A problem for epistemic counter-closure

Papers dealing with the puzzle of epistemic closure have been a cottage industry in philosophy for some decades, but there is another problem for inferential knowledge, epistemic counter-closure. In what follows, I provide an overview of the three major diagnoses of where the problem lies and proposals for solving it, then note that there are test cases from history of science that aren't addressed by the three proposals, specifically the mature nineteenth century version of miasmatic theory developed by Dr William Farr.

1. Introduction

Consider the slogan, ‘inferential knowledge requires known basis beliefs’ (Audi 2003). This is a prima facie plausible slogan to uphold: it would be inappropriate to ascribe inferential knowledge to an epistemic agent if their inferred belief is solely predicated on relevant unknown basis beliefs.

As an illustration of this general diagnosis of the plausibility of this slogan, an analysis of the problems of some pre-Gettier theories of knowledge as justified true belief may pinpoint the error in reasoning as follows: Gettier problems (and other related problems) cast doubt on S inferentially knowing that Q by targeting the appropriateness or relevance of S's purported justification of P. In cases of epistemic double-luck, S inferentially knowing that Q requires a particular type of relevant justified basis belief P, this type of relevant justified basis belief is lacking, and an irrelevant justified belief has masqueraded as relevant. Thus a turning-point in post-1960s epistemology may be due to realising that a condition for knowledge-as-justified-true-belief had not secured the correct relationship between the relevant justified basis beliefs.

Furthermore, other conditions of knowledge may be targeted, such as the belief condition: S must have the doxastic state of belief that P in order for S to inferentially know that Q based on P. If S should hold a degree of credence in Q greater than the conjunction of the relevant basis beliefs P, this would lead to absurdities: the degree of confidence in a logical consequence of a basis belief has somehow become greater than its very basis. By reductio, if the belief R were based on Q and Q based on P, and credence increases down the chain of inference, it would stand to reason that S's degree of credence that R would be similarly greater than P and Q. Therefore, inferential knowledge requires the basis beliefs be believed, rather than merely entertained or considered; similarly, the degree of credence assigned to an inferred belief cannot exceed the degree of credence assigned to the conjunction of the relevant basis beliefs.

Lastly, and most importantly for the following discussion, since all theories of knowledge have a condition of truth, the basis belief P must be true. It is prima facie the case that a singular false relevant basis belief would similarly disqualify ascribing inferential knowledge based on a false basis belief, for knowledge is factive.

There are competing explanations for why inferential knowledge requires true basis beliefs: similar to Gettier problems, the true belief based on a false basis belief may be due, in part, to a ‘happy epistemic accident’. However, all available explanations are unified by the common assumption that there is a failure to preserve desirable features of deductive inferences. For example, the true belief based on false basis beliefs is not due to epistemically valuable aspects of the epistemic agent, the belief or their present environment, and thus due to these deficiencies (wherever the diagnosis should pinpoint the error), while the resultant inferential true belief Q may be approximate to or share similarities with knowledge, S does not automatically inferentially know that Q based on P so long as P is false.

Frederico Luzzi provides one unifying explanation for this intuition: ‘the general thesis is that the epistemic pedigree of a conclusion can seemingly be no better than the epistemic pedigree of the [basis belief]... it is not possible for the premise to have a pedigree that falls short of knowledge yet for the conclusion to have a pedigree sufficient for knowledge’ (Luzzi 2012, p. 32). More colloquially, the belief that epistemic pedigree of inferential knowledge and basis beliefs adheres to the commonsense belief that one cannot squeeze blood from a stone. An epistemic agent cannot get inferential knowledge from basis beliefs that are not justified, not believed or not true. We can set out this commonsense belief as follows:

Epistemic counter-closure: S inferentially knows that Q iff S comes to believe and maintain Q solely on the relevant basis belief(s) P, S knows that P entails Q, and S knows that P.

We can call this thesis the counter-closure thesis: in order for S to inferentially know that Q, the relevant basis belief(s) P must, at minimum, be known. In the case of some early Gettier counter-examples like the case of Smith and Jones each having ten coins in their pockets, for instance, the relevance condition has been violated by introducing irrelevant basis beliefs.

If counter-closure were accepted, deductive inference can do no more than preserve knowledge, deductively transferring the pedigree of each basis belief on down; thus knowledge cannot be generated from false basis beliefs (Luzzi 2012, p. 9), just as gold cannot be generated from dross.

Several philosophers (e.g., Armstrong 1973, 198-9; Nozick 1981, 231; Stanley 2005, 89) explicitly assume this thesis. Furthermore, it is prima facie plausible. In short, acceptance of counter-closure is the establishment view. Can counter-closure be maintained? I suggest it cannot, and for an unexpected reason: certain types of dross in the right epistemic furnace can give us gold. In what follows, I will focus on failures to satisfy the condition of truth that provide grounds for rejecting counter-closure. In order to prevent an epistemic free-for-all, the example I provide is one in which a particular method (namely consulting the predictions derived from miamsatic theory) was exceedingly reliable and surprisingly fecund in a number of areas of public health.

This demonstrated reliability provides a suitable condition for when it would be appropriate to ascribe inferential knowledge to epistemic agents with relevant false basis beliefs. Consequently, as with two following diagnoses for a 'mitigating factor' for a false basis belief that will be discussed in the following sections, the reliability of the method of consulting a a false basis belief provides similar grounds for rejecting counter-closure, but not opening the doors to epistemic anarchy.

2. One case and three diagnoses: perceptual basis beliefs

Logical entailment preserves what Fitelson calls ‘good-making features’ of premises such as truth. Accepting counter-closure appears plausible, in part, because the contrapositive appears plausible as well: entailment preserves the ‘bad-making’ features of premises, such as the preservation of falsity from basis beliefs. However, this is not so: logical entailment fails to preserve ‘bad-making’ features of premises. For example, take the following statements: 'John is in Idaho', 'If John is in Idaho that he is in the United States', 'John is in the United States'. If the first sentence were false, and John were instead to live in Utah, John would still live in the United States. If the first two sentences were set up as an argument, it would not be sound, nevertheless the conclusion would be true.

The kernel of the puzzle facing the acceptance of counter-closure is as follows: if counter-closure is maintained, inferential knowledge cannot ever be gleaned from consulting false basis beliefs; however, consider the following two cases:

Case A: I have the perceptual belief is that that you are five feet tall. I believe that since you are five feet tall, you cannot be taller than eighteen feet. I believe truly that you are not greater than eighteen feet tall, but my perceptual belief is false: you are four-foot-eleven.

Case B: I have the perceptual belief is that you are eleven feet tall. I believe that since you are eleven feet tall, you cannot be taller than eighteen feet. I believe truly that you are not greater than eighteen feet tall, but my perceptual belief is false: you are four-foot-eleven.

One plausible explanation for why it would seem without any issue to ascribe knowledge to me in case A and not case B is that my false perceptual belief in A is ‘approximate to’ or ‘nearby’ the truth, while my false perceptual belief in B is not. There is an important mitigating factor in A not present in B: the closeness of the false perceptual belief to the nearby true belief ‘washes out' any disvalue.

The reasons that underlie this closeness to the truth may be different in extenuating circumstances: had I been in an Ames room, a small room in which there is the optical illusion that a person appears to be taller if they stand in one corner and shorter in the other, if my friend Anna should stand in a corner that makes her appear to me to be eleven feet tall, I would have a perceptual belief that is false, but an explanation that would mitigate the falsity of my belief can be provided as follows: we could also imagine that what is going on is not merely the proximity of my false belief to a nearby true belief, but rather that some local accident prevents me from holding a true basis belief that is nevertheless exculpatory on counterfactual or modal grounds.

In this instance, if we were to consider the counterfactual that had I believed truly rather than falsely, I would have believed truly in most nearby possible worlds: given the relevant and specific environmental conditions in play, my perceptual belief would have been something more like 'Anna is five-foot tall'. In the case of Ames room, had Anna stood anywhere else but in the corner, I would instead have a perceptual belief that is true or more truth-like, and make the same inference that they are not eighteen feet tall, thus preserving the intuition that I still know even if my basis belief were false.

In this first and second diagnoses, the issue is that there exists a number of edge cases near the boundaries of acceptability: this explanation, naturally, comes in a matter of degrees, and shares some similarities with more formalised approaches in truth-likeness; however, it is relevant to more ‘home-grown’ truths such as beliefs about whether one is late for a meeting based on slightly mistimed watches (Warfield 2005, 408), assessing that a child has a fever based on a slightly inaccurate thermometer (Hilpinen 1988, 163-4), inductive inferences based on a false belief about the exact number of confirming instances (Saunders & Champawat 1964), epistemic pranksters (Luzzi 2010, 2) or inaccurate perceptual beliefs about the shape and size of nearby objects. Consequently, while counter-closure is abandoned, inferential knowledge is not opened up to accepting all cases of inferential knowledge based on false basis beliefs--some specific contexts preserve the relevant 'good-making' features of false basis beliefs.

The third diagnosis runs differently than the first two: counter-closure is still accepted; however, a number of relevant true basis beliefs play a key part in the inference (cf. Coffman 2008, 190-1): this targets the 'sole basis belief' condition included in the counter-closure thesis by indicating that a toy example isn't a genuine violation of counter-closure. By analogy, a bad lead actor may ruin a play, but if they are relegated to a secondary or background role and more competent actors take their place, the audience will leave the play having been satisfied, never knowing how the background extra lacked any acting ability whatsoever. The only problem with this third diagnosis, however, is, to extend the analogy further, if the cast is limited to a 'one man show', that is to say, it would not be appropriate to ascribe any other relevant basis beliefs to the epistemic agent.

2.1 A second case and extension of the three diagnoses: basis beliefs in general

As a further illustration, consider the case of Sam the usher:

Sam sits outside an auditorium and, as people enter the auditorium, he records their entry with a tally counter. After the lecture has begun, Sam is contacted by the fire marshal, requesting information as to whether the room is up to safety standards. Sam reads off the tally counter that there are fifty-two people in the auditorium. Since Sam knows the auditorium seats a maximum of two-hundred people, Sam purportedly inferentially knows the fire code has not been violated during a lecture based on the basis belief that fifty-two people entered the room, in conjunction with the belief that the maximum occupancy of the room is two hundred, the tally counter is reliable, and that Sam’s perceptual beliefs are usually reliable.

Had Sam miscounted, and in fact only fifty-one people had entered the room, even though Sam’s belief that fifty-two people entered the room is false, the proximity to the real number of people washes out any ‘bad-making’ features of the false belief. As in the case of the perceptual belief, the basis belief that fifty-two people entered the room is false, but only if truth is considered an all-or-nothing affair.

The second diagnosis for these problems locates the washing out of bad-making features elsewhere, just as in the case of perceptual beliefs. In a number of other cases, such as the perceptual belief in the Ames case, these false basis beliefs are modally ‘fragile’, that is, their falsity in the actual world is mitigated by the large number of proximate possible worlds in which the belief would have been true. In the case of Sam the usher, the explanation given may be that in all similar situations, Sam would have counted correctly, but some ‘fluke’ or case of epistemic bad luck has come into play.

In short, both solutions seek some ‘proximate’ explanation for why a basis belief, although false, may still be treated as ‘close enough’ to the truth to wash out its ‘bad-making’ features. In fact, there seems no problem in choosing either diagnosis over another, or some combination of both diagnoses, in these home-grown situations.

The third diagnosis would conclude that there is a large number of other relevant true basis beliefs that do the epistemic ‘heavy lifting’. In the case of Sam the usher, Sam has many other true basis beliefs: Sam has the basis belief less than one hundred people entered the auditorium, the basis belief approximately fifty people entered the auditorium, and so on. In this diagnosis, counter-closure would be maintained, and a large system of relevant true basis beliefs wash out any singular false basis belief.

3. False theoretical systems

There exist cases that don’t obviously satisfy these three conditions of proximity. This fourth diagnosis focuses on a different type of problem of apparent inferential knowledge based on false basis beliefs: beliefs in the existence of regularities, nonexistent entities and causal explanations that are, within a fairly broad domain, reliable in their predictive accuracy, yet based on faulty causal explanations and prognosticatory models. These cases naturally occur routinely in the history of the natural sciences.

This problem with accepting both counter-closure and the factivity of knowledge can be put as follows: prognosticators in the natural sciences often make reliable predictions based solely on (in retrospect) mostly or entirely false theoretical systems, yet consulting these false theoretical systems when devising concrete and specific plans for action were as reliable in their prognosticatory success as (mostly or approximately) true theoretical systems.

In what follows, I present a historical case-study of the consultation of Dr William Farr's model of miasmatic theory as satisfying these conditions of repeatedly demonstrated reliability in its predictions in a number of fields. Detailing the extent of the reliability of Farr's model shows that it is at least prima facie plausible to ascribe inferential knowledge to epistemic agents that adopt a reliable (but false) system of basis beliefs, even if the core parts of their theoretical system are not proximate to the truth, there are no nearby possible worlds in which the relevant parts of the theoretical system are true, and no relevant truths are available to provide a sufficient system of true basis beliefs to wash out its falsity.

3.1 A historical case-study: nineteenth century miasmatic theory

Miasmatic theory is a prime candidate of a theoretical system in the natural sciences that made numerous specific predictions in health policy that were novel, exact and borne out through repeated testing. Furthermore, nineteenth century health reformers such as Farr, Edwin Chadwick and Florence Nightingale implemented specific health policies based solely on miasmatic theory. This method--adopting health policy based on the predictions of miasmatic theory--was highly reliable in combating the spread of cholera and other courageous diseases in England: one needed only to see under what conditions this mature version of miasmatic theory predicted outbreaks of cholera would occur on the national, city, and district level, and following miamsatic theory, eliminate the purported causes of miasma. This lead to a statistically significant decrease in outbreaks of cholera, all but eliminating outbreaks in London by 1866.

What was miasma?

Miasmatic theory posited that cholera was caused by a nonliving organic poison that was dispersed in the air, arising from decaying organic matter or ‘miasmata’, a mist or vapour that was exuded from decaying material. The pre-theoretical reasoning for the miasmatic theory was intuitive: areas that were purportedly miasmatic had a foul smell, and the smell was either thought to be the poison or an indicator of the presence of poison.

By the mid-1840s, miasmatic theory had developed into a fully-fledged research programme that revolved around the role of smell as either indicating the presence of miasmata or as itself miasmata. Two examples of the era help clarify the central roles of smell and the purported transmission of contagious disease by air within miasmatic theory: Edwin Chadwick, nineteenth century lawyer and social reformer, said, ‘All smell is, if it be intense, immediate acute disease; and eventually we may say that, by depressing the system and rendering it susceptible to the action of all causes, all smell is disease’ (1846, 659) while Neil Arnott wrote in First Royal Commission for Inquiring into the State of Large Towns and Populous Districts, ‘The immediate and chief cause of many of the diseases which impair the bodily and mental health of the people, and bring a considerable proportion prematurely to the grave is the poison of atmospheric impurity arising from the accumulation in and around their dwellings of the decomposing remnants of the substances used for food and from the impurities given out from their own bodies’ (1844, 50).

3.2 A brief reconstruction of Farr’s theoretical system

Farr's version of miasmatic theory was one of the most mature versions of miasmatic theory, involving a number of auxiliary hypotheses that surrounded a central core thesis that the primary vector of transmission of infectious diseases such as cholera was through the air.

One of Farr's major additions to miasmatic theory was the addition of a detailed explanation for the catalysis, structure and development of contagious disease within the body. Farr borrowed heavily from the work of Justus von Liebig's early explanatory model of chemical interaction between living and non-living molecules (Pelling 2002, 27), which had previous success in explaining and predicting certain agricultural practices in Europe (Tulodziecki 2017, 3).

The causal mechanism Farr first proposed for infection of cholera was as follows: Farr reasoned infection occurred when a nonliving organic substance was introduced into the blood in the human body ‘with the lungs as the point of entry to the animal system’ (Eyler 1973, 84).  Upon entry, this organic substance acted as a poison, multiplying in healthy blood and developing the symptoms of disease (ibid. 82). Farr called this process ‘zymosis’, which Farr believed resembled fermentation (1855, 48). This  zymotic theory could 'explain the process of increase of morbid matter, either in the body or outside it' (Pelling 2002, 27).

As Ackerkneckt (2009) and Eyler (1973) note, this molecular theory of zymosis could explain a number of phenomena that earlier theories that attempted to describe the structure of disease with the body could not. Consequently, Liebig's theory produced a number of novel molecular and biological predictions that stood for some decades as a highly predictive causal model of the growth of disease within the body.

Furthermore, Farr presented an explanation for disease-transmission with immense predictive content: the zymotic material, being heavier than air, would hang over cities like a fog, and while suspended in the air, would gradually disperse over an area after it had exited a body through its lungs or object left to putrefy. The degree and magnitude of dispersal was contingent on a number of environmental factors, such as ‘[t]emperature, humidity, wind, precipitation, and barometric pressure', which determined 'an epidemic’s behavior’ (Eyler 1973, 84).

In addition to the zytomic theory of disease and Farr's more predictive version of miasmatic theory, Farr further posited that there was a ‘natural law’ that the mortality of cholera was inversely correlated with elevation: ‘The elevation of the soil in London has a more constant relation with mortality from cholera than any other known element’ (Farr 1852b); ‘the poisonous consequences of filth must be inverse to the elevation of soil’ (Farr 1855, 49); the conditions 'which are so constantly found in alluvial soils, lying on a level with or below the tidal waters' were prime candidate sources for producing miasmata (Farr 1852b, 163).

Consequently, Farr's three interrelated theories predicted that concentrations of miasmata would be closer to rotting and putrid sources, such as London's long-neglected open sewer system and cesspits, and lower as one moved away from these sources of contagion. Since mortality was thought to be directly linked to exposure to miasmata, Farr's miasmatic theory predicted that more people would fall ill and die closer to open sources of miasmata, and fewer at a farther distance.

On the basis of these three theories--zytomic theory, miasmatic theory and Farr's law of elevation--Farr extensively corroborated his findings through statistical analysis. In 1854, Farr, along with Chadwick, was appointed a member of the Scientific Committee for Scientific Inquiries in Relation to the Cholera Epidemic. Given a large degree of control and independence to conduct his studies, he meticulously tested a number of specific predictions of his miasmatic theory using the best available statistical methods available in the mid-19th century. After collecting a large data set of the frequency of infection and morbidity, Farr extensively corroborated these predictions in the report by examining the available evidence on mean temperature correlated to number of reported cases, altitude and location of cases of cholera and the reported degree of air pollution in London's different districts (Langmuir 1961, 173).

Farr also proposed a number of subsidiary hypotheses that were a consequence of his three core theories of disease-transmission. Three such examples are as follows: Farr proposed a 'law of increase and decline' of high-morbidity epidemics over time: 'Rinderpest, or bovilia, has many points of analogy with small-pox, scarlatina, diphtheria, typhus, and influenza, which all follow the law of increase and decline; the increase admitting the retardation, and the decline of acceleration, by judicious measures. These diseases decline because their matter, generated in unhealthy varieties of race, loses some of its virulence by transmission, because all are not susceptible...' (as quoted in Brownlee 1915, 251). Farr also proposed that miasmata could be transmitted through fomites, not just through the air: fomites were objects that absorbed noxious gasses, and would expel the gasses over a period of time (Tulodzicki 2016b, 269). Furthermore, based on Farr's miasmatic theory, Farr presented a mathematical law such that mortality and morbidity were tied to the density of a population.

Further novel predictions of Farr's disease theory

The predictive accuracy of Farr's theoretical system is impressive even today. For example, Farr's theory and subsequent statistical analyses was not limited to cholera epidemics within London, but spanned the entirety of England. He collected data for cholera epidemics outside London and corroborated the novel predictions of his version of the miasmatic theory. As Farr predicted, '[t]he cholera was three times more fatal on the coast than on the interior of the country' and was most fatal in areas 'lying lowest down the river', where the soil was thought to be most miasmatic (1852b, 156).

Furthermore, Farr did not limit himself to explaining and predicting the location of outbreaks of cholera in England, but also the duration and course of a number of other various epidemics and diseases outside England: 'The zymotic theory could explain all of these phenomena through its claims that decomposing material produced miasmas... this was the reason why certain diseases were particularly bad during periods of high temperature and in certain geographical regions (for example, the many fevers in Africa), why urban centres were much more affected than rural areas, and why even specific locations in otherwise more or less healthy areas could be struck' (Tulodziecki 2017, 5).

A more controlled series of experiments conducted Robert Angus Smith, a contemporary of Farr's, measured the air quality of putrefying cesspools, areas around houses, 'closely packed railway carriages' and other locations in great detail (Smith 1859, 218-225). His results corroborated Farr's zymotic theory.

The predictive power of Farr's theory extended to number of precise predictions that focused on an individual's response to infection. The predictive accuracy of Farr's zytomic theory of bodily disease extended to the ability 'for the first time--to figure out the effectiveness of different treatments' of disease (Tulodziecki 2016, 112). For example, Farr predicted how long an individual would likely suffer from smallpox and the odds of recovery or death given their age and environmental factors (ibid., 110).

As Tulodziecki notes, 'all of these were predictions that were impossible before Farr' and extended to, based on his theory of miasmata, producing mathematical models that 'was true both for the nation as a whole and small regions and districts within' (ibid., 110; cf. Eyler 1980, 11).

In fact, Farr made a number of startling predictions in a number of diverse areas that diverged from the predictions made by the medical establishment. This included the course of the rinderpest outbreak of 1865. Rinderpest was a viral disease of cattle with a high infection and morality rate that spread through England over the course of two years. Hundreds of thousands of cattle were infected with the disease and soon began to die. A member of Parliament, Robert Lowe, warned of 'an epizootic of tremendous size' (as quoted in Brownlee 1915, 250) and suggested preparations 'for a calamity beyond all calculation' (251). Farr replied that his 'law of epidemics' implied the reverse of what Lowe, the British Medical Journal and other established medical experts anticipated (ibid., 250-1), specifically that there would be a quick surge in the infection and mortality rates due to rinderpest, which would suddenly decline.

Farr's prediction was correct: the epidemic 'reached its peak a fortnight later than [Farr] had predicted' (Wilkinson 1992, 142). Brownlee, using Farr's available data set, calls Farr's results 'a prophecy which approximates which remarkable closeness to the actual facts' (ibid., 251). Brownlee's conclusions about the impressive predictive capacity of Farr's miasmatic theory were not isolated: due to the repeated accuracy of his predictions on the spread and transmission of infectious diseases, the early twentieth century public health expert Arthur Newsholme said Farr 'must be ranked with William Harvey in Physiology or with Lavoisier in Chemistry' (1927, 203).

A brief summary

In summation, ‘Farr suggested an intimate connection between miasmata and zymotic material; both were, after all, nonliving organic particles capable of suspension in the air. … for all practical purposes squalor might be said to cause disease’ (Eyler 1973, 85). It explained 'why certain localities were particularly prone to (certain kinds of) diseases, while others were spared, even if they were sometimes close by' (Tulodziecki 2016, 107).

Furthermore, Farr's theory explained in great detail why diseases like cholera were seasonal, why certain regions were affected by outbreaks of cholera greater than others, and why certain locations, specifically crowded urban areas, had a higher mortality rate than others. The theory also explained how epidemics would propagate, how past attempts at quarantine had failed and why there were different versions of cholera and other epidemic diseases (Eyler 1973). The explanatory power of the theory was such that it was predicatively successful in a number of areas: '... it would be difficult to improve on the 1849 prediction of the cholera rates that were based on [miasmatic] theory' (Jekel 1996, 68).

3.3 Consequences in public health policy of following the miasmatic theory in combating cholera

While other examples of Farr's predictions based on his theory were successful, none were as influential and highly regarded as his work, as well as the work of public health advocates that accepted miasmatic theory, in changing the landscape of London to combat cholera by the introduction of major policy reforms and a massive public works project to install a sewage system.

After Farr and Chadwick's report of the cholera epidemic was released, Farr communicated with Nightingale and Chadwick (Cook, et al. 2001) on a specific plan to prevent the spread of cholera in London by ending the formation of heavy mists or clouds of miasmata: this involved increased hygiene to remove miasmata clinging to the body, increasing proper ventilation in hospitals to prevent the spread of miasmata, regulating the burial and disposal of the dead, widening city streets, and prohibiting dense living arrangements in slums. Farr's control measures extended to enforcing regulations concerning the assembly of 'large masses of men in pilgrimages' and 'strict sanitary regulation' of boats engaged in international trade (Wilkinson 1992, 142). Most importantly, the miasmatic theorists developing a closed sewer system that disposed faecal matter away from residential areas and potable water (Bally 2010; Eyer 1973, 85).

What is of interest is not merely the novelty of predictions derived from Farr's theory and other mature miasmatic theories, but the success of all these plans in combating the spread of contagious diseases, specifically the spread of cholera.

One example shows the specificity of prediction: Farr’s work, as well as other developments in miasmatic theory, provided the theoretical framework for adopting Nightingale’s ‘pavilion plan’ for hospital ward design (Cook, et al. 2001), which ‘significantly improved the patients’ chance(s) of survival’ by providing ‘a greater degree of separation and segregation’ than ‘earlier designs’ (Cook 2002, 352).

The scope of this massive project to combat cholera was seen in the reshaping of London. After the Great Stink of 1858, Parliament passed an enabling act to construct a sewage system at great cost. The civil engineer Joseph Bazalgette began the public works project of constructing a fully closed sewage system in 1859 and was completed between 1868 and 1870. It was comprised of eighty-two miles of main pipes, over one thousand miles of sewers, the construction of three major embankments on the Thames and several pumping stations and treatment works.

Between 1831 and 1866, approximately forty thousand people died from cholera in London. After the completion of the London sewer system, there were no more cholera epidemics in London.

These reforms that were instigated by miasmatic theorists cannot be understated: as the editors of the British Medical Journal concluded after a vote of medical experts, members of the general public and academic researchers, 'the sanitary revolution' directly inspired by miasmatic theory was rated the greatest medical innovation since 1840 (Ferriman 2007).

The failure of Farr's theoretical system

In retrospect, however, all the key members of Farr’s theoretical system were false: Farr, Nightingale and Chadwick’s work countering cholera and other infectious diseases ‘succeeded despite [their] defective theory of disease transmission’, not because of it (Mackenbach 2007, emphasis added).

Later developments in statistical analysis, specifically developments in logistic regression, show that Farr’s statistical analysis did not corroborate either Farr's ‘natural law’ of elevation or miasmatic theory (Bingham, et al. 2004, 393). Furthermore, these developments in statistical analysis show that Farr’s available data instead corroborated the rival germ theory of disease transmission, indicating that the relevant vector was the water supply, not the air.

Furthermore, while Farr's law of elevation worked exceedingly well in predicting the course of a number of cholera epidemics in England between the years of 1820 and 1860, later epidemics did not conform to his law. Farr's law of elevation, 'despite being responsible for some of Farr's most important novel predictions' (Tulodziecki 2016, 112), was only reliable within a limited range.

Lastly, Farr’s zymotic theory was eventually superseded by the germ theory of disease--germs are living rather than nonliving, and their growth bears little resemblance to fermentation within the body.

In fact, it was the success of the public works project of constructing a closed sewer system that lead Farr to abandon miasmatic theory: in 1866, a number of parishes in London was struck with a cholera epidemic. The East London Water Company supplied drinking water to the area, and had failed to comply with the 1851 and 1852 Metropolis Water Acts: the water company did not isolate their water reservoir from the surrounding groundwater. Bazalgette's sewer system had yet to be completed in this area, and on 27 June, the infected sewage from a labourer's home was discharged into the River Lee, near the East London Water Company's water reservoir, and was carried into the reservoir during high tide (Halliday 1999). After conducting an investigation into the outbreak of cholera, Farr concluded contaminated water was the most important means of transmission of cholera, not contaminated air.

However, even though Farr had changed his position on the material cause of transmission of cholera, he continued to hold on to his law of elevation: ‘We might expect to see the elevation law gracefully abandoned when [Farr] affirmed the primacy of water over air for cholera contagion. But such was not the case.... It seems likely… that he believed a law as clearly set forth as the elevation law in 1849 revealed something fundamental in disease behavior which could not be overlooked, even though the etiological explanation for it might change’ (Eyler 1979, 97-98).

Thus we can examine a number of potential candidate cases of Dr Farr inferentially knowing a large number of propositions based on a large body of false basis beliefs. Consider the following set of basis beliefs:

P1: Cholera, as well as a number of other infectious diseases, such as smallpox, rinderpest, measles, scarlatina, and typhus are caused by the inhalation of miasmata primarily transmitted through the air. The direct cause of these diseases is due to organic matter given off putrefying material, i.e. miasmata.

P2: The spread of diseases in the body are a type of or analogous to decomposition. Zymotic material cause the promulgation of disease within the human body, transmitted from the lungs to the bloodstream, and the zymotic material engages in a process similar or analogous to fermentation or decomposition.

P3: The law of elevation, as well as a number of environmental causes such as barometric pressure, wind direction and humidity, predict the location of cholera outbreaks, as well as outbreaks of a number of infectious diseases.

P4: The available statistics on morbidity by infectious diseases corroborate P1-P3. Furthermore, rival theories, such as Snow's germ theory of disease, could be discarded in favour of Farr's theory on the basis of this evidence (Farr 1855).

And consider the following set of true beliefs that were derived from P1-P4:

Q1: ‘[cholera has] infinite preference for localities that are foetid with organic impurity’ (Farr 1855, 47).

Q2: The number of cattle that were to die from rinderpest in 1865 would quickly rise, and then suddenly drop after a short period of time. Consequently, in Farr's words, as Farr said during the epidemic of rinderpest, 'The epizootic has apparently attained its maximum, and is now going down, as the "law" lead me to believe' (Farr, as quoted in Brownlee 1915, 250).

Q3: Localities closer to water during the periods of a cholera outbreak in England during the decades between 1820 and 1860 would have a greater incidence and morbidity rate than localities farther away from water and at higher elevations.

Q4: The construction of a closed sewer system in London and other large cities that dispose waste far from cities will decrease the frequency of infection by cholera and other infectious diseases.

Q5: Florence Nightingale's 'pavilion plan' for hospital design will have a considerable influence in decreasing the spread of infectious diseases such as smallpox and cholera.

Q6: The 'law of increase and decline' and law of density will model the overall increase and decline and relative frequency of highly contagious and deadly diseases.

Q7: Cholera and other infectious diseases can be transmitted through fomites, necessitating the removal and destruction of objects that may have fomites found in the living and sleeping quarters of people carrying cholera and other highly infectious diseases.

It is not in dispute that Farr genuinely believed both P1-P4 and Q1-Q7. Furthermore, it may be the case that Farr knew Q1-Q7. However, it is difficult to ascribe to Farr any relevant true basis beliefs that entailed Q1-Q7, for Farr rejected the germ theory of disease transmission until 1866, long after his miasmatic theory had made these successful predictions.

Furthermore, P1-P4 are simply not approximate to the truth. A counterfactual or modal account in which Farr would have had true basis beliefs is even more implausible. None of these two diagnoses look applicable in this instance. The miasmatic theory was not merely partially wrong or approximate to the truth, but almost in its entirely false, and false on numerous grounds, such as the causes of cholera, how cholera was transmitted, and how cholera acted in the body. There is no nearby possible world in which cholera is caused by miasma, for miasma does not exist; there is no nearby possible world in which miasmata is transmitted through the air. Cholera is transmitted by the bacterium Vibrio cholerae, and transmitted through water and foodstuffs contaminated with human faeces that contain the bacterium. In sum, both previous diagnoses of approximate truth and counterfactual or modal accounts fail.

Two questions remain: (1) did Farr believe Q1-Q7 solely based on P1-P4 and not some other relevant basis? (2) Did Farr know Q1-Q7 if based on P1-P4? If 1 is answered in the negative and Farr knew Q1-Q7, then Farr knew Q1-Q7 based on a number of true basis beliefs; if 2 is answered in the negative, Farr lacked inferential knowledge of Q1-Q7.

Answering 1, there are at least two different ways to cash out the third diagnosis: the first approach locates the appropriate basis beliefs outside the theoretical system that Farr purportedly believed. The second approach locates the appropriate true basis beliefs in the predictive accuracy of miasmatic theory. Both positions would deny 1. After addressing these two approaches, I will then address 2.

The first approach would diagnose the problem as follows: rather than believing solely that miasmata existed,  it would be appropriate to ascribe to Farr a number of basis beliefs that produced an inductive inference wholly separate from miasmatic theory. For example, Farr may have believed P1*: 'There is cholera at localities T₁...Tn and these locations are foetid with organic impurity'. Farr consulted the available statistical evidence, believed P*, and consequently arrived at Q1 without depending solely on P1-P4. Since Farr knew P1*, Farr inferentially knew Q1, but not based on P1-P4. A similar explanation is available for Q2-Q6: in each case, Farr relied on an inductive inference Pn* that entailed Qn.

This approach, however, falls short for two reasons: Farr's statistical analyses 'were not simply instances of straightforward observations; if they had been, people would have discovered them long before Farr. Rather, a lot of theoretical assumptions went into how to construct even the raw data (such as assumptions about what kind of mortality rate ought to be used), and without Farr's appropriate relating of different data sets and his ensuing interpretation, there would have been just a bunch of numbers' (Tulodziecki 2016, 111).

Furthermore, '[i]t was miasma that was indispensable in explaining why diseases were heavily local, and crucial to Farr's novel predictions about cholera mortality and soil elevation, and also to his predictions about the course and duration of epidemics' (Tulodziecki 2016; cf. Peters 2012, chap 5). The accumulated evidence in favour of miasmatic theory could not have played a role in the construction of miasmatic theory, since the evidence was not formulated by the time of the theory's construction and had been interpreted in light of the theory. It would be appropriate to conclude that, at least on the face of it, Farr's interpretation of the data was dependent solely on his theory of disease transmission, that is, P1-P4 and not P1*-P7*.

The question is whether Farr did in fact believe P or P*. A rational reconstruction of Farr's reasoning process is certainly difficult, but not impossible: Farr approached his statistical analyses of the accumulated data in the 1820s and 1840s having previously accepted miasmatic theory in his published writings, and interpreted his data in light of miasmastic theory; Farr could not have derived the inductive inference that Q1, much less arrived at the inductive inference P1*, without believing P1-P4.

This rational reconstruction, however, can be countered by pointing out that while P1 may have been indispensable in the formation of Q1, that does not rule out the possibility of P1*, although not indispensable, playing some other role behind the scenes. However, appealing to P1* and not P1-P4 proves too much: if it is appropriate to ascribe to Farr P1* that Farr may have possibly believed, but are highly dubious, it is also appropriate to ascribe to any epistemic agent that apparently relied on a false basis belief a large number of relevant true basis beliefs the epistemic agent may have possibly believed, but are equally dubious. Consequently, for any false basis belief, a possible ancillary true belief may have played a role as a basis belief. But mere possibility is not sufficient.

What of the second approach? Assume that in addition to miasmatic theory, Farr also relied on the predictive accuracy of miasmatic theory. Other basis beliefs about the reliability of a model did the epistemic ‘heavy lifting’ behind the scenes, such as P': 'P1-P4 is an empirically adequate and predicatively successful model of disease-transmission'.

This approach, however, has similar defects that faced the first approach: it doesn't reflect Farr's doxastic commitment to P1-P4. It is trivial that for any belief X an epistemic agent S believes is true and provides some prediction Y, S also believes X will be predicatively successful in predicting Y. In other words, the statement 'is predicatively successful' may be appended to any false basis belief X that is predicatively successful in order to produce a true basis belief X': 'X is predicatively successful'. However, what is the basis belief for S to believe X'? In Farr's case, as addressed previously, the basis belief cannot be an inductive inference. Nor are there any true basis beliefs that Farr may have believed. The most plausible explanation for why Farr believed X' is that Farr believed X' on the basis that Farr believed X. Through disquotation, in these similar cases that preclude inductive inferences, if S believes X, S believes X is true. And yet X is stipulated to be false.

In summation, 1 is answered in the affirmative: Farr believed Q1-Q7 based on P1-P4 and not some other basis. What of question 2? Did Farr know Q1-Q7 if based on P1-P4?

Is this a genuine case of inferential knowledge?

The health policies derived from miasmatic theory were, in many important respects, repeatedly corroborated in fine-grained detail by the experimental and statistical methods available at the time. Miasmatic theory was immensely fecund in its predictive capacity for determining where incidents of cholera was likely to occur, as well as prescriptions based on miasmatic theory were successful in combating the contagion.

As a consequence, a reliabilist may have no problem accepting that a reliable method predicated on a series of false basis beliefs produces inferential knowledge so long as ascriptions of knowledge are limited to true beliefs that are formed by said reliable method; however, it is possible that upon reflection on the fact that P1-P4 were false, one may choose to adhere to counter-closure and subsequently deny that Farr inferentially knew Q1-Q7. This may be, however, a bullet too strong to bite.

Here is one potential response to the bullet biter: imagine that tomorrow there is the announcement of a revolutionary medical procedure, such as a pill or injection, that cures a debilitating and highly contagious disease that affects a large percentage of the population. Members of the medical establishment that helped develop this cure believe that a certain highly complex medical theory accurately captured certain causal or explanatory conditions that lead to the development of this procedure.

Through a process of reasoning similar to Farr's, these medical practitioners relied solely on this medical theory in accepting the efficacy of this medical practice or procedure, and the data accumulated to test this theory was interpreted in light of this medical theory. Furthermore, the data lead the medical practitioners to reject other viable medical theories.

Consequently, these medical practitioners sincerely believe that this medical procedure will inoculate individuals from the contagious disease on the basis of this theory. Furthermore, as with Farr's historical case-study, this novel medical procedure is immensely successful in curing the contagious disease, novel predictions are derived from the medical theory that helps successfully treat patients with related diseases, and due to adopting this medical theory,  there are similar developments in treating other contagious diseases. A number of health professionals suggest implementing this medical procedure in a costly public works project, and within a few years have almost eradicated the disease in all its forms wherever the procedure is practiced. Some hundred and fifty years later, medical professionals and lay public alike conclude that this new procedure was, perhaps, one of the greatest medical developments--or even the greatest medical development--in history.

However, if it were the case that a number of core parts of this highly complex imagined medical theory were, unbeknownst to everyone, and remained unknown forever by the medical community, false, these medical professional could not possibly have known the medical procedure will cure this disease, not even after the medical trials had concluded and it had been released to the public, and on the exact same grounds that Farr did not know: the basis beliefs that guided health policy are, unbeknownst to everyone, false.

If we were to live in such a possible world, much of what had purportedly counted as medical knowledge of the efficacy of this treatment had been based on a mistake, even though the procedure would be just as successful had the treatment been based on a true medical theory. This consequence, I would imagine, may be difficult for some philosophers to swallow, but differs only in its specifics in what occurred with Farr's miasmatic theory, and other than simplifying the historical details, does not engage in hyperbole or exaggeration.

3.3 A summation: generalising to historical cases of false theoretical systems

The example of miasmatic theory illustrates a neglected feature of the puzzle surrounding counter-closure: it is prima facie plausible that throughout history of science many scientists did indeed have inferential knowledge based solely on theoretical systems in which the most relevant parts of predictive models were not approximately true. There were no available true beliefs that could replace the relevant parts of the predictive models. The medical practitioners genuinely believed the theoretical systems were not merely accurate in its prognostications or reliable, but were true, and had no other relevant true basis beliefs to replace their false basis beliefs. This problem for epistemic counter-closure, however, does not lead to epistemic anarchy, since as in the first two diagnoses mentioned previously, there exists a mitigating factor, namely the reliability of methods that employed the false theoretical system.

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