On a fuzzy scientific language 2020-06-17 Author: Olle Dahlstedt Supervisor: Sebastian Lutz Thesis project STS - Philosophy Department of Philosophy, Uppsala university 1 Introduction 3 Thesis statement 5 Part 1 6 The justification for a scientific language of fuzzy logic 6 On ‘observation’, ‘measurement’ and ‘vagueness’ 6 On fuzzy logic 8 On fuzzy logic and language 9 On social science and statistics 11 On sets and fuzzy sets in social sciences 13 Part 2 16 Constructing a fuzzy scientific language 16 Language of empirical theories 16 The pragmatics of language 18 The semantics of language 19 The logical syntax of language 20 On the axiomatization of language 21 Primitive signs in the predicate calculus 23 Axiomatization of a fuzzy language 24 Individual variables 25 Primitive connectives 25 Truth constants 26 Definable connectives 26 Axioms and inference rule 27 Truth conditions 27 Fuzzy truth evaluations 28 Conclusion 30 References 33 2 Introduction There is this classic view of science as a continuous process of clever people constantly inventing new theories about the world and rational skeptics constructing elaborate experiments to test those theories. But on what logical basis do we validate the results of scientific research? Typically, to make philosophical statements about scientific results, we do not wish to adjudicate the claims of truth of those results solely on the subjective basis of the social merits of scientists. Inevitably, this rules out the idea that we could accept scientific results on the basis of “because the scientists say so”. A popular approach, favored by philosophers such as Carnap (1939), Przelecki (1969) and Quine (1957), is to discuss the philosophy of science from the perspective of scientific language.1 If scientific results could be analyzed like sentences in a language, it is possible to imagine that other philosophical concepts such as deductions, or accounts of explanations, that work well for logical sentences, similarly could be applied to scientific results. However, one of the major issues with the approach of science as a formal language is that there is no simple way of sewing together the pragmatic aspects of ordinary everyday language with the syntax and semantics of a formal language. A formal language, in this sense, is a language within which we can effectively judge what kind of sentences are logically true or not. The essay will present a more complete description of a formal language later. However, it is only through including the pragmatic aspects of language that we can extend the notion of truth to also be able to include what is factually true, i.e. what can be validated empirically. Beyond the practical concerns of simply finding the appropriate logical system for our language of scientific theories, there are notable difficulties with accurately judging scientific results, even within a formal language system. If we could translate a scientific finding with precision, then there are still problems, such as the grue paradox, that interfere with the validity of logical inferences.2 It is possible that there are some things 1 Foundations of Logic and Mathematics, Carnap (1939), The Logic of Empirical Theories, Przelecki (1969), The Scope and Language of Science, Quine (1957), 2 Goodman, N. (1946), A Query on Confirmation 3 within logical systems that the logical system itself is unable to resolve.3 In spite of this, scientists still use logical inferences, like abduction, to justify the results of their observations as valid with respect to the theory they are testing.4 So it does not follow from the fact that formal languages of science are not perfect, that they would therefore not be useful. It is within the scope of the idea that formal languages can be useful to scientists, that this essay should present a formal language for science, such that it may be useful. By useful, the preferred interpretation is “philosophically interesting”, since the purpose of a scientific language is to have a language within which we can adjudicate truth claims within science. This is therefore not necessarily of any practical concern to scientists, but should be of real interest to philosophers of science. The quality of our language can’t be promised - in order for a language to be considered useful or interesting, the scope of all philosophy of science needs to be addressed. This is plainly impossible in this short essay. However, there are some indicators that this could be productive. Part 1 of this essay will discuss the arguments justifying the introduction of a new type of scientific language. Specifically, we examine the arguments put forth by social scientists regarding the validity of empirical observations within social science. We also try to connect these arguments to deeper problems within the philosophy of science. It is possible to consider this part of the essay an extension of the introduction; in particular, it is not necessary to read part 1 if the reader is already convinced of the reasons why a fuzzy language of science could be useful. In part 2 of the essay, we briefly discuss what a formal language entails, and we introduce this kind of formal language, which is based upon fuzzy logic. Fuzzy set theory is a type of set theory based on set membership degrees that are usually within the real-valued interval [0, 1].5 Fuzzy logic extends fuzzy set theory in the context of making truth claims. Beyond the arguments put forth in part one, the case for 3 While not strictly linked to the new riddle on induction, Gödel’s incompleteness theorems (1931) and Tarski’s undefinability theorem (1939) discuss the limitations of formal systems 4 Johansson, L. (2016, $6.1-6.3). Philosophy of Science for Scientists, pp. 103-109 5 $1, Cintulá, P., Fermüller, C., and Noguera, C., (2017) Fuzzy Logic, Stanford Encyclopedia of Philosophy 4 introducing a “new” type of logic to evaluate truth claims might seem unjustified. As we will discuss in part 1, fuzzy logic is already a suggested methodology within social sciences. This means that any philosophical language of science that is supposed to deal with the truth claims of fuzzy scientific results needs to be formalized in the same logic. Therefore, we already do have a fairly strong reason to admit the necessity of a fuzzy logic language of science. Moreover, a scientific language will inevitably contain sentences which are conventional, such as the proper way to perform a certain kind of experiment, or which theory to use when explaining some result, or how one should label the axes of graphs. Importantly, even which type of logical system to use is in itself a conventional statement. Conventional statements are normative, in the sense that they do reflect the values of the speaker; in this case, the values of scientists. The purpose of this essay is therefore normative, in the sense that when we are providing the logical system of a scientific language, we are engaging in an attempt of trying to describe how instances of “what is” actually ought to be described. The essay is concluded with a brief discussion of the results of part two, as well as considering further arguments for and against this type of language. Thesis statement The first aim of this essay is to provide a justification of introducing vagueness to scientific language. The second aim is to provide a specific alternative for doing so: namely, a language based on fuzzy logic, that is designed to cope with vague truth values, and to prove that it is possible to construct the syntax and evaluate the semantics of such a language. 5 Part 1 The justification for a scientific language of fuzzy logic On ‘observation’, ‘measurement’ and ‘vagueness’ Clearly, it should be stated somewhere in this essay that an analysis of observational evidence in scientific language is something different from the physical act of observation. We should acknowledge that observation in science is a slightly more involved process than opening one’s eyes and just taking a look. It is an important and careful activity whose epistemic value depends on what is observed and in what way. To clarify, we should explain the difference between the words measurement and observation. In this essay, we will use the word observation referring to unaided perceptions by humans. A measurement, meanwhile, can be considered the extension of an observation to include additional content, such as parameters that are not perceivable by humans, arranged in systems of units and ordered in magnitudes, with the help of devices. “The weather seems warm outside” is an observation, whereas “According to my thermometer, the temperature just outside my window is 30 degrees celsius.” is a measurement. Typically, speakers of natural language and philosophers refer to examples of any kind, measurements or not, formulated as ‘observations’, and how these examples are formulated as ‘observational languages’. Later, in the second part of this essay, for the sake of brevity, we will consider a language in which we should be able to describe observations, and in turn use the word observation generally with reference to any kind of measurement or observation. In this part of the essay, when discussing the subtleties of experimental results, the essay uses the word measurement. However, it should be noted that both measurements and observations require fuzzy truth values. From Rosenberg (1975), we propose that vagueness is an intrinsic property of observational language, as he reasons that vagueness cannot be avoided in scientific 6 language in general.6 He points out that observations are by their nature inexact;7 we can convince ourselves of this quite easily. We propose that any observation is inexact because it excludes relevant information. There is no reason we couldn’t decide that there is some causal link between a butterfly flapping its wings and eventually impacting the movement patterns of storm clouds two thousand miles away.