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Lecture 2, MATH 210G.03, Spring 2016: Symbolic Logic
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Silly starter Clicker question
What does the word Proof mean to you
A) Establishing a fact with complete certainty
B) Establishing a fact beyond reasonable doubt
C) What happens to dough when you add yeast
D) None of the above
Silly starter Clicker question
The picture is an example of which fallacy?
A) Straw man
B Slippery slope
C) Argumentum ad hominum
D) Ignoratio Elenchi
E) Circular reasoning
Lecture 2, MATH 210G.03, Spring 2016:Symbolic Logic
Law of the excluded middle• For every proposition, either the proposition is true or its
negation is true
• Either “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…”
A use of the excluded middle• There exist positive, irrational numbers a and b such that
is rational.
• Proof: is irrational (believe me next week)
• If is rational then we are done.
• If is irrational then
• Does the trick.
• Proof is nonconstructive. It does not tell us whether is rational for particular irrational values of a and b.
Problems with the excluded middleMany statements have an element of uncertainty and are subject to an error of equivocation or false dilemma
Four quarters are a dollar
Either it is raining or it is not raining
Either Sophia Vergara is blonde or she is not blonde.
Either NMSU has a better basketball team or UTEP has a better basketball team…
• These examples all suffer from imprecise language.
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.”
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.
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.
Logic, n. The art of thinking and reasoning in strict accordance with the limitations and incapacities of the human misunderstanding. The basic of logic is the syllogism, consisting of a major and a minor premise and a conclusion - thus:
Major Premise: Sixty men can do a piece of work sixty times as quickly as one man.Minor Premise: One man can dig a post-hole in sixty seconds; Therefore-Conclusion: Sixty men can dig a post-hole in one second.
This may be called syllogism arithmetical, in which, by combining logic and mathematics, we obtain a double certainty and are twice blessed.”
ou
Modus ponendo ponens("the way that affirms by affirming”)
• If P, then Q. P. Thus, Q
• If Socrates is a man then Socrates is mortal
• Socrates is a man
• Therefore, Socrates is mortal
Logic and causality
Causality• Plato is a dog.
• all dogs are green
• Plato is green.
Logic and symbol of propositional calculus• P, Q, R etc: propositional variables
• Substitute for statements, e.g., P: Plato is a dog, Q: Plato is Green
• Logical connectives:
• Proposition: If Plato is a dog then Plato is green:
Truth tablesP Q (conjunction)T T T
T F F
F T FF F F
P Q (disjunction)
T T T
T F T
F T T
F F F
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)
Truth table for implication
P Q (implication)
T T T
T F F
F T T
F F T
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.
Truth table for modus ponens• No matter what truth values are assigned to
the statements p and q, the statement is true
Exercise: complete the truth table for modus tollens
P Q
T T T
T F F
F T T
F F T
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.
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:P Q
T T T F
T F F F
F T T F
F F T T
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 casesP Q
T T T F T
T F F F T
F T T F T
F F T T T
Clicker questions:p q
T T T F
T F F F
F T T F
F F T T
• First 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)
Logical equivalence• Two formulas are logically equivalent if they have the
same truth values once values are assigned to the atoms.
• Ex: is equivalent is equivalent to
•
• How to check logical equivalence: verify that the statements always have the same values
Exercise: verify that the statements , and are
logically equivalentp q
T T T T T
T F F
F T T
F F T
Exercise: Verify using truth tables that the following DeMorgan’s laws are logically equivalent
Exercise: Verify using truth tables the following absorption rules and the conditional rules
Match the following logical equivalencies with the corresponding rules of inference
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:
Logical equivalence lawsCommutative 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∧ ∨ ∨ ∧
Show that the following are logically equivalent:(r V p) ^ ( ( ~r V (p^q) ) ^ (r V q) ) and p ^ q
Boole (1815-1864) and DeMorgan (1806–1871)
• De Morgan’s laws:
• not (P and Q) = (not P) or (not Q)
• not (P or Q) = (not P) and (not Q)
• Boolean algebra
Other logical deduction rules
Exercises
Fill in the following truth table
Fill in the following truth table
Fill in the following truth table
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?
Deduce the following using truth tables or deduction rules
Write each of the following three statements in the symbolic form and determine which pairsare logically equivalent
a. If it walks like a duck and it talks like a duck, then it is a duck
b. Either it does not walk like a duck or it does not talk like a duck, or it is a duck
c. If it does not walk like a duck and it does not talk like a duck, then it is not a duck
NAME_______________NAME_____________
NAME_______________NAME_____________
Walks like duck Talks like duck Is duckW Ta DT T TT T FT F TT F FF T TF T FF F TF F F
Walks like duck Talks like duck Is duck
W Ta DT T TT T FT F TT F FF T TF T FF F TF F F
Exercise: Are the statements and are logically
equivalentp q
T T T T
T F F
F T T
F F T
Spatial logic puzzlesSpatial logic puzzles involve deducing certain attributes attached to specific entities by a process of elimination that takes spatial or temporal information into account.
Zebra puzzle
There 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?
Five women bought five different types of flowers for different reasons on different days.Names: Julia, Amy, Bethany, Rachel, and KristenFlowers: Roses, Daisies, Lilies, Tulips, and CarnationsColors: Purple, Yellow, Pink, White, and PeachPlaces or Occasions: Backyard, Park, Office, Wedding, and BirthdayDays: Monday, Tuesday, Wednesday, Thursday, and Friday
1. 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.
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, EthanColors: Red, Green, Blue, Purple, BlackAlias: Captain Dawn, Night Stalker, Dark Elf, McStealth, BillyCapture: Caught at weekly study group, Caught helping friend with car trouble, Ambushed during sleep, Caught on the way to class, WinnerMajor: Economics, Biology, Art History, Sociology, PsychologyMONDAY: 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.
