Lecture2_Arguments.pdf

Presentor Edwin D. Ibańez, Ph.D. Professor, Dept. of Math & Physics Logical Reasoning ARGUMENTS Prepared by: Edwin D. Ibanez, Ph.D. P:rofessor, Math & Physics Dept. Introduction Symbolic logic is mainl...

Presentor Edwin D. Ibańez, Ph.D. Professor, Dept. of Math & Physics Logical Reasoning ARGUMENTS Prepared by: Edwin D. Ibanez, Ph.D. P:rofessor, Math & Physics Dept. Introduction Symbolic logic is mainly a study of arguments; and since argument is made up of propositions, there is a need to determine the truthfulness or falsity of these propositions in order to know the validity or invalidity of an argument. Arguments An argument is a collection of propositions where it is claimed that one of the propositions called conclusion follows from the other propositions called the premises of the argument and is denoted by P1, P2, …, Pn / Q Three ways of validating an argument: 1. by using truth table; 2. by using shortened truth table; and 3. by using diagram. Truth Table Theorem: The argument P1, P2, …, Pn / Q is valid if and only if the proposition (P1  P2 ...  Pn) → Q is a tautology. Example: If a man is a bachelor , he is unhappy. If a man is unhappy, he dies young. So, bachelor dies young. Letting p be the statement “He is a bachelor.”, q is “He is unhappy.”, and r is “He dies young.” Thus, the given argument in symbol can be written as [(p→q)(q→r)] → (p→r) and can be verified by the following truth table. Let: P1 = (p→q), P2 = (q→r), P3 = (p→r), P4 = (p→q)(q→r) P5 = [(p→q)(q→r)] → (p→r) p q r P1 P2 P3 P4 P5 T T T T T T T T T T F T F F F T T F T F T T F T T F F F T F F T F T T T T T T T F T F T F T F T F F T T T T T T F F F T T T T T Shortened Truth Table ❑ The invalidity of an argument may be verified by showing that its propositional form is not a tautology. ❑ Since the propositional form of an argument is an implication, then we should be able to show an instance when the premise is true but the conclusion is false. ❑ We do not have to construct the whole truth table for the propositional form to do this. All we have to do is to determine the combination of values that makes the propositional form of the argument false. ❑ This simplified process of constructing a truth table is called shortened truth table method. Example: Prove the invalidity of the argument p→q r→s qr p  s The propositional form of this argument is [(p → q)(r → s)(q  r)] → (p  s) This is false when the propositional variables have the following truth values: p q r s F T F F Venn Diagram Many verbal statements can be translated into equivalent statements using sets, which can then be described by Venn diagrams. Hence Venn diagram is one of the ways to determine the validity of an argument. Example: Consider this argument Babies are irrational. Nobody is despised who can manage a crocodile. Irrational people are despised. ------------------------------------------------------------ Therefore, babies cannot manage a crocodile. Consider the given example. ❑ By first premise, the set of babies is a subset of irrational people. ❑ By third premise, the set of irrational people is contained in the set of despised people. ❑ By second premise, the set of despised people and the set of people who can manage crocodile are disjoint. Note that the set of babies and the set of people who can manage crocodiles are disjoint. ❑ In other words, “Babies cannot manage crocodiles” is a consequence of the first, second, and third premises, and represent a valid argument. Problem Set 2