new intertheoretic reduction integrative reduction, conﬁrmation,...

Integrative Reduction, Confirmation,and the Syntax-Semantics Map

Kristina Liefke Sanders

Tilburg Center for Logic and Philosophy of Science

Formal Epistemology Workshop, USC, May 21, 2011

Outline

1 Intertheoretic Reduction

2 Montague Grammar

Montague Grammar and Intertheoretic Reduction

Montagovian Rules and Probabilities

3 Bayesian Networks

4 ‘Montague’ Reduction

5 Integrative Reduction

6 Outlook

Reduction Montague Bayes ‘Montague’ R Integrative R Outlook

Motivation

Why Care about Reduction?

Reductive relations are ubiquitous in science:

Thermodynamics (TD) −→ Statistical mechanics (SM)

Chemistry −→ Atomic physics

Psychology −→ Neuroscience

Advantages of reductive relations:

Simplicity

Explanation

Consistency

Confirmation (Dijzadji-Bahmani et al., 2010b)


Nagelian Reduction

The Nagelian Model of Reduction (Nagel, 1961),

cf. (Dijzadji-Bahmani et al., 2010a)

Assume two theories:

TP the reduced, or phenomenological, theory (TD)

TF the reducing, or fundamental, theory (SM)

With each theory, we associate a set of propositions

TP = {T 1P, . . . ,T

nP} resp. TF = {T 1

F , . . . ,TnF}

TP is reduced to TF in three steps:

1 Connect the vocabularies of TF and TP via bridge laws;

2 Substitute terms from TF by their correspondents from TP;

3 Derive every proposition in TP from a proposition in TF.


Nagelian Reduction

The ‘Fundamental’-‘Phenomenological’ Map

TF

TP

(TF) The kinetic theory of gases:Newton’s equations of motion, thedef.s of pressure and kinetic energy

Bridge laws:bridge laws mean kinetic energy ∼ temperature

Auxiliary assumptions:molecules interact only kinetically,the velocity distribution is isotropic

(TP) The thermal theory of the ideal gas:Boyle-Charles Law: pV = kT .

Assumption The connection bw TF- and TP-terms is bijective.


Montague Grammar

Montague’s ‘Two Theories’ Theory (Montague, 1970; 1973)

Categorial Grammar (CG)

A triple �CAT, E ,G�, with

CAT = {n(oun),v(erb), s(entence), . . . } categoriesE = {En, Ev, Es, . . . } expressionsG = {Gs, . . . } syntactic rules

Syntax An algebra ACG = �E ,Gs, . . . � over the set E .

Model-Theoretic Semantics (MS)

A pair �D, S�, with

D = {Dn,Dv,Ds, . . . } objectsS = {Sn, Sv, Ss} semantic rules

Semantics An algebra AMS = �D, Ss, . . . � over the set D.


Montague Grammar

The Syntax-Semantics Map

Dn� �� Dv� �� John��, �run��, �run��,

,�Mary�� walk�� walk�� T,F

En

D�n D��

n D�v

Ev

D��v Ss

Gs Gv Gn

Sv Snhiiiiii hiiiiii

h h

John, run, John runs,Mary walk Mary walks

Gs. If R ∈ Ev and j ∈ En, then [jR �] ∈ Es.

Ss. If �R� ∈ Dv and �j� ∈ Dn, then �R�(�j�) ∈ Ds.


Reduction

A New Type of Intertheoretic Relation?

The syntax-semantics pair instantiates a specific type ofreduction relation.

Commonalities with Nagelian reduction:

The map h connects objects of the two theories.

The map h enables us to derive propositions of one theoryfrom propositions of the other theory.

Differences from Nagelian reduction:

The relation between CG and MS is a directed relation.

We assume Syntax the reduced theory (TP)Semantics the reducing theory (TF)


Reduction

A New Type of Intertheoretic Reduction?

TF

TP

bridge laws

Ss

Gs

the map h−1

=¶S�s, S��s

©


Reduction

Caveats

Our model of the syntax-semantics relation represents only

one particular type of relation:

Nagelian Reduction

↑Nagel-Schaffner Reduction

↑undirected dependency relations

We distinguish ‘Montague’ Reduction (MR) from Integrative

Reduction (IR).

The syntax-semantics relation is a very weak relation:

Syntax is structurally richer than semantics.

Syntax and semantics have distinct domains of application.


Probabilities

Hypothesis Formation

Rules in G, S are obtained by the scientific method:

1 Isolate syntactic structures in a given text sample;

2 Observe their common structural properties;

3 Propose a hypothesis about their formation;

4 Test the hypothesis by analyzing other samples.

An expression supports a hypothesis if

The expression is intuitively well-formed.

Its structure reflects the assumed formation process.


Probabilities

Montagovian Rules and Probabilities

Rules in G, S are subject to probability attributions.

The probability of a rule is informed by frequentist data.

The frequentist probability of a rule influences a linguist’s

psychological confidence in its descriptive adequacy.

Note Only syntactic rules are directly instantiated.

Semantic rules derive their support from

synactic rules via the assumption of the map h.

Caveat We are NOT interested in a probabilistic extension of

Montague Grammar.


Bayesian Networks

We analyze the relation between confirmation and reduction via

Bayesian networks:

Bayesian network A directed acyclical graph, where

Nodes propositional variables,

Arrows probabilistic dependence relations

between variables

Variables can take different values, assigned by P.

We use the following conventions:

Variables H, E ;

Values H, ¬H := ‘the proposition is true/false’;

E, ¬E := ‘the evidence obtains/

does not obtain’.


Bayesian Networks: Illustration

We frame the confirmatory relation between H and E:

H E

P(H) P(E|H),P(E|¬H)

Figure: The dependence between E and H.

The arrow denotes a direct influence of H on E .

To turn the graph into a Bayesian network, we further need:

The marginal probability distribution for every ‘root’ variable;

The conditional prob’y distribution for every ‘child’ variable.


Advantages

Confirmation and Reduction

Gs

Ss

Gv

Sv

Gn

Sn

Es Ev En


Advantages

The Single-Proposition Case

S G E

Figure: Montagovian dependencies of S , G , E .

S G E

Figure: Pre-reductive dependencies of S , G , E .


Advantages

The Pre- vs. Post-Reductive Situation (1)

S G E S G E

P1(S) = σ P2(S) = σ

P1(G) = γ P2(G|S) = 1 , P2(G|¬S) = 0

P1(E|G) = π , P1(E|¬G) = ρ P2(E|G) = π , P2(E|¬G) = ρ

We assume P2(S) = P1(S), P2(G) = P1(G), P2(E|G) = P1(E|G).

Then, iff σ ∈ (0, 1) and π > ρ, the following holds:

P2(S,G) > P1(S,G)

P2(S,G|E) > P1(S,G|E)d2 > d1


Advantages

The Pre- vs. Post-Reductive Situation (1)

S G E S G E

We assume P2(S) = P1(S), P2(G) = P1(G), P2(E|G) = P1(E|G).

Then, iff σ ∈ (0, 1) and π > ρ, the following holds:

P2(S,G) > P1(S,G)

P2(S,G|E) > P1(S,G|E)d2 > d1

‘Montague’ Reduction increases the probabilities and effects a flowof confirmation between syntactic and semantic propositions.


Advantages

The Problem

Triples �Sk ,Gk ,Ek� (with k ∈ {s,v,n}) remain probabilisticallyindependent.

The probability of the truth of propositions in S, Gcorresponds to the product of their individual probabilities:

P2(�

k

�Sk ,Gk�) = P2(Ss,Gs)P2(Sv,Gv)P2(Sn,Gn)

= P2(Ss)P2(Sv)P2(Sn)

Their joint probability converges to zero as their numberincreases.

The improvement of our model requires insight into the mutualdependencies between same-theory propositions.


Advantages

Montague’s Solution (1)

Stipulate a level of types:

Bs Bv Bn

Ds Dv Dnsets of objects

types

sets of expressions


Advantages


Stipulate a level of types:

Bs Bv Bn

Ds Dv Dnsemantic rules

type-formation rules

syntactic rules


Advantages

Types

Types are Montague’s counterparts of bridge laws:Types connect elements from the two theories.

Types introduce a new (onto-)logical level.

We call this new relation Integrative Reduction.

Different ways of implementing types:

1 Assume a separate type for each category, |TY| = |CAT|.2 Assume fewer types than categories, |TY| < |CAT|.

Assume |TY| > 1.Assume |TY| = 1.


Advantages

Case 1: Separate Types

Assume types e, t, p, where

e := the type of objects in Dn, associated with nouns;

t := the type of objects in Ds, associated with sentences;

p := the type of objects in Dv, associated with verbs.


Advantages

Case 1: Separate Types

Gt

Tt

St

Gp

Tp

Sp

Ge

Te

Se

Et Ep Ee


Advantages

The Three-Typed vs. Untyped Situation (1)

St Tt Gt Et Ss Gs Es

P3(Tt) = τ

P3(St |Tt) = 1 , P3(St |¬Tt) = 0 P2(Ss) = σ

P3(Gt |Tt) = 1 , P3(Gt |¬Tt) = 0 P2(Gs|Ss) = 1 ,P2(Gs|¬Ss) = 0

P3(Et |Gt) = π , P3(Et |¬Gt) = ρ P2(Es|Gs) = π,P2(Es|¬Gs) = ρ

If P3(Tt) = P2(Ss), P3(Et |Gt) = P2(Es|Gs), etc., then

P3(Tt , St ,Gt) = P2(Ss,Gs)

P3(Tt , St ,Gt |Et) = P2(Ss,Gs|Es)

d3 = d2


Improvements


Assume fewer types than categories, |TY| < |CAT|:

Some expressions/objects are associated with basic types;

others with constructions out of basic types (derived types).

Derived types establish connections between same-theorypropositions.

Dependencies between same-theory propositions characterizeIntegrative Reduction.


Improvements

Case 2: Two Types

Assume basic types e, t (not p), where

e := the type of objects in Dn, associated with nouns;

t := the type of objects in Ds, associated with sentences;

Type-p objects are represented by functions Dn → {T,F}.

Let w be inhabited by , , and

�is a dog� := {x ∈ Dn | �is a dog�(x) = T} = { }.


Improvements

Case 2: Two Types

Gt

Tt

St

Gp

Sp

Ge

Te

Se

Et Ep Ee

P4(Tt) = τ , P4(Te) = τ �

P4(St |Tt) = 1 , P4(St |¬Tt) = 0

P4(Se |Te) = 1 , P4(Se |¬Te) = 0P4(Sp |Te ,Tt) = 1 , P4(Sp |¬Te ,Tt) = 0

P4(Sp |Te ,¬Tt) = 0 , P4(Sp |¬Te ,¬Tt) = 0...

...

P4(Et |Gt) = π , P4(Et |¬Gt) = ρ

P4(Ee |Ge) = π� , P4(Ee |¬Ge) = ρ�

P4(Ep|Gp) = π�� , P4(Ep|¬Gp) = ρ��

P4(Te ,Tt , Se , St , Sp,Ge ,Gt ,Gp) = τ τ �;

P∗4 =

τ τ � π π� π��

τ τ � π π� π�� + τ̄ τ̄ � ρ ρ� ρ��d4 := P

∗4−P4 =

τ τ � τ̄ τ̄ � (π π� π�� − ρ ρ� ρ��)τ τ � π π� π�� + τ̄ τ̄ � ρ ρ� ρ��


Improvements

The Two- vs. Three-Typed Situation (1)

To compare both situations, we must first obtain

the probabilities P3(�

j�Tj, Sj,Gj�), P3(�

j�Tj, Sj,Gj|Ej�);

the degree of confirmation

d5 := P3(�

j�Tj, Sj,Gj|Ej�)− P3(�

j�Tj, Sj,Gj�).


Improvements


Gt

Tt

St

Gp

Tp

Sp

Ge

Te

Se

Et Ep Ee

P3(Tt) = τ , P3(Te) = τ �

P3(Tp) = τ �� ,

P3(St |Tt) = 1 , P3(St |¬Tt) = 0

P3(Se |Te) = 1 , P3(Se |¬Te) = 0

P3(Sp|Tp) = 1 , P3(Sp|¬Tp) = 0...

...

P3(Et |Gt) = π , P3(Et |¬Gt) = ρ

P3(Ee |Ge) = π� , P3(Ee |¬Ge) = ρ�

P3(Ep|Gp) = π�� , P3(Ep|¬Gp) = ρ��

P3(Te ,Tt ,Tp, Se , St , Sp, . . . ) = τ τ � τ ��;

P∗3 =

Åτ π

τ π + τ̄ ρ

ãÅτ � π�

τ � π� + τ̄ � ρ�

ãÅτ �� π��

τ �� π�� + τ̄ �� ρ��

ã


Improvements


Gt

Tt

St

Gp

Sp

Ge

Te

Se

Et Ep Ee

Gt

Tt

St

Gp

Tp

Sp

Ge

Te

Se

Et Ep Ee

We assume P4(Tj) = P3(Tj) and P4(Ej |Gj) = P3(Ej |Gj).

If Pi (Tt) · Pi (Te) · Pi (Tp) ∈ (0, 1) , Pi (Ep|Gp) > Pi (Ep|¬Gp),then

P4(Tt ,Te , St , Se , Sp,Gt ,Ge ,Gp) > P3

��j

�Tj , Sj ,Gj

��

P∗4 > P∗

3

d4 > d3 iff (P∗4 − P∗

3) > Pi (Tt) · Pi (Te) · Pi (Tp).


Improvements

Integrative Reduction

Integrative Reduction (IR) A type of intertheoretic relation s.t.

1 There exist structural connections betweensame-theory objects and propositions.

2 The prob’y of the conjunction of propositionsafter the establishment of IR relations is higherthan after the establishment of MR relations.

Conjecture

The propositions’ probability and degree of confirmation isinversely proportional to their number of basic types.


Improvements

Case 3: One Type

Gt

Tt

St

Gv

Sv

Gn

Tn

Sn

Et Ev En

We cannot construct non-t rules.

The assumption of a single type isprobabilistically disadvantageous:

P(∗)1 < P

(∗)5 < P

(∗)3

Sn, Sv are not supported by evidence.

The assumption is confirmationallydisadvantageous: d1 < d6 < d5

The introduction of maps Sk → Gk

raises the theories’ probabilities andconfirmation:

P(∗)1 < P

(∗)5 = P

(∗)3

d1 <iid6iiii= d5


Improvements

Case 3: One Type

Let TYm and TYn be different basic-type sets such that TYm ⊆ TYn.

Theorem

If TYm enables the construction of all linguistically relevant types, then

�m �Sm,Gm� has a higher prior probability than

�n�Sn,Gn�.

�m �Sm,Gm� has a higher posterior probability than

�n�Sn,Gn�.

�m �Sm,Gm� may be better confirmed than

�n�Sn,Gn� under the

difference measure.


Note

1 The one-type case is optimal if the basic type is higher-order:

q := ((e → t) → t)

2 The success of Integrative Reduction is not conditional on theuse of types.

3 The IR model can be adapted to accommodate ‘corrected’versions of propositions (Schaffner, 1974), cf. (Nagel, 1977).


Wrap-Up

We have identified a new type of intertheoretic relation, IR,inspired by (Montague, 1973).

We have compared IR to Nagelian reduction.

We have analyzed IR in the framework of confirmation theory.

We have shown that IR is advantageous over NR:

1 It raises the propositions’ prior and posterior probability.

2 It sometimes raises the propositions’ degree of confirmation.

This is due to constructivity relations bw same-theory objects,and dependency relations bw same-theory propositions.

new intertheoretic reduction integrative reduction, conﬁrmation,...

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