Download Probabilistic Theories of Type

Probabilistic Theories of
Type-Free Truth and Probability
Hannes Leitgeb
LMU Munich
March 2014
In the logical and philosophical literature on probability, probabilities are often
assigned to sentences (or formulas) rather than to sets:
‘P (x ) = r ’: the probability of the sentence x is r .
In the logical and philosophical literature on probability, probabilities are often
assigned to sentences (or formulas) rather than to sets:
‘P (x ) = r ’: the probability of the sentence x is r .
Accordingly, the axioms of probability are expressed in terms of syntactic
counterparts of the standard Kolmogorov axioms:
P : L → [0, 1].
For all logically true α in L : P (α) = 1.
For all α, β in L such that |= ¬(α ∧ β):
P (α ∨ β) = P (α) + P (β)
(Sometimes σ-additivity / countable additivity is added—more later.)
In the following, we will be interested in
the probability of sentences saying something about their own truth value
(Part I),
the probability of sentences saying something about their own probability
(Parts II-IV).
Not many people have worked on this—but eventually this area might become
quite relevant to epistemology and philosophy of science, and perhaps even
more so than formal theories of type-free truth and necessity.
Leitgeb, “On the Probabilistic Convention T”, Review of Symbolic Logic, 2008.
Leitgeb, “From Type-Free Truth to Type-Free Probability”, New Waves in Philosophical Logic,
2012.
Christiano et al., “Definability of Truth in Probabilistic Logic”, unpublished draft, 2013.
Caie, “Rational Probabilistic Incoherence”, Philosophical Review, 2013.
Part I: Probabilities of Type-Free Truth
We know from Tarski’s work that we cannot have a type-free truth predicate
and at the same time accept all T-biconditionals (given a reasonable theory of
syntax in the background).
Part I: Probabilities of Type-Free Truth
We know from Tarski’s work that we cannot have a type-free truth predicate
and at the same time accept all T-biconditionals (given a reasonable theory of
syntax in the background).
One way of putting this: there is no probability measure P that assigns to each
formula of the form
Tr (pαq) ↔ α
probability 1 (with α ∈ LTr for LTr semantically closed).
Part I: Probabilities of Type-Free Truth
We know from Tarski’s work that we cannot have a type-free truth predicate
and at the same time accept all T-biconditionals (given a reasonable theory of
syntax in the background).
One way of putting this: there is no probability measure P that assigns to each
formula of the form
Tr (pαq) ↔ α
probability 1 (with α ∈ LTr for LTr semantically closed).
New approach: Instead of accepting that Tr (pαq) and α are equivalent,
we might rather assign the same probability to Tr (pαq) and α!
(This assumption is weaker than the previous one.)
Formally: Let LTr be the first-order language of arithmetic extended by Tr .
(For convenience, we will use the standard model of arithmetic as our “ground
model”; we also fix a recursive coding scheme for LTr .)
Question: Is there a function P : LTr → [0, 1], such that:
P satisfies the axioms of probability.
P satisfies the Probabilistic Convention T:
P (Tr (pαq)) = P (α)
P assigns 1 to all arithmetical truths and to all commutation axioms for Tr
(Tr (p¬αq) ↔ ¬Tr (pαq), Tr (p∀n α[n]q) ↔ ∀nTr (pα[n]q), and so on).
Well, almost!
Well, almost! It all depends on σ-additivity:
for all α in LTr , P (∀n α[n]) = lim P (α[0] ∧ . . . ∧ α[n]).
n→∞
Theorem
1
There is no σ-additive measure P that satisfies all conditions from before.
2
There is a finitely additive measure P that satisfies all conditions from
before (but not σ-additivity).
Well, almost! It all depends on σ-additivity:
for all α in LTr , P (∀n α[n]) = lim P (α[0] ∧ . . . ∧ α[n]).
n→∞
Theorem
1
There is no σ-additive measure P that satisfies all conditions from before.
2
There is a finitely additive measure P that satisfies all conditions from
before (but not σ-additivity).
Proof (Leitgeb 2008):
1. Use McGee’s (1985) fixed-point: µ ↔ ¬∀nTr (pTr n (pµq)q).
Well, almost! It all depends on σ-additivity:
for all α in LTr , P (∀n α[n]) = lim P (α[0] ∧ . . . ∧ α[n]).
n→∞
Theorem
1
There is no σ-additive measure P that satisfies all conditions from before.
2
There is a finitely additive measure P that satisfies all conditions from
before (but not σ-additivity).
Proof (Leitgeb 2008):
1. Use McGee’s (1985) fixed-point: µ ↔ ¬∀nTr (pTr n (pµq)q).
2. Construct a revision sequence (Gupta & Belnap 1993) up to ω;
take the so-called Banach limit of frequencies of truth:
!
!!!!1!
!!!1!
!!!!2/3!
!!3/4!
!3/5!
¬α!
Tr(‘α’)!
α!
Tr(‘¬α’)!
¬α!
Tr(‘α’)!
.....!
α!
α!
Tr(‘α’)!
!Banach!Limit!P!
Example: Consider a “Liar” sentence λ with P (λ ↔ ¬Tr (pλq)) = 1.
P (λ) = P (¬Tr (pλq))
= 1 − P (Tr (pλq))
= 1 − P (λ)
Hence, P (λ) = P (¬λ) = 12 .
This is plausible independently. (Note that this is not three-valued logic!)
Example: Consider a “Liar” sentence λ with P (λ ↔ ¬Tr (pλq)) = 1.
P (λ) = P (¬Tr (pλq))
= 1 − P (Tr (pλq))
= 1 − P (λ)
Hence, P (λ) = P (¬λ) = 12 .
This is plausible independently. (Note that this is not three-valued logic!)
!!!!1!
λ!
!!1/ !
2
¬λ!
!
Tr(‘λ’)!
!!!!2/3!
!!1/2!
!!!λ!
¬λ!
!
!
Tr(‘¬λ’)!
!!!Tr(‘λ’)!
!3/5!
.....!
!!λ!
Tr(‘¬λ’)!
!Banach!Limit!P!
Some further remarks:
The theory can be regarded as a quantitative version of the Revision
Theory of Truth (Gupta & Belnap 1993):
P (α) may be understood as the relative frequency of α’s being true in the
long run (in the given revision sequence).
Some further remarks:
The theory can be regarded as a quantitative version of the Revision
Theory of Truth (Gupta & Belnap 1993):
P (α) may be understood as the relative frequency of α’s being true in the
long run (in the given revision sequence).
Assume P satisfies our desiderata:
The set of sentences which are certain according to P is a superset of the
axiomatic system FS and therefore must be ω-inconsistent (McGee 1985).
Some further remarks:
The theory can be regarded as a quantitative version of the Revision
Theory of Truth (Gupta & Belnap 1993):
P (α) may be understood as the relative frequency of α’s being true in the
long run (in the given revision sequence).
Assume P satisfies our desiderata:
The set of sentences which are certain according to P is a superset of the
axiomatic system FS and therefore must be ω-inconsistent (McGee 1985).
Take the revision sequence that starts with the empty extension of Tr , take
any of the Ps that exist by 2. in our proof of theorem 1:
Is there a sentence α for which P (α) = 1, although α never becomes
stably true in the revision-theoretic sense?
Yes! E.g.,
∃n(∀m(Tr m (p0 = 0q) ↔ m ≤ n) ∧ ¬∃k n = k 2 )
We have seen that the probability of the Liar is pinned down uniquely by our
postulates.
How about a “Truth teller” τ for which τ ↔ Tr (pτq) is derivable arithmetically by
the diagonalization lemma?
We have seen that the probability of the Liar is pinned down uniquely by our
postulates.
How about a “Truth teller” τ for which τ ↔ Tr (pτq) is derivable arithmetically by
the diagonalization lemma?
Theorem
For every r ∈ [0, 1] there is a P, such that
P satisfies our postulates from before (other than sigma-additivity),
P (τ) = r .
Part II: Probabilities of Type-Free Truth and Probability
So far ‘P’ has been on the metalevel.
Now we take this one step further:
New language LTr ,P : add a unary function sign P to LTr .
Part II: Probabilities of Type-Free Truth and Probability
So far ‘P’ has been on the metalevel.
Now we take this one step further:
New language LTr ,P : add a unary function sign P to LTr .
New ground model: extend the standard model of arithmetic to the
standard model of the real closed field (and have a predicate N for ‘natural
number’ available); expand the latter by countably many constants for real
numbers and countably many predicates and function signs for real
relations and functions, as desired.
New coding: fix a recursive coding function for LTr ,P .
Goal:
Find nice axiomatic theories of type-free truth and probability;
determine nice models of such theories.
Here is a suggestion: Theory PT1
Base theory: PA + Real closed field + . . .
Axioms for truth: Commutation axioms for Tr , truth biconditionals
restricted to L .
Axioms for probability:
∀α(LTr ,P (α) → 0 ≤ P (α) ≤ 1)
L
∀α(LTr ,P (α) ∧ ProvBTTr ,P (α) → P (α) = 1)
L
∀α, β(LTr ,P (α) ∧ LTr ,P (β) ∧ ProvBTTr ,P (p¬(α ∧ β)q) →
P (α ∨ β) = P (α) + P (β))
1
Tr ,P (α)
∀α (F
→ P (p∃x α[x ]q) = limn→∞ P (pα[1] ∨ . . . ∨ α[n])q))
Rules for truth:
Rules for probability:
α
Tr (pαq)
Tr (pαq)
α
α
P (pαq) = 1
P (pαq) > 1 − ε
α
ε–Scheme for probability & truth:
∀α(LTr ,P (pαq) → |P (pTr (pαq)q) − P (pαq)| < ε)
Theorem
PT1 is consistent.
Proof (Leitgeb 2012): Determine a revision sequence up to ω, by defining
P (pαq) in model Mn+1 as the relative frequency of α being true in M1 , . . . , Mn .
PT1 is contained in the set of stably true formulas in this sequence.
!
.....!
α!
α!
Tr(‘α’)!
¬α!
Tr(‘α’)!
α!
Tr(‘¬α’)!
¬α!
Tr(‘α’)!
P(‘α’)!=!!3/4!
Here is another suggestion: Theory PT2
Base theory: PA + Real closed field + . . .
Axioms for probability and truth:
∀α(LTr ,P (α) → 0 ≤ P (α) ≤ 1)
∀α(LTr ,P (α) ∧ ProvPT1 (α) → P (α) = 1)
∀α, β(LTr ,P (α) ∧ LTr ,P (β) ∧ ProvPT1 (p¬(α ∧ β)q) →
P (pα ∨ βq) = P (α) + P (β))
∀α(LTr ,P (α) → P (pTr (pαq)q) = P (α))
∀α(LTr ,P (α) ∧ P (pP (pαq) = 1q) > 0 →
P (pTr (pαq)q|pP (pαq) = 1q) = 1)
∀α(LTr ,P (pαq) ∧ ProvPT1 (pP (pαq) ∈ [r , s]q) → P (pαq) ∈ [r , s])
Theorem
PT2 is consistent.
Proof (Leitgeb 2012): Determine a revision sequence up to ω as in the proof of
the last theorem; determine a Banach limit P as in the proof of our first
theorem.
PT2 is satisfied by that P.
.....!
α!
α!
Tr(‘α’)!
¬α!
Tr(‘α’)!
α!
Tr(‘¬α’)!
¬α!
Tr(‘α’)!
P(‘α’)!=!!3/4!
!Banach!Limit!P!
Here is a third suggestion: Christiano et al. (unpublished draft).
They work in a language without truth but with type-free probability: LP .
Base theory: PA + Real closed field + . . .
Axioms for probability:
∀α(LP (α) → 0 ≤ P (α) ≤ 1)
LP
∀α(LP (α) ∧ ProvBT
(α) → P (α) = 1)
LP
∀α, β(LP (α) ∧ LP (β) ∧ ProvBT
(p¬(α ∧ β)q) →
P (pα ∨ βq) = P (α) + P (β))
Probabilistic Reflection:
∀α, a, b(LP (α) ∧ Q(a) ∧ Q(b) ∧ a < P (α) < b → P (pa < P (pαq) < bq) = 1)
E.g., what is the probability of β, where β ↔ P (pβq) < 1 is derivable
arithmetically and has probability 1?
Here is a third suggestion: Christiano et al. (unpublished draft).
They work in a language without truth but with type-free probability: LP .
Base theory: PA + Real closed field + . . .
Axioms for probability:
∀α(LP (α) → 0 ≤ P (α) ≤ 1)
LP
∀α(LP (α) ∧ ProvBT
(α) → P (α) = 1)
LP
∀α, β(LP (α) ∧ LP (β) ∧ ProvBT
(p¬(α ∧ β)q) →
P (pα ∨ βq) = P (α) + P (β))
Probabilistic Reflection:
∀α, a, b(LP (α) ∧ Q(a) ∧ Q(b) ∧ a < P (α) < b → P (pa < P (pαq) < bq) = 1)
E.g., what is the probability of β, where β ↔ P (pβq) < 1 is derivable
arithmetically and has probability 1? By Reflection: P (β) = 1.
Theorem
Christiano et al.’s theory is consistent.
Proof (Christano et al., unpublished): Kakutani’s fixed point theorem.
Part III: A Comparison of Theories of Type-Free Probability
How do the Banach limit theories relate to the Kakutani fixed point theory?
Part III: A Comparison of Theories of Type-Free Probability
How do the Banach limit theories relate to the Kakutani fixed point theory?
Question 1: σ-additivity failed given our Probabilistic Convention T.
What happens given Christiano et al.’s Probabilistic Reflection?
Part III: A Comparison of Theories of Type-Free Probability
How do the Banach limit theories relate to the Kakutani fixed point theory?
Question 1: σ-additivity failed given our Probabilistic Convention T.
What happens given Christiano et al.’s Probabilistic Reflection?
The same: one cannot have σ-additivity given Reflection and the background
assumption that P assigns 1 to certain laws on real numbers (and to the
statement that P assigns real numbers).
For consider β with β ↔ P (pβq) < 1 being derivable arithmetically.
Assume P assigns 1 to the equivalence.
By Reflection: P (β) = 1.
Hence: For all n > 1, 1 − n1 < P (β) < 1 + n1 .
By Reflection: For all n > 1, P (1 − n1 < P (pβq) < 1 + n1 ) = 1.
σ-additivity would entail: P (∀n > 1: 1 − n1 < P (pβq) < 1 + n1 ) = 1.
By P assigning 1 to the respective law on real numbers (and to the statement
that P assigns real numbers): P (P (pβq) = 1) = 1.
And therefore: P (β) = P (P (pβq) < 1) = 0. Contradiction.
(Indeed, σ-additivity fails drastically, as
P (∀n > 1: 1 − n1 < P (pβq) < 1 + n1 ) = 0 !)
Question 2: How much Probabilistic Reflection does one get in the Banach limit
approach?
If the sequence of values of P (pαq) converges (in the normal sense) in
!
.....!
α!
α!
Tr(‘α’)!
¬α!
Tr(‘α’)!
α!
Tr(‘¬α’)!
¬α!
Tr(‘α’)!
!Banach!Limit!P!
P(‘α’)!=!!3/4!
then
if a < P (α) < b, then P (a < P (pαq) < b) = 1.
Question 3: How much Probabilistic Convention T does one get in the Kakutani
fixed point approach?
[Auxiliary question: What would play the role of Tr (pαq) in Christiano et al.?
P (pαq) = 1? ∀n > 1 P (p1 − n1 < P (pαq) < 1 + n1 q) = 1? P (pαq) > 12 ?]
If P (α) = 1, then for all n > 1:
P (α) = P (1 −
1
n
1
< P (pαq) < 1 + )
n
Part IV: σ-additivity as the Linchpin of Type-Free Probability
A little new (and useful) theorem:
Theorem
The following theory (in the language LP ) of type-free probability is consistent
but ω-inconsistent:
Robinson arithmetic.
The axioms of probability for P (except for σ-additivity).
σ-additivity of P.
Necessitation:
α
P (pαq) = 1
Proof: Apply McGee (1985) to ‘P (p.q) = 1’.
So one of the central questions when studying and applying type-free
probability in the future will be:
What shall we give up?
Will it be σ-additivity or probabilistic necessitation?
(Or even finite additivity?)