6 Lecture 2, MATH 210G.03, Spring 2014: Symbolic Logic
7 Law of the excluded middle For every proposition, either the proposition is true or its negation is trueEither “Socrates is a man” or “Socrates is not a man”Either “It is true that Socrates is a man” or “It is true that Socrates is not a man”What about “This sentence is neither true nor false”Problem of “self-reference” or implied “it is true that…”
8 Problems with the excluded middle Many statements have an element of uncertainty:Either it is raining or it is not rainingEither Sophia Vergara is blonde or she is not blonde.Either NMSU has a better football team or UTEP has a better football team…
9 Logical arguments I: The syllogism Aristotle, Prior Analytics: a syllogism is "a discourse in which, certain things having been supposed, something different from the things supposed results of necessity because these things are so.”
10 Syllogism cont.A categorical syllogism consists of three parts: the major premise, the minor premise and the conclusion.Major premise: All men are mortal.Minor premise: Socrates is a man.Conclusion: Socrates is mortal.Major premise: All mortals die.Minor premise: All men are mortals.Conclusion: All men die.
11 Identify the major premise: All dogs have four legs Milo is a dog ________________________ People who solve problems can get jobs. Students good in math can solve problems. _______________________ Women like a man with a prominent chin. Robert Z’dar has a prominent chin.
12 Modus ponendo ponens ("the way that affirms by affirming”) If P, then Q. P. Therefore, QIf Socrates is a man then Socrates is mortalSocrates is a manTherefore, Socrates is mortal
16 Logic and symbol of propositional calculus 1616Logic and symbol of propositional calculusP, Q, R etc: propositional variablesSubstitute for statements, e.g., P: Plato is a dog, Q: Plato is GreenLogical connectives:Proposition: If Plato is a dog then Plato is green:
18 Clicker question P: Socrates is a man Q: Socrates is mortal. If Socrates is a man then Socrates is mortal.Suppose that Socrates is not a man.Is the whole statement: true or false?Clicker: True (A) or False (B)
20 Why is “If p then q” true whenever p is false? Your mom always tells the truth…right? Your mom makes a promise: “: if you clean your room then we can go for ice cream” Suppose you don’t clean your room. If you don’t go for an ice cream, then your mom has not broken her promise. If you do go for ice cream, she still has not broken her promise. P: you clean your room; Q: go for ice cream holds either way.
21 Truth table for modus ponens No matter what truth values are assigned to the statements p and q, the statement is true
22 Exercise: complete the truth table for modus tollens QTF
23 Simple and compound statements A simple statement is sometimes called an atom. E.g., Milo is a dog; Socrates is a man; Men are mortal. A compound statement is a string of atoms joined by logical connectives (and, or, then, not) Logical equivalence: vs Truth value of a compound statement is inherited from the values of the atoms.
24 For compound statements with conjunctions (∧) to be true, the elements on both sides of ∧ must have the value “T” so the fourth column is as follows:PQTF
25 For condition statements or “implications” with “ ->” to be true, either the statement to the left of the implication has to be false or the statement to the right of the implicationhas to be true. The statement (p->q)∧-q is false in the first three cases and the statement –p is true in the last, so the fourth column has value “T” in all casesPQTF
26 Clicker questions: First row: True (A) or False (B) pqTFFirst row: True (A) or False (B)Second row: True (A) or False (B)Third row: True (A) or False (B)Fourth row: True (A) or False (B)
27 Logical equivalenceTwo formulas are logically equivalent if they have the same truth values once values are assigned to the atoms.Ex: is equivalent is equivalent toHow to check logical equivalence: verify that the statements always have the same values
28 Exercise: verify that the statements , and are logically equivalent pqTF
29 Exercise: Verify using truth tables that the following DeMorgan’s laws are logically equivalent
30 Exercise: Verify using truth tables the following absorption rules and the conditional rules
31 Match the following logical equivalencies with the corresponding rules of inference
32 Tautology and contradiction: T or C A logical statement that is always true, independent of whether each of the symbols is true, is called a tautology. A logical statement that is always true, independent of whether each of the symbols is true, is called a contradiction. Note:
33 Logical equivalence laws Commutative laws: p ∧ q = q ∧ p; p ∨ q = q ∨ p Associative laws: (p ∧ q) ∧ r = p ∧ (q ∧ r), (p ∨ q) ∨ r = p ∨ (q ∨ r) Distributive laws: p ∧ (q ∨ r) = (p ∧ q) ∨ (p ∧ r) p ∨ (q ∧ r) = (p ∨ q) ∧ (p ∨ r) Identity, universal bound, idempotent, and absorption laws: p ∧ t = p, p ∨ c = p p ∨ t = t, p ∧ c = c p ∧ p = p, p ∨ p = p p ∨ (p ∧ q) = p, p ∧ (p ∨ q) = p De Morgan’s laws: ~(p ∧ q) = ~p ∨ ~q, ~(p ∨ q) = ~p ∧ ~q
34 Show that the following are logically equivalent: (r V p) ^ ( ( ~r V (p^q) ) ^ (r V q) ) and p ^ q ( ~r V (p^q) ) ^ (r V q) =( (~rVp) ^ (~rVq) )^(rVq) (DL2) = (~rVp) ^( (~rVq) )^(rVq)) (AL1) = (~rVp) ^( (~r^r)Vq) (DL1) = (~rVp) ^(cVq) (definition of c) = (~rVp) ^q (ID2) so (r V p) ^ ( ( ~r V (p^q) ) ^ (r V q) ) =(r V p) ^ (~rVp) ^q =((r^~r)Vp)^q=(cVp)^q=p^q
40 Exercise 1: Complex deduction • Premises: – If my glasses are on the kitchen table, then I saw them at breakfast I was reading the newspaper in the living room or I was reading the newspaper in the kitchen – If I was reading the newspaper in the living room, then my glasses are on the coffee table – I did not see my glasses at breakfast – If I was reading my book in bed, then my glasses are on the bed table – If I was reading the newspaper in the kitchen, then my glasses are on the kitchen table • Where are the glasses?
41 Deduce the following using truth tables or deduction rules
42 a. If it walks like a duck and it talks like a duck, then it is a duck NAME_______________ NAME_____________Write each of the following three statements in the symbolic form and determine which pairs are logically equivalenta. If it walks like a duck and it talks like a duck, then it is a duckb. Either it does not walk like a duck or it does not talk like a duck, or it is a duckc. If it does not walk like a duck and it does not talk like a duck, then it is not a duck
43 Walks like duckTalks like duckIs duckWTDFWalks like duckTalks like duckIs duckWTDF
44 Zebra puzzleThere are 5 houses each with a different color. Their owners, each with a unique heritage, drinks a certain type of beverage, smokes a certain brand of cigarette, and keeps a certain variety of pet. None of the owners have the same variety of pet, smoke the same brand of cigarette or drink the same beverage. Clues: The Brit lives in the red house. The Swede keeps dogs as pets.The Dane drinks tea. Looking from in front, the green house is just to the left of the white house. The green house's owner drinks coffee. The person who smokes Pall Malls raises birds. The owner of the yellow house smokes Dunhill. The man living in the center house drinks milk. The Norwegian lives in the leftmost house. The man who smokes Blends lives next to the one who keeps cats. The man who keeps a horse lives next to the man who smokes Dunhill. The owner who smokes Bluemasters also drinks beer. The German smokes Prince. The Norwegian lives next to the blue house.The man who smokes Blends has a neighbor who drinks water. Who owns the pet fish?
46 Five women bought five different types of flowers for different reasons on different days. Names: Julia, Amy, Bethany, Rachel, and Kristen Flowers: Roses, Daisies, Lilies, Tulips, and Carnations Colors: Purple, Yellow, Pink, White, and Peach Places or Occasions: Backyard, Park, Office, Wedding, and Birthday Days: Monday, Tuesday, Wednesday, Thursday, and Friday The flowers were purchased in the following order: tulips, the flowers for the office, the purple flowers, the roses for the park, and the white flowers bought by Julia. 2. Bethany loves flowers but is allergic, so she would never have them indoors. 3. It rained on Wednesday and Friday, because of this, the wedding and birthday party had to be moved indoors. 4. Amy bought her flowers after Rachel, but before Kristen. 5. Rachel needed something more to add to her office, so she chose peach flowers to match her curtains. 6. On Wednesday the only purple flowers available at the flower shop were daisies. 7. The pink flowers were bought after the carnations, but before the lilies. 8. The flowers for the birthday were bought after the flowers for the office, but before the flowers for the wedding.
47 Assassin is a popular game on college campuses Assassin is a popular game on college campuses. The game consists of several players trying to eliminate the others by means of squirting them with water pistols in order to be the last survivor. Once hit, the player is out of the game. Game play is fair play at all times and all locations, and tends to last several days depending on the number of participants and their stealth. At Troyhill University, 5 students participated in a game that only lasted four days. Can you determine each player's first name, their color, their assassin alias, how they were eliminated, and their major? Names: Liam, Anabel, Bella, Oliver, Ethan Colors: Red, Green, Blue, Purple, Black Alias: Captain Dawn, Night Stalker, Dark Elf, McStealth, Billy Capture: Caught at weekly study group, Caught helping friend with car trouble, Ambushed during sleep, Caught on the way to class, Winner Major: Economics, Biology, Art History, Sociology, Psychology MONDAY: Liam, the girl named Captain Dawn, and the person in purple avoided any action that day. The psychology major was able to easily catch Ethan because she already had a study group meeting with him that day. Since it was a weekly engagement, he didn't suspect a thing. Goodbye red player. TUESDAY: Everyone tried to get in on the action today. The girl masquerading as the Dark Elf (who was wearing either black or red) and the sociology major lived to see another day. The purple player was able to catch the obliging yet naive green player by calling her and pretending he had car trouble. WEDNESDAY: The biology major (who was still "alive") was surprised to hear that the Psychology major, who wasn't Anabel the art history major, ambushed Night Stalker as he slept in his dorm. THURSDAY: The black player was declared the victor after luckily spotting "Billy" on his way to Mammalian Physiology, a class required by his major.