logic

1. Degrees of truth as objective probabilities
Rossella Marrano
Scuola Normale Superiore
Joint work with Hykel Hosni
London, 6 June 2014
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2. Motivation
The calculus of probability can be considered as a many-valued logic, and
this point of view is the best one for elucidating the fundamental concept
and logic of probability. But this end is far from being achieved by the
mere conclusion, of a purely formal nature, that the calculus of
probabilities is a many-valued logic; such a conclusion is useful only as a
point of departure, it does not constitute a way of solving the problem, but
only an apt way of expressing it distinctly. (de Finetti, 1935)
I Degrees of truth Vs Degrees of belief
Perplexing observations:
1. theory of probability as a many-valued logic
2. real valued valuation functions as probability functions
Overall aim
Justifying the formal overlapping between degrees of truth and belief from a
conceptual point of view by providing a unified framework
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3. Classical probabilistic logic
Language
I L = fp1; p2; : : : g
I ?
I :, !
I SL
Classical logic
I v : SL ! f0; 1g with truth-tables
I j= () 8v v() = 1
Defined connectives
I := :?
I _ := : !
I ^ := :(: _ :)
A probability function over L is a map P : SL ! [0; 1] satisfying for all
; 2 SL
(P1) if j= then P() = 1,
(P2) if j= :( ^ ) then P( _ ) = P() + P():
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4. Real-valued Łukasiewicz logic
I v : SL ! [0; 1]
1. v(?) = 0
2. v(:) = 1 v()
3. v( ! ) =
1; se v() v();
1 v() + v(); otherwise.
4. v( _ ) = minf1; v() + v()g
5. v( ^ ) = maxf0; v() + v() 1g
I j=1 ( j=)
For all ; 2 SL
(P1) if j=1 then v() = 1,
(P2) if j=1 :( ^ ) then v( _ ) = v() + v():
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5. Degrees of truth Vs degrees of belief
Our proposal
Looking at the corresponding qualitative notions: more or less true/probable
I more fundamental level
I intuitive appeal
I axioms as properties
I independence from the mathematical apparatus
Aim: shedding light on the quantitative side by means of representation
theorems
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6. Ordinal foundations
I comparative judgments
I pairwise evaluation
I X2
I numerical analysis
I point-wise evaluation
I f : X ! R
Representation theorems
If satisfies certain conditions then there exists f such that for all x; y 2 X
x y () f(x) f(y):
I Utility [von Neumann Morgenstern (1947), Savage (1954), Debreu (1954)]
I Probability [de Finetti (1931), Savage (1972), Fine (1973)]
I Truth [Ongoing work with H. Hosni and V. Marra]
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7. No less true than
(T.1) SL2 is complete and transitive
(T.2) , ?
(T.3) j=1 =)
(T.4) 1 2; 1 2 =) 1 _ 1 2 _ 2
(T.5) =) : :
Theorem
If satisfies axioms (T.1)–(T.5) then there exists a unique Łukasiewicz
valuation v : SL ! [0; 1] such that for all ; 2 SL:
=) v() v():
I each ordering satisfying (T.1)–(T.5) is a model for sentences
I ‘objective’ ordering (agent-independent)
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8. No less probable than – de Finetti (1931)
I E; ;;E; [;;
(P.1) is complete and transitive
(P.2) E Ei ;
(P.3) If E1 E2 = ;, F1 F2 = ; and E1 F1,E2 F2 then
E1 [ E2 F1 [ F2
? there are always n incompatible cases equally probable
Theorem
If SL2 satisfies axioms (P.1)–(P.3) and (?) then there exists a probability
function P : SL ! [0; 1] such that for all ; 2 SL:
=) P() P():
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9. Comparison between comparisons
(P.1) is complete and transitive
(P.2) , ?
(P.3) j= =)
(P.4) j= :(1 ^1); j= :(2 ^2); 1 2; 1 2 ) 1 _1 2 _2
(P.5) =) : :
Crucial differences:
I the underlying semantics!
1. restriction on incompatible events
2. interpretation: agent-independence
Claim
Two-step path toward objectivity
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10. From subjective to objective: a first step
Preadditivity:
j= :(1 ^ 1); j= :(2 ^ 2); 1 2; 1 2 =) 1 _ 1 2 _ 2
I The restriction on incompatible events corresponds to the lack of full
compositionality of probability functions [subjective element]
I Removing the restriction while retaining compatibility with classical
logic leads to binary assignments
I Removing the restriction and having Łukasiewicz tautologies as
underlying semantics
New family of probability orders
1. formally equivalent to the family of truth orders
2. more objective (smaller) but still subjective
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11. Interpretation: the second step
Strict Subjectivism additivity is the only constraint that can be normatively
imposed on a rational agent’s degrees of belief
Empirical Subjectivism prior degrees of belief should also be calibrated with
with physical probabilities
Objective Bayesianism degrees of belief should be probabilities, calibrated
with evidence and should otherwise equivocate
All the Bayesian positions accept the fact that selection of degrees of
belief can be a matter of arbitrary choice, they just draw the line in
different places as to the extent of subjectivity. [. . . ] Objectivity is a
matter of degree. (Williamson, 2010)
Achieving full objectivity
If we fix a language and a ultimate knowledge base we finally end up with a
ultimate, unique, objective probability order
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12. Objective probabilities as degrees of truth
If is an objective probability order over Lukasiewicz logic then there is a
Łukasiewicz valuation function v : SL ! [0; 1] representing it.
Philosophical feedback:
I many-valued events? not vagueness but objective uncertainty (chances)
I future events and determinism
Either there will be or there will not be a sea battle tomorrow.
Tertium non datur. (Łukasiewicz)
I degrees of truth as ultimate degrees of belief
I intersubjectivity: consensus-based notion of truth
The opinion which is fated to be ultimately agreed to by all who
investigate, is what we mean by the truth. (Peirce)
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13. Conclusion
Truth Belief
Truth values Degrees of truth Degrees of belief
Valuations functions Probability functions
More or less true More or less probable
I From subjective to objective
1. compositionality
2. agent-independence
I Degrees of truth as objective probabilities
Not exactly a rehabilitation
“Probability does not exist.” (de Finetti, 1974)
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14. References
B. de Finetti.
Sul significato soggettivo della probabilità.
Fundamenta Mathematicae, 17:289–329, 1931.
B. de Finetti.
The Logic of Probability.
Philosophical Studies, 77:181–190, 1935.
B. de Finetti.
Theory of Probability. Vol I.
John Wiley Sons, New. York, 1974.
D. Dubois and H. Prade.
Possibility theory, probability theory and multiple-valued logics: A Clarification.
Annals of Mathematics and Artificial Intelligence, 32:35-66, 2001.
T. L. Fine
Theories of Probability. An Examination of Foundations.
Academic Press, New York and London, 1973.
P. Hájek.
Metamathematics of Fuzzy Logic.
Kluwer Academic Publishers, 1998.
J.B. Paris.
The uncertain reasoner’s companion: A mathematical perspective.
Cambridge University Press, 1994.
J. Williamson
In Defence of Objective Bayesianism.
Oxford University Press, Oxford, 2010.
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