Gradational Accuracy and nonclassical logic:

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Gradational Accuracy and nonclassical logic:
a graphical guide
J. R. G. Williams, University of Leeds
Worlds and probability space
NB: Until the generalized to nonclassical probabilitiesall the material here is an informal report of ideasand proofs from Joyce “A nonpragmatic vindicationof probabilism”, Philosophy of Science (1998).
Possible worlds correspond in a oneone way with “truth value assignments”.
(I’ll use the black/greencircles for functions from each proposition to a number in [0,1])
What is a (classical) probability?• Think of dividing total credence (=1) amongst a number of complete,
consistent, possible situations. • Probabilities map oneone onto assignments of credences to worlds. • The probability of a proposition P
= the sum of the credence assigned to worlds where P is true.
Interesting fact to remember: • Probabilities are “weighted averages” of truth value assignments.• Take a “weighted average” of two probabilities; you get another probability • …because weighted averages of weighted averages are weighted averages!
The space of (classical) probabilities is the “closure under mixing of (classical) truth value assignments”
Distances in belief spaceWe have a measure of
“how accurate an arbitrary belief state is, given the way the world is”.
This is the accuracy score.
For our purposes, we will need to work with a more general notion: “how far apart two arbitrary belief states are”.
The reductio argument for domination
We’ve given a definition of the candidate probability function that is supposed to “accuracy dominate” B.
G=The closest probability function to B (we can prove this is unique).
Can we show it accuracy dominates B, or are we being fooled by pictures?
Required to prove: for arbitrary W, GW is less than BW.
Anything “on the line” between two probabilityfunctions, WG, is a probability function.
(Probabilities closedunder “mixing”).
G
W
B
Geometrically, since WG is longer than WB,we can see there’s a point on the line WGwhich is closerto B than is G.
(Main task of rigorous proof:check this to be so, givenonly the structure induced by axioms for score).
B G
W
Putting this together, we find a probability function G* closer to B than G is.
This contradicts the construction of G.
GB
G*
W
Reductio!
So WG must be shorter than WB. I.e. G is closer to W than B is.
What our proof depends on:
1. The fact we can think of worlds as truth value assignments 2. … and hence as (extreme) probabilities. 3. The fact that accuracy scores induce a distance among arbitrary
probabilities. 4. The fact that any mixture (weighted average) of two probability functions
is a probability function. 5. The fact that this “distance” behaves geometrically as you’d expect.
But accuracy domination argument relied only on closure property, not anything to do with classicality.
B*
G*
Any set “closed under averages” will be such that “accuracy domination” holds.
“Generalized probabilities” are the minimal set(i) containing each (generalized) world
(ii) closed under weighted averages
B*
G*
• We have a characterization of generalized probabilities, relative to a generalized notion of truthvalue assignment.
• We can prove an accuracydomination theorem for these, just as for classical probabilities/classical truthvalue assignments.
• Can we find an illuminating axiomatization? • I know how to generalize the classical probability axioms, such that any
weighted average as above must satisfy these axioms.
Generalized logic AB iff on no truth value assignment, A>B.
Axioms for generalized probability (mostly):1. If B then P(B)=12. If A then P(A)=03. If AB then not: P(A)>P(B)4. P(A)+P(B)=P(A&B)+P(AvB)
In fact, (4) depends on our truth value assignments satisfying:A+B=A&B+AvBWhere e.g. we have inequality on TVs, we get inequality version of (4).
Applications
Take a logic characterized via designated value (e.g. Kleene, LP, supval,int). • If a model assigns X a designated value, its truth value is 1 (“true”)• If a model assigns X a nondesignated value, its truth value is 0 (“untrue”)The probability theory axiomatized via this logic is related via accuracy
domination to such truthvalue assignments
Take a logic with characterized via linear [0,1] “no drop in truth value” (Lukasiewicz,DegLog)
• Let truth value assignments simply be the values assigned by models. The probability theory axiomatized via this logic is related via accuracy
domination to such truthvalue assignments