A logical and algebraic treatment of conditional
probability
Tommaso Flaminio,Franco Montagna
Department of Mathematics and Computer Science
Pian dei Mantellini44,53100Siena
Italy e-mail:{flaminio,montagna}@unisi.it
Abstract
This paper is devoted to a logical and algebraic treatment of condi-tional probability.The main ideas are the u of non-standard probabili-
ties and of some kind of standard part function in order to deal with the
ca where the conditioning event has probability zero,and the u of a
many-valued modal logic in order to deal probability of an eventϕas the
truth value of the ntenceϕis probable,along the lines of H´a jek’s book
[H98]and of[EGH].To this purpo,we introduce a probabilistic many-
valued logic,called F P(S LΠ),which is sound and complete with respect
a class of structures having a non-standard extension[0,1] of[0,1]as t
of truth values.We also prove that the coherence of an asssment of con-
ditional probabilities is equivalent to the coherence of a suitably defined
theory over F P(S LΠ)who proper axioms reflect the asssment itlf.
1Introduction
There are two important mathematical theories of uncertainty:the theory of probability and the theory of fuzzy ts.(Strictly speaking,the theory of fuzzy ts is a mathematical theory of vagueness,but here we are using the word un-certainty in a very broad n).Although difference,the theories are related each other.Their logic counterparts are probability logics and fuzzy logics.The book[CS]c
真人cs作文>哭哭啼啼的意思
ontains an attempt to treat the t-norm s of many-valued logic by means of conditional probability.On the other hand,H´a jek’s book[H98]con-tains a treatment of probabilistic logic inside a modal fuzzy logic.The idea is that the probability of an eventϕmay be regarded as the truth’s degree of the modal formulaϕis probable,denoted by P r(ϕ).Of cour,the modal logic of probability is not bad on classical logic,becau intermediate values are needed,but rather on many-valued logic.The most appropriate many-valued logic ems to be L ukasiewicz Logic,becau in such logic it is possible to ex-press additivity of the probability operator P r.Moreover,it is uful to have some constants corresponding to rational truth-values.Thus H´a jek in[H98] introduces a probabilistic many-valued logic with rational truth values,denoted
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by F P(RP L).Since this logic is crucial in order to understand H´a jek’s many-valued approach to probability,we describe it since now.
The logic F P(RP L)(Fuzzy Probabilistic Rational Pavelka Logic)has infinitely many propositional variables p0,...,p n,...,propositional constants¯q for every rational q∈[0,1],the connectives→and&of L ukasiewicz Logic,and the modal operator P r.Formulas split into two class,Boolean formulas and modal formulas,which are inductively defined as follows:
(i)The t BF of Boolean formulas is the smallest t containing all propo-
sitional variables and clod under the connectives∨,∧and¬.(In L u-kasiewicz logic,the connectives are definable byϕ∧ψ=ϕ&(ϕ→ψ),ϕ∨ψ=(ϕ→ψ)→ψ,and¬ϕ=ϕ→¯0,but in the definition of Boolean formulas we can u them as primitive symbols).
(ii)The t MF of modal formulas is the smallest t containing all formulas of the form P r(ϕ),ϕis a Boolean formula,as well as all formulas of the form¯q,q is a rational in[0,1],and clod under→and&.
The axioms and rules of F P(RP L)are as follows:
(L)All axioms and rules of L ukasiewicz Logic,e[H98],restricted to modal formulas.
(C)All axioms of Classical Logic restricted to Boolean formulas.
(R)All axioms for rational constants,namely:
All axioms of the form¯r↔(¯p→¯q),(whereϕ↔ψstands for(ϕ→
ψ)&(ψ→ϕ))for all p,q,r∈Q∩[0,1]such that min{1,1−p+q}=r.
练字帖楷书All axioms of the form¯r↔(¯p&¯q)for all p,q,r∈Q∩[0,1]such that
r=max{0,p+q−1}.
(Pr)The axiom schemes P r(¬ϕ∨ψ)→(P r(ϕ)→P r(ψ)),P r(¬ϕ)↔¬P r(ϕ) and P r(ϕ∨ψ)↔((P r(ϕ)→P r(ϕ∧ψ))→P r(ϕ)).
for any Boolean formulaϕ.
(N)The Necessitation Rule:ϕ
P r(ϕ)
A Boolean evaluation is a map v from the t BF of Boolean formulas into {0,1}such that v(¬A)=1−v(A),v(A∧B)=min{v(A),v(B)},v(A∨B)= max{v(A),v(B)}.
The mantics for F P(RP L)is constituted by the class of probabilistic Kripke models.The structures are systems W,µ,e where:
W is a non-empty t,who elements are called nodes,and e is a map from W×BF into{0,1}such that for all w∈W,e(w,·)is a Boolean evaluation.
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µ(but possibly not countably additive)probability measure on a Boolean subalgebra of the powert of W which takes values in[0,1],such that for every Boolean formulaϕ,the t Wϕ={w∈W:e(w,ϕ)=1}is µ-,µ(Wϕ)is defined).
Given a probabilistic Kripke model M= W,µ,e and a node w∈W,the truth value ϕ M,w of a formulaϕin M at the node w is defined inductively as follows:
Ifϕis Boolean,then ϕ M=e(w,ϕ).
Ifϕis a propositional constant¯q,then ϕ M,w=q.
P r(ϕ) M,w=µ(Wϕ).
ϕ&ψ M,w=max{0, ϕ M,w+ ψ M,w−1},and
ϕ→ψ M,w=min{1,1− ϕ M,w+ ψ M,w}.
Ifϕis a modal formula,then ϕ M,w is independent of w,so we will omit the subscript w.We say that M i
s a model of a modal formulaϕ(denoted by M|=ϕ)if ϕ M=1.IfΓis a t of formulas,then we say that M is a model ofΓ(denoted by M|=Γ)if for allψ∈Γ,M|=ψ.
In[H98]it is proved that F P(RP L)is sound and complete with respect to the class of probabilistic Kripke models.This means the following:as usual,for every logic L,for every tΓof ntences of L and for every ntenceϕof L, say that L∪Γ ϕiffthere is afinite quenceϕ1,...,ϕn of ntences of L such thatϕn=ϕ,and for i=1,...,n,eitherϕi is an axiom of L,orϕi∈Γ,orϕi can be derived by a rule of L from afinite number of formulas of the formϕj with j<i.Also,letΓ|=LϕL∪Γ(in a n which will be specified for every theory we will deal with)is a model ofϕ.Then:
Theorem1.1LetΓbe a t of modal formulas of F P(RP L),and letϕbe a modal formula of F P(RP L).Then,F P(RP L)∪Γ ϕiffΓ|=F P(RP L)ϕ. Thus F P(RP L)is adequate for a treatment of simple probability.
Another fundamental contribution to the study of the relationship between prob-ability and many-valued logic is the book chapter[MR],where the Caratheodory-von Neumann approach to algebraic probability theory is extended to a many-valued framework.However,this(very deep)investigation is carried on in a purely algebraic tting,and it ems very hard to translate the sophisticated tools involved into a logical language.
A crucial concept in probability theory is that of conditional probability.Thus a natural problem related to the connections between probability and many-valued logic is the investigation of logics which are suitable for the treatment of conditional probability.
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This problem is afforded in[EGH].The central idea is that product conjunction &Π(who truth function is ordinary product in[0,1])has a residual→Πwho truth function⇒Πbehaves like a truncated division,namely:
x⇒Πy= y关于想象的名言
x
if y≤x 1otherwi
Thus if the truth value of P r(ψ)is non-zero,we can express the conditional probability P r(ϕ|ψ)ofϕgivenψby(the truth value of)P r(ψ)→ΠP r(ϕ∧ψ).In[EGH],Esteva,Godo and H´a jek introduce a probabilistic logic which is bad on LΠ,a logic introduced by Esteva,Godo and Montagna in[EGM] which includes both L ukasiewicz and product conjunctions and their residuals. Roughly speaking,su
ch a probabilistic logic results from LΠby the adding of a probabilistic modality P r and the axioms and rules of F P(RP L).
Following the(very smart)ideas,the authors obtain a logical treatment of conditional probability which works well in all cas,except the one where the conditioning event has probability0.Indeed,if P r(ψ)has value0,then the formula P r(ψ)→ΠP r(ψ∧ϕ)has always truth-value1,and this fact does not correctly reflect the mathematical concept of conditional probability.
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In order to overcome such difficulty,we u a non-stadard approach to prob-ability([N],[K]).The(not new)idea is the following:wefirst introduce a non-standard probability P r such that only the impossible event may have probablity0,but a non-impossible event may have an infinitesimal probability. Then the non-standard conditional probability P r (ϕ|ψ)may be expresd as P r (ϕ∧ψ)
P r (ψ)
,which is the truth value of the formula P r(ψ)→ΠP r(ψ∧ϕ).More-over,we can go back to standard probability(thus allowing for non-impossible events of probability0)by taking the standard part,that is defining a standard probability P r by P r(ϕ)=St(P r (ϕ)),where St is the standard part function which associates to every non-standard realα∈[0,1]the unique standard real βsuch that the distance|α−β|b
etweenαandβis infinitesimal.
We illustrate this idea by an example:the unit square[0,1]2has measure0in [0,1]3.Thus if we choo a point at random in[0,1]3(with uniform distribution), the probability of choosing an element of[0,1]2is0.Still,it is reasonable to say
that the probability of choosing a point in[0,1
2]2given that the point belongs
to[0,1]2is1
4.This can be expresd by means of non-standard probability
as follows:think of the unit square as a parallelepiped of dimensions1,1and
ε,whereεis a positive infinitesimal.Similarly,[0,1
2]2can be regarded as a
parallelepiped with dimensions1
2,1
2
andε.Letϕdenote the event:we choo a
point in[0,1]2,and letψdenote the event we choo a point in[0,1
2]2.Then the
non-standard probability ofϕis given by P r (ϕ)=ε,and similarly P r (ψ)=
1 2·1
2
·ε=ε
4
,therefore,taking the standard parts,we get P r(ϕ)=P r(ψ)=0,
and P r(ψ|ϕ)=St(ε
4ε)=St(1
4
)=1
4
.
The aim of the prent paper is to express all the ideas in a logical and algebraic formalism.To this purpo,we introduce a class of algebras which are
roughly LΠ1
2-algebras A( LΠ-algebras added by a constant for1
2
)with a unary
4
operationσwhich is an idempotent endomorphism of the PMV-reduct of
of the reduct of A relative to the L ukasiewicz operations and constants⊕,∼,0
and1,and to product·.(σis suppod to reprent the function standard part,
but this function is not evenfirst-order axiomatizable,so what we really get is
only some approximation of it).The algebras obtained in this way are called
S LΠ-algebras.
The logic counterpart of such algebras is a logic,S LΠ,which is roughly LΠ1
2 with an additional operator S satisfying some axioms which are just the logic
translation of the axioms forσin the definition of S LΠ-algebras.
Then we introduce a probabilistic logic F P(S LΠ)who language is that
of F P(RP L)plus product conjunction&Πand product implication→Π,and
who axioms and rules are tho of S LΠplus tho of F P(RP L).Here,the
truth values of P r(ϕ)and P r(ψ)→ΠP r(ϕ∧ψ)are suppod to reprent the
non-standard probability ofϕand the non-standard conditional probability ofϕ
givenψrespectively.The corresponding standard probabilities are reprented
by S(P r(ϕ))and by S(P r(ψ)→ΠP r(ϕ∧ψ)).
The paper is organized as follows:in Section2we collect some known results
which will be ud throughout the paper.
In Section3,we introduce the logic S LΠand its algebraic mantics,namely,the
variety of S LΠ-algebras.We also prove that every S LΠ-algebra is isomorphic唐朝的诗人
to a subdirect product of linearly ordered S LΠ-algebras,and we derive some
uful conquences of this result.
In Section4,we prove some kind of non-standard completeness for S LΠ.That
is,we prove that S LΠis complete with respect to interpretations in ultrapow-
ers[0,1] of the LΠ1
2-algebra[0,1] L
Π1
2
on[0,1]equipped with an idempotent评价功能
endomorphismσof the PMV-reduct of[0,1] .
In Section5we introduce the probabilistic variant F P(S LΠ)of S LΠ,and we prove that it is complete with respect to the class of Kripke models in which the probabilistic measure takes values in a non-standard extension[0,1] of
全国疟疾日[0,1] L
Π1
2,and S is interpreted as an idempotent endomorphismσof the PMV-
reduct of[0,1] .
Finally,in Section6we apply the results of the previous ctions to the problem of the coherence of an asssment of conditional probabilities.We prove that probabilistic coherence is equivalent to the logical coherence of a theory over F P(S LΠ)who proper axioms reflect the probability asssment.
2Preliminary notions
In this ction we introduce some algebraic concepts which will be ud in order to describe the mantics of the probabilistic logics taken into consideration. Definition2.1([BF00]).A hoop is an algebra H, ,⇒,1 such that H, ,1 is a commutative monoid,and⇒is a binary operation such that the
5
following identities hold:
x⇒x=1,x⇒(y⇒z)=(x y)⇒z and x (x⇒y)=y (y⇒x).
A Wajsberg hoop is a hoop satisfying the equation(x⇒y)⇒y=(y⇒x)⇒x.A bounded hoop is a hoop equipped by a constant0such that the identity 0⇒x=1holds.A MV-algebra is a structure A,⊕,∼,0,1 such that letting x y=∼(∼x⊕∼y)and x⇒y=∼x⊕y,the algebra A, ,⇒,0,1 is a bounded Wajsberg hoop.
In the quel,in any MV-algebra we put x y=∼(∼x⊕y),x y=x (x⇒y), x y=(x⇒y)⇒y,and x≤y iffx⇒y=1.Note that≤is a distributive lattice order,and and are the corresponding operations of join and meet ([CMO]).
A typical MV-algebra,which generates the whole variety of MV-algebras([CMO]), is[0,1]MV= [0,1],⊕,∼,0,1 ,where[0,1]is the unit real interval,x⊕y= min{x+y,1},and∼x=1−x.
Let·denote the restriction of ordinary product to[0,1],and let for all x,y∈
[0,1],
x⇒Πy= y
x
if y≤x 1otherwi
Then⇒Πis the residuum of·,i.e.,one has x·y≤z iffx≤y⇒Πz.Let
[0,1] L
Π= [0,1],⊕,∼,·⇒Π,0,1 ,and let[0,1] L
Π1
2
= [0,1],⊕,∼,·⇒Π,0,1,1
2
.
In[EGM]it is shown that the elements of the variety generated by[0,1] L
Π,
called LΠ-algebras,can be described as follows:
Definition2.2A LΠ-algebra is an algebra A= A,⊕,∼,·,⇒Π,0,1 such that, letting∼Πx=x⇒Π0,andδ(x)=∼Π∼x,the following conditions hold: A,⊕,∼,0,1 is a MV-algebra.
A,·,⇒Π,0,1 is a bounded hoop.
For all x,y,z∈A one has:x·(y z)=(x·y) (x·z).
δ(x⇒y)≤x⇒Πy≤x⇒y.
δ(x)≤δ(x⇒y)⇒δ(y)
δ(x) ∼δ(x)=1
δ(x)≤x
δ(δ(x))=δ(x)
δ(1)=1
δ(x y)=δ(x) δ(y)
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