Abstract
Inconsistency toleration is the phenomenon of working with inconsistent information without threatening one’s rationality. Here I address the role that ignorance plays for the tolerance of contradictions in the empirical sciences. In particular, I contend that there are two types of ignorance that, when present, can make epistemic agents to be rationally inclined to tolerate a contradiction. The first is factual ignorance, understood as temporary undecidability of the truth values of the conflicting propositions. The second is what I call “ignorance of theoretical structure”, which is lack of knowledge of relevant inference patterns within a specific theory. I argue that these two types of ignorance can be explanatory of the scientists’ rational disposition to be tolerant towards contradictions, and I illustrate this with a case study from neutrino physics.
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Notes
A logic is paraconsistent if it invalidates the Principle of Explosion and allows the avoidance of logical triviality despite the use of inconsistent information. A logic L is paraconsistent if the L-closure of some inconsistent premise sets is non-trivial.
Some of the formal resources that resulted from the development of paraconsistent logics have been. used to describe and explain historical episodes that illustrate both the presence of contradictions in the sciences and the absence of logical triviality.
These commitments to classical logic include, at least, the spirit of bivalence that permeates the whole argument and the assumption of contradictions being false (as expressed in 1).
Lakatos understood, in a very different way from Popper and Hempel, that the role of contradictions in scientific dynamics is different from simply refuting hypotheses or justifying a simplistic form of theory rejection. As a matter of fact, not only research programs often proceed legitimately in an ocean of anomalies, but also to be tolerant towards the presence of anomalies is a decisive factor of progress in mathematics.
I thank an anonymous referee for giving a better phrasing to my ideas on this point.
The most compelling exemplar of a dialetheistic empirical theory is the Hegelian theory of motion discussed in Priest (1987). This theory says that, in light of Zeno’s paradoxes, contradictions are necessary for explaining two important physical phenomena: the nature of the instant of change and the nature of motion—change of location with respect to time (Priest 1987: p. 165, 166).
According to Priest, in this case, contradictions are both explanatory and the best option that scientists had at hand. However, Boccardi and Macías-Bustos (2018) have recently argued that contradictions are dispensable when developing a successful theory of motion and that a Russellian account, a consistent available alternative, is more explanatory than the Hegelian theory. Therefore, not even for this exemplar, one should grant contradictions to be true.
I am greatly indebted to Otávio Bueno and the referees for helping me to give a better phrasing to my ideas here.
In recent years, another type of ignorance has been put forward: absence of answers to questions, which has been commonly labeled as “erotetic ignorance”. Nonetheless, as this type of ignorance remains very much unexplored, there is no clarity regarding its status compared to the other three subcategories; in particular, nowadays there is an open debate on whether erotetic ignorance reduces to factual ignorance. Here I do not focus on erotetic ignorance; however, for more detailed analyses on the subject see: Rescher (2009), Nottelmann (2016) and Peliš (2017).
At the time, Pouchet was one of the most important supporters of the theory of heterogenesis, which was a variation of the theory of spontaneous generation, which contented that matter is capable of generating living organisms that do not need to be biologically related to each other, for example, new living organisms came from inanimate objects, such as air. Nowadays it is well known that this theory is false but in the early 19th Century it was still a candidate for explaining the origin of life.
If interested in the connections between this conception of undecidability for statements from the empirical sciences see Gutiérrez-Ramírez (2015: Chap. 1. In Spanish).
A partial relation R\(_{i}\) over D is a relation that is not necessarily defined for all n-tuples of elements of D (see da Costa and French 1990: p. 255).
If \(\ulcorner \)All P are Q \(\urcorner \) is derivable from the theory, then to derive \(\ulcorner \)Qa \(\urcorner \) from \(\ulcorner \)Pa \(\urcorner \) is an inference pattern.
Solar Neutrinos are subatomic particles that are generated from solar fusion; it was believed, that this type of particles did have neither electric charge nor mass.
The SSM is a
theoretical framework derived from the application of laws about energy conservation and transport(...) The SSM consists of a set of assumptions both theoretical and empirical, that—depending on the interpretation of the SSM that is used- could efficiently describe a unique empirical domain, in this case, the Sun. It has also the capability of giving descriptions of specific phenomena, predictions and guidance for experiments on the phenomena it describes, one its applications is to describe and allow to make predictions regarding the flux of solar neutrinos. (Martínez-Ordaz 2017: p. 133.)
Which are determined by the domain of application of the theory, for instance: while \(A_{NeuProd}\) is constructed by taking all elements from the SSM that were needed to predict the production of solar neutrinos, \(A_{Heliosmg}\) is build by taking all SSM elements that are needed for the study of wave oscillations in the stars. These substructures can shared elements between them but what does large part of the work is that they also have non-shared elements.
This is, information about the process in which four protons are combined to produce two protons, two neutrons, two positrons; information about nuclear reactions and cross sections of elements such as \(Ar^{37}\) and \(Cl^{37}\), among others.
Thanks to the anonymous reviewer who helped me to give a better phrasing of my ideas on this point.
Partial overcoming of ignorance of theoretical structure means that, when tolerating a contradiction, scientists need not to identify the ultimate or the total structure of their theory, but that they can provide a set of inference patterns that allow them to successfully use the theory in question while avoiding logical triviality.
Options such as simple rejection or (dialetheist) acceptance are ruled out for the reasons discussed in Sect. 2.
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Acknowledgements
I am indebted to Diderik Batens, Miguel Ángel Fernández-Vargas and Rafal Urbaniak for the fruitful discussions on these issues. I owe special thanks to Joke Meheus for her help with the case study. Thanks also to Otávio Bueno, Moisés Macías-Bustos, Xavier de Donato Rodríguez, Ana Rosa Pérez-Ransanz, Cristian Gutiérrez-Ramírez, Luis Estrada-González and the anonymous referees for their valuable comments and suggestions. Thanks to the audience at the workshop “Inconsistency in Factual Science” (Creativity2019). Without the help of Gabrielle Ramos-García this file would have never compiled.
This research was supported by the Programa Nacional de Pós-Doutorado PNPD/CAPES (Brazil) and by the UNAM-PAPIIT Projects: RG100316 “Explorando la gravitación con la cuántica y viceversa”, IG400219 “Conciencia y normatividad” and IN403719 “Intensionalidad hasta el final: un nuevo plan para la relevancia lógica”.
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Martínez-Ordaz, M.d.R. The ignorance behind inconsistency toleration. Synthese 198, 8665–8686 (2021). https://doi.org/10.1007/s11229-020-02593-4
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DOI: https://doi.org/10.1007/s11229-020-02593-4