CT3340/CD5540 Research Methodology for Computer Science and Engineering Theory of Science LOGIC AND CRITICAL THINKING Gordana Dodig-Crnkovic Department of Computer Science and Engineering Mälardalen University 2004 1 Theory of Science Lectures 1. SCIENCE, KNOWLEDGE, TRUTH 2. SCIENCE, RESEARCH, TECHNOLOGY AND SOCIETY 3. LOGIC AND CRITICAL THINKING 4. FORMAL LOGICAL SYSTEMS LIMITATIONS, LANGUAGE AND COMMUNICATION, BRIEF RETROSPECTIVE OF SCIENTIFIC THEORY 5. PROFESSIONAL ETHICS 2 2 LOGIC AND CRITICAL THINKING • LOGICAL ARGUMENT • DEDUCTION • INDUCTION – Empirical Induction – Mathematical Induction – Induction vs. Deduction, Hypothetico-deductive Method • REPETITIONS, PATTERNS, IDENTITY • CAUSALITY AND DETERMINISM • FALLACIES 3 LOGIC Logic is the science of reasoning, proof, thinking, or inference. Logic allows us to analyze a piece of reasoning and determine whether it is correct or not. To use the technical terms, we determine whether the reasoning is valid or invalid. 4 LOGIC This lecture deals only with simple Boolean logic. Other sorts of mathematical logic, such as fuzzy logic, obey different rules. When people talk of logical arguments, though, they generally mean the type being described here. 5 What is an Argument? "An argument is a connected series of statements intended to establish a proposition". (Michael Palin i Monty Python’s Argument Clinic.) See Monty Python's Argument Clinic under: http://www.duke.edu/~pms5/humor/argument.html 6 What is an Argument? There are three stages to an argument: – premises – inference and – conclusion. 7 JUDGEMENT Now, the question, What is a judgement? is no small question, because the notion of judgement is just about the first of all the notions of logic, the one that has to be explained before all the others, before even the notions of proposition and truth, for instance. Per Martin-Löf On the Meanings of the Logical Constants and the Justifications of the Logical Laws; Nordic Journal of Philosophical Logic, 1(1):11 60, 1996. 8 What is an Argument? An argument is thus a statement logically inferred from premises. Neither an opinion nor a belief can qualify as an argument! Two sorts of arguments: – deductive – inductive 9 What is an Argument? The building blocks of a logical argument are propositions, also called statements. A proposition is a statement which is either true or false; for example: "The first programmable computer was built in Cambridge." "Dogs cannot see color." 10 1. Premises One or more propositions are necessary for the argument to continue. They must be stated explicitly. They are called the premises of the argument. They are the evidence (or reasons) for accepting the argument and its conclusions. 11 2. Inference The premises of the argument are used to obtain further propositions. This process is known as inference. In inference, we start with one or more propositions which have been accepted. We then derive a new proposition. There are various forms of valid inference. The propositions arrived at by inference may then be used in further inference. Inference is often denoted by phrases such as implies that or therefore. 12 3. Conclusion Finally, we arrive at the conclusion of the argument, another proposition. The conclusion is often stated as the final stage of inference. It is affirmed on the basis the original premises, and the inference from them. Conclusions are often indicated by phrases such as therefore, it follows that, we conclude and so on. 13 Deductive inferences: general → particular Inductive inferences: particular → general 14 DEDUCTION A deductive argument is defined as: • constructed according to valid rules of inference • the conclusion necessarily follows from the premises. 15 Modus Ponens All humans are mortal. (premise) Kevin is human. (premise) Thus, Kevin is mortal. (conclusion) 16 Modus Tollens All birds have wings. (premise) Kevin has no wings. (premise) Kevin is not a bird. (conclusion) 17 Disjunctive Syllogism The baby can either be a boy or a girl. (premise) The baby is not a girl. (premise) The baby is a boy. (conclusion) 18 Hypothetical Syllogism If Karro is a terrier, Karro is a dog. (premise) If Karro is a dog, Karro is a mammal. (premise) If Karro is a terrier, Karro is a mammal. (conclusion) 19 NON-STANDARD LOGICS • Categorical logic • Many-sorted logic • Combinatory logic • Many-valued logic • Conditional logic • Modal logic • Constructive logic • Non-monotonic logic • Cumulative logic • Paraconsistent logic • Deontic logic • Partial logic • Dynamic logic • Prohairetic logic • Epistemic logic • Quantum logic • Erotetic logic • Relevant logic • Free logic • Stoic logic • Fuzzy logic • Substance logic • Higher-order logic • Substructural logic • Infinitary logic • Temporal (tense) logic • Intensional logic • Other logics • Intuitionistic logic • Linear logic 20 NON-STANDARD LOGICS http://www.earlham.edu/~peters/courses/logsys/nonstbib.htm http://www.math.vanderbilt.edu/~schectex/logics/ 21 INDUCTION • Empirical Induction • Mathematical Induction 22 EMPIRICAL INDUCTION The generic form of an inductive argument: • Every A we have observed is a B. • Therefore, every A is a B. 23 An example of inductive inference • Every instance of water (at sea level) we have observed has boiled at 100° C. • Therefore, all water (at sea level) boils at 100° C. Inductive argument will never offer 100% certainty! A typical problem with inductive argument is that it is formulated generally, while the observations are made under specific conditions. ( In our example we could add ”in an open vessel” as well. ) 24 An inductive argument have no way to logically (with certainty) prove that: • the phenomenon studied do exist in general domain • that it continues to behave according to the same pattern According to Popper, inductive argument supports working theories based on the collected evidence. 25 Counter-example Perhaps the most well known counter-example was the discovery of black swans in Australia. Prior to the point, it was assumed that all swans were white. With the discovery of the counter-example, the induction concerning the color of swans had to be re-modeled. 26 MATHEMATICAL INDUCTION In the empirical induction we try to establish the law. In the mathematical induction we have the law already formulated. We must prove that it holds generally. The basis for mathematical induction is the property of the well-ordering principle for the natural numbers. 27 THE PRINCIPLE OF MATHEMATICAL INDUCTION Suppose P(n) is a statement involving an integer n. Than to prove that P(n) is true for every n ≥ n0 it is sufficient to show these two things: 1. P(n0) is true. 2. For any k ≥ n0, if P(k) is true, then P(k+1) is true. 28 THE TWO PARTS OF INDUCTIVE PROOF • the basis step • the induction step. • In the induction step, we assume that statement is true in the case n = k, and we call this assumption the induction hypothesis. 29 THE STRONG PRINCIPLE OF MATHEMATICAL INDUCTION (1) Suppose P(n) is a statement involving an integer n. In order to prove that P(n) is true for every n ≥ n0 it is sufficient to show these two things: 1. P(n0) is true. 2. For any k ≥ n0, if P(n) is true for every n satisfying n0 ≤ n ≤ k, then P(k+1) is true. 30 THE STRONG PRINCIPLE OF MATHEMATICAL INDUCTION (2) A proof by induction using this strong principle follows the same steps as the one using the common induction principle. The only difference is in the form of induction hypothesis. Here the induction hypothesis is that k is some integer k ≥ n0 and that all the statements P(n0), P(n0 +1), …, P(k) are true. 31 Example. Proof By Strong Induction • P(n): n is either prime or product of two or more primes, for n ≥ 2. • Basic step. P(2) is true because 2 is prime. • Induction hypothesis. k ≥ 2, and for every n satisfying 2 ≤ n ≤ k, n is either prime or a product of two or more primes. 32 • Statement to be shown in induction step: If k+1 is prime, the statement P(k+1) is true. • Otherwise, by definition of prime, k+1 = r·s, for some positive integers r and s, neither of which is 1 or k+1. It follows that 2 ≤ r ≤ k and 2 ≤ s ≤ k. • By the induction hypothesis, both r and s are either prime or product of two or more primes. • Therefore, k+1 is the product of two or more primes, and P(k+1) is true. 33 The strong principle of induction is also referred to as the principle of complete induction, or course-of-values induction. It is as intuitively plausible as the ordinary induction principle; in fact, the two are equivalent. As to whether they are true, the answer may seem a little surprising. Neither can be proved using standard properties of natural numbers. Neither can be disproved either! 34 This means essentially that to be able to use the induction principle, we must adopt it as an axiom. A well-known set of axioms for the natural numbers, the Peano axioms, includes one similar to the induction principle. 35 PEANO'S AXIOMS 1. N is a set and 1 is an element of N. 2. Each element x of N has a unique successor in N denoted x'. 3. 1 is not the successor of any element of N. 4. If x' = y' then x = y. 5. (Axiom of Induction) If M is a subset of N satisfying both: 1 is in M x in M implies x' in M then M = N. 36 INDUCTION VS DEDUCTION, HYPOTHETICO-DEDUCTIVE METHOD Deduction and induction occur as a part of the common hypothetico-deductive method, which can be simplified in the following scheme: • Ask a question and formulate a hypothesis/educated guess ( induction) • Make predictions about the hypothesis (deduction). • Test the hypothesis (induction). 37 INDUCTION & DEDUCTION: AN ETERNAL GOLDEN BRAID • Deduction, if applied correctly, leads to true conclusions.