> {}z'` bjbj 0d
: """"""""$ h7"""""""".""""`BP0 f"""""" """"::::::Entailment in Conditional Probability Logics and Its Relation to
Conditional Event Algebra
I. R. Goodman
Space & Naval Warfare Systems Center, Pacific
Donald Bamber
University of California, Irvine
The calculus of probabilities, in addition to being a numeric tool, is a kind of logic; it falls into the category of multivalued, nontruthfunctional logics. However, this logic has its limitations. There is a variety of questions that arise in both everyday reasoning and in science that the probability calculus cannot answer. Some of these questions are explicitly probabilistic, some only implicitly so. For example: If the conditional probability of A given B is 0.8, what is the best estimate of the conditional probability of notB given notA? Another example: If nearly all As are Bs and nearly all Bs are Cs, should we expect that nearly all As are Cs? To answer such questions, new probability logics are needed.
In this talk, we present both general background and specific new results for one branch of probabilistic reasoning, namely, entailment (of conclusions from premises) in conditional probability logics. We also present both old and new significant results in a littleknown and underdeveloped field at the juncture of probability theory and algebra conditional event algebra (CEA). CEA has already proven very useful in both formulating and deriving results in our ongoing efforts in conditional probability logics.
In summary, our presentation will cover all, or parts of, the following topics of relevance:
1. The meaning of entailment schemes and associating possible validity / invalidity, using various criteria based on probability or other concepts. Development of conditional probability entailments logics, including: High Probability Logic (HPL), Certainty Probability Logic (CPL), (Unscaled) Near Surety Logic (UNSL), Entropy Probability Logic (EPL), and the related nonprobability Material Conditional Logic (MCL).
2 Use of second order probabilities and a fundamental modification of MCLModified Material Conditional Logic (MMCL)leading to the theorem that entailment validity for UNSL coincides with that for MMCL.
3. Introduction, brief history, and motivation for developing CEA as an underpinning for conditional probability, and therefore for analysis of entailment in conditional probability logics. Response to Lewis triviality result and construction of a boolean / sigmaalgebra form of CEA, with a new characterization theorem.
4. New extension of quantitative aspects of UNSL in terms of two additional modifications of MMCL / UNSL, together with estimation of posterior conditional probabilities. Applications to patterned entailment schemes and comparisons of various logics, including use of a new characterizing theorem for EPL in terms of MCL.
\kv
n+v
&
hFh6hD\hCJhD\h6hULh6hhULh5B\]k
$a$gd$a$gd.:p/ =!"#$%<@<NormalCJaJmH sH tH DADDefault Paragraph FontRiRTable Normal4
l4a(k(No ListB\]k !!!!!!!!! !!^!!^!!^!!^!!^!!^!B\]k 08000000000000000000000
08 k@T=XgP@UnknownGz Times New Roman5Symbol3&z Arial Ah(f(ffk k 24>2QP]2AEntailment in Conditional Probability Logics and Its Relation to GaIL BamberkilOh+'0, 8D
dp
DEntailment in Conditional Probability Logics and Its Relation to GaIL BamberNormal.dotkil2Microsoft Office Word@F#@=L<@B@Bk GVT$m nZ&" WMFC[ ^~lUT#m EMF~M$I*U"
Rp@Times New RomanGz Times ew Roman8R+PcR"0RL^0Rdv%TXk/@@ALEntailment in Conditional Probability Logics and Its Relation to C8!2R,8!8H288!282<,2828!2C22,'288'!'H,2!28!2TTlX/@@lLP 6T4</@@'LConditional Event AlgebraH288!282C2,8!H2,8,2TT5j</@@5'LP 6Rp@Times New RomanGz Times ew RomanRR=RR"0RL^0Rdv%TT >"
/@@ LP T"/@@
LhI. R. Goodman CH222N,2TTK"/@@LP .Rp@Times New RomanGz Times ew RomanRR<RR"0RL^0Rdv%%T\.$
/@@.~LSpace & Naval Warfare Systems Center, Pacific222,,KC3,2R2'2',2,',H'C,2,'=2,,%TT
$
/@@
~LP .TT "
/@@ LP T
2{/@@d
LhDonald BamberH22,2C,N2,!TT3
`{/@@3dLP .%T}e/@@ LUniversity of California, IrvineH2,,'',2C22'22!',2,TTf}/@@fLP 5%TTa/@@JLP Tdc/@@LTThe =2,%Tc/@@Lcalculus of probabilities,,,22'%2!%2!23,2,'TXc/@@LP, %T cu
/@@#Lin addition to being a numeric tool2%,2222%2%2,21&,%22N,!%22TTv
c
/@@v
LP,Td
c/@@
LT is %'%Tc/@@Lta kind of logic; it ,%222%2!%31,%TG/@@0 Lfalls into the category of multi!,'"22"2,",,13#0"2""N2TTG/@@0LP!T G/@@0Ldvalued, non2,2,2"222TT G/@@ 0LP!Tl
G/@@ 0LXtruth!22TT
G/@@
0LP"T
G/@@
0Lpfunctional logics.!22,22,"21,'T
G/@@
0Lx However, this logic ""H2I2,!"2'"21,TI/@@6Lhas its limitations. There is a variety of questions 2,' ' N,22' =2,", '  2!,0 2" 22,'23' TI/@@Ltthat arise in both e2, !', 2 223 ,TI/@@Lpveryday reasoning 2#120!",,'2231T/@@Ltand in science that,22 2 ',,2, 2,Td/@@LT the 2T /@@Lx probability calculus 3!22,20!,,,22'T B/@@ Ll cannot answer.!,,223 ,2'H,!TC/@@CL Some of these questions are 82N, 3! 2,( 22,'22' ,",%Tx//@@2Lexplicitly probabilistic, some only implicitly so.,32,/<2!23,2',<'2N,<220<N2&" WMFC >~,0<(2T/
/@@
Lh For example<<72!=,3,N2,TX
//@@
LP: <%TT/0/@@LPI!T//@@(Lpf the conditional <2,<,222222TD/ /@@)Lprobability of A given B is 0.8, what is 2'2222,2=2,,2='22C22'T`0 /@@0 LTthe2,TT /@@ LP Tl
/@@ LXbest 2,'T
/@@
Llestimate of the,(H2,22,Tk /@@
L` condition,22222Tl /@@lLpal probability of 22'2222,2%T` o /@@o LTnot22TTn /@@no LP!T X /@@o LdB given not=2,,322TTW w /@@Wo LP!TXr /@@xo LPA?=2%TX /@@o LP T /@@o LpAnother example: H222,!,3,N2,%T 1 /@@ o LlIf nearly all A!3,2',2=TT2 R /@@2o LP !TxS n /@@So L\s are B'2'=TTo /@@oo LP !T Y /@@o Lps and nearly all B'2232,2'3=TTZ z /@@Zo LP !Tx{ /@@{o L\s are C'2',CTT /@@o LP !T` /@@o LTs, '%T K /@@ "Lshould we expect that nearly all A'2222C,,,3,,2222',2=TTL l /@@L LP !Txm /@@m L\s are C'2',CTT /@@ LP !TX
/@@ LPs?'3%TX
B
/@@
LP THC
/@@C
*LTo answer such questions, new probability =3,2'H,"'2,223,'22'2,H2!23,21T l
/@@U
Lplogics are needed.21,'!,2,2,2TT l
/@@U
LP .Rp@Times New RomanGz Times ew RomanRR=RR"0RL^0Rdv%TTn
/@@
LP %T
%+/@@LlIn this talk, w 22',2HT%
/+/@@%
Lhe present bot,2!,(,222T0
+/@@0=Lh general background and specific new results for one branch 21,2!,2,31!2222 ,22'2,,!,2H!,'2'!2!22,2!,3,3%T$% /@@$Lof probabilistic reasoning, namely, 2!E2!22,2',E!,'2221E3,N,0E%T& 
/@@& L`entailmen,22H,2TT

/@@
LPtT
i/@@
L (of conclusions from premises)F 2E,22,2'22'E'2IE2',H',( Tdj/@@jLT in E3%T/@@Lconditional probability logics,2222222'2222,22,'%TT/@@LP.T/@@<L We also present both old and new significant results in a _,,'32!,(,222222,223,H'12!,,2",'2'2,%Tp/@@mLXlittle,TT/@@mLP!T/@@mLlkn&" WMFC ~own and underd222H2,22222,!2T/@@mALeveloped field at the juncture of probability theory and algebra c2,22,2!,2,2,22,2!,2!2!22,202,3#1,22,1,3!,TT/@@mLP 2TT/@@mLP o%T;/@@Lconditional event algebrap,222222>,,,2>22,3'2%T</@@<4L (CEA). CEA has already proven very useful in both >!C=H!?>C=H>2,(>,!,40>2!23,2>2#0?2',!2>2>223Tbj
/@@S
Lformulating and deriving res!2!N2,21,222,!232!,'Tcj
/@@cS
Lults in our ongoing efforts in 2'222!221231"!2!'2Tj
/@@S
Lconditional probability logics.,22222,2!222/31,'TTj
/@@S
LP .%TTl
/@@
LP %T
*/@@LpIn summary, our pr 2'2NN#022!2"T
*/@@JLesentation will cover all, or parts of, the following topics of relevance:,',222H,22,!,2!2,!'2!2,!22H2122,'2!!,,3,2,,TT
*/@@LP .%TT,w/@@hLP %Tpy/@@[L1. The meaning of entailment schemes and associating possible validity / invalidity, using 2=2,N,,2212!,2,N,2',2,N,',22,''2,,2122''2,2,2/22,202'22TD]/@@FTLvarious criteria based on probability or other concepts. Development of conditional 2,!22'4,!,!,42,',243242!22,2042!422!5,22,,2'4I,2,22N,252!4,22222,Tl_/@@LXproba2!22,T _/@@NLbility entailments logics, including: High Probability Logic (HPL), Certainty 2/@,2,N,2'?21,'?2,2221?H22?8!22,2/@<31,?!I9;!?C,!,21T,C/@@,LProbability Logic (CPL), 8!22,2/.<31,!C9<"TTMC/@@,LP(!TNC/@@N,L\UnscaledH2',,,2TTC/@@,LP)!ThC/@@,/L Near Surety Logic (UNSL), Entropy Probability H,!82!,0/<31,!HH9<!=2!3308!22,20TE/@@ LLogic (EPL), and the related non<31,!=9<!,222,!,,2222TTE2/@@LP!Th3E/@@3/Lprobability Material Conditional Logic (MCL). 2!22,20Z,!,C22222,;31,"YD;!TTE/@@LP .%TT/@@LP %TPu/@@^Lh2 Use of secon2 H', 2! ',,22TQ$u/@@Q^;Ld order probabilities and a fundamental modification of MCL2 2!3,! 2"22,2,' ,22 , !222,O,2, N22!,,22 2! YD;TT$u/@@%^LP dTu/@@^ L`Modified Y22!,3Tw/@@!LMaterial Conditional Logic (MMCL)Y,,!,C22222,;31,!YYD<!TTw* /@@LP d&WMFC~T* w
/@@* Ldleading to ,2312T`
wN/@@
LTthe2,TXOw/@@OLP tT$w/@@$Lheorem that entailment validity for 2,2!,N2,,2,N,22,2/!2!T[/@@D"LUNSL coincides with that for MMCL.HH9;,22,2,'H22!2!YYD;TT [/@@DLP .%TT]/@@LP %TM
/@@AL3. Introduction, brief history, and motivation for developing CEA22!223,222!,!2'2"0,22N22,22!2!3,2,2221C=HTN
/@@N
L as an underpinning for ,',2232,!22221!2!T/@@w]Lconditional probability, and therefore for analysis of entailment in conditional probability ,22222,2!22,20222,!,!3!,!3!,2,0''2!,2,N,22,22222,2!22,21T,/@@PLlogics. Response to Lewis triviality result and construction of a boolean / 21,'++C,'222',+2,<.H'!+,!2,0,,!,'2+22+,22'!2,22+2!+,+232,,2++Tl/@@LXsigma'1NTT/@@LP!Tt/@@]5Lalgebra form of CEA, with a new characterization theo,2,2!,!2!N2!C=HH2,2,H,3,!,,!,222,2Td5t/@@]LTrem.!,NTT6ct/@@6]LP .%TTv/@@LP %T4/@@LL4. New extension of quantitative aspects of UNSL in terms of two additional 2EH,HF,3,2'22E2!F22,2,2,E,'2,,'E3"EHH9<E2E,!N'E3!EH2E,2222,T,6/@@PLmodifications of MMCL / UNSL, together with estimation of posterior conditional N22!,,22'62!6YYB<66HH9;631,2!6H26,'N,2262!622&,!2!6,22222,T\/@@XLprobabilities. Applications to patterned entailment schemes and comparisons of various 2!22,2,'!!H22,,12'!2!2,,!2,2!,2,N,2!',2,N,'!,22!,2N2,!'22'!2!!2,!22'Tp/@@vLXlogics21,'Tt/@@vHL, including use of a new characterizing theorem for EPL in terms of MCL.2,22312',2!2,H,2!,,,!212,2",N!2!=9<2,!N'2!YD;TTu/@@uvLP .%666666666666666666666666666666666666 6 66 6
6
66
6
66666666
6
66
6
66666666666666666666~.@Times New Roman m2
^A~Entailment in Conditional Probability Logics and Its Relation to o
2
^~ 12
m~Conditional Event Algebra
2
m~ @Times New Roman
2
}Y~ 2
1
~I. R. Goodman
2
~ @Times New RomanO2
~Space & Naval Warfare Systems Center, Pacific

2
~
2
Y~ 2
.
~Donald Bamber
2
~ ;2
~University of California, Irvine
2
~ 
2
e~ 2
e~The 12
~calculus of probabilities 2
~, @2
#~in addition to being a numeric tool
2
~,2
~ is )2
~a kind of logic; it ;2
e ~falls into the category of multi
2
~2
~valued, non
2
T~2
Y~truths
2
r~&2
w~functional logics.,2
~ However, this logic
\2
e6~has its limitations. There is a variety of questions )2
~that arise in both e&2
~veryday reasoning (2
e~and in science thate2
~ the+2
~ probability calculus "2
R~ cannot answer. 82
~ Some of these questions are V2
(e2~explicitly probabilistic, some only implicitly so.
2
(
~ For exampler
2
(~: 
2
(~I&2
(~f the conditional I2
8e)~probability of A given B is 0.8, what is 2
8>~the
2
8N~ 2
8R~best s"2
8l~estimate of the
2
8
~ condition&2
8~al probability of 2
Ge~not
2
Gw~2
G{~B given not
2
G~2
G~A? 2
G~ %2
G~Another example:
"2
G8~If nearly all A
2
G~2
G~s are Bd
2
G~&2
G~s and nearly all B
2
G~2
G~s are Cd
2
G?~2
GC~s, >2
We"~should we expect that nearly all A
2
W~2
W$~s are Cd
2
WK~2
WP~s?2
W\~ J2
Wc*~To answer such questions, new probability
&2
fe~logics are needed.
2
f~ @Times New Roman
2
re~ "2
e~In this talk, w
2
~e present bot g2
=~h general background and specific new results for one branch
A2
e$~of probabilistic reasoning, namely,
2
= ~entailmen
2
r~t:2
v~ (of conclusions from premises)
2
8~ in 82
e~conditional probability logics
2
~.e2
<~ We also present both old and new significant results in a
2
e~little
2
~~#2
~known and underd
m2
A~eveloped field at the juncture of probability theory and algebra o
2
E~
2
L~ 12
e~conditional event algebraoY2
4~ (CEA). CEA has already proven very useful in both
52
e~formulating and deriving res:2
~ults in our ongoing efforts in :2
~conditional probability logics.
2
A~ 
2
e~ &2
e~In summary, our pr
z2
J~esentation will cover all, or parts of, the following topics of relevance:
2
J~ 
2
e~ 2
e[~1. The meaning of entailment schemes and associating possible validity / invalidity, using
2
eT~various criteria based on probability or other concepts. Development of conditional
2
!e~probae2
!N~bility entailments logics, including: High Probability Logic (HPL), Certainty
12
1e~Probability Logic (CPL), o
2
1~(2
1~Unscaled
2
1/~)R2
13/~ Near Surety Logic (UNSL), Entropy Probability
;2
Ae ~Logic (EPL), and the related non
2
A~R2
A/~probability Material Conditional Logic (MCL).
2
A"~ 
2
Le~  2
Ze~2 Use of secon
d2
Z;~d order probabilities and a fundamental modification of MCL
2
Z~2
Z ~Modified =2
je!~Material Conditional Logic (MMCL)
2
j0~
2
j=~leading to 2
jw~the2
j~ tA2
j$~heorem that entailment validity for
>2
ze"~UNSL coincides with that for MMCL.
2
z5~ 
2
e~ m2
eA~3. Introduction, brief history, and motivation for developing CEA /2
~ as an underpinning for 2
e]~conditional probability, and therefore for analysis of entailment in conditional probability 2
eP~logics. Response to Lewis triviality result and construction of a boolean /
2
'~sigmal
2
H~[2
e5~algebra form of CEA, with a new characterization theo
2
~rem.
2
~ 
2
e~ }2
eL~4. New extension of quantitative aspects of UNSL in terms of two additional
2
eP~modifications of MMCL / UNSL, together with estimation of posterior conditional
2
eX~probabilities. Applications to patterned entailment schemes and comparisons of various
2
e~logicsw2
H~, including use of a new characterizing theorem for EPL in terms of MCL.
2
~ "System~~~~~~~~~~~~}}}}}}}}}}}}}}}}՜.+,0Hhp
*$University of California, Irvine'BEntailment in Conditional Probability Logics and Its Relation to Title
!"#$%&'()*+,./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqstuvwxyRoot Entry F0!B~1TableWordDocument0SummaryInformation(DocumentSummaryInformation8rCompObjq
FMicrosoft Office Word Document
MSWordDocWord.Document.89q