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A portion of a workshop presentation by Catrin Campbell-Moore on the Revision Theory of Probability. The theory aims to provide a coherent framework for understanding probability and its relationship to truth. the basics of the theory, some results, and conclusions.
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Catrin Campbell-Moore
Corpus Christi College, Cambridge
Bristol–M¨unchen Workshop on Truth and Rationality June 2016
Language with a type-free truth predicate (T) and type-free probability function symbol (P).
λ ↔ ¬Tpλq π ↔ ¬Ppπq > 1 / 2
Want to determine:
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
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
Says how true a sentence is. This semantic probability assigns:
Add in additional probabilistic information to a usual truth construction.
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.
Revision sequence for 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
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.
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
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:
Gupta and Belnap (1993) give a general revision theory.
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
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
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
Reformulated:
If a property of the hypotheses is brought about by the sequence beneath μ then it should be reflected in the μth^ stage.
Applying revision to probability
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.