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An Overview of Conditionals and Biconditionals in Probability An Overview of Conditionals and Biconditionals in Probability
Nataniel Greene
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An Overview of Conditionals and Biconditionals in Probability
Nataniel Greene
Kingsborough Community College, CUNY
Department of Mathematics and Computer Science
2001 Oriental Boulevard Brooklyn, NY 11235
USA
email: [email protected]
Abstract: Conditional and biconditional statements are a standard part of symbolic logic but they have only recently
begun to be explored in probability for applications in articial intelligence. Here we give a brief overview of the
major theorems involved and illustrate them using two standard model problems from conditional probability.
Key–Words: probability of material implication; probability of a conditional; probability of a biconditional
1 Introduction
Elementary treatments of symbolic logic include a
discussion of the logical operations of negation, con-
junction, disjunction, the conditional and the bicondi-
tional. Elementary treatments of probability, however,
will discuss probabilities of a negation, conjunction,
and disjunction but will leave out probabilities of con-
ditionals and biconditionals: statements of the form
“if Q then P ” and “P if and only if Q.” Our treat-
ment here follows the work of Nguyen, Mukaidono,
and Kreinovich [6] where the topic is developed for
applications in articial intelligence. The authors
there show that an analog of Bayes' theorem involv-
ing probabilities of conditionals called Logical Bayes'
theorem, yields comparable results to the standard
version of Bayes' theorem.
In this presentation we outline a treatment of this
topic suitable for undergraduates. We show how
these probabilities are computed and apply them to
two standard model problem. Our contention is that
once an instructor has developed the laws of probabil-
ity of involving the logical operations of “and,” “or”
and “not” and once an instructor has covered condi-
tional probability, the student is fully prepared for a
discussion on the probability of a conditional, bicon-
ditional, and related issues such as the corresponding
logical analog of Bayes' theorem.
We begin with an experiment which produces a
sample space S, the set of all possible outcomes of the
experiment. Any subset of the sample space is known
as an event. Let P and Q be any two events. Then
we recall the denition of conditional probability for
the discrete case as
P (P jQ) =
# outcomes in (P \ Q)
# outcomes in Q
;
from which it readily follows that
P (P jQ) =
P (P \ Q)
P (Q)
:
This may be interpreted as given certain knowledge
that Q occurred, our sample space can be restricted
from S to Q, and as a result the event P is restricted
to P \ Q: We call this the probability of P given Q,
or P given knowledge that Q occurred.
Next we recall the logical denitions of the condi-
tional (P implies Q) and the biconditional (P implies
Q and Q implies P ). We do so using truth tables.
P Q P ! Q P $ Q
T T T T
T F F F
F T T F
F F T T
The operation P ! Q is known as material im-
plication or the conditional and it is seen to be logi-
cally equivalent to P [Q: It is also logically equiv-
alent to (P \ Q) : It is often read "P implies Q"
or "if P then Q:" It is pointed out, for example in
Copi [2] p.17, that the symbol for material implication
should not be thought of as representing the unique
meaning of if-then, rather it should be thought of as
the partial meaning which all uses of the term if-then
have in common. One interesting usage of P ! Q
is a statement like "If the team wins, then I'm a mon-
key's uncle." No real connection is being made be-
tween the two statements. Since the statement as a
whole P ! Q is asserted by the speaker to be true,
and the precedent Q is meant to be absurd, it can only
be that the antecedent P is false. This is precisely the
speaker's intent in asserting this implication: to assert
that P is false.
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2 The Probability of a Conditional
We now wish to consider the probability of a condi-
tional, otherwise known as or material implication,
and see how this probability relates to the more well
known concept of conditional probability.
2.1 Probability of a Conditional
Denition 1 Dene the event
Q ! P = Q [ P
= (Q\ P )
The following theorem describes the probability
of a conditional in terms of conditional probabilities.
Theorem 2
P (Q ! P ) = P ( Q) + P (P \ Q)
Proof.
P (Q ! P ) = P ( (Q\ P ))
= 1 P (Q\ P )
= 1 P ( P jQ) P (Q)
= 1 (1 P (P jQ)) P (Q)
= 1 P (Q) + P (P jQ) P (Q)
= P ( Q) + P (P \ Q)
The next theorem and its corollary can be thought
of as describing, among other things, a paradox of
small sets: from a very small set one can draw any
conclusion with a high degree of probability. Or, one
may describe this as a paradox of improbable events:
from a very unlikely event, one can draw any conclu-
sion with a high degree of probability. Not being
aware of this paradox in one's reasoning can lead to
misleading conclusions. An example of this para-
dox might be: if a plane crashed (a very unlikely
event) then the eet is unsafe. Technically this is
a highly probable inference. An underlying cause
might be that whatever caused the plane to crash could
be shared by most of the planes. But we can also
make the highly probable inference: if a plane crashed
then the eet is safe. The underlying reason being
that whatever the hidden causes were which affected
the one plane do not pertain to the rest of the eet.
Corollary 3
P (? ! P ) = 1
P (S ! P ) = P (P )
P (Q ! ?) = P ( Q)
P (Q ! S) = 1
P (Q ! ?) = P ( Q)
P (Q ! Q) = 1
P (Q ! Q) = P ( Q)
P ( Q ! Q) = P (Q)
Now we express standard conditional probability
in terms of the probability of a conditional.
Theorem 4
P (P jQ) =
P (Q ! P ) P ( Q)
P (Q)
From here we may now formulate a version of
Bayes' theorem. Recall the simplest version of
Bayes' theorem which states
P (P jQ) =
P (QjP ) P (P )
P (Q)
:
From this and the previous theorem one can derive the
logical analog of Bayes' theorem.
Theorem 5 Logical Analog of Bayes' Theorem (1).
P (Q ! P ) = P (P ! Q) + P (P ) P (Q)
Corollary 6 When P and Q are equiprobable, i.e.
P (P ) = P (Q), then P (Q ! P ) = P (P ! Q) :
Now we recall the law of total probability and the
corresponding more general version of Bayes' theo-
rem. The law of total probability states that given a
partition of the sample space S into a disjoint union of
events Q
i
and any event P; we may decompose P (P )
as:
P (P ) =
n
X
i=1
P (P jQ
i
) P (Q
i
)
Substituting for P (P jQ
i
) using the previous theorem
yields the following reformulations of the law of total
probability.
Theorem 7 Reformulation: Law of Total Probability.
P (P ) =
n
X
i=1
(P (Q
i
! P ) P ( Q
i
))
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Finally we recall Bayes' Theorem in its usual
general formulation, namely:
P (Q
i
jP ) =
P (P jQ
i
) P (Q
i
)
P (P )
and reformulate it in terms of the probability of an
implication.
Theorem 8 Logical Analog of Bayes' Theorem (2).
P (P ! Q
i
) = P (Q
i
! P ) + P (Q
i
) P (P )
where P (P ) =
n
X
i=1
(P (Q
i
! P ) P ( Q
i
)) or
any other equivalent expression to the law of total
probability.
We now conclude this section with an important
theorem. Conditional probabilities and probabilities
of conditionals share the same order relations.
Theorem 9 Let P; Q and R be events. Then
P (P jQ) > P (RjQ)
if and only if
P (Q ! P ) > P (Q ! R) :
Proof. We have
P (Q ! P ) > P (Q ! R)
P ( Q) + P (P jQ) P (Q) > P ( Q)
+P (RjQ) P (Q)
P (P jQ) > P (RjQ)
and vice-versa.
A practical application of this theorem is the fact
that probabilities of conditionals and the logical ver-
sion of Bayes' theorem can be used instead of the
standard version of Bayes' theorem, allowing one to
draw similar conclusions. We illustrate this in our
model problem.
3 Probability of a Biconditional
The probability of the biconditional P if and only
if Q, P (Q $ P ) ; may be thought of intuitively
as the probability that P and Q are either simul-
taneously true or simultaneously false. The prob-
ability of the negated biconditional P (Q = P ) =
P (Q $ P ) = P ( Q $ P ) ; is the probability
that P and Q are neither simultaneously true nor si-
multaneously false.
Denition 10 Dene the event
Q $ P = (P \ Q) [ ( P \ Q)
= ( Q [ P ) \ ( P [ Q) :
and the corresponding probability
P (Q $ P ) = P ((P \ Q) [ ( P \ Q)) :
The following theorem describes the probability
of a biconditional in terms of unions and intersections.
Theorem 11
P (Q $ P ) = P (P \ Q) + P ( P \ Q)
P (Q $ P ) = 1 P (P [ Q) + P (P \ Q)
P (Q $ P ) = 1 P (P ) P (Q) + 2P (P \ Q)
Corollary 12
P (P $ P ) = 1
P (P $ P ) = 0
P (P $ S) = P (P )
P (P $ ?) = P ( P )
The rst two statements can be interpreted as say-
ing that P correlates with itself perfectly and P corre-
lates with its negation not at all. The second two state-
ments say in words that the probability of an event P
measures the logical correlation of the event with cer-
tainty. The probability of P not occurring measures
the logical correlation of the event with impossibility.
The negated biconditional, denoted by P = Q; is
described in set theory as symmetric difference where
it is often denoted by P M Q: It is also described
in logic as exclusive or (xor), where it is denoted by
P xor Q or by P t Q: Different notations may be
employed depending on the context. Since the con-
text here views the biconditional as primary, we will
employ the negated biconditional symbol here. We
are interested in the probability of the negated bicon-
ditional. The following equivalences can be veried:
P = Q = (P $ Q)
= (P 9 Q) [ (Q 9 P )
= (P \ Q) [ (Q\ P )
= Q $ P
= Q $ P
Theorem 13 The Negated Biconditional or Exclusive
Or.
P (Q = P ) = P (P [ Q) P (P \ Q)
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P = Q = (P $ Q)
= (P 9 Q) [ (Q 9 P )
= (P \ Q) [ (Q\ P )
= Q $ P
= Q $ P
Finally we describe the probability of a bicondi-
tional in terms of the probability of a conditional.
Theorem 14
P (Q $ P ) = P (Q ! P ) + P (P ! Q) 1
P (Q $ P ) = 2P (P ! Q) + P (P ) P (Q) 1
P (Q $ P ) = 2P (Q ! P ) + P (Q) P (P ) 1
We leave the proof of these statements to the
reader.
4 A Logical Correlation Coefcient
When P (P $ Q) is near 1 we can say that P and
Q are nearly simultaneously true or simultaneously
false. We may also say that the events are nearly
the same. This corresponds to P and Q exhibiting
a strong direct (positive) logical correlation. When
P (P $ Q) is near 0 we can say that P and Q are
nearly simultaneously true or simultaneously false,
which corresponds to P and Q exhibiting a strong in-
verse (negative) logical correlation. We may also say
that the events are nearly complementary. It follows
that P (P $ Q) t
1
2
corresponds to P and Q ex-
hibiting little logical correlation. Based on this we
may dene an effective logical correlation coefcient
for two events.
Denition 15 The logical correlation coefcient for
the events P and Q is given by
(P; Q) = 2P (P $ Q) 1:
It follows that
P (P $ Q) =
(P; Q) + 1
2
and that
1 (P; Q) 1:
The next theorem is quite fundamental in that it
gives conceptual justication for the preceding den-
ition. It says that the logical correlation coefcient
of two events is the probability that those events are
the same minus the probability that those events are
different.
Theorem 16
(P; Q) = P (P $ Q) P (P = Q)
Proof. By denition
(P; Q) = 2P (P $ Q) 1
= P (P $ Q) + P (P $ Q) 1
= P (P $ Q) (1 P (P $ Q))
= P (P $ Q) P (P = Q)
Corollary 17
(P; Q) = 1 2P (P ) 2P (Q) + 4P (P \ Q)
The following are some algebraic properties of logical
correlation which the reader can verify.
Theorem 18
(P; Q) = (Q; P )
( P; Q) = (P; Q)
(P; P ) = 1
( P; P ) = 1
(P; S) = 2P (P ) 1
(P; ?) = 1 2P (P )
(P; Q) = 0 if and only if P (P $ Q) =
1
2
5 Logical and Independence
We now describe a notion of logical independence
which is analogous but distinct from the standard no-
tion of stochastic or statistical dependence. Re-
call that two events P and Q are stochastically in-
dependent if P (P jQ) = P (P ) or alternatively if
P (P \ Q) = P (P ) P (Q) :
Denition 19 Logical independence. The events P
and Q are called logically independent if
P (Q ! P ) = P (P ) :
It follows from this denition and the logical ver-
sion of Bayes' theorem that P (Q ! P ) = P (P )
whenever P (P ! Q) = P (Q) :
Theorem 20 The events P and Q are logically inde-
pendent if and only if
P (P $ Q) = P (P \ Q)
or
P (P [ Q) = 1
or
P ( P \ Q) = 0:
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We see that logically independent events cover
the sample space. Equivalently, we see that logically
independent events cannot be simultaneously false.
Recall that P and Q are statistically independent if
P (P jQ) = P (P ) or alternatively if P (P \ Q) =
P (P ) P (Q) : We may ask: in which circumstance do
the two notions of logical independence and statistical
independence coincide? The next theorem answers
this question.
Corollary 21 The events P and Q are both logically
independent and statistically independent whenever:
P (P ) = 1 or P (Q) = 1:
That is, the only way for two events to be both logi-
cally and statistically independent is if one of them is
certain to occur.
Proof. We begin with the formula
P (P [ Q) = P (P ) + P (Q) P (P \ Q)
which reduces in our case to
1 = P (P ) + P (Q) P (P ) P (Q)
0 = (1 P (P )) (1 P (Q))
from which the result follows.
Corollary 22 If the events P and Q are logically in-
dependent and statistically independent then
P (P $ Q) = P (P ) P (Q) :
Corollary 23 If the events P and Q are logically in-
dependent and mutually exclusive then
Q =s P:
We recall that
P (P $ Q) = P (P \ Q) + P ( P \ Q) :
Dividing both sides by P (P $ Q) yields
1 =
P (P \ Q)
P (P $ Q)
+
P ( P \ Q)
P (P $ Q)
:
When P and Q are logically independent the rst term
is 1 and the second term is 0. Therefore the rst term
can be thought of as measuring the degree to which P
and Q are logically independent and the second term
can be thought of as measuring the degree to which P
and Q are logically dependent.
Denition 24 The logical independence of P and Q
is measured by
I
L
(P; Q) =
P (P \ Q)
P (P $ Q)
=
P (P \ Q)
P (P \ Q) + P ( P \ Q)
and the logical dependence of P and Q is measured by
D
L
(P; Q) =
P ( P \ Q)
P (P $ Q)
=
P ( P \ Q)
P (P \ Q) + P ( P \ Q)
:
In terms of our notation we have I
L
(P; Q) +
D
L
(P; Q) = 1:
6 Applications
We apply here probabilities of conditionals, bicondi-
tionals and the logical analog of Bayes' theorem to
two standard model problems which illustrates many
of the concepts we have described.
We stress again that strong conditional and bicon-
ditional relationships (logical correlations) do not nec-
essarily imply that there is cause-and-effect. Proba-
bilities of conditionals and biconditionals are simply
ways of measuring association between two events.
6.1 Application 1: Medical Testing
Example 1 A patient takes a diagnostic test to check
for a disease. Let D = a patient has the disease, P =
a patient tests positive for the disease. Suppose the
following information is known:
P(D) = 10% P( D) = 90%
P (P jD) = 95% P ( P jD) = 5%
P (P j D) = 7% P ( P j D) = 93%
A standard exercise in total probability and
Bayes' theorem would be to use this information to
calculate the probability that a patient tests positive
and the inverse conditional probabilities. We do so in
the next tables. Using the Law of Total Probability
P(P ) = 16% P( P ) = 84%
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and we can compute using Bayes' theorem
P(DjP ) = 60% P(Dj P ) = 1%
P( Dj P ) = 99% P( DjP ) = 40%
Now we are interested in comparing and contrasting
this standard exercise with the analogous probabilities
of material implications and biconditionals. We show
the results of the computations in the following tables.
P (D ! P ) = 100% P (D ! P ) = 91%
P ( D ! P ) = 7% P ( D ! P ) = 94%
P(P ! D) = 94% P( P ! D) = 7%
P( P ! D) = 100% P(P ! D) = 91%
:
We begin by observing that the probability a pa-
tient has the disease, given that the test was positive
P (DjP ) ; is only 60%. This is often surprising to
those who rst encounter this, and it is explained by
the fact that the number of those who have the dis-
ease and test positive is only moderately larger than
the number of those who don't have the disease and
test positive. The surprise is sometimes explained as
due to people confusing P (DjP ) with P (P jD).
We contrast P (DjP ) with P (P ! D) = 94%
which would seem to provide the result which
people initially expect, except for the fact that
P (P ! D) = 91%. Apparently, testing posi-
tive for the disease implies with about 94% proba-
bility that a person has the disease, and it also im-
plies with about 91% probability that a person does
not have the disease! One may explain this appar-
ent paradox by recalling that P (? ! Q) = 1, for
any Q: From here we see that since P is a small
set, one is able to infer both a proposition D and
its negation D with high probability. We note
however that P (P ! D) > P (P ! D) just as
P (DjP ) > P(Dj P ):
Arguably, it is the probability of the biconditional,
or probable logical correlation, which coincides best
with the intuition that people mistakenly apply to
P (DjP ) initially. The biconditional probabilities are
as follows:
P(P $ D) = 93% P(P $ D) = 7%
:
We may compute their logical correlation coefcients
as:
(P; D) = 0:86 (P; D) = 0:86
We see that testing positive for the disease and ac-
tually having the disease have a strong direct logical
correlation, as one would expect. Therefore, testing
positive and not having the disease, or testing negative
and having the disease, have a logical correlation with
the same magnitude but the opposite sign.
6.2 Application 2: Manufacturing and Qual-
ity Control
Example 2 A company manufactures items using
three machines. 40% of the manufactured items come
from Machine 1, 40% from Machine 2 and 20% from
Machine 3. Of those items coming from machine 1,
95% work properly, of those items coming from Ma-
chine 2, 90% work properly, and of those items com-
ing from Machine 3, 93% work properly.
Let M 1; M2; and M 3 denote the events that a
manufactured item comes from Machines 1, 2, and 3
respectively. Let W denote the event that a manufac-
tured item works properly. Then we have
P(M1) = 40% P( M1) = 60%
P(M2) = 40% P( M2) = 60%
P(M3) = 20% P( M3) = 80%
and
P(W jM1) = 90% P( W jM 1) = 10%
P(W jM2) = 95% P( W jM 2) = 5%
P(W jM3) = 93% P( W jM 3) = 7%
:
Furthermore we can compute using the Law of Total
Probability:
P(W ) = 93% P( W ) = 7%
and we can compute using Bayes' theorem
Credit Blame
P(M1jW ) = 39% P(M 1j W ) = 57%
P(M2jW ) = 41% P(M 2j W ) = 29%
P(M3jW ) = 20% P(M 3j W ) = 19%
Total = 100% Total = 100%
:
At this point a very important but standard text-
book exercise is completed. Given the knowledge
that the item is working properly we can apportion the
credit according to the rst column and given that the
item is defective we can apportion the blame accord-
ing to the second column.
While this analysis is standard and correct, we are
not entirely satised. Which machine correlates most
with a working item, (direct logical correlation) and
which machine correlates most with a defective item
(inverse logical correlation)? To answer this let us
begin with the analogs of the above computations for
the probability of a conditional.
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P(M1 ! W ) = 96% P(M 1 ! W ) = 64%
P(M2 ! W ) = 98% P(M 2 ! W ) = 62%
P(M3 ! W ) = 99% P(M 3 ! W ) = 81%
We may compute, either directly or using the logical
analog of Bayes' theorem the reverse probabilities of
conditionals.
Credit Blame
P(W ! M 1) = 43% P( W ! M1) = 97%
P(W ! M 2) = 45% P( W ! M2) = 95%
P(W ! M 3) = 26% P( W ! M3) = 94%
Although these probabilities are less intuitive, their
relative ranking remain the same. That is, we would
come to the same decision as to how to rank the ma-
chines in terms of credit and blame whether we used
conditional probabilities or probabilities of condition-
als.
However, it is the logical correlations, or the
probabilities of biconditionals, that we are most in-
terested in. We compute them here.
Biconditionals
P(W $ M 1) = 39% P( W $ M1) = 61%
P(W $ M 2) = 43% P( W $ M2) = 57%
P(W $ M 3) = 25% P( W $ M3) = 75%
We can express these results equivalently in terms of
their effective correlation coefcients:
Logical Correlations
(W; M 1) = 0:22 ( W; M 1) = 0:22
(W; M 2) = 0:14 ( W; M 2) = 0:14
(W; M 3) = 0:5 ( W; M 3) = 0:5
The fact that all the logical correlations are negative
indicates that for a given machine, it is more likely
than not that an item was either defective and pro-
duced by it, or working and produced by another ma-
chine. This does not speak well for the manufacturer.
The logical correlations also lead us to rank the ma-
chines in the following order: M3 < M1 < M2.
7 Conclusion
Probabilities of conditionals can be used in addition
to, or even instead of conditional probabilities in any
problem involving Bayes' theorem.
The probability of a biconditional provides a
highly intuitive measure of relatedness, or logical cor-
relation, between two events. It measures the de-
gree to which two events are simultaneously true or
simultaneously false. One may also dene a logical
correlation coefcient for two events via the formula
(P; Q) = 2P(P $ Q) 1:
It is our opinion that selections from the topics we
have discussed here can be incorporated into under-
graduate treatments of probability once the standard
material has been developed. Probabilities of condi-
tionals, and especially probabilities of biconditionals,
can then provide additional insights into many stan-
dard problems.
References
[1] Adams E., “The Logic of Conditionals.” Inquiry,
8, (1965), pp. 166-197.
[2] Copi, I. M., Symbolic Logic, Macmillan Publish-
ing Co., Inc., New York, 1979
[3] Fulda, Joseph S., Material Implication Revisited,
The American Mathematical Monthly, Vol. 96,
No.3 (Mar., 1989), pp. 247-250.
[4] Lewis, D., Probabilities of Conditionals and Con-
ditional Probabilities, The Philosophical Review,
Vol. 85, No. 3 (Jul., 1976), pp. 297-315.
[5] Milne, P., Bruno de Finetti and the Logic of Con-
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232.
[6] Nguyen, Mukaidono, and Kreinovich. Probability
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ISSN: 1790-5117 ISBN: 978-960-6766-47-3 177
