# Logic: normative model

Alice is to the left of Betty.
Betty is to the left of Carol.
So, Alice is to the left of Carol.

An X can be a Y.
A Y can be a Z.
Therefore, an X can be a Z.

Are these valid?

# Some logic problems

All As are Bs. All Bs are Cs.

Some As are Bs. All Bs are Cs.

No As are Bs. All Bs are Cs.

Some As are not Bs. All Bs are Cs.

All As are Bs. Some Bs are not Cs.

# Descriptive model: Johnson-Laird's theory

```All As are Bs    Some As are Bs
All Bs are Cs    All Bs are Cs

a = b = c       a = b = c
(b)= c      (a) (b)= c
(c)             (c)

Conclusion:
All As are Cs    Some As are Cs
```

# More mental models

```No As are Bs     Some As are not Bs    All As are Bs
All Bs are Cs      All Bs are Cs    Some Bs are not Cs

a                 a                 a = b
-----------       -----------       ------------
b = c        (a)  b = c        (a) = b  = c
(c)               (c)            (b) = c

a
a      (c)        a      (c)       (b)
-----------       -----------       ------------
b = c        (a)  b = c        (a) = b    c
a   = b    c
a      (c)
-----------
b = c

Some Cs are not As  No conclusion   No conclusion
```

# A hard one (from Johnson-Laird)

Suppose that only one of the following assertions is true about a specific hand of cards:

1. There is a king in the hand or there is an ace in the hand, or both.

2. There is a queen in the hand or there is an ace in the hand, or both.

Which is more likely to be in the hand, the king or the ace?

# The solution

There is a king in the hand or there is an ace in the hand, or both.
 k a k a

There is a queen in the hand or there is an ace in the hand, or both.
 q a q a

~k and ~a
OR
~q and ~a

# Wason's four-card problem (modified)

 K
 L
 2
 4

Rule to test:
If K, then 2.
Which cards must you turn over?

# Multi-card task (Beattie and Baron)

 A1 A2 A3 B1 B2 B3 C1 C2 C3

Test: If A, then 2.
Which cards would you ask about to find out whether this rules holds for all the cards?

# A response to the multi-card task.

(S is asked whether an A I and an A3 would be informative in the unnegated multi-card task.)
S: No. Because... I was looking for A2 only.
E: But your goal is to find out ii the rule is true or false. If there was A I or A3 in there, would that be informative...?
S: No.
E: If the agent came back and said there was an A 1 and an A3 in the pouch, do you think that the rule would be true?
S: Yes.
E: You think it would?
S: Yes.
E: Even though there are As that don't have 2s on them.
S: Yes. Because.. the rule says if there is an A then there is also a 2. It doesn't say there couldn't be an Al or an A3.

# Logical attitudes

An engineer, a physicist and a mathematician were driving in the country when they came upon pasture after pasture with only black cows in them. The engineer said, "There seem to be only cows colored black in this area." The physicist said, "That doesn't quite follow. The only cows we have seen are black." Then the mathematician said, "I don't think you are right there. The only cows we have seen are black on one side."
(Submitted by Ed Howland, ehowland@cyber.net)

Theorem:
A cat has nine tails.

Proof:
No cat has eight tails.
A cat has one tail more than no cat.
Therefore, a cat has nine tails.