Transcript Law of the excluded middle

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:
•
If
is rational then we are done.
•
If
is irrational then
•
Does the trick.
•
is irrational (believe me next week)
Proof is nonconstructive. It does not tell us whether
rational for particular irrational values of a and b.
is
Problems with the excluded middle
Many 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
Major premise: All mortals die.
three parts: the major premise,
the minor premise and the
Minor premise: All men are
conclusion.
mortals.
Conclusion: All men die.
Major premise: All men are
mortal.
Minor premise: Socrates is a
man.
Conclusion: Socrates is mortal.
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.
ou
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; ThereforeConclusion: 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.”
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.
Universe of discourse
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 tables
P
Q
(conjunction)
T
T
T
T
F
F
F
T
F
F
F
F
P
Q
T
T
T
T
F
T
F
T
T
F
F
F
(disjunction)
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:
•
Clicker: True (A) or False (B)
true or false?
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 cases
P
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 equivalent
p
q
T
T
T
T
F
F
F
T
T
F
F
T
T
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.
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
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
NAME_______________
NAME_______________
NAME_____________
NAME_____________
Write each of the following three statements in the symbolic form and
determine which pairs
are 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
Walks like duck
Talks like duck
Is duck
W
Ta
D
T
T
T
T
T
F
T
F
T
T
F
F
F
T
T
F
T
F
F
F
T
F
F
F
Walks like duck
Talks like duck
Is duck
W
Ta
D
T
T
T
T
T
F
T
F
T
T
F
F
F
T
T
F
T
F
F
F
T
F
F
F
Exercise: Are the statements
and
are logically
equivalent
p
q
T
T
T
T
F
F
F
T
T
F
F
T
T
Spatial logic puzzles
Spatial 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 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
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, 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.
More logic grid puzzles