RossellaMarrano_ReasoningClub3

Degrees of truth as objective probabilities
Rossella Marrano
Scuola Normale Superiore, Pisa
Joint work with Hykel Hosni
24 June 2014
Rossella Marrano (SNS) Degrees of truth as objective probabilities 24/06/2014 1 / 17

Motivation
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
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)
Overall aim
Justifying the formal overlapping between degrees of truth and belief from a
conceptual point of view by providing a unified framework

Classical probabilistic logic
Language
I L = fp1; p2; : : : g
I :, !
I SL
I ?
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():

Real-valued Łukasiewicz logic
I v : SL ! [0; 1]
1. v(?) = 0
2. v(:) = 1 v()
3. v( ! ) =

1; if 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():

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

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]

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():

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():

Comparison between comparisons
Qualitative probability reformulated:
(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) =) : :
Differences:
1. interpretation: agent-independence
2. the underlying semantics
3. restriction on incompatible events

Interpretation
Truth orders agent-independent, objective
Probability orders agent ordering her beliefs, subjective
Standard interpretation:
I events are in themselves more or less true (vagueness, truthlikeness . . . )
A new interpretation:
I degrees of truth only arise as ultimate degrees of belief
Main claim
Each agent has her (subjective) probability order. If rational agents are forced
to agree by imposing
1. compositionality
2. norms on beliefs
the resulting order is a truth order

Compositionality I
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
I arbitrariness in the choice: no constraints on the probability of
compound events when propositional variables are compatible
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 [working hypothesis]

Compositionality II
Interpretation:
[. . . ] the lack, up to this time, of an accepted, or even acceptable,
semantics for truth functional belief, despite its frequent and continuing
presence in expert systems is, to say the least, “unfortunate.” (A New
Criterion for Comparing Fuzzy Logics for Uncertain Reasoning, A.D.C.
Bennett, J.B. Paris, and A. Vencovská)
I intermediate step
New family of probability orders
1. formally equivalent to the family of truth orders
2. smaller than before: reducing freedom to disagree

Imposing norms on beliefs
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)

Objective probabilities as degrees of truth
I Achieving full objectivity:
1. truth-functional belief
2. imposing norms on beliefs
3. fix a language
4. ultimate knowledge base
I We finally end up with a ultimate, unique, objective probability
order
I this is a truth order!
If is an objective probability order over Łukasiewicz logic then there is a
Łukasiewicz valuation function v : SL ! [0; 1] representing it.
I objective probabilities as degrees of truth

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)
Degrees of truth as ultimate degrees of belief
I bottom-up notion of truth
We’ll never know reality. But insofar we know something,
that’s reality for us. (The old lady I met on the plane)
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)

Conclusion
Truth Belief
Truth values Degrees of truth Degrees of belief
Valuations functions Probability functions
More or less true More or less probable
I construct an objective probability order
I an objective probability order is a truth order
I degrees of truth can be interpreted as objective probabilities
Is this a rehabilitation?
“Probability does not exist.” (de Finetti, 1974)

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.

RossellaMarrano_ReasoningClub3

Recommended

Recommended

More Related Content

Similar to RossellaMarrano_ReasoningClub3

Similar to RossellaMarrano_ReasoningClub3 (16)

Recently uploaded

Recently uploaded (20)

RossellaMarrano_ReasoningClub3