Applications of Mathematics in Philosophy: Four Case Studiesrzach/static/banff/leitgeb.pdf · B. Russell, R. Carnap: generalized abstraction; abstracting “similarity classes”

Applications of Mathematics in Philosophy:Four Case Studies

Hannes Leitgeb

University of Bristol

February 2007

Hannes Leitgeb (University of Bristol) Applications of Mathematics in Philosophy: Four Case StudiesFebruary 2007 1 / 22

Introduction

Many philosophers still deny that mathematical methods can play asubstantive role in philosophy.

Paradigmatic case: Kant in the Critique of Pure Reason.

According to Kant’s Transcendental Doctrine of Method, philosophycannot be developed according to the definitions-axioms-proofs scheme.

This is because mathematics deals with pure intuitions, whereasphilosophy deals with pure concepts (or so Kant says). However,

in the meantime, mathematics has developed into a theory of abstractstructures in general

the progress in logic shows that the “space of concepts” has itself anintricate mathematical structure

if Platonists such as Godel are right, there is a rational intuition ofconcepts!?


Introduction




This is because mathematics deals with pure intuitions, whereasphilosophy deals with pure concepts (or so Kant says).

However,





Introduction




This is because mathematics deals with pure intuitions, whereasphilosophy deals with pure concepts (or so Kant says). However,





Claims:

When philosophical theories get sufficiently complex, they are in need ofmathematics.

The necessary bridge between philosophy and mathematics is oftensupplied by logic broadly understood.

Plan of the talk: Examples

Similarity, Properties, and Hypergraphs

Nonmonotonic Logic and Dynamical Systems

Belief Revision for Conditionals and Arrow’s Theorem

Semantic Paradoxes and Probability

(If time permits: Meaning Similarity and Compositionality)


Claims:










Claims:










Claims:











G. Frege, A.N. Whitehead & B. Russell: definition by logical abstraction;abstracting equivalence classes from equivalence relations

B. Russell, R. Carnap: generalized abstraction;abstracting “similarity classes” from similarity relations↪→ abstraction for the empirical domain (where transitivity often fails)

• 〈S ,∼〉 is a similarity structure on S :iff∼⊆ S × S is reflexive and symmetric.

E.g.: metric similarity (for colours,. . .)

r r≤ ε• 〈S ,P〉 is a property structure on S :iffP ⊆ ℘(S), ∅ /∈ P, andfor every x ∈ S there is an X ∈ P, s.t. x ∈ X .

E.g.: (colour) spheres r r&%'$

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B. Russell, R. Carnap: generalized abstraction;abstracting “similarity classes” from similarity relations

↪→ abstraction for the empirical domain (where transitivity often fails)





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• 〈S ,P〉 is a property structure on S :iffP ⊆ ℘(S), ∅ /∈ P, andfor every x ∈ S there is an X ∈ P, s.t. x ∈ X .


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Every property structure on S determines a similarity structure on S :(cf. Leibniz: “Peter is similar to Paul” reduces to “Peter is A now and Paulis A now”)

Definition (Determined similarity structure)

〈S ,∼P〉 is determined by 〈S ,P〉 :ifffor all x , y ∈ S : x ∼P y iff there is an X ∈ P, such that x , y ∈ X .

Every similarity structure on S determines a property structure on S :

X ⊆ S is a clique of 〈S ,∼〉 :iff for all x , y ∈ X : x ∼ y .

Definition (Determined property structure)

〈S ,P∼〉 is determined by 〈S ,∼〉 :iffP∼ = {X ⊆ S |X is a maximal clique of 〈S ,∼〉}.

Special case: equivalence classes equivalence relations X






































Example

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But this method of abstraction does not work in all cases. . .(as N. Goodman and others observed)

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Example

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QUESTION: If similarity is determined by properties, under whichconditions can the latter be reconstructed from the former?


Example


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↪→ Hypergraph theory!

Theorem (Gilmore; cf. Berge 1989)

Let 〈S ,∼P〉 be determined by 〈S ,P〉, with S finite:

1 〈S ,∼P〉 is faithful with respect to 〈S ,P〉 iff(a) for all A,B,C ∈ P there is an X ∈ P, such that

(A ∩ B) ∪ (A ∩ C ) ∪ (B ∩ C ) ⊆ X

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3 〈S ,∼P〉 is faithful & full with respect to 〈S ,P〉 iff (a)+(b).

Proof: By induction over |S |.

Russell, Our Knowledge of the External World: X (Helly’s Theorem)Carnap, The Logical Structure of the World: 6XIf determination starts with similarity, then “reconstruction” works.If n = |S | and |P| >

( nbn/2c

)then 〈S ,∼P〉 is not full. (Sperner’s Theorem)

(See Leitgeb 2007, JPL. In work: Monograph on a new Logischer Aufbau.)






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If determination starts with similarity, then “reconstruction” works.If n = |S | and |P| >

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Russell, Our Knowledge of the External World: X (Helly’s Theorem)Carnap, The Logical Structure of the World: 6XIf determination starts with similarity, then “reconstruction” works.

If n = |S | and |P| >( nbn/2c








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Here is a problem in the philosophy of mind:

On the one hand, (human, animal, robot) brains seem to be physicalsystems that can be described in terms of differential or differenceequations, i.e., as dynamical systems.

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On the other hand, these brains seem to have beliefs, draw inferences,and so forth, the contents of which can be expressed by sentences:

x believes that ¬ϕx infers that ϕ ∨ ψ from ϕ

We take these to be distinct but compatible perspectives on the samephenomenon.

So we have to associate system states with propositions:system states carry information that can be expressed linguistically!

What might a theory of such interpreted dynamical systems look like?





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Definition

A triple S = 〈S , ns,6〉 is an ordered discrete dynamical system :iff

1 S is a non-empty set (the set of states)

2 ns : S → S (the internal next-state function)

3 6 ⊆ S × S is a partial order (the information ordering) on S , s.t.for all s, s ′ ∈ S there is a supremum sup(s, s ′) ∈ S with respect to 6.

E.g.: S = {s|s : N → {0, 1}} (with N = {n1, n2, n3, n4})

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The internal dynamics of such systems is given by the iterationns(= ns1), ns2, ns3, . . . of the mapping ns.

Now we add an input which is regarded to activate a fixed state s∗ ∈ S :the “next” state of the system is given by the superposition of s∗ with thenext internal state ns(s), i.e., we actually iterate

Fs∗(s) := sup(s∗, ns(s))

Fixed points sstab of Fs∗ are the “answers” which the system gives to s∗.


Definition






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Definition






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3 6 ⊆ S × S is a partial order (the information ordering) on S ,

s.t.for all s, s ′ ∈ S there is a supremum sup(s, s ′) ∈ S with respect to 6.


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Now we add an input which is regarded to activate a fixed state s∗ ∈ S :

the “next” state of the system is given by the superposition of s∗ with thenext internal state ns(s), i.e., we actually iterate




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Finally, we assign formulas to states of these systems.

Let L be a propositional language:

Definition

A quadruple SI = 〈S , ns,6, I〉 is an interpreted ordered system :iff

1 〈S , ns,6〉 is an ordered discrete dynamical system

2 I : L → S (the interpretation mapping) has “nice” properties, such as

for all ϕ,ψ ∈ L: I(ϕ ∧ ψ) = sup(I(ϕ), I(ψ))

for every ϕ ∈ L: there is a unique I(ϕ)-stable state...

Definition1 SI � ϕ⇒ ψ :iff

if sstab is the unique I(ϕ)-stable state, then I(ψ) 6 sstab2 T H⇒(SI) = {ϕ⇒ ψ |SI � ϕ⇒ ψ}

(the conditional theory corresponding to SI).




Definition












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for every ϕ ∈ L: there is a unique I(ϕ)-stable state

...







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if sstab is the unique I(ϕ)-stable state, then I(ψ) 6 sstab

2 T H⇒(SI) = {ϕ⇒ ψ |SI � ϕ⇒ ψ}(the conditional theory corresponding to SI).




Definition










Theorem (Representation)

For all SI: T H⇒(SI) is closed under the rules of the systemC(umulativity) of nonmonotonic logic (cf. KLM 1990).

Furthermore, for all consistent T H that are closed under these rulesthere is an interpreted system SI, such that T H = T H⇒(SI).

Proof: (i) induction over formulas; (ii) construction of systems.

There is a corresponding representation theorem for hierarchicalinterpreted systems and the nonmonotonic system C+L(oop).

ϕ0 ⇒ ϕ1, ϕ1 ⇒ ϕ2, . . . , ϕj−1 ⇒ ϕj , ϕj ⇒ ϕ0

ϕ0 ⇒ ϕj(Loop)

Application: Layered neural networks are particularly nice examples ofhierarchical systems. Distributed representation can be computational.

(Leitgeb 2001, Artificial Intelligence.Leitgeb 2004, Inference in the Low Level, Kluwer-Springer.Leitgeb 2005, Synthese & French edition of Scientific American.)







ϕ0 ⇒ ϕ1, ϕ1 ⇒ ϕ2, . . . , ϕj−1 ⇒ ϕj , ϕj ⇒ ϕ0

ϕ0 ⇒ ϕj(Loop)









ϕ0 ⇒ ϕ1, ϕ1 ⇒ ϕ2, . . . , ϕj−1 ⇒ ϕj , ϕj ⇒ ϕ0

ϕ0 ⇒ ϕj(Loop)









ϕ0 ⇒ ϕ1, ϕ1 ⇒ ϕ2, . . . , ϕj−1 ⇒ ϕj , ϕj ⇒ ϕ0

ϕ0 ⇒ ϕj(Loop)









ϕ0 ⇒ ϕ1, ϕ1 ⇒ ϕ2, . . . , ϕj−1 ⇒ ϕj , ϕj ⇒ ϕ0

ϕ0 ⇒ ϕj(Loop)









ϕ0 ⇒ ϕ1, ϕ1 ⇒ ϕ2, . . . , ϕj−1 ⇒ ϕj , ϕj ⇒ ϕ0

ϕ0 ⇒ ϕj(Loop)





Belief revision theorists (cf. AGM 1985) are interested in how to reviseone’s belief set K rationally in the face of new evidence α, i.e.:For which β is it the case that β ∈ K ∗ α?

Grove 1988: representation theorem for all operators ∗ which satisfy theAGM axioms – any ∗ can be put in one-to-one correspondence to a rankedmodel of truth value assignments for a propositional language L, i.e.,

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Ramsey test α⇒ β ∈ K iff β ∈ K ∗ α








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Ramsey test α⇒ β ∈ K iff β ∈ K ∗ αHannes Leitgeb (University of Bristol) Applications of Mathematics in Philosophy: Four Case StudiesFebruary 2007 14 / 22

Theorem (Gardenfors)

The axioms of ∗ are inconsistent with the Ramsey test for conditionals(given weak non-triviality assumptions).

Here is a new take on this result:

The logic of subjunctive conditionals was proved valid (by D. Lewis) withrespect to a semantics that is similar to the semantics for ∗:

“If you left Max Bell Room 159 now︸︷︷︸α

, then I would be sad︸︷︷︸β

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So for a language with Lewis-type conditionals, ranked models for ∗ ratherlook like. . .

rehj rrehj rrehjrehj rehjrehj rehj rehj� ��

. . .and Gardenfors’ Impossibility Theorem expresses the fact that the“small” ranking in the least layer cannot correspond to the “big” rankingas suggested by the Ramsey test.

But this sounds familiar:

Arrow’s theorem (1951): There is no function that extends any given set ofindividual rankings 6i of alternatives (for fixed individuals i ∈ {1, . . . , n})to a social ranking 6, such that certain axioms are satisfied.

E.g., Pareto:

(P) If x 6i y for all i , then x 6 y .







E.g., Pareto:








E.g., Pareto:



Once can show: the assumptions in Arrow’s theorem can be formulated onthe basis of ∗ and ⇒, where Arrow’s individuals and alternatives are“coded” by L-worlds.

It follows that there is a belief revision counterpart of Arrow’s theorem:

Theorem

K∗1–K∗8, IIA, P, ND are jointly inconsistent (given backgroundassumptions on belief sets and on the number of possible worlds involved).

Proof: Translation of Arrow’s proof.

The Pareto condition P is just the left-to-right direction of the Ramseytest for conditionals:

(P) For all L⇒-consistent belief sets K , for all α, β ∈ L:if α⇒ β ∈ K , then β ∈ K ∗ α.

(Leitgeb 2005, unpublished manuscript.Leitgeb& Segerberg 2007, Synthese KRA.Leitgeb 2007, “Beliefs in Conditionals vs. Conditional Beliefs”,Topoi.)




Theorem









Theorem









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In view of Tarski’s result, we know that we cannot have a type-free truthpredicate and at the same time accept all T-biconditionals.

Indeed, there is no subjective probability measure P that assigns to eachformula of the form

Tr(‘α’) ↔ α

probability 1 (with α ∈ L for L semantically closed).

New approach: Instead of believing that Tr(‘α’) and α are equivalent, wemight rather assign the same degree of belief to Tr(‘α’) and α!

Formalized: let L be the first-order language of arithmetic extended by Tr .

QUESTION: Is there a function P : L → [0, 1], such that

P satisfies the analogues of standard probability axioms.

P satisfies:P(Tr(‘α’)) = P(α)

P assigns 1 to all commutation axioms for Tr .





Tr(‘α’) ↔ α












Tr(‘α’) ↔ α












Tr(‘α’) ↔ α







P assigns 1 to all commutation axioms for Tr .Hannes Leitgeb (University of Bristol) Applications of Mathematics in Philosophy: Four Case StudiesFebruary 2007 18 / 22

Well, almost!

Theorem

1 There is no σ-additive measure P that satisfies all the above.

2 There is a finitely additive measure P that satisfies all the conditionsabove except for σ-additivity.

Proof: 1. McGee’s fixed-point formula. 2. Hahn-Banach.

What about a “Liar” sentence λ with P(λ↔ ¬Tr(‘λ’)) = 1?

P(λ) = P(¬Tr(‘λ’))

= 1− P(Tr(‘λ’))

= 1− P(λ)

Hence, P(λ) = P(¬λ) = 12 .

(Leitgeb 2005, unpublished manuscript.)

Similar methods can be used to investigate type-free probability.


Well, almost!

Theorem





P(λ) = P(¬Tr(‘λ’))

= 1− P(Tr(‘λ’))

= 1− P(λ)

Hence, P(λ) = P(¬λ) = 12 .




Well, almost!

Theorem





P(λ) = P(¬Tr(‘λ’))

= 1− P(Tr(‘λ’))

= 1− P(λ)

Hence, P(λ) = P(¬λ) = 12 .




Well, almost!

Theorem





P(λ) = P(¬Tr(‘λ’))

= 1− P(Tr(‘λ’))

= 1− P(λ)

Hence, P(λ) = P(¬λ) = 12 .




Well, almost!

Theorem





P(λ) = P(¬Tr(‘λ’))

= 1− P(Tr(‘λ’))

= 1− P(λ)

Hence, P(λ) = P(¬λ) = 12 .




Well, almost!

Theorem





P(λ) = P(¬Tr(‘λ’))

= 1− P(Tr(‘λ’))

= 1− P(λ)

Hence, P(λ) = P(¬λ) = 12 .




Conclusions

We found:

Hypergraph theory can be used to determine under which conditionsproperties can be reconstructed from similarity.

Dynamical systems theory can be used to justify systems ofnonmonotonic logic. Both together throw new light on the symboliccomputationalism vs. connectionism debate.

Social choice theory can be used to improve our understanding oflimitative results on belief revision with conditionals.

Functional analysis can be used to support a doxastic account oftype-free truth.

Even in philosophy, calculemus!


Meaning Similarity and Compositionality

Goodman (1972): Linguistic expressions are hardly ever synonymous butmuch more often merely similar in meaning.

Churchland vs. Fodor&Lepore on Meaning Similarity vs. Compositionality

QUESTION: Which general postulates does a semantic resemblancerelation ∼ on a language L satisfy?












SIM ∼⊆ L× L is reflexive and symmetric.

CONN ∼ is connected.

So: for all α, β ∈ L there is a sequence

α = γ1 ∼ γ2 ∼ . . . ∼ γn = β

NON-TRIV No tautology stands in the ∼-relation to any contradiction.

COM¬ For all α, β ∈ L: if α ∼ β then ¬α ∼ ¬β.

COM∧ For all α, β, γ, δ ∈ L: if α ∼ β and γ ∼ δ then α∧ γ ∼ β ∧ δ.COM∨ For all α, β, γ, δ ∈ L: if α ∼ β and γ ∼ δ then α∨ γ ∼ β ∨ δ.

Theorem

No relation satisfies SIM, CONN, NON-TRIV, COM¬, COM∧, COM∨.

Proof: “Shrink” ∼-chains up to logical equivalence.

Conclusion: Either meaning similarity has to go or compositionality has tobe “softened”.

(Leitgeb 2006, unpublished draft.)





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Theorem









α = γ1 ∼ γ2 ∼ . . . ∼ γn = β




Theorem






Documents

Applications of Mathematics in Philosophy: Four Case Studiesrzach/static/banff/leitgeb.pdf · B. Russell, R. Carnap: generalized abstraction; abstracting “similarity classes”