THE (PROBABILISTIC) LOGICAL CONTENT OF CADIAG2

Rule-based Probabilistic Approach

David Picado Mui˜no

Institut f¨ur Diskrete Mathematik und Geometrie, Wiedner Hauptstrasse 8 - 104, Vienna, Austria

Keywords:

Rule-based expert systems, Medical expert systems, Probabilistic inference.

Abstract:

Cadiag2 is a well-known rule-based expert system that aims at providing support for medical diagnose in

internal medicine. Cadiag2 consists of a knowledge base in the form of a set of if-then rules that relate medical

entities, in this paper interpreted as conditional probabilistic statements, and an inference engine constructed

upon methods of fuzzy set theory. The aim underlying this paper is the understanding of the inference in

Cadiag2. To that purpose we provide a (probabilistic) logical formalization of the inference of the system and

check its adequacy with probability theory.

1 INTRODUCTION

Cadiag2 (Computer Assisted DIAGnosis) is a well-

known rule-based expert system aimed at providing

support in diagnostic decision making in the ﬁeld of

internal medicine. Its design and construction was

initiated in the early 80’s at the Medical University

of Vienna by K.P. Adlassnig – see (Adlassnig et al.,

1986), (Adlassnig et al., 1985), (Adlassnig, 1986) or

(Leitich et al., 2002) for more on the origins and de-

sign of Cadiag2.

Cadiag2 consists of two fundamental pieces: the

inference engine and the knowledge base. The in-

ference engine is based on methods of approximate

reasoning in fuzzy set theory, in the sense of (Zadeh,

1965) and (Zadeh, 1975). In fact Cadiag2 is pre-

sented in some monographs as an example of a fuzzy

expert system, (Klir and Folger, 1988), (Zimmer-

mann, 1991). The knowledge base, Φ

Cad

, consists of

a set of if-then rules intended to represent relation-

ships between distinct medical entities: symptoms,

ﬁndings, signs and test results on the one hand and

diseases and therapies on the other. The number of

rules in Φ

Cad

is approximately 50.000. The vast ma-

jority of them are binary (i.e., they relate single medi-

cal entities) and only such rules are considered in this

paper. The rules in Φ

Cad

are deﬁned along with a

certain degree of conﬁrmation which intuitively ex-

presses the degree to which the antecedent conﬁrms

the consequent. For example,

IF suspicion of liver metastases by liver pal-

pation THEN may be pancreatic cancer with

degree of conﬁrmation 0.30.

As mentioned in (Adlassnig, 1986) we can iden-

tify such degrees of conﬁrmation with probabilities

and the rules themselves with conditional probabilis-

tic statements. In (Adlassnig, 1986) it is stated that

such degrees of conﬁrmation can be interpreted as

frequencies. An interpretation in terms of degrees of

belief of the doctor (or doctors) on the truth of the

consequent given that the antecedent of the rule holds

is also possible though. This fact motivates a prob-

abilistic interpretation of Cadiag2’s inference. Such

an interpretation leads to the primary aim of this pa-

per: formalise the inference inCadiag2 on probabilis-

tic grounds and check its adequacy with probability

logic (Halpern, 2003) or, more generally, with prob-

ability theory. We shall not expect big surprises in

this respect. The inference mechanism in Cadiag2

proceeds in a compositional way and thus it is bound

to be probabilistically unsound (as will be clariﬁed

later). This was soon observed in earlier studies con-

cerning the celebrated expert system MYCIN – see

(Buchanan and Shortliffe, 1984) or (Shortliffe, 1976)

for a description of MYCIN and (Hajek, 1988), (Ha-

jek, 1989), (Hajek and Vald´es, 1994), (Heckerman,

1986), (Vald´es, 1992) for probabilistic approaches to

it. How far is Cadiag2’s inference from probabilistic

soundness remains to be seen though.

It is worth mentioning here that, although the in-

terest among theoretical AI researchers in rule-based

This rule is mentioned as an example in (Adlassnig

et al., 1986).

Picado Muiño D. (2010).

THE (PROBABILISTIC) LOGICAL CONTENT OF CADIAG2 - Rule-based Probabilistic Approach.

In Proceedings of the 2nd International Conference on Agents and Artiﬁcial Intelligence - Artiﬁcial Intelligence, pages 28-35

DOI: 10.5220/0002707700280035

 SciTePress

expert systems seems to be lesser today than some

years ago, rule-based expert systems are still very

popular among AI engineers. Many Cadiag2-like

systems are in use and more are being built for fu-

ture implementation. Is is mainly for this reason that

we believe that further analysis and understanding of

Cadiag2-like systems is of relevance.

This paper is in some way a continuation of (Cia-

battoni and Vetterlein, 2009). In (Ciabattoni and Vet-

terlein, 2009) the inference mechanism of Cadiag2 is

formalised by means of a logical calculus, CadL, and

compared to t-norm-based formalisms (Hajek, 1998).

It is shown that CadL does not respond to any t-norm-

based (or to any fragment of a t-norm-based) logic.

As far as we know, (Ciabattoni and Vetterlein, 2009)

constitutes the ﬁrst attempt at formalising and under-

standing Cadiag2 in a logical way. The present paper

is the second.

This paper is structured as follows. In Section 2

we give some basic deﬁnitions and introduce most of

the notation used later in the other sections. In Sec-

tion 3 the inference process in Cadiag2 is brieﬂy de-

scribed. In Section 4 the formal system CadPL is de-

ﬁned and analysed in the light of probability logic.

CadPL is a formalization of the inference mechanism

of Cadiag2 based on a probabilistic interpretation of

it.

2 PRELIMINARY DEFINITIONS

AND NOTATION

Throughout we will be working with a ﬁnite proposi-

tional language, L = {p

, ..., p

}. We will denote by

SL its closure under classical connectives. Within the

context of Cadiag2 the language L represents the set

of medical entities in the system.

Let L

Lit

= {p , ¬p | p ∈ L} ⊂ SL, the set of literals

of the language L.

Let ∆ = {φ

, ..., φ

} ⊆ SL. We will denote by

∆

the sentence φ

∧ ... ∧ φ

Deﬁnition 1. Let w : SL −→ [0, 1]. We say that w is a

probability function on L if the following two condi-

tions hold, for all θ, φ ∈ SL:

• If |= θ then w(θ) = 1.

• If |= ¬(θ∧ φ) then w(θ∨ φ) = w(θ) + w(φ).

We deﬁne conditional probability from the notion

of unconditional probability in the conventional way.

For w a probability function on L and φ, θ ∈ SL,

w(φ|θ) =

w(φ∧ θ)

w(θ)

Here (and throughout) |= is classical entailment.

The statements we will be dealing with are primar-

ily of the form ’the probability of θ given φ is equal

to η’. Let F L

be the set of all the statements of

the form P(θ|φ) = η, for θ,φ ∈ SL and η ∈ [0, 1]. Oc-

casionally we will refer to the set F L

≥

, deﬁned like

F L

but with ’≥’ in place of ’=’.

We will refer to φ in a statement of the form

P(θ|φ) = η as the evidence and to θ as the uncertain

entity or event.

We will denote by F L

the subset of conditional

statements of F L

where both the evidence and the

uncertain entity are literals, i.e. sentences in L

Lit

. By

F L

we will denote the subset of conditional state-

ments of F L

where the uncertain entity is a literal

and the evidence consists of a conjunction of literals

(we deﬁne F L

≥

and F L

≥

analogously).

The binary fragment of Cadiag2’s knowledge

base, Φ

CadBin

, will be in principle regarded as a subset

of F L

. That is arguably the most natural interpreta-

tion of Φ

CadBin

when interpreting the rules probabilis-

tically.

Let Θ ∈ F L

and w a probability function on L.

We deﬁne satisﬁability of Θ by w (denoted |=

Θ) in

the obvious way. More speciﬁcally, for η ∈ [0, 1] and

θ, φ ∈ SL,

P(θ|φ) = η ⇐⇒ w(θ|φ) = η.

Satisﬁability for statements in F L

≥

is deﬁned

analogously. Such notion of satisﬁability is extended

to subsets in F L

and F L

≥

in its trivial way. We

will sometimes identify the notion of consistency of a

set of probabilistic statements with that of satisﬁabil-

ity.

Deﬁnition 2. Let  be the partial ordering relation

on [0, 1] deﬁned as follows: For a, b ∈ [0, 1], a  b if

and only if 0 < a ≤ b or 0 < a < 1 and b = 0.

We deﬁne ≺ from  in the conventional way.

As we will see later, the deﬁnition of ≺ responds

to the use of both 0 and 1 as maximal values in

Cadiag2. The value 0 denotes certainty in the non-

occurrence of an event or falsity of a statement and

the value 1 denotes certainty in its occurrence or its

truth.

For the next deﬁnition let

D = [0, 1] × [0, 1] − {(0, 1), (1, 0)}.

Deﬁnition 3. The function max

∗

: D −→ R is deﬁned

as follows, for (a, b) ∈ D:

max

∗

(a, b) =



a if b ≺ a

b otherwise

The deﬁnition of max

∗

is extended to more than

two arguments in its trivial way.

THE (PROBABILISTIC) LOGICAL CONTENT OF CADIAG2 - Rule-based Probabilistic Approach

3 THE INFERENCE IN CADIAG2

In this section we describe very brieﬂy a generaliza-

tion of the inference mechanism in Cadiag2. A more

detailed description and analysis of it can be found in

(Ciabattoni and Vetterlein, 2009).

Cadiag2 formally distinguishes between three dif-

ferent types of rules: type conﬁrming to the degree d

(for d ∈ [0, 1]), type mutually exclusive and type al-

ways occurring – see (Adlassnig et al., 1986), (Ad-

lassnig et al., 1985), (Adlassnig, 1986) or (Daniel

et al., 1997) for more on Cadiag2’s rules. The last

two types mentioned are classical in the sense that

the degree of conﬁrmation for the rules of these types

is 1 and that the antecedent of such rules (or evidence

in our settings) needs to be fully true (degree of pres-

ence or of truth 1, see below) in order for these rules

to be triggered by the system. Such a distinction is not

taken into consideration in this paper and it is in this

sense that we say that our description of the inference

mechanism of Cadiag2 is actually a generalisation of

the real inference process. The inference engine in

Cadiag2 gets started with a set of symptoms, ﬁndings,

signs and diseases occurring in Φ

CadBin

present in the

patient. Let Γ be the set of such medical entities.

Cadiag2 starts with an assignment w

on Γ that

gives a value in the interval [0, 1] to each entity in

Γ. Such value is intended to represent the degree to

which the entity is present in the patient. The in-

tended interpretation of such values is based, in prin-

ciple, on fuzzy set theory. However, other interpreta-

tions can also be suitable, at least to some extent. In

fact, when deﬁning the system CadPL, the interpre-

tation to which we will commit will be probabilistic.

The assignment w

is then extended to negative state-

ments and logical equivalents according to the follow-

ing rule:

If w

(φ) = η then w

(¬φ) = 1− η, for φ ∈ SL

and η ∈ [0, 1].

After the initial assignment the inference rules in

CadBin

come into play. All the rules triggered by the

sentences in Γ are used during the inference process.

At each step in the inference process a rule is ap-

plied (that is done, in principle, in no particular order).

At the ﬁrst step in the inference a rule of the form

P(θ|φ) = η in Φ

CadBin

is triggered, with η ∈ [0, 1] and

θ, φ ∈ L

Lit

. In order for that to happen φ or its negation

needs to be in Γ and the value w

(φ) has to be strictly

positive. The application of the rule P(θ|φ) = η gen-

erates a new assignment, w

, on {θ}. The value as-

signed to θ by it is calculated as the minimum be-

tween η and w

(φ) and the value assigned to ¬θ and

logical equivalents (if necessary for the inference) is

calculated from w

(θ) as mentioned above for w

At the n

step in the inference process a new rule

of the form P(θ|ψ) = η in Φ

CadBin

will be triggered,

for η ∈ [0, 1] and θ, ψ ∈ L

Lit

. In order for P(θ|ψ) = η

to be triggered ψ must have been assigned at least one

value in (0, 1] either by the initial assignment, w

, or

by any other assignment on {ψ} deﬁned during the

inference process at some previous step. At the n

step the application of this new rule will generate a

new assignment on {θ} that will give θ the minimum

between η and the value of ψ considered for trigger-

ing the rule at this step in the inference (as above, this

value needs to be strictly positive). If the strictly posi-

tive values generated for ψ before the n

step are mul-

tiple the inference mechanism inCadiag2 will call the

rule P(θ|ψ) = η again in further steps, if it has not

done so previously,until all the valuesfor ψ havebeen

used and all the possible values for θ generated. The

assignment w

is deﬁned to ¬θ as mentioned above.

The inference process goes on until all the rules

triggered by all the sentences in Γ and its negations

have been used and all the possible assignments for

the sentences involved in the inference have been gen-

erated. Cadiag2 yields as an outcome of the inference

the set of medical entities in L occurring in the rules

triggered by the evidence in Γ along with the maxi-

mal value (with respect to the ordering  deﬁned in

Section 2) assigned to them during the inference. If

a sentence is assigned both value 0 and 1 along the

inference process the system generates an error mes-

sage.

It is worth mentioning that the original inference

process in Cadiag2 works in a slightly different way.

The update in the value of the distinct sentences in-

volved in the inference is done as soon as two differ-

ent values for the same sentence are produced by the

system. The value chosen in the update for atomic

sentences in L is the maximal one (with respect to

the ordering ). Notice though that this feature has

a highly undesirable result (unless further restrictions

on the rules or on the order in which the rules are ap-

plied are imposed), which is that the outcome of a run

of the inference mechanism can depend on the order

in which the rules are applied.

Such a drawback is easily avoided by assuming

that the update is only done at the end of the pro-

cess. There are other several undesirable features in

Cadiag2’s inference engine, most of them related to

the maximal value 0 and negated propositions. Maybe

the most evident concerning the maximal value 0 is

that a medical entity that at some step along the in-

ference process is assigned value 0 (that is to say, it

is considered false with certainty or impossible) trig-

gers any rules in which it occurs as evidence if any

other value other than 0 is assigned to it along the in-

ICAART 2010 - 2nd International Conference on Agents and Artificial Intelligence

ference process. For a deeper analysis of such aspects

of the inference process in Cadiag2 see (Ciabattoni

and Vetterlein, 2009).

We represent sentences together with the assign-

ments generated for them at each step in the inference

by pairs in SL× [0, 1] along with a subscript indicat-

ing the step in the process at which such pairs have

been generated. As mentioned above, a step in the in-

ference process is given by the application of a rule in

CadBin

and the new assignments that it generates for

the sentences involved in the rule.

Let p ∈ L and η ∈ [0, 1] be the highest assignment

to p in a run of the inference mechanism in Cadiag2.

We will use the subscript max

∗

on the pair (p, η) –

that is to say, (p, η)

max

∗

– to denote that η is the maxi-

mal value assigned during the inference process for p

(with respect to the ordering ).

4 THE FORMAL SYSTEM CADPL

Some medical entities that occur in the rules of

Cadiag2 represent statements that are vague. For ex-

ample, in Cadiag2 we have a medical entity given by

the following statement: ’reduced glucose in serum’.

In such a statement the adjective ’reduced’ is vague.

Cadiag2 tackles vagueness by assigning values to

medical entities in the interval [0, 1]. Such values

stand in principle for fuzzy membership within the

context of fuzzy set theory – see (Adlassnig et al.,

1986), (Adlassnig et al., 1985) or (Adlassnig, 1986).

In this paper we consider the possibility of interpret-

ing such values as probabilities, which can be done in

quite intuitive ways given the nature of the statements

we are dealing with.

Let us consider again the statement ’reduced glu-

cose in serum’. Let us assume that the value assigned

by the evaluation system in Cadiag2 to the statement

’Patient A has reduced glucose in serum’ out of the

evidence given by the corresponding measurement of

the amount of glucose in Patient A is η, for some η ∈

[0, 1]. As an example, we could interpret such value

as the degree of belief that a medical doctor has in the

truth of the statement given the evidence. As such η

could be interpretedas a probability. The probabilistic

interpretation is certainly favoured by the discretiza-

tion applied to medical concepts in Cadiag2 (for ex-

ample, the concept ’glucose in serum’ generates ﬁve

distinct medical entities in Cadiag2: ’highly reduced

glucose in serum’, ’reduced glucose in serum’, ’nor-

mal glucose in serum’, ’elevated glucose in serum’

and ’highly elevated glucose in serum’). Notice that

This example is extracted from (Adlassnig et al., 1986).

such an interpretation places us within the subjective

probabilistic frame and thus, for the sake of coher-

ence, the knowledge base Φ

CadBin

should also be in-

terpreted subjectively. Other interpretations are also

possible though. For example, one could regard such

values as the ratio given by the number of doctors that

agree on the truth of the statement out of all the doc-

tors involved in the assessment. In order to accommo-

date such values into a coherent probabilistic frame

along with the statements in Φ

CadBin

one could justify

them as being subjective probabilities assessed by a

group of experts – see (Genest and Zidek, 1986) or

(Osherson and Vardi, 2006) for an analysis and justi-

ﬁcation of such concept.

Let φ ∈ L

Lit

represent a medical entity present in

the patient and assume that η ∈ [0, 1] is the initial

value assigned to it at the start of a run of Cadiag2’s

inference process. We can formalise this by means

of a probabilistic conditional statement of the form

P(φ|κ) = η in F L

, where κ ∈ SL is the evidence

that supports the presence of φ in the patient. For sim-

plicity the sentence κ will be assumed to be a literal

in L

Lit

Next we are going to deﬁne the formal system

CadPL. Recall that the ultimate goal when doing so is

to deﬁne a system which represents the inference pro-

cess in Cadiag2 when interpreted from a probabilistic

point of view. Although the inference in Cadiag2 can

be closely related to probability theory (given the na-

ture of the rules of inference in Φ

CadBin

) it is not based

on probabilistic methods and so the degree of free-

dom when choosing the rules of the system CadPL is

high. We have chosen the rules by interpreting in the

most natural way the steps along the inference pro-

cess within a probabilistic frame. The main idea be-

hind such interpretation consists of the identiﬁcation

of the inference process with the propagation of evi-

dence facilitated by the rules in Φ

CadBin

. For example,

from P(φ|κ) = η, where k ∈ L

Lit

is evidence support-

ing the presence of φ in the patient, and P(θ|φ) = ζ

in Φ

CadBin

we would infer P(θ|κ) = min(η, ζ), where

min(η, ζ) is the value (probability) assigned to θ given

the evidence κ. We would have a propagation process

of this nature for each single piece of evidence. The

evidence would then be brought together in Cadiag2

by what we call the Right conjunction rule: given

two outcomes of Cadiag2’s inference process, say

P(p|κ

) = η and P(p|κ

) = ζ, for p ∈ L and κ

, κ

∈

Lit

, Cadiag2 combines the evidence given by κ

and

by computing P(p|κ

∧ κ

) = max

∗

(η, ζ). The in-

ference rules of CadPL that we next present formalise

this interpretation.A theory T in CadPL is a ﬁnite

subset of sentences in F L

THE (PROBABILISTIC) LOGICAL CONTENT OF CADIAG2 - Rule-based Probabilistic Approach

For what follows let T = Ω ∪ Φ, with

Ω = {P(φ

|κ

) = η

, ..., P(φ

|κ

) = η

for some m ∈ N.

Let K

Ω

= {κ

, ..., κ

} and Γ = {φ

, ..., φ

The set Ω is intended to represent the initial as-

signment in the inference process, Φ the set of rules of

the system, Γ the initial set of medical entities present

in the patient and K

Ω

the evidence in support of the

presence of the corresponding medical entities in Γ.

The formal system CadPL is deﬁned by the fol-

lowing inference rules:

Inference rules

• Reﬂexivity rule

For φ ∈ L

Lit

, κ ∈ K

Ω

and η ∈ [0, 1],

P(φ|κ) = η ∈ Ω

T ⊢ P(φ|κ) = η

• Negation rule

For φ ∈ L

Lit

, ψ ∈ SL and η ∈ [0, 1],

T ⊢ P(φ|ψ) = η

T ⊢ P(¬φ|ψ) = 1− η

• Equivalence rule

For ψ, φ, θ ∈ SL and η ∈ [0, 1],

ψ ≡ φ T ⊢ P(φ|θ) = η

T ⊢ P(ψ|θ) = η

• Minimum rule

For θ,φ ∈ L

Lit

, κ ∈ K

Ω

, η ∈ (0, 1] and ζ ∈ [0, 1],

T ⊢ P(θ|κ) = η P(φ|θ) = ζ ∈ Φ

T ⊢ P(φ|κ) = min(η, ζ)

• Right conjunction rule

For p ∈ L, K

, K

⊆ K

Ω

and η, ζ ∈ [0, 1],

T ⊢ P(p|

) = η T ⊢ P(p|

) = ζ

T ⊢ P(p|

∪ K

}) = max

∗

(η, ζ)

• Exhaustivity rule

For p ∈ L, κ ∈ K

Ω

, K ⊆ K

Ω

and η ∈ [0, 1],

T ⊢ P(p|

K) = η ∀ζ ∈ [0, 1] T 0 P(p|κ) = ζ

T ⊢ P(p|κ∧

K) = η

Notice that the Exhaustivity rule does not have any

bearing on the decidability of whether P(p|κ) = ζ is

provable from T or not for ζ ∈ [0, 1], p ∈ L and κ ∈

Ω

. The Exhaustivity rule can only be applied after

its provability or non-provability has been decided.

Given a theory T of CadPL and a statement Θ ∈

F L

, a proof of Θ from T is deﬁned as a ﬁnite se-

quence of sequents of the form

T ⊢ Θ

, ..., T ⊢ Θ

with Θ

= Θ and where, for i ∈ {1, ..., n}, each Θ

T ⊢ Θ

follows from T by the application of one of

the rules above, from Θ

in a previous sequent (with

j < i) or from Θ

, Θ

in previous sequents (with j, k <

i) by one of the rules above.

Let Θ be the statement P(θ|φ) = η, for some η ∈

[0, 1] and θ, φ ∈ SL. We say that there exists a maximal

proof of Θ from T if there exists a proof of Θ from

T and there is no proof from T of P(θ|φ) = ζ with

η ≺ ζ.

We say that Θ follows maximally from T (denoted

by T ⊢

CadPL

Θ) if there exists a maximal proof of Θ

from T .

For the next proposition let T = Ω∪Φ

CadBin

, with

Ω = {P(φ

|κ

) = η

, ..., P(φ

|κ

) = η

Ω

= {κ

, ..., κ

} ⊂ L

Lit

and Γ = { φ

, ..., φ

} a subset

of literals occurring in Φ

CadBin

Proposition 4. Let p ∈ L and η ∈ [0, 1]. We have that

T ⊢

CadPL

P(p|

Ω

) = η

if and only if (p, η)

max

∗

is the outcome of a run of

Cadiag2’s inference process on T .

Proof.

In order to prove the left implication let us

consider a run of Cadiag2’s inference mechanism on

T . The inference starts from pairs of the form (φ, η)

and (¬φ,1 − η)

for some η ∈ [0, 1] for all φ ∈ Γ. In

CadPL a pair of the form (φ, η)

, for φ ∈ Γ, corre-

sponds to a sequent of the form T ⊢ P(φ|κ) = η, for

κ ∈ K

Ω

. The pair (¬φ, 1− η)

corresponds to the se-

quent T ⊢ P(¬φ|κ) = 1 − η. The former corresponds

to an application of the Reﬂexivity rule. The latter fol-

lows from the ﬁrst one by an application of the Nega-

tion rule.

Let us assume now that we are at the n

step of the

inference process and that a rule of the form P(θ|ψ) =

ζ is triggered, for some ζ ∈ [0, 1] and θ, ψ ∈ L

Lit

. Let

us suppose that we have (ψ, µ)

n−t

, the pair that trig-

gers the rule at the n

step of the process, for µ ∈ (0, 1]

and t ≤ n − 1. In CadPL that would correspond

to a sequent of the form T ⊢ P(ψ|κ) = µ derived

from a previous step in the inference, for κ ∈ K

Ω

The inference mechanism in Cadiag2 produces the

pairs (θ, min(ζ, µ))

and (¬θ, 1 − min(ζ, µ))

which,

in CadPL, corresponds to the sequents T ⊢ P(θ|κ) =

min(ζ, µ) and T ⊢ P(¬θ|κ) = 1 − min(ζ, µ) respec-

tively, which follow by an application of the Minimum

rule and, for the latter, an application of the Negation

rule on the former.

For the sake of brevity we will deal with sentences as

if they were equivalence classes. If anything applies to a

sentence of the form ¬φ, with φ ∈ L

Lit

, we also assume that

it applies to any logical equivalent of φ without mentioning

it.

ICAART 2010 - 2nd International Conference on Agents and Artificial Intelligence

At the end of the process Cadiag2 generates the

pair (p, η)

max

∗

for each sentence p ∈ L involved in the

inference, where η is the maximal value (with respect

to the ordering  ) among those assigned to p along

the inference. This maximization process is achieved

in CadPL by means of repeated applications of the

Right conjunction rule. Instances of the Exhaustivity

rule (if necessary) complete the inferential counter-

part of Cadiag2 in CadPL.

In order to prove the right implication let us sup-

pose that we have a maximal proof of the form

T ⊢ Θ

, ..., T ⊢ Θ

where Θ

is the statement P(p|

Ω

) = η, for some

η ∈ [0, 1] and p ∈ L.

The ﬁrst sequent of the proof needs to respond to

an instance of the Reﬂexivity rule, T ⊢ P(φ|κ) = η, for

some φ ∈ Γ, κ ∈ K

Ω

and η ∈ [0, 1]. The correspond-

ing counterpart of this sequent in Cadiag2 is the pair

(φ, η)

Let us move now to the n

sequent, with n ≤ m.

The n

sequent can be an instance of the Reﬂexivity

rule, T ⊢ P(φ|κ) = η, for some φ ∈ Γ, η ∈ [0, 1] and

κ ∈ K

Ω

. The counterpart for this sequent in Cadiag2

is the pair (φ, η)

The n

sequent can follow from a previous one in

the proof by an instance of the Negation rule. Let us

suppose that the n

sequent is T ⊢ P(¬θ|ψ) = 1 − η

for some η ∈ [0, 1], θ ∈ L

Lit

and ψ ⊆ L

Lit

and that

there is a sequent T ⊢ Θ

, for some i < n, of the form

T ⊢ P(θ|ψ) = η. The latter corresponds to a pair of

the form (θ,η)

in Cadiag2 and the former to the pair

(¬θ, 1 − η)

, where t is the step in the inference pro-

cess at which such pairs have been generated.

The n

sequent can follow from a previous one

by an instance of the Minimum rule. Let us assume

that the n

sequent is T ⊢ P(θ|κ) = min(η,ζ), for

some θ ∈ L

Lit

, κ ∈ K

Ω

, η ∈ [0, 1] and ζ ∈ (0, 1], that

T ⊢ P(ψ|κ) = ζ is a previous sequent in the proof and

that P(θ|ψ) = η ∈ Φ

CadBin

. The latter corresponds in

Cadiag2 to the pair (ψ, ζ)

and the former to the pair

(θ, min(η, ζ))

t+k

, where t,t + k indicate the steps at

which the pairs have been generated by the inference

process.

The n

sequent can follow from previous sequents

by an application of the Right conjunction rule. The

counterpart in Cadiag2 of such an outcome consists

of the maximization process at the end of the infer-

ence. Instances of the Exhaustivity rule are irrelevant

to the inference in Cadiag2.

This completes the proof. 

It is worth commenting on some features of the in-

ference rules ofCadPL in connection with probability

theory.

Soundness with respect to probabilistic semantics

of the Reﬂexivity, Negation and Equivalence rules is

clear. The Minimum rule is certainly not sound with

respect to such semantics. The Right conjunction rule

is not sound and it can generate probabilistic conse-

quences that are inconsistent with its premises and

the theory T (in the sense that such consequences

along with the premises and the theory are not si-

multaneously satisﬁable by a probability function).

The Exhaustivity rule assumes some probabilistic in-

dependence among sentences that may not actually

be independent. Overall, CadPL does not score well

within probability theory. This is no surprise. The

computation of conditional probabilistic statements in

a compositional way, as done by Cadiag2 primarily

by means of the min and max

∗

operators, is clearly

bound to be probabilistically unsound. One may won-

der though what could be done in order to improve

the inference on probabilistic grounds from a knowl-

edge base like Φ

CadBin

. The answer seems to be ’not

much’. Certainly a Φ

CadBin

-like knowledge base (i.e.,

a knowledge base given by some binary probabilistic

conditional statements) is not the most convenient for

inferential purposes in probability theory for medical

applications like Cadiag2. As is well known, there

are other knowledge-base structures better suited for

that purpose, Bayesian networks being the most cel-

ebrated among them, see (Castillo et al., 1997) or

(Pearl, 1988).

In terms of consistency, it is worth noting that

CadPL satisﬁes what we can call weak consistency

– called weak soundness in (Hajek, 1988) –, deﬁned

as follows: if there is a maximal proof in CadPL of a

statement of the form P(φ|

∆) = 1 (or P(φ|

∆) = 0)

from a certain theory T , with φ ∈ SL and ∆ ⊆ SL then,

if there is a maximal proof in CadPL of a statement

of the form P(φ|

∆

∗

) = η, with ∆ ⊂ ∆

∗

, then η = 1

(or η = 0 respectively). That is to say, if CadPL con-

cludes certainty about the occurrenceof some eventor

about the truth or falsity of some sentence then adding

new evidence does not alter this certainty. Weak con-

sistency is provided in CadPL and so in Cadiag2’s in-

ference mechanism by the operator max

∗

deﬁned over

the ordering .

It is also worth noting that one could guaran-

tee consistency (i.e., satisﬁability) by considering

CadBin

a subset of F L

≥

(in place of F L

, regarding

the valuesof the conditional statements as lower prob-

ability bounds rather than as exact probabilities) and

by restricting the system to a positive fragment of L

Lit

(i.e., only one of p, ¬p can occur in Φ

CadBin

). This

way consistency is trivially guaranteed for Φ

CadBin

together with any outcomes produced by the system

during the inference process.

In terms of soundness there does not seem to be

THE (PROBABILISTIC) LOGICAL CONTENT OF CADIAG2 - Rule-based Probabilistic Approach

much that one can do in order to improve the infer-

ence mechanism for knowledge bases like Φ

CadBin

, or

at least not much that one can do that does not come

at the price of generating probabilistic statements

with very low probabilistic bounds (when working

in F L

≥

), which would make Cadiag2 potentially

useless for practical purposes. There is some room

for improvement for some steps in the inference that

come by the addition of some independence assump-

tions among some of the medical entities in Φ

CadBin

Under such independence assumptions the product

operator in place of the min operator could yield

soundness for the inference steps referred.

5 CONCLUSIONS

Cadiag2 is a reasonably well-performing medical ex-

pert system (Adlassnig et al., 1986), but how it is so is

far from clear. The inference engine of Cadiag2 was

built with methods of approximate reasoning in fuzzy

set theory but, as such, it was not based on any logical

formalism or theory embeddedwith a clear semantics.

This fact motivated the main aim of this paper, which

was no other than the understanding of Cadiag2 in a

logical way.

The natural interpretation of the inference rules of

Cadiag2 (i.e., probabilistic) placed us upon the at-

tempt of interpreting the inference itself probabilis-

tically. We formalised this interpretation by means of

the system CadPL, the logical (probabilistic) coun-

terpart of the inference engine of Cadiag2. The un-

soundness of some of the rules of CadPL (and thus

of some inference steps in Cadiag2) and the inconsis-

tency of the calculus (and thus of the inference pro-

cess in Cadiag2) was made clear. Apart from these

drawbacks, otherwise expected, some other aspects of

CadPL were also stressed and analysed. At the end of

the paper some possibilities for an improvement of

Cadiag2 in terms of soundness and consistency were

also mentioned.

ACKNOWLEDGEMENTS

This research was supported by the Vienna Science

and Technology Fund (WWTF), Grant MA07− 016.

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