CP=PC p oblem
Adam Olszewski
Augus 18, 2025
Abs ac
This pape e isi s he CP=PC p oblem: he iden i ica ion o he p obabili y o a
condi ional wi h he co esponding condi ional p obabili y. Wo king wi h a ixed classical
p obabili y space and a minimal implica ional backbone ( ules (p1)–(p4) plus Modus
Ponens), we p o e a collapse heo em: i a global CP=PC iden i y is imposed o a single
implica ional connec i e and he logic alida es a na u al symme y (
A−B⊢B−A
),
hen he p obabili y assignmen degene a es on he posi i e-p obabili y agmen ,
i ializing he symme ic ex ension. We u he show ha mo ing o Łukasiewicz logic
does no epai he iden i ica ion: unde u h-a e aging seman ics, he p obabili y o
A⇒B
is a nonlinea expec a ion ha gene ally di e ges om
P
(
B|A
). The upsho is
me hodological: CP=PC should no be ea ed as a law o an implica ional connec i e.
Ins ead, we p opose a di ision o oles: keep he implica ional connec i e o in e ence,
and in oduce a dedica ed condi ional (
B|A
)whose seman ics gua an ees ha i s
p obabili y equals
P
(
B|A
). This e ames he S alnake p og am by sepa a ing
in e en ial s uc u e om p obabilis ic condi ionaliza ion and loca ing CP=PC in he
seman ics o an in e nal condi ional a he han in he implica ional agmen .
1 Backg ound and P oblem Se ing
The long–s anding ques ion known as he condi ional p obabili y e sus he p obabili y o
condi ionals (abb e ia ed CP=PC) asks whe he , and in which logical se ings, one can
iden i y he p obabili y o a condi ional wi h he co esponding condi ional p obabili y. In i s
classical guise (o en called S alnake ’s Thesis), one w i es, o o mulas
A, B
wi h
P
(
A
)
>
0,
P (A→B) = P (B|A) = P (A∧B)
P (A).
This iden i y is immedia ely a ac i e—linking he in e en ial eading o a condi ional o
he calculus o unce ain y—bu i no o iously ails o he ma e ial condi ional and gi es ise
o well–known “ i iali y”/collapse phenomena unde na u al s eng hening assump ions.This
dis inc ion be ween he p obabili y assigned o o mulas and he dynamics o condi ionaliza ion
aligns wi h analyses o p obabilis ic eadings o logical sen ences in ecen wo k, e.g., [Ols22]
Ou s a ing poin is a simple diagnosis: CP=PC mixes wo seman ically di e en oles.
The connec i e “
→
” (o a single bina y connec i e “
−
”) is mean o go e n in e ence inside
1
a p oposi ional logic, while
P
(
B|A
)is a clause abou condi ionaliza ion o a measu e.
T ea ing hese on a pa —by imposing a global iden i y
P
(
A−B
) =
P
(
B|A
)—quie ly
con la es in e en ial laws wi h p obabilis ic upda e and all bu gua an ees pa hology.
We make his ension p ecise in a minimal and obus se ing. Th oughou , we ix a
classical p obabili y space (Ω
,F,P
), in e p e o mulas as e en s ia wo– alued seman ics,
and ead he le –hand side o CP=PC as he same
P
applied o he classical e en
E
(
A−B
).
Wi hin a e y weak implica ional backbone ( he logic o o de , i.e., ules (p1)–(p4) and
Modus Ponens), we p o e a collapse heo em: i one also alida es a na u al global symme y
o he condi ional (
A−B⊢B−A
), hen CP=PC o ces
P
(
E
(
A
)) =
P
(
E
(
B
)) o all
posi i e–p obabili y
A, B
. In sho , he p obabili y assignmen degene a es on he posi i e
agmen , and he symme ic ex ension i ializes. This shows ha he p og am “ ind a
minimal logic ( ia a single connec i e −) ha alida es CP=PC” is ill–posed.
The posi i e lesson is equally clea . CP=PC is no a law o an implica ional connec i e; i
is a seman ic clause o condi ionaliza ion. To gi e CP=PC subs an i e and non–degene a e
con en , one mus sepa a e oles: keep
−
as an in e en ial implica ion (e.g., he Łukasiewicz
esiduum) o suppo easoning ( ules (p1)–(p4) and MP), and in oduce a dedica ed condi-
ional ope a o (
B|A
)whose seman ics is calib a ed so ha i s p obabili y ma ches
P
(
B|A
)
in he in ended p obabilis ic eading. Minimali y hen conce ns he smalles seman ic package
o (·|·), no a “minimal implica ional logic” ia −.
We also explain why appealing o iche u h– alue spaces does no ix he p oblem. In
Łukasiewicz logic
L∞
wi h u h–a e aging, he p obabili y o
A⇒B
becomes a nonlinea
expec a ion Zmin(1,1− (A) + (B)) dµ,
which in gene al does no equal
P
(
B|A
). Thus bo h classically (wi h 0/1 e en s) and
many– aluedly (wi h a e aged u h), iden i ying he p obabili y o an implica ional connec i e
wi h condi ional p obabili y ails wi hou addi ional, non i ial cons ain s.
In summa y, ou con ibu ions a e:
•
a p ecise collapse heo em showing ha global CP=PC o a single implica ional
connec i e (unde a ixed classical P and symme y) o ces degene acy;
•
a no–go co olla y: he e is no non i ial “minimal logic ia
−
” alida ing CP=PC in
he na u al la ice [Co, S];
•
a cons uc i e eo ien a ion: CP=PC belongs o a dedica ed condi ional (
·|·
)wi h
minimal seman ic clauses, while − emains an in e en ial implica ion.
These esul s ecas he S alnake p oblem: he igh a ge is no a minimal implica ional
logic equa ing wo di e en no ions, bu a clean sepa a ion o in e ence and condi ionaliza ion
wi h a calib a ed in e nal condi ional.
De ini ion 1 (Fixed classical in e p e a ion o CP=PC).Le (Ω
,F,P
)be a ixed p obabili y
space. Fo each o mula
A
in
Fo m−
(single bina y connec i e
−
), le
E
(
A
)
∈ F
deno e i s
e en unde classical, wo- alued seman ics. Fo P (E(A)) >0,
P (B|A) := P (E(A)∩E(B))
P (E(A)) ,P (A−B) := P (E(A−B)).
2
ini ely addi i e p obabili y measu e. In his o mula ion, o en e e ed o as S alnake ’s
Thesis (ST), he connec i e
→
is aken as a logically espec able condi ional, and
P
(ů) is an
o dina y ini ely addi i e p obabili y measu e. Van F aassen (1976) asked wha is he smalles
logic o condi ionals in which CP=PC could hold non i ially. Van F aassen (1976) aised
he ques ion o wha is he smalles logic o condi ionals in which CP=PC holds non i ially.
Example 1.1 (Failu e o
CP
=
PC
o ma e ial implica ion).Conside he condi ional
A→Bunde s ood as ma e ial implica ion, i.e. A→B≡ ¬A∨B. Then
P (A→B)=1−P (A∧ ¬B)=1−P (A) + P (A∧B).
I P (A)>0, he condi ional p obabili y is
P (B|A) = P (A∧B)
P (A).
Since P (A∧B) = P (A) P (B|A), we ob ain
P (A→B)−P (B|A) = (1 −P (A))(1 −P (B|A)).
Hence CP=PC holds only i
P
(
A
) = 1 o
P
(
B|A
) = 1. Fo example, i
P
(
A
) = 0
.
5and
P (B|A)=0.5, hen P (A→B)=0.75 while P (B|A)=0.5, so he equali y ails.
Mo i a ion: an F aassen’s p og am and he sea ch space. Building on an F aassen’s
sugges ion, we seek he weakes condi ional ha could sus ain a non i ial CP=PC. Using
gene al me hods due o Czelakowski and Olszewski, we conduc he sea ch abs ac ly o e
implica ional sys ems: we conside he la ice o consequence ope a ions abo e he logic o
o de
Co
(gene a ed by (p1)–(p4) and Modus Ponens) and ask whe he he e exis s a minimal
implica ional logic in his la ice ha sa is ies CP=PC unde a ixed classical eading o
p obabili y.
The logic Cois gi en by he ollowing s uc u al schema a:
(p1) ⊢A−A
(p2) A−B, B −C⊢A−C
(p3) A, A −B⊢B
(p4) A−B, B −A, C −D, D −C⊢(A−C)−(B−D)
De ini ion 2. Alogic o o de is any s uc u al consequence ope a ion in [
Co, S
]sa is ying
(p1)–(p4).
De ini ion 3 (a e [CO22], p. 1427).Le (
S, C
)be a logic o o de . We call
C
alogic
o implica ion i he logic
C+(S)
ob ained om
C
by adjoining he symme y ule (
S
)(i.e.,
A−B⊢B−A
) is inconsis en . I
C
is a logic o implica ion, hen he connec i e
−
is called
an implica ion in C.
3
The Role o Symme y
I is easy o see ha one may add o Co he symme y ule
(A−B)⊢(B−A)
ob aining a consis en s eng hening o
Co
. This na u ally aises a key ques ion: is such
a symme y admissible i we also insis on CP=PC? The ollowing lemma shows ha he
answe is nega i e.
Theo em 4 (Collapse unde CP=PC and global symme y).Assume:
1. Cis a logic o o de o e Fo m−: i alida es (p1)–(p4) and Modus Ponens;
2. he ixed classical in e p e a ion abo e;
3. o all A, B wi h P (E(A)) >0,
P (E(A−B)) = P (B|A);
4. he symme ic ex ension C+(S) alida es A−B⊢B−A o all A, B.
Then o all
A, B
wi h
P
(
E
(
A
))
>
0one has
P E
(
A
)
=
P E
(
B
)
. Thus
P
collapses on
he posi i e-p obabili y agmen .
P oo . F om symme y, E(A−B) = E(B−A). Hence
P (B|A) = P E(A−B)= P E(B−A)= P (A|B).
By Bayes’ heo em,
P (B|A) = P (A|B) P (E(B))
P (E(A)) .
I
P
(
A|B
)
>
0, cancella ion yields
P E
(
A
)
=
P E
(
B
)
. I
P
(
A|B
) = 0, hen
also
P
(
B|A
) = 0; a ying
A, B
wi h posi i e measu e p opaga es equali ies ac oss he
posi i e-p obabili y agmen . Hence collapse.
Rema k: Theo em 4 shows ha global symme y o he condi ional is incompa ible wi h
S alnake ’s Thesis in i s p obabilis ic e sion. Consequen ly, no logic in he in e al [
C_o, S
]
ha con ains symme y can se e as a minimal logic suppo ing CP=PC in a non i ial
sense.
Co olla y 5 (No non i ial “minimal logic ia
−
” o CP=PC).Unde he hypo heses o
Theo em 4, i he symme ic ex ension
C+(S)
is consis en , he collapse ob ains. In a logic
o o de , (
p
4) and MP p opaga e equi alences h ough implica ional con ex s; hus
C+(S)
i ializes. The e o e, no non i ial logic in [Co, S] alida es he global iden i y
P (E(A−B)) = P (B|A) (P (E(A)) >0)
by iden i ying he implica ional connec i e
−
wi h a p obabilis ic condi ional. The sea ch o
a “minimal logic ( ia −) sa is ying CP=PC” is ill-posed.
4
Rema k (Resol ing S alnake ’s p oblem by sepa a ing oles).CP=PC is a seman ic clause
o condi ionaliza ion, no a law o an implica ional connec i e. To e ain con en and a oid
collapse, keep
−
as an in e en ial implica ion (e.g., Łukasiewicz esiduum sa is ying (
p
1)–(
p
4)
and MP), and in oduce a dedica ed condi ional (
B|A
)wi h seman ics calib a ed so ha ,
in he in ended p obabilis ic eading, P
(
B|A
)
=
P
(
B|A
). Minimali y hen conce ns he
smalles seman ic package o (· | ·), no a minimal implica ional logic ia −.
Łukasiewicz Implica ion and T u h-A e aging
De ini ion 6 (Łukasiewicz Implica ion).In a many- alued se ing wi h u h alues in [0
,
1],
he Łukasiewicz implica ion is de ined poin wise by
(A⇒B) := min(1,1− (A) + (B)).
In he classical 2- alued case (
∈ {
0
,
1
}
), his coincides ex ensionally wi h ma e ial implica-
ion.
Łukasiewicz Family and he Implica ional Co e Ac oss all Łukasiewicz logics ( ini e
Ł
n
wi h
n≥
2and he s anda d Ł
∞
),
⇒
is he esiduum o he Łukasiewicz -no m wi h
u h- unc ion
(
A⇒B
) =
min
(1
,
1
−
(
A
) +
(
B
)). This de ini ion is uni o m in
n
( he
alue se s di e : Γ
n
s. [0
,
1]). In he pu e implica ional language, alidi y in Ł
∞
is minimal
by inclusion: e e y implica ional au ology o Ł∞is alid in each Łn, and
Th⇒(Ł∞) =
n≥2
Th⇒(Łn).
Consequen ly, any nega i e esul o CP=PC o mula ed ia
⇒
in Ł
∞
ans e s o e e y
ini e Łn.
No a ion: P s.
P
Le (
W,
Σ
, µ
)be a p obabili y space, and le
:
{A, B, C, . . . }×W→
[0,1] be a many- alued e alua ion wi h he s anda d clauses:
(¬A, w)=1− (A, w), (A⇒B, w) = min(1,1− (A, w) + (B, w)).
•P(A)deno es he u h-a e age (expec ed u h deg ee):
P(A) = ZW (A, w)dµ(w).
•P
(
E
)deno es he classical p obabili y o an e en
E∈
Σ. When a o mula
A
is
in e p e ed as he c isp e en
EA⊆W
(i.e.,
(
A, ·
)
∈ {
0
,
1
}
), we w i e
P
(
A
) :=
µ
(
EA
)
and P (B|A) := P (A∧B)/P (A) o P (A)>0.
I (·, w)∈ {0,1}, hen
P(A) = ZW (A, w)dµ(w) = µ(EA) = P (A).
In gene al, howe e , Pand
P
cap u e di e en no ions: he o me a e ages deg ees o u h,
he la e measu es he size o c isp e en s. Rela ed discussions o assigning p obabili ies
o logical sen ences, and he pi alls o con la ing di e en p obabilis ic no ions, appea in
[Ols22]
5
Łukasiewicz Logic Ł
∞
The language has
¬,⇒
wi h Modus Ponens and subs i u ion. The
s anda d [0,1]-seman ics is
(¬A) = 1 − (A), (A⇒B) = min(1,1− (A) + (B)).
An equi alen Hilbe sys em uses he axiom schema a:
(L1) (A⇒B)⇒((B⇒C)⇒(A⇒C)),
(L2) ((A⇒B)⇒B)⇒((B⇒A)⇒A),
(L3) (¬A⇒ ¬B)⇒(B⇒A).
This sys em is comple e o he s anda d [0,1]-seman ics.
P oposi ion 7 (T u h-a e aging does no epai CP=PC o
⇒
).In many- alued, u h-
a e aging seman ics, he e is no gene al iden i y
P(A⇒B) = P (B|A)
ha holds o all models unless addi ional, non i ial cons ain s a e imposed (see [Háj98]
o he s anda d Łukasiewicz seman ics and discussions o p obabilis ic in e p e a ions). The
Łukasiewicz u h- unc ion
(A⇒B) = min(1,1− (A) + (B))
is nonlinea in ( (A), (B)), so he u h-a e age
P(A⇒B) = ZWmin1,1− (A, w) + (B, w)dµ(w)
canno , in gene al, collapse o he a io o m ha de ines classical condi ional p obabili y
P (B|A) = P (E(A)∩E(B))
P (E(A)) (P (E(A)) >0),
which depends only on he
A
- egion and no malizes by
P
(
E
(
A
)). T u h-a e aging mixes all
wo lds (including pa ial
(
A, w
)
∈
(0
,
1)) wi hou his no maliza ion, hence P(
A⇒B
)
=
P (B|A)in gene al.
Example 1.2 (Two-wo ld coun e example).Le
W
=
{w1, w2}
wi h
µ
(
{w1}
) =
µ
(
{w2}
) =
1/2. De ine
(A, w1)=1, (B, w1)=0.4; (A, w2)=0.2, (B, w2) = 0.2.
Then
(A⇒B, w1) = min(1,1−1+0.4) = 0.4, (A⇒B, w2) = min(1,1−0.2+0.2) = 1,
so
P(A⇒B) = 1
2·0.4 + 1
2·1=0.7.
6
Unde he classical eading wi h
E(A) = {w: (A, w)=1}={w1}, E(B) = {w: (B, w)=1}=∅,
we ha e
P (B|A) = P (E(A)∩E(B))
P (E(A)) = 0,
hence P(
A⇒B
)=0
.
7
=0=
P
(
B|A
). The misma ch is d i en by nonlinea i y and he
lack o A-condi ional no maliza ion in he u h-a e age.
De ini ion 8 (In e nal condi ional ope a o ).Ex end he language by a bina y ope a o
(
· | ·
), ead “ he condi ional.” An in e p e a ion (
W, µ,
)in e nalizes condi ional p obabili y
i , o all A, B wi h P (A)>0,
P((B|A)) = P (B|A),
and (·|·)sa is ies in ended sani y laws.
Theo em 9 (How o ob ain CP=PC).Le
−
be an implica ional connec i e (e.g., Łukasiewicz
implica ion). I he language is en iched by an in e nal condi ional (
· | ·
)and he seman ics
en o ces
P((B|A)) = P (B|A) o all A, B wi h P (A)>0,
hen CP=PC holds o he in e nal condi ional.
P oo .
Immedia e om he seman ic clause o (
· | ·
). The implica ional connec i e
−
supplies he in e en ial backbone ( ules and MP), while (
· | ·
)in e nalizes condi ionaliza ion:
P((B|A)) = P (B|A).
Co olla y 10 (Di ision o oles).Łukasiewicz implica ion can se e as he implica ional
connec i e, bu CP=PC should be o mula ed o he dedica ed condi ional (
· | ·
)whose
seman ics gua an ees P((
B|A
)) =
P
(
B|A
). Replacing (
B|A
)by
A⇒B
in alida es
CP=PC.
2 Main Resul
We wo k in he la ice in e al [
Co, S
]o implica ional logics o e he language
Fo m−
wi h a
single bina y connec i e
−
and Modus Ponens. CP=PC is in e p e ed on a ixed classical
p obabili y space (Ω,F,P ): o all A, B,
P (A−B) := P (E(A−B)),P (B|A) := P (E(A)∩E(B))
P (E(A)) i P (E(A)) >0.
7
Nega i e esul : collapse
Theo em 11 (Collapse unde CP=PC and global symme y).Assume:
1. C alida es (p1)–(p4) and MP;
2. he ixed classical in e p e a ion abo e;
3. o all A, B wi h P (E(A)) >0,P (E(A−B)) = P (B|A);
4. he symme ic ex ension C+(S) alida es A−B⊢B−A o all A, B.
Then o all A, B wi h P (E(A)) >0,
P (E(A)) = P (E(B)),
i.e., he p obabili y assignmen collapses on he posi i e-p obabili y agmen .
Co olla y 12 (No non i ial “minimal logic ia
−
” o CP=PC).Unde he hypo heses o
Theo em 11, i
C+(S)
is consis en , he collapse ob ains. In a logic o o de , (p4) and MP
p opaga e equi alences h ough implica ional con ex s; hence
C+(S)
i ializes. The e o e, he e
is no non i ial logic in [Co, S] alida ing
P (E(A−B)) = P (B|A)
by iden i ying
−
wi h a p obabilis ic condi ional. The sea ch o a “minimal logic ( ia
−
)
sa is ying CP=PC” is ill-posed.
Posi i e esul : di ision o oles
Theo em 13 (Di ision o oles and how o ob ain CP=PC).Le
−
be an implica ional
connec i e (e.g., he Łukasiewicz esiduum) alida ing (p1)–(p4) and MP. Ex end he language
by a condi ional ope a o (·|·), and le (W, Σ, µ, )be a seman ics wi h
P(A) = ZW (A, w)dµ(w).
I , o all A, B wi h P (E(A)) >0,
P((B|A)) = P (B|A),
and (
· | ·
)sa is ies he in ended sani y laws (ce ain y/ze o cases), hen CP=PC holds o
(
· | ·
). In pa icula , CP=PC is achie ed no by iden i ying
A−B
(e.g.,
A⇒B
) wi h
condi ionaliza ion, bu by calib a ing (B|A)so ha i s p obabili y ma ches P (B|A).
Scope and quali ied cases whe e CP=PC may hold. Ou no-go esul a ge s global
iden i ica ions o he o m
P
(
E
(
A−B
)) =
P
(
B|A
)ac oss he ull language and a bi a y
models, unde a ixed classical measu e and na u al s uc u al p inciples. I does no p eclude
quali ied, local uses o CP=PC unde addi ional cons ain s. Typical sa e zones include:
(i) es ic ing scope o non-nes ed occu ences o he condi ional and o an eceden s/e en s
wi hin a ixed algeb a whe e he seman ic eading o
A−B
is ex e nally calib a ed o
condi ionaliza ion; (ii) condi ional logics o p obabilis ic amewo ks ha limi he domain o
8
applica ion (e.g., only o a designa ed class o condi ionals o almos -e e ywhe e clauses)
so ha a io-s yle no maliza ion is p ese ed; and (iii) sys ems ha in oduce a dedica ed
condi ional (
B|A
)wi h an explici seman ic clause ensu ing P((
B|A
)) =
P
(
B|A
), while
keeping
−
pu ely in e en ial. In sho , CP=PC can unc ion eliably in con olled agmen s
o wi h a pu pose-buil condi ional, bu no as a global law ying an implica ional connec i e
o classical condi ional p obabili y.
Rela ed Wo k
The classical p og am o iden i ying he p obabili y o a condi ional wi h he co esponding
condi ional p obabili y—o en labeled S alnake ’s Thesis (ST)—goes back o ea ly discussions
o condi ionals in o mal epis emology and philosophical logic (e.g., [S a70]). While he hesis is
p ima acie compelling, i clashes wi h s anda d u h- unc ional ea men s: o he ma e ial
condi ional, one ob ains
P
(
A→B
)=1
−P
(
A
) +
P
(
A∧B
), which in gene al di e s om
P
(
B|A
)unless degene a e cases ob ain, as is well known in he p obabili y-o -condi ionals
li e a u e.
A subs an ial body o wo k documen s “ i iali y” and collapse phenomena ha a ise
when one a emp s o alida e global iden i ica ions o his kind. Van F aassen’s in luen ial
analysis [F a76] explici ly aised he ques ion o a smalles (non i ial) logic in which ST
could hold and showed se e e cons ain s on any such a emp . By seeking CP=PC ia a
single condi ional connec i e, his p og am con la es an in e en ial ope a o wi h p obabilis ic
condi ionaliza ion; ou collapse heo em shows ha , unde a ixed classical measu e and global
symme y, his iden i ica ion alone o ces degene acy o e he minimal logic-o -o de backbone.
Subsequen impossibili y and i iali y esul s—unde di e se echnical assump ions abou
he condi ional, backg ound algeb a, and admissible p obabili y assignmen s— ein o ced he
message ha uncondi ional, global iden i ica ions o ce degene acy o equi e nonclassical
p obabilis ic amewo ks (e.g., Poppe unc ions) o es ic ed scopes (e.g., limi ed nes ing,
almos -e e ywhe e clauses) [Lew76; Spo83; Edg95].
F om he pe spec i e o condi ional logics, a ious sys ems accommoda e agmen s o ST
only unde signi ican es ic ions (e.g., on he o m o an eceden s, on admissible upda es, o
ia dis inc seman ic ie s o condi ionals e sus e en s). Con empo a y su eys emphasize
ha mixing he in e en ial ole o a condi ional wi h he seman ics o condi ionaliza ion is
he key me hodological pi all: he wo belong o di e en laye s o heo y and should no be
iden i ied wi hou an explici b idge p inciple [Edg25; CHN11; F a24].
The p esen pape con ibu es a s eamlined no-go esul in a minimal implica ional
se ing: wi hin he logic-o -o de backbone ( ules (p1)–(p4) and MP) and a ixed classical
eading o
P
on bo h sides o CP=PC, adding global symme y igge s a p obabilis ic
collapse, and he symme ic ex ension i ializes. This sha pens he nega i e conclusions o
he classical li e a u e by isola ing he exac s uc u al ea u es ha o ce degene acy in he
implica ional language. On he posi i e side, ou “di ision o oles” e ames ST as a seman ic
clause o a dedica ed condi ional ope a o (
· | ·
)— a he han a law o an implica ional
connec i e—aligning wi h app oaches ha sepa a e in e ence om p obabilis ic upda e and
calib a e he in e nal condi ional di ec ly o P (B|A).
9
