A LOGICAL ILLUSION - part 3

There is NO King in the hand !!

Here's why:

Consider the following statements about a hand of cards:

1. If there is a Jack in the hand, then there is a King in the hand, or else if there's no Jack in the hand, then there is a King in the hand.

2. There is a Jack in the hand.

These 2 sentences can be represented in propositional logic as follows:

(Where "J" = There's a Jack in the hand, and "K" = There's a King in the hand, and "->" = material conditional, "v" = inclusive disjunction, "&" = conjunction, and "~" = negation.)

1. [(J -> K) v (~J -> K)] & ~[(J -> K) & (~J -> K)]

   (i.e., Either: if there's a Jack, then there's a King,
               or: If there's no Jack, then there's a King,
                     but not both.

2. J

JK
~JK

But if you do a truth-table analysis of (1) and (2) (just assign "true" or "false" to J and K--there are 4 possible combinations--then compute the truth values of (1) and (2)), you will see that:

• If both J and K are true (which corresponds to the first mental model), then the "not both" clause of (1) is false; i.e., the disjunction is really an inclusive one, not an exclusive one. Consequently, (1) is false in this case! But that means that the conjunction of (1) and (2) is also false. So we can't conclude anything about K.

• It is possible for J to be true but K to be false. But this case doesn't get represented in the mental model (which is why the mental-model theory predicts that most people will think that there is a King!).

• If J is false and K is true (which corresponds to the second mental model), we get the same analysis as for both J and K being true. In other words, this plus the first case, which "match" the mental-model analysis, are in fact impossible situations.

• It is possible for both J and K to be false. Again, this is not shown in the mental-models representation, which is why it predicts that most people will think there can be a King in the hand.

Consequently, only 2 situations are really possible, and in both, K is false. Moreover, neither of the possible situations are represented in the mental-model theory.

For more information, see:

1. Laird-Johnson, P.N.; Girotto, Vittorio; & Legrenzi, Paolo (1998), "Mental Models: A Gentle Guide for Outsiders"

2. Selmer Bringsjord, Ron Noel, Elizabeth Bringsjord (1998), "In Defense of Logical Minds"
   • A nice overview on the psychology of reasoning.

3. Johnson-Laird, Philip N. (1983), Mental Models: Towards a Cognitive Science of Language, Inference, and Consciousness (Cambridge, MA: Harvard University Press).

4. Rips, Lance J. (1994), The Psychology of Proof: Deductive Reasoning in Human Thinking (Cambridge, MA: MIT Press).

5. Johnson-Laird, Philip N. (1997), "Rules and Illusions: A Critical Study of Rips's The Psychology of Proof", Minds and Machines 7(3): 387-407.

6. Rips, Lance J. (1997), "Goals for a Theory of Deduction: Reply to Johnson-Laird", Minds and Machines 7(3): 409-424.

7. Johnson-Laird, Philip N. (1997), "An End to the Controversy? A Reply to Rips", Minds and Machines 7(3): 425-432.

8. Fetzer, James H. (1999), "Deduction and Mental Models", Minds and Machines 9(1): 105-110.

9. Johnson-Laird, Philip N., & Byrne, Ruth M.J. (1999), "Models Rule, OK? A Reply to Fetzer", Minds and Machines 9(1): 111-118.

10. Fetzer, James H. (1999), "Mental Models: Reasoning without Rules", Minds and Machines 9(1): 119-126.