distances between formal...

General idea Axiomatic distance Conceptual distance

Distances between Formal TheoriesJoint work with

Mohamed Khaled (Bahcesehir University) andMichelle Friend (George Washington University)

arXiv:1807.01501

Koen Lefever1 Gergely Szekely2

1Vrije Universiteit Brussel

2Alfred Renyi Institute for Mathematics

Makkai80 Conference, Budapest 2019-06-20

Koen Lefever, Gergely Szekely Distances between Formal Theories

http://arxiv.org/abs/1807.01501


General idea Axiomatic distance Conceptual distance Motivating example General framework

Traditionally, empirical adequacy is the criterion to comparecompeting theories.

However, this is not always straightforward:

Early Copernican heliocentrism provided less accuratepredictions than the old Prolemaic geocentric model; see[Kuhn 1957].One theory may be better suited to explain one part of theempirical data, another theory may be better suited to explainother parts of reality.

We will here present another way to approach the comparisonof theories, independent of empirical adequacy.



A motivating example



Kin:={AxEField,AxEv,AxSelf,AxSymD,AxLine,AxTriv,AxNoAcc}

ClassicalKin:=Kin ∪ {AxEther,AbsTime,AxThExp+}

SpecRel:= Kin ∪ {AxPhc,AxThExp}

Theorem:

ClassicalKin ` Worldview transformations are Galileantransformations.

Theorem: (Andreka–Madarasz–Nemeti, 1998)

SpecRel ` Worldview transformations are Poincare transformations.



Definition

A translation is a function between formulas of languagespreserving the logical connectives, i.e. Tr(φ ∧ ψ) = Tr(φ) ∧ Tr(ψ),etc.

Definition

An interpretation of theory T1 in theory T2 is a translation Trwhich translates all tautologies and all axioms of T1 into theoremsof T2.

Definition

A definitional equivalence exists between two theories if thosetheories can be interpreted in each other and if all formulas fromboth theories translated into the other theory and back are logicalequivalent to the original formulas.



Theorem:

There is an interpretation Tr of SpecRel in ClassicalKin.

SpecRel

Ether

FTL-IOb

ClassicalKinTr



Theorem:

There is no definitional equivalence between SpecRel andClassicalKin.

SpecRel

EtherFTL-IOb

ClassicalKinTr



How can we make SpecRel and ClassicalKin equivalent?



Adding and removing concepts to make both theoriesequivalent:

Removing FTL observers from ClassicalKin

Adding a “primitive ether” to SpecRel



ClassicalKinSTL:=Kin ∪ {AxEther,AbsTime,AxThExpSTL,AxNoFTL}

AxNoFTL :

All inertial observers move slower than light with respect to theether frames.

¬∃m(IOb(m) ∧ ∃e[Ether(e) ∧ SpeedCK

e (m) ≥ ce]).

AxThExpSTL :

Inertial observers can move with any speed which is in the etherframe slower than that of light.

∃h(IOb(h)

)∧ ∀ex y

(Ether(e) ∧ space(x , y) < ce · time(x , y)

→ ∃k IOb(k) ∧W (e, k , x) ∧W (e, k , y)).



SpecRele :=SpecRel ∪ {AxPrimitiveEther}

AxPrimitiveEther :

There is a non-empty class of ether observers, stationary withrespect to each other, which is closed under trivial transformations.

∃e(E (e) ∧ ∀k

[[IOb(k) ∧ (∃T ∈ Triv)wSR

ek = T ]↔ E (k)])



Theorem:

There is an interpretation Tr ′+ of ClassicalKinSTL in SpecRele .SpecRele ` Tr ′+(ClassicalKinSTL)

SpecRel

SpecRele

E Ether

ClassicalKin

FTL-IOb

Tr

Tr ′+

4≡

Id

Tr+|SpecRel = Tr

Tr+(E (x)

)= Ether(x)

Tr+



Theorem:

Tr+ and Tr ′+ are a definitional equivalence between SpecRele andClassicalKinSTL.

Tr′+ ◦ Tr+(SpecRele )⇔ SpecRele and Tr+ ◦ Tr′+(ClassicalKinSTL)⇔ ClassicalKinSTL

SpecRel

SpecRele

E Ether

ClassicalKin

FTL-IOb

Tr

Tr ′+

4≡

Id

Tr+|SpecRel = Tr

Tr+(E (x)

)= Ether(x)

Tr+



Theorem:

There is an interpretation Tr∗ of ClassicalKinSTL in ClassicalKin.ClassicalKin ` Tr∗(ClassicalKin

STL)

Theorem:

There is an interpretation Tr ′∗ of ClassicalKin in ClassicalKinSTL.ClassicalKinSTL ` Tr ′∗(ClassicalKin)



Theorem:

Tr∗ and Tr ′∗ are a definitional equivalence between ClassicalKinSTL

and ClassicalKin.

Tr ′∗ ◦ Tr∗(ClassicalKinSTL)⇔ ClassicalKinSTL and

Tr∗ ◦ Tr ′∗(ClassicalKin)⇔ ClassicalKin

Ether

ClassicalKin

FTL-IOb

Ether

ClassicalKin

FTL-IOb

Tr ′∗4≡

Tr∗ClassicalKinSTL



Theorem:

Tr∗ ◦ Tr+ and Tr ′+ ◦ Tr ′∗ are a definitional equivalence betweenSpecRele and ClassicalKin.

SpecRel

SpecRele

E Ether

ClassicalKin

FTL-IOb

Tr

Tr ′+

4≡

Id

Tr+|SpecRel = Tr

Tr+(E (x)

)= Ether(x)

Tr+

Ether

ClassicalKin

FTL-IOb

Tr ′∗

4≡

Tr∗

ClassicalKinSTL



Equivalence relations:

There are several ways in which theories can be equivalent to eachother:

Logical Equivalence (≡)⇓6⇑

Definitional Equivalence (4≡)

⇓6⇑Conceptual Equivalence (Morita Equivalence,definitional equivalence with different sorts)

⇓6⇑Categorical Equivalence

See [Barrett & Halvorson 2016].



Equivalence relations gives us a trivial (discrete) distance:

Let X -be any set of theories and E any equivalence relation on X .The discrete distance on (X ,E ) is the following:

d(x , y) =

{0 if E (x , y),

1 if ¬E (x , y).



We are going to refine these trivial distances given byequivalence relations between theories.



We define distances between two theories based on the minimumnumber of things (e.g., axioms or concepts) which need to beadded to or substracted from one theory to make it (logically,definitionally, etc.) equivalent to the other theory.

Two theories which are equivalent have a distance of zero.



Definition

A cluster network is a triple (X ,E ,S), where X is a class with anequivalence relation E and a symmetric relation S .

E (x , x ′): x and x ′ are basically the same objects.

S(x , x ′): x or x ′ can be reached from the other in one “step.”



A path between x and x ′ in cluster network (X ,E ,S) is a finitesequence of joining E -edges and S-edges connecting x and x ′.

The length of a path is the number of S-edges in the path.

Objects x , x ′ ∈ X are connected in (X ,E , S) iff there is a pathfrom one of them to the other.



Let X = (X ,E ,S) be a cluster network.

Definition

The step distance on X is the function dX : X × X → N ∪ {∞}defined as:

dX (x , x ′)def≡ min{k ∈ N : ∃ a path from x to x ′ of length k}

if x and x ′ are connected in X , and

dX (x , x ′)def≡ ∞ otherwise

for each x , x ′ ∈ X .



Theorem:

Let dX : X × X → N ∪ {∞} be the step distance on clusternetwork X = (X ,E ,S). Then for each x , y , z ∈ X ,

(a) dX (x , y) ≥ 0, and dX (x , y) = 0 ⇐⇒ x E y .

(b) dX (x , y) = dX (y , x).

(c) dX (x , y) ≤ dX (x , z) + dX (z , y).



Axiomatic distance



Adding one axiom:

T ↽ T ′ iff there is ϕ ∈ Fm such that T ∪ {ϕ} ≡ T ′.

One axiom difference:

T T ′ iff either T ↽ T ′ or T ′ ↽ T .

Definition

Let T be a class of theories. We call the step distance on thiscluster network (T ,≡,) the axiomatic distance AdT on T .



Example

Let T be a class of theories. Let T ,T⊥ ∈ T be two theoriesformulated in the same language. Suppose that T is consistentwhile T⊥ is inconsistent, then AdT (T ,T⊥) = 1.

Consequently, if T ,T ′ ∈ T are formulated on the same languageand an inconsistent theory T⊥ of that language is in T , thenAdT (T ,T ′) ≤ 2 since we have T ⇀ T⊥ ↽ T ′.



Let CON be the class of all consistent theories.

Proposition:

Let T ,T ′ ∈ CON having the same language L. Thenthe axiomatic distance AdCON(T ,T ′) ≤ 3.

Moreover, AdCON(T ,T ′) = 2 if T 6≡ T ′ and they are complete.

Corollary:

Let T ,T ′ ∈ CON. Then,

AdCON(T ,T ′) =∞

m

T and T ′ are formulated in different langauges.



Theorem:

Fix a language L. Let T ∈ CON be any theory on language L andlet ∅L be the empty theory on L. Then:

(1) AdCON(∅L,T ) = 0 iff T is trivial (T ≡ ∅L).

(2) AdCON(∅L,T ) = 1 iff T is finitely axiomatizable andnon-trivial.

(3) AdCON(∅L,T ) = 2 iff T is not finitely axiomatizable, but hasa finitely axiomatizable consistent extension.

(4) AdCON(∅L,T ) = 3 iff T has no finitely axiomatizableconsistent extension.

Example

AdCON(∅L,PA) = 3, because PA (Peano Arithmetic) has no finitelyaxiomatizable consistent extension, see [Ryll-Nardzewski 1952].



By giving weights to the axiom adding steps, e.g., considering theaddition of certain kinds of axioms as two or more steps, one couldincrease the resolution of axiomatic distance.



Conceptual distance



Definition

Theory T ′ of language L′ is a one-concept-extension of theoryT of language L:

T T ′def⇐⇒ L′ = L ∪ {R} and (∀ϕ ∈ L)(T ′ |= ϕ iff T |= ϕ)

for some relation symbol R.

T ! T ′def⇐⇒ T T ′ or T ′ T



Definition

Let T be a class of theories. The conceptual distance CdT is the

step distance on cluster network (T , 4≡,!), where4≡ is the

definitional equivalence.

If T is the class of all theories of FOL, we omit the subscript Tand simply write Cd.

Theorem:

For every n ∈ N ∪ {∞}, there are theories T and T ′ such thatCd(T ,T ′) = n.



Definition

The spectrum of theory T , in symbols I(T , κ), is the number ofits different models of T (up to isomorphism) of cardinality κ.



Theorem:

If T1 and T2 are formulated in countable languages, then:

Cd(T1,T2) <∞

⇓

(∀ cardinal κ)[I(T1, κ) 6= 0 ⇐⇒ I(T2, κ) 6= 0

].

Corollary:

There are infinitely many theories that are, in terms of conceptualdistance, infinitely far from each other.

Proof.: If A and B are two finite models of different cardinality,then Cd

(Th(A),Th(B)

)=∞. Q.E.D.



Theorem:

If T1 and T2 are formulated in finite languages, then:

Cd(T1,T2) <∞

m

(∀ cardinal κ)[I(T1, κ) 6= 0 ⇐⇒ I(T2, κ) 6= 0

].

Corollary:

There are infinitely many theories that are, in terms of conceptualdistance, infinitely far from each other.

Proof.: If A and B are two finite models of different cardinality,then Cd

(Th(A),Th(B)

)=∞. Q.E.D.



Theorem:

If T1 and T2 are formulated in finite languages, then[Cd(T1,T2) <∞ and I(T1, 1) = I(T2, 1) = 0

]⇓

Cd(T1,T2) ≤ 2.



Theorem:

Cd(ClassicalKin, SpecRel) = 1

Proof.:

SpecRel SpecRele4≡ ClassicalKinSTL

4≡ ClassicalKin

Q.E.D.



References:H. Andreka, J. X. Madarasz and I. Nemeti, with contributions from A. Andai,G. Sagi, I. Sain, and Cs. Toke: “On the logical structure of relativity theories”,Alfred Renyi Institute of Mathematics, Hungar. Acad. Sci., (2002).

H. Andreka, J. X. Madarasz and I. Nemeti: “Defining new universes inmany-sorted logic”, Research Report, Alfred Renyi Institute of Mathematics,Hungar. Acad. Sci. (2008).

T. W. Barrett and H. Halvorson: “Morita Equivalence”, The Review ofSymbolic Logic 9 (3): 556-582 (2016).

M. Khaled, G. Szekely, K. Lefever and M. Friend: “Distances between FormalTheories”, arXiv:1807.01501 (2018).

T. Kuhn. “The Copernican Revolution: Planetary Astronomy in theDevelopment of Western Thought” Harvard University Press (1957).

K. Lefever: “Using Logical Interpretation and Definitional Equivalence toCompare Classical Kinematics and Special Relativity Theory”PhD Dissertation, Vrije Universiteit Brussel (2017).

K. Lefever and G. Szekely: “Comparing Classical and Relativistic Kinematics inFirst-Order Logic”, Logique et Analyse, ISSN: 2295-5836, p. 57-117, Vol 61, Nr241 (2018).

C. Ryll-Nardzewski, “The role of the axiom of induction in elementaryarithmetic”, Fundamenta Mathematicae, 39:239263 (1952).



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