Revision Theory of Probability: Catrin Campbell-Moore's Approach, Slides of Construction

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Revision Theory of Probability

Catrin Campbell-Moore

Corpus Christi College, Cambridge

Bristol–M¨unchen Workshop on Truth and Rationality June 2016

Introduction

Language with a type-free truth predicate (T) and type-free probability function symbol (P).

λ ↔ ¬Tpλq π ↔ ¬Ppπq > 1 / 2

Want to determine:

• (^) Which sentences are true,
• What probabilities different sentences receive. or at least some facts about these. E.g.
• (^) T(Tp0 = 0q) = t,
• p(λ) = 1 / 2 ,
• p(ϕ) + p(¬ϕ) = 1.

Outline

Introduction Probability Revision Theory of Truth Revision sequence for truth Revision Theory of Probability Applying revision to probability Strengthening the limit clause Some results Showing they exist Properties of the revision sequences Horsten Conclusions

Outline

Introduction Probability Revision Theory of Truth Revision sequence for truth Revision Theory of Probability Applying revision to probability Strengthening the limit clause Some results Showing they exist Properties of the revision sequences Horsten Conclusions

Semantic Probability

Says how true a sentence is. This semantic probability assigns:

• 1 if ϕ is true
• (^) 0 if ϕ is false. E.g. p(H) = 0 or p(H) = 1. But can give additional information about problematic sentences.

Add in additional probabilistic information to a usual truth construction.

• Kripkean: how many fixed points the sentence is true in.
• (^) Revision: how often the sentence is true in the revision sequence.

Connection to Degrees of Truth

So it’s very similar to a degree of truth. Often Lukasiewicz logic used; to study vagueness.

Difference semantic probability and usual degrees of truth: Compositionality.

• p(ϕ) and p(ψ) don’t fix p(ϕ ∨ ψ) unless ϕ and ψ are logically incompatible.
• DegTruth(ϕ ∨ ψ) = min{DegTruth(ϕ), DegTruth(ψ)}. Though, e.g. Edgington (1997) for degrees of truth as probabilities.

Revision sequence for truth

Revision Theory of Truth

Fix M a model of L. Construct a model of LT by considering the extensions of truth.

T 0 (λ) = f M 0 |= λ

T 1 (λ) = t M 1 6 |= λ

T 2 (λ) = f M 2 |= λ

T 3 (λ) = t M 3 6 |= λ Tn+1(ϕ) = t ⇐⇒ Mn |= ϕ.

At each finite stage some

︷ ︸︸^ n ︷ TpTp... Tp0 = 0q.. .qq is not satisfied. So extend to the infinite stage and get Mω |= ∀nTnp0 = 0q

In fact just going to ω isn’t enough (E.g. Tp∀nTnp0 = 0qq) so need to go to the transfinite.

Revision sequence for truth

Limit stage

At a limit stage α, one “sums up” the effects of earlier revisions: if the revision process up to α has yielded a definite verdict on an element, d,... then this verdict is reflected in the αth^ hypothesis; Gupta and Belnap (1993)

If a definite verdict is brought about by the revision sequence beneath μ then it should be reflected in the μth^ stage.

Outline

Introduction Probability Revision Theory of Truth Revision sequence for truth Revision Theory of Probability Applying revision to probability Strengthening the limit clause Some results Showing they exist Properties of the revision sequences Horsten Conclusions

Applying revision to probability

Applying revision to probability

A revision sequence is a sequence of hypotheses: (T 0 , p 0 ), (T 1 , p 1 ), (T 2 , p 2 )... (Tω, pω), (Tω+1, pω+1)...

To give a revision theory we need to say:

• (^) How to revise (Tα, pα) to get (Tα+1, pα+1).
• (^) How to sum up these into (Tμ, pμ) for limits μ.

Gupta and Belnap (1993) give a general revision theory.

• (^) The revision rule gives the (Tα, pα) 7 → (Tα+1, pα+1).
• (^) For the limit step, one uses:

If a definite verdict is brought about by the sequence beneath μ then it should be reflected in the μth^ stage.

I.e. If p(ϕ) = r is stable beneath μ then pμ(ϕ) = r.

Applying revision to probability

Limit stage

If a definite verdict is brought about beneath μ then it should be reflected at μ.

If p(ϕ) = r is stable beneath μ then pμ(ϕ) = r.

0 = 0.

0 1 2 3 4 5 n

1

pnH.L

So we get: pω(0 = 0) = 1

Applying revision to probability

It’s not powerful enough here

Converging probability

Desired: If pn(ϕ) −→ r as n −→ ω then pμ(ϕ) = r.

Tp0 = 0q.

0 1 2 3 4 5 n

1

pnH.L

Want: pω(Tp0 = 0q) = 1 But this isn’t yet a “definite verdict”.

Applying revision to probability

The limit stage

Reformulated:

If a property of the hypotheses is brought about by the sequence beneath μ then it should be reflected in the μth^ stage.

• Brought about:
  • Stable beneath μ
• Properties to focus on:
  • All definite verdicts: p(ϕ) = r , T(ϕ) = t/f.

Applying revision to probability

The limit stage

Extended to capture convergence:

If a property of the hypotheses is brought about by the sequence beneath μ then it should be reflected in the μth^ stage.

• Brought about:
  • Stable beneath μ
• Properties to focus on:
  • All intervals: r 6 p(ϕ) 6 q, and T(ϕ) = t/f.