the context on the left:

C p iff C(p) = T , for every atomic formula p,C ¬ϕ iff C 1 ϕ,C ϕ ∧ ψ iff C ϕ and C ψ,C ϕ ∨ ψ iff C ϕ or C ψ,C ϕ→ ψ iff Cϕ ψ,

whereCϕ = {D ∈ C | D ϕ}.

Validity in Logic of Semantic Relativity (LSR)

Definitionϕ1, . . . , ϕn/ψ is LSR-valid iff for any context Csuch that max{d(ϕ1), . . . ,d(ϕn),d(ψ)} ≤ d(C)if C supports ϕ1, . . . , ϕn then C supports ψ.

Contradiction, possibility and necessity

I ⊥ =def p ∧ ¬p d(⊥) = 0I ♦ϕ =def ¬(ϕ→ ⊥) d(♦ϕ) = d(ϕ) + 1I �ϕ =def ¬♦¬ϕ d(�ϕ) = d(ϕ) + 1

Dependence

The relation of truth-value dependence dep(ϕ,ψ) is defined asI ((ϕ→ ψ) ∨ (ϕ→ ¬ψ)) ∧ ((¬ϕ→ ψ) ∨ (¬ϕ→ ¬ψ))

d(dep(ϕ,ψ)) = max{d(ϕ) + 1,d(ψ)}

Variations

I A variation of the formulas ϕ1, . . . , ϕn is any formulaχ1 ∧ . . . ∧ χn where for any i , χi = ϕi or χi = ¬ϕi .

I The set of the variations of ϕ1, . . . , ϕn will be denoted asΣ(ϕ1 . . . ϕn)

I E.g. Σ(p,q) = {p ∧ q,p ∧ ¬q,¬p ∧ q,¬p ∧ ¬q}

General definition of dependence

The relation of truth-value dependence dep(ϕ1, . . . , ϕn, ψ) isdefined asI

∧χ∈Σ(ϕ1,...,ϕn)((χ→ ψ) ∨ (χ→ ¬ψ))

d(dep(ϕ1, . . . , ϕn, ψ)) = max{max{d(ϕ1), . . . ,d(ϕn)}+1,d(ψ)}

Some observations

Proposition

1. ∅ ϕ, for every formula ϕ.

Assume d(C) = d(ϕ) + 1. Then2. C ♦ϕ iff for some D ∈ C, D ϕ,3. C �ϕ iff for all D ∈ C, D ϕ.

Assume d(C) = 1. Then4. C dep(p1, . . . ,pn,q) iff for any v ,w ∈ C, if v(pi) = w(pi),

for any i, then v(q) = w(q).

Failure of substitution

I q → r ,p → q/p → r validI q → ♦r ,p → q/p → ♦r invalid

(Example: If Bob is in Munich, he is in Germany. If Bob is inGermany, he might be in Berlin. Therefore, if Bob is in Munich,he might be in Berlin.)I p → ¬q,q/¬p validI p → ¬♦q,♦q/¬p invalid

(Example: If Bob is in Munich, it is not possible that he is inBerlin. It is possible that Bob is in Berlin. Therefore, Bob is notin Munich.)

A counterexample to Peirce’s law: ((p → q)→ p)→ p

• • • • p q

p q

p1 q

1 p q


• • • • p q

p q

p1 q

1 p q

p → q p

1 p → q1 p


• • • • p q

p q

p1 q

1 p q

p → q p

1 p → q1 p


• • • • p q

p q

p1 q

1 p q

p → q p

1 p → q1 p

(p → q)→ p1 p


• • • • p q

p q

p1 q

1 p q

p → q p

1 p → q1 p

(p → q)→ p1 p

1 ((p → q)→ p)→ p

Classical fragment

Conditional-free formulasα ::= p | ¬α | α ∧ α | α ∨ α

Classical formulasϕ ::= α | α→ ϕ | ϕ ∧ ϕ

PropositionFor any classical formulas ϕ1, . . . , ϕn, ψ:I ϕ1, . . . , ϕn/ψ is LSR-valid iff ϕ1, . . . , ϕn/ψ is CL-valid.

Decidability

PropositionLSR-validity (restricted to finite number of premisses) isdecidable.

Functional Completeness

PropositionAssume that the number of atomic formulas is finite. Then forany number n and any set of n-contexts S, there is ann-formula ϕ such that ||ϕ|| = S (where ||ϕ|| is the set ofn-contexts that support ϕ).

Conclusion

I I have introduce a semantics for the basic propositionallanguage (¬,∧,∨,→) that reflects the principles ofsemantic relativity and syntactic sensitivity.

I The dependence operator can be defined in the basiclanguage and its peculiar properties stem from the twoprinciples.

I Future work: A completeness result

Documents

A hierarchy of dependencies - Helsinki