*This paper continues the series Philosophy of Diagnosis. See also Part 1 and Part 2.*

The most influential account of diagnostic reasoning in the philosophical literature comes from Kazem Sadegh-Zadeh (2000a; 2011). His account combines fuzzy logic and a criterion of causal relevance derived from Suppes (1970) to attempt to analyse diagnostics. Sadegh-Zadeh’s account is the most well-developed philosophical account of diagnostics and has been widely deployed in computer-aided diagnostics. As such, a detailed philosophical analysis of his account is needed. Unfortunately, though much of his machinery may be valuable, his equation of diagnosticity with causal relevance proves unsustainable. Indeed, not only is diagnosticity not equivalent to probabilistic causal relevance as Sadegh-Zadeh claims, but positive causal relevance turns out to be strictly speaking unnecessary for a statement to be a diagnosis of a patient.

## Diagnostic Form

Sadegh-Zadeh offers a clear answer to the question of diagnosis structure. Consider a patient *p*, who has a set of symptoms, *S*. In order for a statement *D *to be a diagnosis for *p*, according to Sadegh-Zadeh, it must be the case that:

(1)Dis a declarative statement aboutp, such that every atomic proposition withinDhas the formQp, whereQis an abnormality predicate.

*D *may be an individual subject-predicate statement (i.e. *Qp*) or a complex statement composed only of subject-predicate atomic statements (e.g. *Qp & Rp* or *Qp v (Rp & Sp)*), as long as every predicate is an abnormality predicate. “Abnormality” here is a blanket phrase designed to capture diseases, disorders, injuries, malformations, etc. (see Part 4). So, to have the form of a diagnosis-statement, D must be a declarative sentence about a patient which ascribes one or more abnormality predicates to that patient. For the purposes of this paper, we can accept without endorsing Sadegh-Zadeh’s account of diagnostic structure pending further analysis in Part 5.

## Causal Relevance

But not every statement ascribing an abnormality to a patient is a candidate diagnosis of their symptoms. Sadegh-Zadeh adds two further conditions designed to capture the idea that *D *is a diagnosis for *p *to the extent that the *D* is positively causally relevant to *p*’s symptom-set *S*. These conditions are:

(2)DprecedesS(that is, the abnormalities ascribed topbyDprecede the occurrence of the symptoms inp’s symptom-setS).

(3)Dis positively causally relevant toS.

The specific meaning of ‘positive causal relevance’ per Sadegh-Zadeh will be explicated below. Sadegh-Zadeh’s intuition that diagnostic candidacy requires a putative causal relationship between abnormalities and symptoms has surface merit. However, his criteria both prove too strong. Criteria (2) and (3) capture an intuition that should be preserved, albeit in a different form. But we will show here that both Sadegh-Zadeh’s stated versions of these criteria, and an adapted form which avoids some of the initial problems posed by Sadegh-Zadeh’s versions, will fail. Indeed, neither precedence nor causal relevance is strictly necessary for a statement to be a diagnosis.

### Precedence

Condition (2) is too strong because Sadegh-Zadeh requires that *all *abnormalities ascribed to *p* by *D* must precede *all *of *p*’s symptoms *S*. In practice, this is often not the case in legitimate diagnoses. First, symptoms sometimes form part of the diagnosis. For instance, suppose *p*’s symptom set *S* is congestive heart failure (CHF), dyspnea (shortness of breath) and hypertension (i.e. *S *= {CHF, dyspnea, hypertension}). Hypertension is a symptom but may also form part of a diagnosis of the other symptoms. One legitimate diagnosis for *p* would invoke Ischemic Heart Disease (IHD). IHD is a prominent cause of CHF but does not in itself directly cause hypertension or dyspnea. However, CHF can cause dyspnea, while hypertension is a risk-factor for both IHD and CHF. Thus, a complete diagnostic statement for *p* might be: “*p* has hypertension leading to IHD, leading to CHF, which caused dyspnea”. This statement ascribes three abnormality predicates to *p* (hypertension, IHD and CHF), but none precede every symptom in set *S*, as none precede hypertension. Another diagnosis might attempt to explain the hypertension too. But the above diagnosis would generally be considered a legitimate attempt to diagnose *p*, and is certainly a candidate diagnosis, so stands as a counterexample to condition (2). Moreover, symptoms do not have to be admitted into diagnoses for this problem to arise; if we just give “*p* has IHD” as our diagnosis, and hypertension precedes IHD, then IHD cannot count as diagnostic per Sadegh-Zadeh, even though it accounts for the symptom of CHF (and indirectly for the dyspnea), and is quite probable given the patient’s hypertension.

We could attempt to do justice to the intuition that something in the diagnosis must precede something in the symptom set through the weaker constraint:

(2*) Some abnormality ascribed topinDprecedes some symptomsϵS.

This constraint is satisfied in the IHD diagnosis. The IHD abnormality precedes the CHF symptom, so the constraint is met. This criterion does have the consequence that a symptom cannot diagnose itself. If *p*’s only symptom is hypertension, then ‘*p* has hypertension’ is not a candidate diagnosis for *p*. This may ultimately be acceptable. Objections arise because we think of conditions like hypertension as causing or consisting of a number of other symptoms—headaches, fatigue, chest pain, irregular heart rhythms and so forth—which, were they included as symptoms, would satisfy the constraint. However, in the case of constitutive matching (see Part 2), the problem is more severe. A purely constitutive syndrome is one which is extensionally defined as the presence of a certain set of symptoms. If we accept even the modified precedence condition, we must accept that diagnosing such a condition cannot count even as a candidate diagnosis. The Constitutive Matching Constraint and the *Sine qua non* Constraint do not *require* that purely constitutive syndromes *can be *candidate or legitimate diagnoses. Thus far, this account has left the possibility open. But (2*) rules them out entirely. So, if we wish to make reference to any such syndromes, then even (2*) is too strong.

(2*) also excludes a null diagnosis from being a candidate diagnosis. “There is nothing medically wrong with *p*” is a legitimate outcome of many diagnostic processes. Sadegh-Zadeh’s focus is on clinical encounters (which themselves may sometimes result in null diagnoses), but we should also consider other diagnostic roles in which null diagnosis is commonplace. For instance, the specialism of public health foregrounds the epidemiological role of diagnostics. We might test individuals to monitor the spread of an infectious disease. If a patient’s tests negative, the warranted diagnosis is null: the patient does not have the disease. But “*p *does not have Covid-19”, for example, cannot be the output of a diagnostic process in Sadegh-Zadeh’s framework because *not having* Covid-19 is not an abnormality (nor does it meaningfully precede anything in the symptom set). For the patient with the clean test, the diagnosis is null. But a null diagnosis cannot satisfy (2*) trivially, as there are no abnormalities so none can precede any symptoms. Similarly, in the academic role of diagnostics, diagnosis may be a part of research: for example, researchers might do a complete work-up of a clinical trial volunteer to ensure that they are healthy before admitting them to a phase-I clinical trial. Again, it seems “There is nothing medically wrong with *p*”is a legitimate outcome of a diagnostic process here. For Sadegh-Zadeh, it is trivially true that a patient with no symptoms cannot be diagnosed at all. But, again, the patient who is screened for a disease or given a clean bill of health in a physical is excluded from the scope of diagnostics on this account. This stands even if the diagnostic process involves removing all symptoms from the symptom set (perhaps by exposing them as misleading signs or incorrect test results).

While Sadegh-Zadeh could potentially bite the bullet on all of these concerns, it seems that the precedence constraint may be misguided or unhelpful. Later, we will see how a different model of diagnostic structure overcomes this limitation, particularly by understanding diagnosis as a process rather than as a relationship between a statement and a symptom set.

### Fuzzy diagnosticity

Sadegh-Zadeh’s causal relevance criterion (3) and subsequent equation of causal relevance with diagnosticity is the most important and contentious part of his account. Sadegh-Zadeh’s approach is to fuzzify the concepts of diagnostic candidacy and legitimacy (although Sadegh-Zadeh (2000a; 2011) does not distinguish the two: we can refer to these as *diagnosticity* when conflated together—diagnosticity is the extent to which a statement *is a diagnosis* for a patient). A fuzzy property admits of degrees. Using the framework of fuzzy set theory, we can define a *fuzzy membership function *μ which maps the domain into real numbers in the interval [0,1], and a fuzzy property P as a set of ordered pairs of the form (*x*, μ_{p}(*x*)) for all *x* in the domain. The membership function μ gives the *degree of membership* of the set P for every *x*—that is, the degree to which *x* possesses the property P. If μ_{p}(*x*)=1, then *x *fully possesses the property P (i.e. *x* is a full member of set P). If μ_{p}(*x*)=0, then *x* does not possess the property at all. If μ_{p}(*x*) is a real number 0 < *r *< 1, then* x* possesses the property P to degree *r*.

For example, the property of *tallness* is a fuzzy property. Some people are tall, some are not, others are somewhat tall. So if Adam is 7 feet tall, Ben 6 feet tall, and Charles 5 feet tall, we would say that μ_{tall}(Adam)=1 (i.e. Adam is tall), μ_{tall}(Charles)=0 (he is not at all tall), and μ_{tall}(Ben) is somewhere in between, perhaps 0.4 (he partially possesses the property of tallness). Sadegh-Zadeh has prominently used this fuzzy theoretic framework to attempt to capture several medical concepts, including health, patienthood and disease (see e.g. 1999; 2000a; 2000b; 2001; 2011).

### Diagnosticity as Causal Relevance

Supposing that criteria (1) and (2) are met, Sadegh-Zadeh equates the degree of diagnosticity of a diagnosis statement *D* for a patient *p* with its degree of causal relevance to *p*’s symptom set *S*. He employs an account of causal relevance derived from Suppes (1970). For events A and B, in a population P, the function cr(A,B,P) gives the causal relevance of event A to event B as follows:

cr(A,B,P) = p(B|A&P) – p(B|P) (Sadegh-Zadeh, 2011, p.368)

That is, the causal relevance of event A to event B in population P equals the probability of B occurring given that A has occurred, minus the probability of B occurring. Intuitively, the causal relevance of A to B in P is the degree to which event A changes the probability of event B in P. cr(A,B,P) can take values in the interval [-1,1]. A positive value indicates positive causal relevance, a negative value indicates negative causal relevance (prevention), and a zero-value indicates that there is no relationship between A and B.

This account of causal relevance is in many ways a deliberate simplification of that developed by Suppes (1970) and similar accounts (e.g. Reichenbach, 1956; Cartwright, 1979; Skyrms, 1980), as Sadegh-Zadeh acknowledges. However, the important element of the account for our purposes is the use of positive causal relevance as a minimal prerequisite for diagnosticity. Sadegh-Zadeh stipulates that any diagnosis must be positively causally relevant to the patient’s symptoms. So, for a diagnosis *D*, symptom set *S* and population *P*, *D *is a diagnosis (to any degree) only if:

(3) cr(D,S,P) > 0

While Suppes (as Reichenbach, Cartwright and others) had claimed positive causal relevance as necessary for causation, they do not view it as sufficient, and indeed offer considerably more complex accounts, usually drawing on further “screening off” conditions or “background contexts”. Sadegh-Zadeh, though, offers a relatively straightforward equation of diagnosticity with degree of positive causal relevance, such that for any *D*, *S* and *P *which satisfies condition (3):

μ_{Diag}(D) = cr(D,S,P) (Sadegh-Zadeh, 2011, p.371)

So, the extent to which *D* is a diagnosis for *p* is the extent to which *D* is positively causally relevant to *p*’s symptom set *S* in a relevant population *P*. If the causal relevance of *D* to *S* is negative or zero, then the degree of diagnosticity is zero and *D *is *not a diagnosis* for *p*. Sadegh-Zadeh uses this mechanism to rule out non-diagnostic statements like “*p *has blonde hair” on the grounds of causal irrelevance. So positive causal relevance is necessary for diagnosticity, and moreover a putative diagnosis is diagnostic for a patient to the extent that it is positively causally relevant to their symptoms.

An advantage of Sadegh-Zadeh’s account is that because causal relevance, as he defines it, is not necessarily an attempt to capture or define causal relationships, he does not need to worry about the problem of common causes. A confounding cause which explains the probabilistic dependency between *D *and* S* in *P* does not undermine the diagnosticity of *D *for *S* in his framework or prevent *D* from being a diagnosis. Thus, he can sidestep much of the complexities of the accounts of probabilistic causation offered by Suppes and others, and allow for a larger diagnostic space. Sadegh-Zadeh does not require that a diagnosis *actually causes* the symptoms, only that there is some probabilistic dependency between the diagnosis and symptoms. This could turn out to be due to common causes or even due to a causal relationship from symptom to diagnosis (though this is intentionally excluded by the precedence requirement).

This account might seem initially plausible if a little permissive. If the patient’s symptoms are more likely given the abnormalities in the diagnosis, then the diagnosis is worth considering. Sadegh-Zadeh assembles a set Δ of diagnoses which meet the preconditions, which can be ranked or sorted according to the degree of diagnosticity and used to inform subsequent clinical interactions. However, there are several flaws, at least one of which is fatal to this account. I will describe only the most important problems here. As the criticism offered here primarily relates to requirement (3) in demonstrating that positive causal relevance is not necessary for *D* to be a diagnosis for *S*, we can similarly sidestep the more detailed interpretation of causal relevance. Probabilistic causation in general, and Suppes account in particular (see e.g. Otte, 1981) has been widely criticised, but this criticism relates only and specifically to its application to diagnosticity. In showing that positive causal relevance is unnecessary for diagnosticity, this criticism will also invalidate Sadegh-Zadeh’s further claim that μ_{Diag}(*D*) = cr(*D*,*S*,*P*).

## Causal relevance is unnecessary

To analyse Sadegh-Zadeh’s causal relevance criterion, let us imagine a simplified model case. There is a rare symptom called Blue Hair in which the patient’s hair suddenly and persistently turns a specific shade of blue. There are exactly two conditions which cause Blue Hair; call them A and B. A causes Blue Hair through a known mechanism which also causes high blood pressure (hypertension). A-sufferers always have hypertension and always have blue hair. So, both are individual *sine qua non* of A (see Part 2).

B also causes Blue Hair through a known mechanism and this mechanism too plays havoc with blood pressure. In the vast majority of patients (99.99%) this causes dangerously low blood pressure (hypotension) while in a small minority it causes hypertension (0.01%, i.e. 1 in 10,000). Again, both Blue Hair and *some form of* abnormal blood pressure are *sine qua non* for B. In the language of Part 2’s *Sine qua non *Constraint (Blunt, 2020), there are two *sine qua non *sets for B: {Blue Hair} and {hypertension, hypotension}.

The final thing we need to know to apply Sadegh-Zadeh’s machinery to this case is the prevalence of the symptoms in the population P. We will stipulate that 10% suffer hypertension and 10% suffer hypotension overall in the population (inclusive of A and B sufferers). Hypo- and hyper-tension are mutually exclusive. When afflicted by A or B, any existing blood pressure complaint is overridden by the disease (so, even if a person would be suffering hypotension, if they then contract A, then they will suffer hypertension—and furthermore, there is no connection between pre-existing blood pressure complaints and whether a B-sufferer develops hypo- or hyper-tension).

What about the prevalence of Blue Hair? We stipulated that Blue Hair is pathognomonic for A or B. The pathognomy set for Blue Hair is {A, B}. So, the prevalence of Blue Hair in the general population is simply the sum of the prevalence of A and B. Let us stipulate that both A and B have a prevalence of 1 in 1,000. So, the probability of Blue Hair is 1 in 500, i.e. 0.002.

Patient *p* presents with symptom set *S* = {hypotension, Blue Hair}. We look at our set of conditions. By the pathognomy constraints, we know that any diagnosis for *p* must include either A or B. No diagnosis which excludes both can be a candidate or legitimate diagnosis for *p*. By the *sine qua non *constraint, we also know that A cannot be a candidate or legitimate diagnosis for *p* because *S* omits one of A’s *sine qua non*, namely hypertension. So, the candidacy and legitimacy of A as a diagnosis is zero.

Sadegh-Zadeh’s account does justice to this. We calculate the causal relevance of A to S as follows:

P(S|A&P) = 0

P(S|P) = 0.0009999 (as the only way a person in P can haveSis if they have B, hence the probability of S in P just is the probability of B (0.001) multiplied by the probability of hypotension given B (0.9999) in P).

Hence:

cr(A,S,P) = 0 – 0.0009999 = -0.0009999

So, A is negatively causally relevant to S, and therefore A cannot be a diagnosis for *p* according to Sadegh-Zadeh’s criteria.

But now we have established for certain that *p* has B, by virtue of the pathognomy constraint. Hence, the candidacy and legitimacy of B for *p* must be equal to 1. Let us calculate this according to Sadegh-Zadeh’s causal relevance formula:

P(S|B&P) = 0.9999

P(S|P) = 0.0009999

Hence:

cr(B,S,P) = 0.9989001

There is a high degree of causal relevance of B to S according to this calculation, but despite the fact that patient *p* must have B given that they have S, it remains the case that according to Sadegh-Zadeh, B is only diagnostic for *p* to degree 0. 9989001. It is not *fully* diagnostic for *p*. This seems patently false given that by construction we knowfor sure that *p* has B.

Indeed, this is a general problem with Sadegh-Zadeh’s causal relevance definition. It is demonstrably impossible for any diagnosis to have full causal relevance to any set of symptoms unless neither the diagnosis nor the symptoms ever occur in the population. This can be seen from the causal relevance definition. As cr(D,S,P) = p(S|D&P) – p(S|P), it is evident that cr(D,S,P) = 1 if and only if p(S|D&P) = 1 and p(S|P) = 0. That is, the symptoms always occur when D occurs, but the symptoms never occur. Hence, D must never occur also. Even for a very simple case, the causal relevance definition creates an unacceptable result. Suppose that patient *q* presents with Purple Hair, which is both pathognomonic and a *sine qua non* for condition C. It follows that *q* has C for sure. Suppose C occurs in 1 in 1000 patients. Hence:

cr(C,S,P) = 1 – 0.001 = 0.999

This is a high degree of causal relevance, no doubt, but is unsatisfactory insofar as it does not render C fully diagnostic for *q*.

Consider the situation of condition C. The causal relevance of C to its defining symptom of Purple Hair is given by:

cr(C,S,P) = 1 – p(C|P)

In other words, the condition is causally relevant to its distinctive symptom in inverse proportion to the condition’s prevalence in the population, because p(S|P) = p(C|P).

From this, it follows that not only does the causal relevance account fail to rule in a guaranteed diagnosis as fully diagnostic, it creates a counterexample to (3) in which a diagnosis which is guaranteed by virtue of pathognomy and *sine qua non* is ruled to be entirely non-diagnostic! This would occur precisely when everyone has condition C. Suppose an epidemic of C turns everyone’s hair purple. Then:

cr(C,S,P) = 0

Because everyone has the symptom and everyone who has the condition has the symptom, C is not causally relevant to S *even though it caused S for every person in P*! As such, not only does this account have the highly undesirable consequence that a guaranteed condition does not have diagnosticity of 1, it also follows that positive causal relevance is not even necessary for diagnosticity. C is obviously a fully accurate, legitimate candidate diagnosis for anyone with Purple Hair, yet in this limit has zero causal relevance and thus is not even a diagnosis per Sadegh-Zadeh’s account. The account as it stands is therefore indefensible.

### A modified causal relevance criterion

There is a reasonably straightforward remedy for this problem which has been defended in the literature on probabilistic causal relevance (see e.g. Eells, 1991; Skyrms, 1980). We revise the definition of causal relevance as follows:

cr_{2}(D,S,P) = p(S|D&P) – p(S|¬D&P)

Here “¬D” is the absence of D. Rather than deducting the prevalence of the symptoms in the general population, we deduct the probability of the symptoms in the population who are not suffering from D. By removing the D-sufferers from P for the purpose of the calculation, we circumvent the problem. Consider the case of C and Purple Hair again:

cr_{2}(C,S,P) = 1 – p(S|¬C&P) = 1

Because S never occurs in P without C, p(S|¬C&P) = 0, so the causal relevance on this definition is 1, as expected. This persists in the extreme case where C afflicts everyone as long as we can stipulate that the probability of a symptom in an empty set of people is trivially zero.

So, to continue to defend a causal relevance account for diagnostics, we must move to cr_{2} and abandon cr. Note, though, that this modification will not fully resolve the problem as seen in the Blue Hair case. Under the revised definition, as P(S|B&P) = 0.9999, cr_{2}(B,S,P) = 0.9999. So even though we can conclusively show that *p *has B, B is only diagnostic to degree 0.9999 for *p*. Sadegh-Zadeh may be able to bite the bullet on this consequence of the account. However, even in the modified form a counterexample can still be constructed, as we shall see.

### Negative causal relevance as diagnostic

Now, return to the case of A, B and the Blue Hair conundrum. This time, consider what happens when patient *r* presents with symptom set *S* = {Blue Hair, hypertension}. We can calculate the causal relevance of A and B respectively to S as follows:

cr_{2}(A,S,P) = p(S|A&P) – p(S|¬A&P) = 1 – 0.0000001 = 0.9999999

p(S|A&P) = 1 because every A-sufferer has symptom set S as Blue Hair and hypertension are both *sine qua non* of A. Meanwhile, p(S|¬A&P) requires us to calculate the probability of the symptom set in the remaining population of non-A-sufferers. The only people who can possibly have Blue Hair and hypertension are those who have B in its rare form which causes hypertension. B occurs in 1 in 1,000 patients, and hypertension in 1 in 10,000 of those. Hence, p(S|¬A&P) is 1 in 10 million, i.e. 0.0000001. This seems quite reasonable: there is a high probability that S was indeed caused by A (indeed, only 1 in 10,000 cases are not caused by A), so its causal relevance is rightly very high.

But what is the causal relevance of B to S?

cr_{2}(B,S,P) = p(S|B&P) – p(S|¬B&P) = 0.0001 – 0.001 =-0.0009

We see that p(S|B&P) = 0.0001 because the probability of Blue Hair in B-sufferers is 1, and the probability of hypotension is 0.0001. But p(S|¬B&P) = 0.001 because these are the exact symptoms of A, so the probability of S in the non-B-suffering population is just the probability of A in that population, i.e. 0.001. Hence, it follows that for cr_{2}, B is *negatively causally relevant *to S. This is despite the fact that B is known to cause both of the symptoms in S albeit rarely. Due to its negative causal relevance, according to Sadegh-Zadeh’s account, B is not a diagnosis for *r*.

This does not directly contradict the pathognomy and *sine qua non *constraints, but it does clash very strong with our intuitions. From the patient’s symptom set, we can deduce for certain that the patient has either A or B, and that A is much more likely than B. But we cannot rule B out.

Moreover, this is diagnostically relevant. Suppose that the impact of suffering Blue Hair and hypotension on daily life is relatively mild, but worth treating if we can do so at low risk. There is a treatment for A which cures the disease, but which has the side-effect of killing anyone with B who receives it. We can deduce that there is a 1 in 10,000 chance that our treatment will kill patient *r*. This is despite the fact that according to Sadegh-Zadeh’s account, the only possible diagnosis for *r* is A on the grounds that B is *not even a diagnosis* for *r*.

## Conclusion

We have seen that Sadegh-Zadeh’s precedence criterion is too strong, and that for either version of the causal relevance criterion, positive causal relevance is not necessary for a condition to be a candidate diagnosis for a patient. Sadegh-Zadeh’s account demonstrably excludes some candidate diagnoses which it should not exclude. Moreover, it renders some diagnoses which conclusively are candidate and legitimate diagnoses as only partially diagnostic. As such, we can conclude that this causal relevance account is not fit for purpose and is in need of replacement. Because diagnoses which are negatively causally relevant to symptoms can be candidate diagnoses, an expanded account of diagnosticity will need to incorporate more than just probabilistic dependency in order to adequately explicate the logic of diagnosis. We turn next to the structure of such a replacement.

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