Pure inductive logic

Pure inductive logic (PIL) is the area of mathematical logic concerned with the philosophical and mathematical foundations of probabilistic inductive reasoning. It combines classical predicate logic and probability theory (Bayesian inference). Probability values are assigned to sentences of a first-order relational language to represent degrees of belief that should be held by a rational agent. Conditional probability values represent degrees of belief based on the assumption of some received evidence.

PIL studies prior probability functions on the set of sentences and evaluates the rationality of such prior probability functions through principles that such functions should arguably satisfy. Each of the principles directs the function to assign probability values and conditional probability values to sentences in some respect rationally. Not all desirable principles of PIL are compatible, so no prior probability function exists that satisfies them all. Some prior probability functions however are distinguished through satisfying an important collection of principles.

## Contents

* 1 History
* 2 Framework
* 2.1 General case
* 2.2 Unary case
* 3 Central question
* 4 Rational principles
* 4.1 General rational principles
* 4.2 Further rational principles for unary PIL
* 5 Relationships between principles
* 5.1 General Case
* 5.2 Unary case
* 6 Important probability functions
* 6.1 General probability functions
* 6.2 Further probability functions (unary PIL)
* 7 Representation theorems
* 8 Notes
* 9 References

## History

Inductive logic started to take a clearer shape in the early 20th century in the work of William Ernest Johnson and John Maynard Keynes, and was further developed by Rudolf Carnap. Carnap introduced the distinction between pure and applied inductive logic,[1] and the modern Pure Inductive Logic evolves along the lines of the pure, uninterpreted approach envisaged by Carnap.

## Framework

### General case

In its basic form, PIL uses first-order logic without equality, with the usual connectives ∧,∨,¬,-> (and, or, not and implies respectively), quantifiers ∃,∀, finitely many predicate (relation) symbols, and countably many constant symbols a1,a2,a3,….

There are no function symbols. The predicate symbols can be unary, binary or of higher arities. The finite set of predicate symbols may vary while the rest of the language is fixed. It is a convention to refer to the language as L and write

L={R1,R2,…,Rq}

where the Ri list the predicate symbols. The set of all sentences is denoted SL. If a sentence is written with constants appearing in it listed then it is assumed that the list includes at least all those that appear. 𝒯L is the set of structures for L with universe {a1,a2,a3,…} and with each constant symbol ai interpreted as itself.

A probability function for sentences of L is a function w with domain SL and values in the unit interval [0,1] satisfying the following conditions:

– any logically valid sentence θ has probability 1: w(θ)=1

– if sentences θ and ϕ are mutually exclusive then w(θ∨ϕ)=w(θ)+w(ϕ)

– for a formula ψ(x) with one free variable the probability of ∃xψ(x) is the limit of probabilities of ψ(a1)∨ψ(a2)∨…∨ψ(an) as n tends to ∞.

This last condition, which goes beyond the standard Kolmogorov axioms (for finite additivity) is referred to as Gaifman's Axiom and it is intended to capture the idea that the ai exhaust the universe.

For a probability function w and a sentence ϕ with w(ϕ)>0, the corresponding conditional probability function w(.|ϕ) is defined by

w(θ∣ϕ)=w(θ∧φ)w(φ) (θ∈SL).

Unlike belief functions in many valued logics, it is not the case that the probability value of a compound sentence is determined by the probability values of its components. Probability respects the classical semantics: logically equivalent sentences must be given the same probability. Hence logically equivalent sentences are often identified.

A state description for a finite set of constants is a conjunction of atomic sentences (predicates or their negations) instantiated exclusively by these constants, such that for any eligible atomic sentence either it or its negation (but not both) appears in the conjunction.

Any probability function is uniquely determined by its values on state descriptions. To define a probability function, it suffices to specify nonnegative values of all state descriptions for a1,…,an (for all n) so that the values of all state descriptions for a1,…,an,an+1 extending a given state description for a1,…,an sum to the value of the state description they all extend, with the convention that the (only) state description for no constants is a tautology and that has value 1.

If Θ is a state description for a set of constants including ai,aj then it is said that ai,aj are indistinguishable in Θ, ai∼Θaj, just when upon adding equality to the language (and axioms of equality to the logic) the sentence Θ∧ai=aj is consistent. ∼Θ is an equivalence relation.

### Unary case

In the special case of Unary PIL, all the predicates R1,…,Rq are unary. Formulae of the form

β(x)=±R1(x)∧±R2(x)∧…∧±Rq(x)

where ±R stands for one of R, ¬R, are called atoms. It is assumed that they are listed in some fixed order as β1,β2,…,β2q.

A state description specifies an atom for each constant involved in it, and it can be written as a conjunction of these atoms instantiated by the corresponding constants. Two constants are indistinguishable in the state description if it specifies the same atom for both of them.

## Central question

Assume a rational agent inhabits a structure in 𝒯L but knows nothing about which one it is. What probability function w should s/he adopt when w(θ) is to represent his/her degree of belief that a sentence θ is true in this ambient structure?

## Rational principles

### General rational principles

The following principles have been proposed as desirable properties of a rational prior probability function w for L.

The constant exchangeability principle, Ex. The probability of a sentence θ(a1,a2,…,am) does not change when the a1,a2,…,am in it are replaced by any other m-tuple of (distinct) constants.

The principle of predicate exchangeability, Px. If R,R′ are predicates of the same arity then for a sentence θ,

w(θ)=w(θ′)

where θ′ is the result of simultaneously replacing R by R′ and R′ by R throughout θ.

The strong negation principle, SN. For a predicate R and sentence θ,

w(θ)=w(θ′)

where θ′ is the result of simultaneously replacing R by ¬R and ¬R by R throughout θ.

The principle of regularity, Reg. If a quantifier-free sentence θ is satisfiable then w(θ)>0.

The principle of super regularity (universal certainty), SReg. If a sentence θ is satisfiable then w(θ)>0.

The constant irrelevance principle, IP. If sentences θ,ϕ have no constants in common then w(θ∧ϕ)=w(θ)⋅w(ϕ).

The weak irrelevance principle, WIP. If sentences θ,ϕ have no constants nor predicates in common then w(θ∧ϕ)=w(θ)⋅w(ϕ).

Language invariance principle, Li. There is a family of probability functions wJ, one on each language J, all satisfying Px and Ex, and such that wL=w and if all predicates of J belong also to K then wJ and wK agree on sentences of J.

The (strong) counterpart principle, CP. If θ,θ′ are sentences such that θ′ is the result of replacing some constant/relation symbols in θ by new constant/relation symbols of the same arity not occurring in θ then

w(θ∣θ′)≥w(θ).

(SCP) If moreover θ″ is the result of replacing the same and possibly also additional constant/relation symbols in θ by new constant/relation symbols of the same arity not occurring in θ then

w(θ∣θ′)≥w(θ∣θ″)≥w(θ).

The Invariance Principle, INV. If F is an isomorphism of the Lindenbaum-Tarski algebra of sentences of L supported by some permutation μ of 𝒯L in the sense that for sentences θ,ϕ,

F([θ])=[ϕ] just when M⊨θ⟺μ(M)⊨ϕ

then w(θ)=w(ϕ).

The Permutation Invariance Principle, PIP. As INV except that F is additionally required to map (equivalence classes of) state descriptions to (equivalence classes of) state descriptions.

The Spectrum Exchangeability Principle, Sx. The probability w(Θ) of a state description Θ depends only on the spectrum of Θ, that is, on the multiset of sizes of equivalence classes with respect to the equivalence relation ∼Θ.

Li with Sx. As the Language Invariance Principle but all the probability functions in the family also satisfy Spectrum Exchangeability.

The Principle of Induction, PI. Let Θ be a state description and ak a constant not appearing in Θ. Let Φ, Ψ be state descriptions extending Θ to include (just) ak. If ak is ∼Φ-equivalent to some and at least as many constants as it is ∼Ψ-equivalent to then w(Φ∣Θ)≥w(Ψ∣Θ).

### Further rational principles for unary PIL

The Principle of Instantial Relevance, PIR. For a sentence θ, atom β and constants ak,am not appearing in θ,

w(β(ak)∣β(am)∧θ)≥w(β(ak)∣θ).

The Generalized Principle of Instantial Relevance, GPIR. For quantifier-free sentences ψ(ak),ϕ(am),θ with constants ak,am not appearing in θ, if ψ(x)⊨ϕ(x) then

w(ψ(ak)∣ϕ(am)∧θ)≥w(ψ(ak)∣θ).

Johnson Sufficientness Principle, JSP. For a state description Θ for n constants, atom β and constant ak not appearing in Θ, the probability

w(β(ak)∣Θ)

depends only on n and on the number of constants for which Θ specifies β.

The Principle of Atom Exchangeability, Ax. If τ is a permutation of {1,2,…,2q} and Θ is a state description expressed as a conjunction of instantiated atoms then w(Θ)=w(Θ′) where Θ′ obtains from Θ upon replacing each βi by βτ(i).

Reichenbach's Axiom, RA. Let βhi for i=1,2,3,… be an infinite sequence of atoms and β an atom. Then as n tends to ∞, the difference between the conditional probability

w(β(an+1)∣βh1(a1)∧βh2(a2)∧…∧βhn(an))

and the proportion of occurrences of β amongst the βh1,βh2,…,βhn tends to 0.

Principle of Induction for Unary languages, UPI. For a state description Θ, atoms βi,βj and constant ak not appearing in Θ, if Θ specifies βi for at least as many constants as βj then

w(βi(ak)∣Θ)≥w(βj(ak)∣Θ).

Recovery. Whenever Ψ(a1,a2,…,an) is a state description then there is another state description Φ(an+1,an+2,…,ah) such that w(Φ∧Ψ)≠0 and for any quantifier-free sentence θ(ah+1,ah+2,…,ah+g),

w(θ(ah+1,ah+2,…,ah+g)|Φ∧Ψ)=w(θ(ah+1,ah+2,…,ah+g)).

Unary Language Invariance Principle, ULi. As Li, but with the languages restricted to the unary ones.

ULi with Ax. As ULi but with all the probability functions in the family also satisfying Atom Exchangeability.

## Relationships between principles

### General Case

Sx implies Ex, Px and SN.

PIP + Ex implies Sx.

INV implies PIP and Ex.

Li implies CP and SCP.

Li with Sx implies PI.

### Unary case

Ex implies PIR.

Ax is equivalent to PIP.

Ax+Ex implies UPI.

Ax+Ex is equivalent to Sx.

ULi with Ax implies Li with Sx.

## Important probability functions

### General probability functions

Functions VM. For a given structure M∈𝒯L and θ∈SL,

VM(θ)={1ifM⊨θ,0otherwise.

Functions ωΨ. For a given state description Ψ(a1,a2,…,aK), ωΨ is defined via specifying its values for state descriptions as follows. ωΨ(Θ(a1,a2,…,an)) is the probability that when ah1,ah2,…,ahn are randomly picked from {a1,…,aK}, with replacement and according to the uniform distribution, then Ψ(a1,…,aK)⊨Θ(ah1,ah2,…,ahn).

Functions  ∘(ωΨ). As above but employing a non-standard universe (starting with a possibly non-standard state description Ψ) to obtain the standard  ∘(ωΨ).

∙ The  ∘(ωΨ) are the only probability functions that satisfy Ex and IP.

Functions up‾. For a given infinite sequence p‾=⟨p0,p1,p2,p3,…⟩ of non-negative real numbers such that

p1≥p2≥p3≥…≥0 and ∑i=0∞pi=1,

up‾ is defined via specifying its values for state descriptions as follows:

For a sequence c→=⟨c1,c2,…,cn⟩ of natural numbers and a state description Θ(a1,a2,…,an), Θ is consistent with c→ if whenever cs=ct≠0 then as∼Θat. C(c→) is the number of state descriptions for a1,a2,…,an consistent with c→. up‾(Θ) is the sum over those c→ with which Θ is compatible, of

C(c→)−1∏s=1npcs.

∙ The up‾ are the only probability functions that satisfy WIP and Li with Sx. (The language invariant family witnessing Li with Sx consists of the functions up‾,J with fixed p‾, where up‾,J is as up‾ but defined with language J.)

### Further probability functions (unary PIL)

Functions wc→. For a vector c→=⟨c1,c2,…,c2q⟩ of non-negative real numbers summing to one, wc→ is defined via specifying its values for state descriptions as follows:

wc→(Θ)=∏j=12qcjmj

where mj the is number of constants for which Θ specifies βj.

∙ The wc→ are the only probability functions that satisfy Ex and IP (they are also expressible as  ∘(wΨ) ).

Carnap continuum functions cλ. For λ>0, the probability function cλ is uniquely determined by the values

cλ(βj(an+1)∣Θ)=mj+λ2−qn+λ

where Θ is a state description for n constants not including ak and mj is the number of constants for which Θ specifies βj.

Furthermore, c∞ is the probability function that assigns 2−nq to every state description for n constants and c0 is the probability function that assigns 2−q to any state description in which all constants are indistinguishable, 0 to any other state description.

∙ The cλ are the only probability functions that satisfy Ex and JSP.

∙ They also satisfy Li – the functions cλJ with fixed λ, where cλJ is as cλ but defined with language J provide the unary language-invariant family members.

Functions wδ. For −(2q−1)−1≤δ≤1, wδ is the average of the 2q functions wc→ where c→ has all but one coordinate equal to each other with the odd coordinate differing from them by δ, so

wδ=2−q∑i=12qwei→

where ei→=⟨γ,γ,…,γ,γ+δ,γ,…,γ⟩, (γ+δ in ith place) and γ=2−q(1−δ).

For 0≤δ≤1, the wδ are equal to up¯ for

p¯=⟨1−δ,δ,0,0,0,…⟩

and as such they satisfy Li.

∙ The wδ are the only functions that satisfy GPIR, Ex, Ax and Reg.

∙ The wδ with 0≤δ<1 are the only functions that satisfy Recovery, Reg and ULi with Ax.

## Representation theorems

A representation theorem for a class of probability functions provides means of expressing every probability function in the class in terms of generic, relatively simple probability functions from the same class.

Representation Theorem for all probability functions. Every probability function w for L can be represented as

w=∫𝒯LVMdμ(M)

where μ is a σ-additive measure on the σ-algebra of subsets of 𝒯L generated by the sets

{M∈𝒯L∣M⊨θ}(θ∈SL).

Representation Theorem for Ex (employing non-standard analysis and Loeb Integration Theory[2]). Every probability function w for L satisfying Ex can be represented as

w=∫A∘(ωΨ)dμ(Ψ)

where A is an internal set of state descriptions for a1,a2,…,aν (with ν a fixed infinite natural number) and μ is a σ-additive measure on a σ-algebra of subsets of A .

Representation Theorem for Li with Sx. Every probability function w for L satisfying Li with Sx can be represented as

w=∫𝔹up‾dμ(p‾)

where 𝔹 is the set of sequences

p‾=⟨p0,p1,p2,p3,…⟩

of non-negative reals summing to 1 and such that p1≥p2≥p3≥…≥0 and μ is a σ-additive measure on the Borel subsets of 𝔹 in the product topology.

de Finetti's Representation Theorem (unary). In the unary case (where L is a language containing q unary predicates), the representation theorem for Ex is equivalent to:

Every probability function w for L satisfying Ex can be represented as

w=∫𝔻wx→dμ(x→).

where 𝔻 is the set of vectors x→=⟨x1,x2,…,x2q⟩ of non-negative real numbers summing to one and μ is a σ-additive measure on 𝔻.

## Notes

1. ↑ Rudolf Carnap (1971). A Basic System of Inductive Logic, in Studies in Inductive Logic and Probability, Volume 1, pp 69-70.
2. ↑ Cutland, N.J., Loeb measure theory, in Developments in Nonstandard Mathematics, Eds. N.J.Cutland, F.Oliveira, V.Neves, J.Sousa-Pinto, Pitman Research Notes in Mathematics Series, Vol. 336, Longman Press, 1995, pp151-177.

## References

* Jeff Paris, Alena Vencovská (2015). Pure Inductive Logic, Cambridge University Press.
* Jeff Paris (2010). Guangzhou Winter School Course on Pure Inductive Logic (PDF).
* Jeff Paris (2012). Munich Formal Epistemology Workshop Slides (PDF).
* Alena Vencovská (2017). Prague Pure Inductive Logic Course Notes (PDF), with exercises.

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