Topic 2 - Laws of Thoughts, Claims and Arguments.pdf

CS/SE 2211 Deductive Reasoning & Logic Programming Topic 2 : Laws of Thoughts, Claims and Arguments by K H Navoda [B.Sc. (Hons), MCS, BCS HEQ] Bachelor of Science (Hons) in CS / SE / IT Department of Computing Faculty...

CS/SE 2211 Deductive Reasoning & Logic Programming Topic 2 : Laws of Thoughts, Claims and Arguments by K H Navoda [B.Sc. (Hons), MCS, BCS HEQ] Bachelor of Science (Hons) in CS / SE / IT Department of Computing Faculty of Computing & Technology Saegis Campus Nugegoda. Saegis Campus  Laws of thought The laws of thought are principles that are considered fundamental to logical reasoning. They've been around for centuries and are debated by philosophers, but they provide a groundwork for clear and reasoned thinking. ◦ Law of Identity ◦ Law of Non-contradiction ◦ Law of Excluded Middle ◦ Law of Community for Conjunction ◦ The index law K H NAVODA [B.SC. (HONS), MCS, BCS HEQ] Saegis Campus 2  Law of Identity This law says, everything is identical to itself. Examples ◦ If anything is A, then it is A. ◦ A square is a square. K H NAVODA [B.SC. (HONS), MCS, BCS HEQ] Saegis Campus 3  Law of Non-contradiction This law states that nothing can be both A and not A. Example ◦ A square cannot be at the same time both a square and not a square with everything remaining the same. K H NAVODA [B.SC. (HONS), MCS, BCS HEQ] Saegis Campus 4  Law of Excluded Middle This law states that for everything, it has to be either A or not A. There is no middle ground. (It cannot be ‘not either’). Example ◦ A thing has to be either a square or not a square. K H NAVODA [B.SC. (HONS), MCS, BCS HEQ] Saegis Campus 5  Law of Commutativity for Conjunction If two statements are joined by ‘and’, then the order in which the statements are placed is immaterial. The truth or the falsity of the conjunction remains unaffected. Example ◦ Being a metallic object and a water carrier is equivalent to being a water carrier and a metallic object. K H NAVODA [B.SC. (HONS), MCS, BCS HEQ] Saegis Campus 6  The index law Asserting a statement is equivalent to its assertion in conjunction with itself. Example ◦ Being a metallic object and a metallic object is being a metallic object. K H NAVODA [B.SC. (HONS), MCS, BCS HEQ] Saegis Campus 7  Claim vs. Argument CLAIM ARGUMENT A claim is a statement that can be determined as An argument is a set of claims that has a either true or false. central or principle claim which is at issue and is to be argued for, and other claims that are Example offered as reasons or supporting evidence for ◦ Watching TV from a close distance harms eyesight. accepting or believing in that principal claim. ◦ Thailand is not is Asia. Example Grammatically there are expressions such as ◦ We are in for an unusually hot summer because commands, questions and greetings which does the data from Meteorological Office shows lower not consider as claims. temperature in the region in the previous year. Example ◦ What time is it? K H NAVODA [B.SC. (HONS), MCS, BCS HEQ] Saegis Campus 8  More on Arguments Arguments are the most interest to logicians ◦ The principal claim that is argued for this is known as the conclusion. ◦ Other claims that are offered as supporting reasons in the argument for conclusion are called premises. K H NAVODA [B.SC. (HONS), MCS, BCS HEQ] Saegis Campus 9  Characteristics of Arguments Arguments are not claims Every set of claims is not an argument No fixed number on how many premises There may be unstated premises Arguments have a standard format K H NAVODA [B.SC. (HONS), MCS, BCS HEQ] Saegis Campus 10  Standard format of an Argument Premises must be separated from the conclusion. In the standard format, premises need to be mentioned first. Then state the conclusion with a conclusion marker, such as the ∴ 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∴ 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 Example 𝐴𝐴𝑙𝑙𝑙𝑙 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 ℎ𝑎𝑎𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑖𝑖𝑖𝑖 𝑎𝑎 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 ∴ 𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ℎ𝑎𝑎𝑎𝑎 𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 K H NAVODA [B.SC. (HONS), MCS, BCS HEQ] Saegis Campus 11  Types of Arguments DEDUCTIVE ARGUMENTS INDUCTIVE ARGUMENTS If all the premises are true, then its conclusion These are arguments that do not rely on upon must be true. merely explicating the information that might be implicitly contained in the premises. The joint truth of all the premises of a good Inductive arguments have conclusions which go deductive argument is supposed to guarantee beyond that which covered, explicitly and the truth of the conclusion. implicitly by the premises. Example It is also known as inductive gap. ◦ All Humans are mortal. Example ◦ All Sri Lankan voters are humans ◦ Amal has asked for a fruit after lunch every day for ◦ Therefore, All Sri Lankan voters are mortal the last 6 days. Therefore, today also he will ask for a fruit after lunch. K H NAVODA [B.SC. (HONS), MCS, BCS HEQ] Saegis Campus 12  More on Deductive Arguments Expectation of a deductive argument is that all the premises will provide conclusive support for the conclusion. Conclusive support means that it will not leave any doubt for the truth of the conclusion. If an argument fails to demonstrate the support, then there’s a possibility that it will create a defect in the deductive argument. Those statements which are fail to demonstrate the support conclusively are considered as bad deductive arguments. Example ◦ All human beings are mortal ◦ All whales are mortal. ◦ Hence, all whales are human beings K H NAVODA [B.SC. (HONS), MCS, BCS HEQ] Saegis Campus 13  More on Inductive Arguments In a inductive argument, between premises and target conclusion, there’s always a gap. Joint truth of all the premises can at best provide partial support in terms of probability of the truth of the conclusion. At best, a good inductive argument can provide a very high degree of probability, but can never yield conclusive support. K H NAVODA [B.SC. (HONS), MCS, BCS HEQ] Saegis Campus 14  More on Inductive Arguments Example 1 99% of the residents of this town have credit cards. Hence, Sunil who is a resident of this town, will also likely to have a credit card. Example 2 Two snakes that I saw were brown. So, snakes in this locality are all brown. Example 1 is an example of strong inductive reasoning, where as examples 2 is an example of weak inductive reasoning. Strongness or the weakness depends on the probability of it happening. K H NAVODA [B.SC. (HONS), MCS, BCS HEQ] Saegis Campus 15  Diagramming the arguments A diagram of an argument helps to reveal the inner logical relationships within the arguments. Steps in constructing the logical diagram is as follows 1. Identify the components (Premises and Conclusions) 2. Use a mark such as a number 1, 2, etc. 3. Use arrow heads to indicate the relationship among components 4. Properly represent the logical relationship K H NAVODA [B.SC. (HONS), MCS, BCS HEQ] Saegis Campus 16  Example 1 Because every object’s temperature is above absolute zero, motion at the atomic level is always present. K H NAVODA [B.SC. (HONS), MCS, BCS HEQ] Saegis Campus 17  Example 1 Because every object’s temperature is above absolute zero, motion at the atomic level is always present. K H NAVODA [B.SC. (HONS), MCS, BCS HEQ] Saegis Campus 18  Exercise 1 She seems happy. Therefore is a spark in her eyes and her face looks calm and rosy. K H NAVODA [B.SC. (HONS), MCS, BCS HEQ] Saegis Campus 19  Exercise 2 All physical objects have mass and this table is a physical object. Therefore it has mass. K H NAVODA [B.SC. (HONS), MCS, BCS HEQ] Saegis Campus 20  Validity and Invalidity An argument is valid if and only if it is not possible for all its premises to be true and its conclusion to be false. Example ◦ Paul is in France ◦ France is in Europe ◦ Therefore, Paul is in Europe. K H NAVODA [B.SC. (HONS), MCS, BCS HEQ] Saegis Campus 21  Validity and Invalidity An argument is invalid, if and only if it is possible for its conclusion to be false even when all its premises are true. Example ◦ A is taller than C ◦ B is taller than C ◦ Therefore, A is taller than B K H NAVODA [B.SC. (HONS), MCS, BCS HEQ] Saegis Campus 22  Validity and Invalidity Its also important to notice that even if there are arguments with false premises, does not mean that it is invalid. A valid argument can have false premises. Example The sum of three angles of a triangle is 190 degrees. Hence, if one of the angles it is 90 degrees, then the some of other two angles must be 100 degrees. It is obvious that the sum of 3 angles in a triangle is 180. Therefore, the first premise is false. But if we considers that to be true then its clear that upon that one could conclude that sum of the other two angles is 100 degrees. K H NAVODA [B.SC. (HONS), MCS, BCS HEQ] Saegis Campus 23  Validity and Invalidity Example All cats have 6 legs. All 6-legged animals have wings. Therefore, all cats have wings. In the above example both premises are false. Yet if both were considered in to be true, then the conclusion becomes true. K H NAVODA [B.SC. (HONS), MCS, BCS HEQ] Saegis Campus 24  Validity and Invalidity Premises Conclusion Valid Invalid True True Possible Possible True False Not Possible Possible False True Possible Possible False False Possible Possible K H NAVODA [B.SC. (HONS), MCS, BCS HEQ] Saegis Campus 25  Soundness and Unsoundness A sound argument requires both ◦ The argument must be valid ◦ The premises of the argument must be true. An argument which violates any of the above two conditions or both is known as unsound. K H NAVODA [B.SC. (HONS), MCS, BCS HEQ] Saegis Campus 26  Consistency A set of statements is consistent if and only if there are at least a possible situation in which every member of that set is true. It is inconsistent if and only if no such possibility exists for the members of the set. K H NAVODA [B.SC. (HONS), MCS, BCS HEQ] Saegis Campus 27  As a summary Argument Succeed Fail Valid Sound Unsound [All Premises True] [One or more premises false] Check for consistency Among the premises K H NAVODA [B.SC. (HONS), MCS, BCS HEQ] Saegis Campus 28  Fallacies An error in a premise is known as a factual error. A fallacy is a logical error. Fallacies can belong to two broad groupings of arguments, ◦ Deductive Fallacies ◦ Inductive Fallacies K H NAVODA [B.SC. (HONS), MCS, BCS HEQ] Saegis Campus 29  Deductive Fallacies There are two well-known patterns of deductive fallacies available ◦ Denying the antecedent If someone has diabetes, his eyes may be affected Saman’s eyes are affected Therefore, Saman has diabetes. ◦ Affirming the consequent If someone has diabetes, his eyes may be affected. Ajith does not have diabetes. Hence, Ajith’s eyes cannot be affected K H NAVODA [B.SC. (HONS), MCS, BCS HEQ] Saegis Campus 30  Inductive Fallacies These usually occur when the premises of inductive arguments fail to provide desirable amount of probabilistic support for their conclusions. ◦ Illicit or Hasty generalization I have met two Americans, and both were dishonest, so all Americans are dishonest. My grandmother at 82 can walk faster than me, can still remember everyone’s name and is healthy. Therefore, all people who are of that age must be very strong and healthy. ◦ Fallacy of generalization based on unrepresentative samples The tiger that I saw was white, so all tigers must be white. K H NAVODA [B.SC. (HONS), MCS, BCS HEQ] Saegis Campus 31  Inductive Fallacies ◦ False analogy Tables have four legs. Elephants have four legs. Hence tables, just like elephants, are animals. ◦ Fallacy of exclusion Amali is a 35-year-old woman, and most women of her age are married, hence she too must be married. K H NAVODA [B.SC. (HONS), MCS, BCS HEQ] Saegis Campus 32  Questions K H NAVODA [B.SC. (HONS), MCS, BCS HEQ] Saegis Campus 33 K H NAVODA [B.SC. (HONS), MCS, BCS HEQ] 34