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Remarks on Mathematical Logic:
Paradoxes and Puzzles
AAG
1.11.01
Selmer Bringsjord
The Minds & Machines Laboratory
Department of Philosophy, Psychology & Cognitive Science
Department of Computer Science
RPI Troy NY USA
[email protected]
www.rpi.edu/~brings
What is Logic?
• The science of reasoning.
• The only invincible subject there is.
• The basis for all things intellectual, not only the basis of
mathematics, but also of engineering to computer
science to philosophy.
• The most challenging subject there is.
• A key to riches.
• The key to divining the meaning of life (and other such
big questions)
• One of two fundamental approaches to studying minds,
and replicating/simulating minds in machines…
• …
• The thing many detectives have mastered – have you (as
a New Yorker)?…
Encapsulating the
Minds & Machines Lab & Program
Number Sense
5
1
3
11
7
9
4
Vicky’s secret number is inside the triangle. It is
outside the square. It is greater than 7 but less than 10.
Vicky’s secret number is… 9
This sentence is false.
The sentence below is false.
The sentence above is true.
We can handle “Liar-based” puzzles:
You travel through space and arrive on the planet
Trekia, upon which reside aliens each of which are
in one of two cultures: the Larpals, who always
lie, and the Tarsals who always tell the truth. You
ask three aliens which culture they belong to. The
first alien murmurs something you can’t make out.
The second alien says: “It said it was a Larpal.”
The third alien says to the second alien: “You are
a liar!” To which culture does the third alien
belong?
Proof:
A3 is either a Larpal or a Tarsal. If a Larpal,
then A2 is a Tarsal, which implies that A1 said:
“A1 is a Larpal,” i.e., “A1 is a liar.” But then if
A1 is a liar, since on this assumption A1 said it is
a liar, A1 is truth-teller. On the other hand, if A1
is a truth-teller, then since on this assumption A1
said it is a liar, A1 is a liar. In other words: A1 is
a truth-teller if and only A1 is a liar – which is a
contradiction! Ergo, A3 is not a Larpal and the
other possibility must be right: A3 is a Tarsal.
Simple Selection Task
E
T
4
7
Suppose I claim that the following rule is true.
If a card has a vowel on one side, it has an even number on
the other side.
Which card or cards should you turn over in order to try to decide
Whether the rule is true or false?
NYS 1
Given the statements
a  b
b
ca
which one of the following statements must also be true?
c
b
c
h
a
none of the above
NYS 2
Which one of the following statements is logically equivalent to the
following statement: “If you are not part of the solution, then you
are part of the problem.”
If you are part of the solution, then you are not part of the problem.
If you are not part of the problem, then you are part of the solution.
If you are part of the problem, then you are not part of the solution.
If you are not part of the problem, then you are not part of the
solution.
“NYS 3”
Given the statements
c
ca
a  b
bd
(d  e)
which one of the following statements must also be true?
c
e
h
a
all of the above
J-L 1
Suppose that the following premise is true:
If there is a king in the hand, then there is an ace
in the hand, or else if there isn’t a king in the hand,
then there is an ace.
What can you infer from this premise?
NO!
There is an ace in the hand. NO!
In fact, what you can infer is that there isn’t an ace in the hand!
The Dreadsbury Mansion Mystery
Someone who lives in Dreadsbury Mansion killed Aunt Agatha.
Agatha, the butler, and Charles live in Dreadsbury Mansion, and
are the only people who live therein. A killer always hates his
victim, and is never richer than his victim. Charles hates no one
that Aunt Agatha hates. Agatha hates everyone except the butler.
The butler hates everyone not richer than Aunt Agatha. The
Email
your answer
with supporting
butler hates
everyone
Agatha hates.
No one hatesproof…
everyone.
Agatha is not the butler.
Now, given the above clues, there is a bit of disagreement
between three (competent?) Norwegian detectives. Inspector
Bjorn is sure that Charles didn’t do it. Is he right? Inspector
Reidar is sure that it was a suicide. Is he right? Inspector Olaf is
sure that the butler, despite conventional wisdom, is innocent. Is
he right?
Do my stuents at RPI really learn
logic through paradoxes, and
what kind of system would they
use?
Barber Paradox in Hyperproof
• In a certain village in England, there was a barber
who claimed to shave all and only those men who
did not shave themselves.
• Is this possible?
• What if the barber is a man?
• Let’s use
–
–
–
–
dodecs to represent men
tets to represent women
and Likes(x,y) to represent Shaves(x,y)
b to represent the barber
• Now let’s look at a model of our village…
Last Brain Teaser
Is the following assertion true or false? Prove that you are correct.
There exists something which is such that if it’s a bird, then
everything is a bird.
Email your answer with supporting proof w/i
-- for a free dinner at River Street Cafe