Richer Geometry of Probabilities and Propositions
Explore the rich geometry of probabilities and propositions, aiming to represent Bayesian credences accurately through images and discussions. Delve into serious considerations around accepting propositions and the intricate relationships between probabilities and propositions.
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{ Kevin T. Kelly , Hanti Lin } Carnegie Mellon University This work was supported by a generous grant by the Templeton Foundation.
Propositions B A C
Propositions Probabilities (0, 1, 0) (1/3, 1/3, 1/3) B A C (0, 0, 1) (1, 0, 0)
Propositions Probabilities (0, 1, 0) ? (1/3, 1/3, 1/3) B A C (0, 0, 1) (1, 0, 0)
Propositions Probabilities (0, 1, 0) Acpt (1/3, 1/3, 1/3) B A C (0, 0, 1) (1, 0, 0)
You condition on whatever you accept (Kyburg, Levi, etc.) Very serious business! Does it ever happen? It s not Sunday, so let s buy beer at the super market . You would never bet your life against nothing that what you say to yourself in routine planning are true.
The geometry of probabilities is much richer than the lattice of propositions. Aim: represent Bayesian credences as aptly as possible with propositions.
The geometry of probabilities is much richer than the lattice of propositions. Aim: represent Bayesian credences as aptly as possible with propositions. Acpt B B A C
The geometry of probabilities is much richer than the lattice of propositions. Aim: represent Bayesian credences as aptly as possible with propositions. Acpt B v C B C A
Suppose you accept propositions more probable than 1/2. Consider a 3ticket lottery. For each ticket, you accept that it loses. That entails that every ticket loses. (Kyburg) 1/2 -A -B -C
(0, 1, 0) (1/3, 1/3, 1/3) (1, 0, 0) (0, 0, 1)
A p(A) = 0.8 B C
A p(A v B) = 0.8 A v B B C
A A v C A v B B C
A A Closure under conjunction A v C A v B B C
A A v C A v B B B v C C
A A v C A v B T B B v C C
A t A v C A v B 2/3 T B B v C C
A t A v C A v B 2/3 T B B v C C
A A v C A v B 2/3 T t B B v C C
LMU CMU A A A v C A v C A v B A v B T T B C B C B v C B v C (Levi 1996)
LMU CMU A A A v C A v B A v C A v B T T B C B v C B B v C C (Levi 1996)
LMU CMU A A A v C A v B T B v C B C B C (Levi 1996)
Thats junk! I want a smoother ride! I want tighter handling! designer consumer
Grow up. We can optimize one or the other but not both. designer consumer
Grow up. We can optimize one or the other but not both. LMU steady CMU responsive designer consumer
= Change what you accept only when it is logically refuted. LMU steady = Track probabilistic conditioning exactly. CMU responsive
LMU Steadiness steady CMU responsive Responsive -ness
