THOG problem


The THOG problem is one of cognitive psychologist Peter Wason's logic puzzles, constructed to show some of the weaknesses in human thinking.
You are shown four symbols
  1. a black square
  2. a white square
  3. a black circle
  4. a white circle
and told by the experimenter "I have picked one colour and one shape definitely a THOG, b) undecidable, or c) definitely not a THOG?"
Presented in this form, the task is quite difficult, because much information must be held in working memory at the same time.

Solution 1 (Analysis of symbol properties)

The chosen symbol of the experimenter is not a black square, since it shares both properties with the black square, and so the black square would not be a THOG.
The chosen symbol of the experimenter is not a white circle, since it shares 0 properties with the black square, and so the black square would not be a THOG.
So the experimenter could have chosen either a black circle or a white square. Since the colours and shapes of these two possibilities are opposites, it means:
  1. Any symbol that shares exactly one property with one of the possibilities must necessarily share exactly one property with the other possibility.
  2. Any symbol that shares both properties with one of the possibilities must necessarily share 0 properties with the other possibility.

Solution 2 (Case Analysis of the Experimenter's choice)

This solution analyzes the four possible hidden choices of the experimenter.
Experimenter's hidden choiceRelation to black squarerelation to white squarerelation to black circlerelation to white circleNote
Black squarenot a THOGTHOGTHOGnot a THOGImpossible choice, since the black square is not a THOG
White squareTHOGnot a THOGnot a THOGTHOG
Black circleTHOGnot a THOGnot a THOGTHOG
White circlenot a THOGTHOGTHOGnot a THOGImpossible choice, since the black square is not a THOG

From the table we see that the only valid choice for the experimenter are the white square or the black circle. In both of those rows, all of the information is the same.
So the conclusions are:
  1. The black square is definitely a THOG
  2. The white square is definitely not a THOG
  3. The black circle is definitely not a THOG
  4. The white circle is definitely a THOG