Transcript Logic - Cognitive Science Department

How Do You Stack Up Against
Spock, Holmes, Wolf, Moriarity…?
Selmer Bringsjord
Professor of Logic, Philosophy, Cognitive Science, Computer Science
Department of Cognitive Science (Chair)
Department of Computer Science
Director, Rensselaer AI & Reasoning Laboratory
Rensselaer Polytechnic Institute
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, from 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).
• The better way to program computers (9.20 lunch).
• One of two fundamental approaches to studying minds,
and replicating/simulating minds in machines…
• The thing many creatures of fiction have mastered – have
you (as a New Yorker)?…
• …
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
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!
Problem Given Selmer on Train
Everyone loves anyone who loves someone.
Alvin loves Bill.
Can you infer that everyone loves Bill?
ANSWER:
JUSTIFICATION:
Problem Given Selmer on Train
Everyone loves anyone who loves someone.
Alvin loves Bill.
Can you infer that everyone loves Bill?
ANSWER: Yup.
JUSTIFICATION: See proof just given.
Harder Variants
Everyone loves anyone who loves someone.
Alvin loves Bill.
Can you infer that Cath loves Bill?
Can you infer that Cath loves Dave?
ANSWER:
JUSTIFICATION:
Harder Variants
Everyone loves anyone who loves someone.
Alvin loves Bill.
Can you infer that Cath loves Bill?
Can you infer that Cath loves Dave?
ANSWER: Yup
JUSTIFICATION: ??
Bringsjord 1
(1) The following three assertions are either all true or
all false:
If Billy helped, Doreen helped.
If Doreen helped, Frank did as well.
If Frank helped, so did Emma.
(2) The following assertion is definitely
true: Billy helped.
Can it be inferred from (1) and (2) that Emma helped?
YUP!
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
See Hyperproof...
butler hates everyone Agatha
hates. No one hates 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?
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.
Explorations...
“Show me the $” Problem
If one of the following assertions is true then so is the
other:
(1) There is a king in the hand if and only if there is an
ace in the hand.
(2) There is a king in the hand.
Which is more likely to be in the hand, if either: the king
or the ace?