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Inductive Reasoning Applied Logic Lecture 4 part 1 – Inductive reasoning Marcin Szczuka Institute of Informatics, The University of Warsaw Monographic lecture, Spring semester 2018/2019 Marcin Szczuka (MIMUW) Applied Logic 2019 1 / 30 Lecture plan 1 Introduction 2 Incomplete inductive reasoning Types of inductive reasoning 3 Towards Inductive Logic Marcin Szczuka (MIMUW) Applied Logic 2019 2 / 30 Marcin Szczuka (MIMUW) Applied Logic 2019 3 / 30 Deduction vs. induction All of the reasoning systems (logics) we have seen so far were based on deduction. The worked with use of certain inference rules that allowed derivation of consequences from axioms. In particular, deductive systems are closed with respect to creation of new notions and inference of true consequences. Reasoning based on pure deduction is very rare in real life. In most cases they are limited to precise, mathematical models of reality (theoretical physics, mathematics, informatics, ..). A great example of work based on rigorous deductive reasoning is the monumental work by Euclid of Alexandria, the Elements. In real-life scenarios it is usually hard to preserve the strict rules of deduction and the inferred conclusions are not always absolutely true. Therefore, in complement to deductive reasoning in real-life situation we frequently use also induction and/or abduction. Marcin Szczuka (MIMUW) Applied Logic 2019 4 / 30 Inductive reasoning In simplest terms, inductive reasoning can be seen as inference from particular to general or from examples to rules. Inductive reasoning is by nature imprecise. Inductive reasoning is built on human capability of finding patterns and rules on the basis of observation of a finite (and possibly incomplete and imperfect) sample. For example, on the basis of obserwation any “reasonable” thinker will consider the following sentence to be empirically proven. All crows are black, but not all cats are black. Marcin Szczuka (MIMUW) Applied Logic 2019 5 / 30 Nuances of induction The inference on the basis of empirical observation is as old as scientific reasoning itself. However, due to the lack of formalised approach, up to the end of middle ages the Aristotelean deductive reasoning was considered the only “true” method for proving. While Aristotle considered briefly the inductive method, but limited only to the primitive complete enumerative induction. Nowadays the reasoning by induction is much more precise and regularised. First important questions regarding validity and practicality of simplified, enumerative approach to induction were posed by Francis Bacon (1561-1626). Bacon proposed more practical eliminative induction (induction by elimination). The principle of eliminative induction was formulated more precisely by American philosopher John Stuarta Mill in A System of Logic, Ratiocinative and Inductive (1843) in the form of five so called Mill’s methods (cannons). Marcin Szczuka (MIMUW) Applied Logic 2019 6 / 30 Nuances of induction After Bacon, eliminative induction principle was disputed by David Hume (1748). Immanuel Kant pitched in too. Hume, in his philosophical works about cognition and causality, proposed a novel view on induction and provided a constructive critical Hume’s propositions became a cornerstone of the modern understanding of inductive reasoning. Hume divides all reasoning into demonstrative, by which he means deductive, and probabilistic, by which he means the generalization of causal reasoning. Today the idea that our knowledge of the world is not complete and precise is commonplace. However, in Hume’s era it sounded like a shocking paradox, especially in light of recently formulated foundations of Newtonian physics. Contemporary understanding of inductive reasoning system diverted from the Hume’s and Kant’s ideas towards inductive logic. Nowadays, instead of answering the question “what justifies truthfulness of the statement?” the researchers concentrate on the question “why a given statement is possible/probable?”. A prominent representative of this kind of approach is (amongst others) Rudolf Carnap. Marcin Szczuka (MIMUW) Applied Logic 2019 7 / 30 Types of inductive reasoning Complete induction Complete induction (complete enumerative induction, exhaustive induction) is a reasoning method that establishes truthfulness of a rule (proposition) by checking all possible cases when the rule applies. Complete induction is in fact a certain, deductive method. It eliminates possible contradiction by exhaustive enumeration of all positive cases. In most non-trivial situations the complete induction method is highly inefficient. A special case on complete induction is the mathematical induction commonly used to prove mathematical theorems, especially in discrete domains. Somewhat against its name, mathematical induction is a deductive proof technique. Marcin Szczuka (MIMUW) Applied Logic 2019 8 / 30 Eliminative induction “...when you have eliminated all which is impossible, then whatever remains, however improbable, must be the truth.” Sherlock Holmes ŹAfter: Arthur Conan Doyle, The Blanched Soldier Marcin Szczuka (MIMUW) Applied Logic 2019 9 / 30 Types of inductive reasoning Eliminative induction The simplest eliminative induction scheme (Bacon) is based on building a list of mutually exclusive hypotheses and then eliminating all but one of them through empiric experiment. J.S. Mill extended the eliminative induction principle by setting five rules for elimination of hypotheses (Mills methods). Mill’s methods allow for (partial) formalisation of inference by establishing relationships of the type “ Cause A yields result a” on the basis of series of observations. For example, the direct method of agreement (first method) facilitates the following reasoning: Situation 1: We record appearance of causes A, B, C and results a, b, c. Situation 2: We record appearance of causes A, D, E and results a, d, e. Conclusion: We eliminate non-repeating (not directly agreeing) observations and we get “Cause A yields result a”. Marcin Szczuka (MIMUW) Applied Logic 2019 10 / 30 Types of inductive reasoning Incomplete induction Incomplete induction (incomplete enumerative induction) is the method that establishes a general rule (proposition) on the basis of a finite number of statements (observations) that confirm the rule. We reason from sample about general regularities. Incompleteness of this reasoning scheme is a manifestation of the nature of reality that we attempt to describe. In real life we almost never are able to observe sufficiently many (or all) possible situations. Incompleteness also means, that previously constructed theories may need to be modified (completed) as new observations emerge. For example, Einstein’s relativity theory extends Newtonian mechanics. Incomplete induction is one of the most basic tools for all experimental sciences. Many disciplines have developed frameworks for dealing with uncertainty introduced through use of incomplete induction, for example error calculus, sta Marcin Szczuka (MIMUW) Applied Logic 2019 11 / 30 Lecture plan 1 Introduction 2 Incomplete inductive reasoning Types of inductive reasoning 3 Towards Inductive Logic Marcin Szczuka (MIMUW) Applied Logic 2019 12 / 30 The issue of induction The issue of incomplete inductive reasoning has been considered by researchers for centuries. The discussion about validity and necessity of its use in describing the universe can be traced back to Sextus Empiricus (3-2 century B.C.). Throughout the ages some of the most illustrious minds, including Bacon, Cartesius, Kant, Newton, Mill, Hume and others, addressed the issue. In contemporary discourse on induction as a mean to discovery important additions were made by eminent philosophers of sciece, including Karl Popper, Wesley C. Salmon and David Miller. Construction of a logical system based on incomplete inductive reasoning poses a challenge. On the one hand, such logic should extend deductive systems by addition of a mechanism for deriving conclusion that may not be absolutely true. On the other hand we are not willing to part with the essential property of deductive systems: True premises guarantee true conclusions. Marcin Szczuka (MIMUW) Applied Logic 2019 13 / 30 Criterion of Adequacy In order for inductive logic to be considered useful we usually expect that it has a mechanism for establishing a level of support for conclusions expressed in it. This mechanism measures the degree of influence of premises’ truthfulness on validity of the conclusion. We expect that the measure used by this mechanism satisfies the Criterion of Adequacy (CoA). CoA - Criterion of Adequacy As evidence accumulates, the degree to which the collection of true evidence statements comes to support a hypothesis, as measured by the logic, should tend to indicate that false hypotheses are probably false and that true hypotheses are probably true. Marcin Szczuka (MIMUW) Applied Logic 2019 14 / 30 Peculiarities of induction To be able to use inductive reasoning principles (inductive logic) properly it is necessary to eliminate the possibility of creating paradoxical results or sophismata. Inductive proof of immortality Fact 1 – n Many times (n 1) I heard that somebody died. Fact n + 1 Every time I heard that somebody died – it was not me. conclusion There are no observations supporting the fact of me dying. Hence, I am immortal. Obviously, this reasoning is a fallacy. It does not take into account any negative information as well as considers only a part of positive information that by no “decent” measure can be considered complete. However, in practical
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