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CmSc 175 Discrete Mathematics Lesson 07: Arguments with Quantified Statements 1. Rule of universal instantiation If some property is true of everything in a domain, then it is true of any particular thing in the domain. This is the fundamental tool of deductive reasoning. 2. Universal Modus Ponens The rule of universal instantiation can be combined with modus ponens to obtain the rule called universal modus ponens: x, if P(x) then Q(x) P(a) for a particular a Q(a). (the symbol is read "therefore" The argument consists of two premises at least one of which is universally quantified The first and the second premises are called major and minor premises respectively. Example: 1. If a number is even, then its square is even 2. 6 is a particular number that is even 62 is even As a formal argument: x, if even(x) then even (x2) even(6) even(62) 3. Universal Modus Tollens When we combine universal instantiation with modus tollens we have universal modus tollens, the heart of proof by contradiction. x, if P(x) then Q(x) Q(a) for a particular a P(a). 1 Example: 1. If a number is divisible by 6 then it is divisible by 2. 2. 15 is a particular number that is not divisible by 2 15 is not divisible by 6 As a formal argument: x, if div_by_6(x) then div_by_2 (x) not div_by_2(15) not div_by_6(15) 4. Converse and Inverse errors Converse Error (Quantified Form): When we are given the premises "For all x, if P(x) then Q(x)”, and "Q(a)" -- then P(a) is an invalid conclusion. Inverse Error (Quantified Form): When we are given the premises "For all x, if P(x) then Q(x)" and " ~P(a)" -- then ~Q(a) is an invalid conclusion. Venn Diagrams: A particularly helpful technique for evaluating the validity of an argument having quantified statements is to use Venn Diagrams. Depict each predicate P(x) as a statement of membership of element x in some circular region P; and depict "if P(x) then Q(x)" by sketching circle P strictly inside of circle Q; etc. If the conclusion statement holds true for every value of x that makes the premise statements (geometrically) true, then the argument is valid. Examples a. Converse error All human beings are mortal Felix is mortal Felix is a human being <<<<<< Invalid conclusion Venn diagram: Felix mortal mortal Felix human human 2 Felix is somewhere in the circle representing the mortal beings. However Felix is not necessarily within the circle representing humans b. Inverse error All human beings are mortal Q is not human Q is not mortal <<<<<< Invalid conclusion Q Q mortal human mortal human Q is somewhere outside the circle of human beings, however Q is not necessarily outside the circle of mortal beings Diagrams for valid arguments a. Universal modus ponens All human beings are mortal Socrates is human Socrates is mortal mortal Socrates human 3 Socrates is in the circle of human beings. The circle of human beings is inside the circle of mortal beings. Therefore Socrates is in the circle of mortal beings b. Universal modus tollens All human beings are mortal Zeus is not mortal Zeus is not human Zeus mortal human Zeus lies somewhere outside the circle of mortal beings. Since the circle of humans is inside the circle of mortal beings, Zeus is outside the circle of human beings. 5. Exercises Identify the type of the argument if valid, or the type of the error in case of invalid argument All nerds are good at math. Buffy is good at math. Buffy is a nerd. Animals at the bottom of the food chain are very nervous. People are not at the bottom of the food chain. People are not nervous 4 All trees have leaves. Roses have leaves. Roses are trees. Pigs can’t fly. Wilbur is a pig. Wilbur can’t fly. Pigs can’t fly. Tweety can fly. Tweety isn’t a pig. Every adult is eligible to vote. John is eligible to vote. John is an adult. Any odd integer x can be written in the form x = 2k+1 for some integer k y is an odd integer y = 2k+1 for some integer k My professors are happy when I pay close attention to their lectures. My professors are not happy today I don’t pay close attention to their lectures today. People who are good at logic make good programmers Buffy would not be a good programmer Buffy is not good at logic 5