AbstractThis paper introduces ’pessimistic meta-deduction,’ a thought experiment that synthesizes Bayesian inference with the infinite monkey theorem and applies it to the concept of fallibilism. The initial (and wrong) blend advocated for a rationalistic analogue of a pessimistic meta-induction. Upon critical self-analysis, a partial refutation of the thought experiment has been revealed. The critique diminished the impact of pessimistic deduction, making it a subtly refined form of fallibilism. However, and most importantly, the implication thereof seems to indicate a potentially mathematically indeterminable problem in the context of any hypothetically true theory, fallibilistic conundrum. IntroductionIn an attempt to create a way to measure the probability of a counterintuitive paradigm (in plant stress physiology, so outside the scope of philosophy) being true, multiple ideas were combined. This particular idea, like many others, did not turn out to be fruitful from the standpoint of what was intended. It synthesized Bayesian probability with the infinite monkey theorem and applied it to the philosophy of science revealing new insights on fallibilism.Infinite monkey theoremIn exploring the philosophical implications of scientific knowledge and discovery, a turn to the illustrative thought experiment known as the Infinite Monkey Theorem was made. This theorem, in its popular form, is philosophical rather than mathematical and serves as a conceptual tool to elucidate ideas of randomness and probability in the context of an unlimited number of attempts.The theorem posits a scenario in which a monkey, engaged in the random act of striking keys on a typewriter, is allotted an infinite duration of time. Under these conditions, the thought experiment demonstrates that the monkey would reproduce any given text, such as a work of Shakespeare (the cited source used Hamlet as an example). In the thought experiment, each keystroke is presumed to be independent and random, devoid of any pattern. [1]In this paper, the infinite monkey theorem assumes metaphorical significance. It suggests that the endless pursuit of scientific inquiry, at least some of which is random, holds the potential to yield an inexhaustible spectrum of theories, data points, and ideas, much like the infinite keystrokes leading to every possible text.Introduction to Bayesian inferenceBayesian inference is a statistical method. It is based on Bayes’ theorem. Unlike frequentist statistics, Bayesian probability is inherently subjective. It measures the degree of belief. Bayesian inference is the process of updating beliefs in light of new evidence. It begins with a prior probability, which represents an initial degree of belief in a hypothesis. As new evidence is observed, this prior probability is updated, leading to a revised probability, known as the posterior probability. The theorem mentioned provides a formula to update beliefs about a hypothesis when presented with new evidence. It calculates the posterior probability by considering the prior probability, the likelihood of the evidence under the hypothesis, and the overall probability of the evidence. Prior beliefs can vary between individuals. With new evidence, however, the posterior beliefs of different individuals tend to converge, albeit that is subject to a similar interpretation of the new evidence. [2]Bayesian probability can be compared to several philosophical positions, notably empiricism and fallibilism. It is in accordance with empiricism by emphasizing evidence in shaping beliefs while also supporting the fallibilist notion that knowledge is a subject to revision.Pessimistic meta-inductionPessimistic meta-induction in the philosophy of science posits that, given the fact that most scientific theories have been falsified, so too will the ones currently accepted. The argument attempts to falsify the notion of scientific realism (with the scientific method, no less, thus it can be viewed as both a deductive and an inductive argument), which holds that successful scientific theories accurately describe reality. Historical examples bolstering it include the phlogiston theory and the ether theory. The phlogiston theory was completely replaced by the discovery of oxygen and the development of modern chemistry. Similarly, the ether theory, which posited a medium for light waves to travel through space, was rendered obsolete by the theory of relativity. [3]Numerous philosophical efforts went into refuting the conclusion, at least one of them claimed that few scientific theories were rejected and that the inductive evidence favors optimism. Ironically, in the first table of the cited paper, among the „uncontroversial“ scientific theories, there are at least three mutually exclusive theories: VSEPR, valence bond theory, and molecular orbital theory. [4] Each predicting the shape of the molecule in a very different way and each reaching, usually, the same conclusion.Contemporary significance of pessimistic meta-induction (chemistry as an example)The gross theoretical incompatibility when it comes to valence bond and molecular orbital theories is well known. [5] The valence bond theory is usually utilized in organic chemistry. It focuses on the concept of electron pairs shared between atoms, forming covalent bonds. It explains molecular shapes through the overlapping of the atomic orbitals between the molecules and through hybridization of the orbitals within one atom, where the atomic orbitals of an atom mix shapes to form new orbitals. Molecular orbital theory is usually preferred in inorganic chemistry. It conceptualizes electrons in molecules as occupying molecular orbitals that extend over the entire molecule. This theory explains not just the shape of molecules but also their magnetic properties and colors, which is unlike the valence bond theory. It should be noted that the source previously cited heterodoxically disagrees with the failure of the valence bond theory to gauge the magnetic properties of oxygen, which is the most commonly used example for its faillure to gauge magnetic property of a compound [5].This dichotomy between theoretical incompatibility and methodological complementarity of these two theories goes to show that pessimistic meta-induction is not irrelevant in the modern context.Materials and methodsThe methods were based on thought experiments. The research, if it could be called that, was unplanned and the insight in the paper was made in error. The goal was to create a statistical tool that would reduce the subjectivity of Bayesian inference. Though the subjective elements cannot entirely be removed therefrom.The original thought experimentThe thought experiment, pessimistic meta-deduction, begins by applying Bayesian inference to the realm of scientific theories. A well-tested theory would still yield a small probability of falsehood due to the Bayesian theorem never truly reaching. The infinite monkey theorem, in its application, generates an infinite number of attempts at falsification. The preliminary conclusion of the deduction is therefore that our current scientific theories are likely to end up falsified, as we are multiplying with a number that approaches infinity.Partial refutation due to Bayesian updatingThe thought experiment’s reliance on Bayesian probability introduces a critical weakness therein: while scientific theories are indeed fallible and subject to change, the likelihood of a well-established theory being completely overturned decreases with ever-more supportive evidence. The partial refutation brings the thought experiment’s conclusion closer to a quantitative analogue of fallibilism. The outcome of the experiment, therefore, does not substantially alter the concept of fallibilism but barely refines it. It should be mentioned that the contemporary fallibilism has many forms already [6].Implications of partial refutation: a mathematically undefinable outcome in true theoriesDeeper examination reveals that the partial refutation of the ’pessimistic meta-deduction’ thought experiment reveals a potentially unresolvable mathematical problem, at least when applied to a hypothetical true paradigm.As the evidence accumulates, the Bayesian probability of a paradigm’s veracity continually increases, asymptotically approaching certainty (a probability of 1). Consequently, the probability of falsification per attempt correspondingly diminishes, ever-nearing zero.On the other hand, due to the infinite amount of time, the opportunity for potential falsification extends without bounds. In other words, the second factor approaches infinity as the first factor approaches zero. The mathematical product, which is the probability of falsification, of these two factors, one diminishing towards zero and the other expanding towards infinity, results in a situation here named the fallibilistic conundrum.DiscussionThere is a possibility that the fallibilistic conundrum can be further refined if a mathematical formulation capable of resolving it is made. There are two components that create the conundrum. The number of opportunities for falsification goes towards infinity, and the perpetual updating drives the probability of falsification per opportunity towards zero as the Bayesian probability approaches one.Since the same variable (time) drives both phenomena, a good mathematical formulation might resolve the conundrum, possibly with the use of limits or calculus. A mathematical resolution of the conundrum, if at all possible, would yield the probability that the theory will withstand the test of time, which would be of importance in science in the narrower sense of the term and in data science as well.It should not go unsaid that any real or hypothetical ability to update the estimate of probability, including the intuitive sense of truth value, would go on to generate a fallibilistic conundrum. Bayesian probability was simply used as an example.Had any probabilistic assessment that was not immune to the Duch book been used, it would have synthesized the outcome of the first thought experiment.ConclusionThe Bayesian probability raises with evidence acquired through time towards one if the theory is true (and the data and the interpretation thereof are not faulty). Consequently, the probability of falsification per attempt falls with it, towards zero. But if there is an infinite amount of time, the number of attempts starts approaching infinity. So it incepts the conundrum where the (subjective) probability of falsification per attempt drops towards zero, but the number of attempts rises towards infinity. The two would have to be multiplied to get the probability of falsification, which possibly cannot be done. Since time as the common variable drives both probabilities, one directly and one indirectly, it might possibly (uncertainly) be resolvable with limits or calculus in the event of a good mathematical formulation of the problem. A potential resolution might not only further refine fallibilism and maybe other aspects of the philosophy of science but also advance data science.LiteratureBanerji, C.R.S.; Mansour, T.; Severini, S. A Notion of Graph Likelihood and an Infinite Monkey Theorem. Journal of Physics A: Mathematical and Theoretical 2013, 47, 035101, doi:10.1088/1751-8113/47/3/035101.Bolstad, W.M.; Curran, J.M. Introduction to Bayesian Statistics, Third Edition; 2016;Laudan, L. A Confutation of Convergent Realism. Philosophy of Science 1981, 48, 19–49, doi:10.1086/288975.Mizrahi, M. The Pessimistic Induction: A Bad Argument Gone Too Far. Synthese 2012, 190, 3209–3226, doi:10.1007/s11229-012-0138-3.Shaik, S.; Danovich, D.; Hiberty, P.C. Valence Bond Theory—Its Birth, Struggles with Molecular Orbital Theory, Its Present State and Future Prospects. Molecules 2021, 26, 1624, doi:10.3390/molecules26061624.Reed, B. Fallibilism. Philosophy Compass 2012, 7, 585–596, doi:10.1111/j.1747-9991.2012.00502.x.Affiliations:At the time of the writing and submission, the author was a PhD candidate at Megatrend University, Faculty of Biofarming, and an analogue of CEO (in the anglosphere) that in Serbia is titled „direktor“ of ecoera doo, Belgrade. The word „doo“ has roughly the same meaning as „LLC“ in the anglosphere or GmbH in Germany, Austria and Switzerland.Conflicts of interest:The author declares no conflicts of interest.