Gradational Accuracy and non-classical logic:

a graphical guide

J. R. G. Williams, University of Leeds

Worlds and probability space

NB: Until the generalized to non-classical probabilitiesall the material here is an informal report of ideasand proofs from Joyce “A non-pragmatic vindicationof probabilism”, Philosophy of Science (1998).

Ways things might have been

Possible worlds correspond in a one-one way with “truth value assignments”.

(I’ll use the black/greencircles for functions from each proposition to a number in [0,1])

Probability assignments

We can find the truth-valueassignments here! (extremalprobability functions)

What is a (classical) probability?• Think of dividing total credence (=1) amongst a number of complete,

consistent, possible situations. • Probabilities map one-one 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”.

Score=n

Score=m

A

B

W

C=W+(B-A)

Score=k

Distance(A,B):=Score(W,C)=Score(W,W+(A-B))=k

B

A

C

The idea of accuracy domination

B

G

Let G be the closest probability function to B

No matter how the world is, G is more accurate than B

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.

W

B G

Suppose for reductio that G is further from W than is B(GW>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.

Extending the result

We now allow “non-classical” possible worlds(in addition to classical ones).

We can represent them as truth-value assignments, once again.

“Weighted averages” of classical worlds gave classical probabilities

Weighted averages of all worlds give new functions: “non-classical probabilities”.

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 truth-value assignment.

• We can prove an accuracy-domination theorem for these, just as for classical probabilities/classical truth-value 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|+|AvB|Where 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 non-designated value, its truth value is 0 (“untrue”)The probability theory axiomatized via this logic is related via accuracy

domination to such truth-value 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 truth-value assignments