Immanuel Kant (1724-1804)
There is one anecdote of Kant’s life which suggests the possibility of a bizzarre case of self-recursion or of a paradox. The story is the following: later in his life, Kant realised that his servant (Lampe), that he so much trusted, had actually been robbing him steadily throughout the years. Hence Kant dismissed Lampe and wrote in his diary:
“Remember to forget Lampe”
If we rephrase the sentence in the following way, we might find ourselves trapped in a paradox:
“Once you have forgotten Lampe, scratch this sentence”
Let’s immagine an hypothetical person (Mr. Kant) writing this sentence on his notebook. Would Mr. Kant be able to fulfil the task presented in the question?
The answer is no. Indeed, I will show that a paradox will arise- a paradox we shall call ‘Lampe’s paradox’.
There are two cases here: case 1 where Mr. Kant forgets Lampe (this case also has a sub-case), and case 2 where Mr. Kant fails to forget Lampe.
CASE 1): if Mr. Kant succeeds in forgetting Lampe then he would not be able to scratch the sentence because surely he must have completely forgotten about Lampe’s existence, hence he will not go back to the notebook to scratch the sentence.
SUB-CASE 1.1): if by any circumstance Mr. Kant is somehow prompted to read his notebook because he remembers that there was a task to fulfil (despite having forgotten that the task is related to Lampe) then he would not be able to scratch the sentence because by reading the very sentence twice he would be reminded again of the existence of Lampe!
CASE 2: if Mr. Kant did not forget Lampe then he would not ne able to scratch the sentence.
There is a possible variation of this paradox, whose ambiguity depends more on the vagueness of ordinary language than on a logical inconsistency.
So let’s turn Kant’s original notation into an order:
“You must remember to forget Lampe”
Will Mr. Kant be able to fulfil the order? This paradox will show that you cannot fulfil the order and forget Lampe at the same time: strangely enough, the two conditions are mutually exclusive!
These are three possible scenarios once Mr. Kant has read the notation:
CASE 1: if Mr. Kant forgets Lampe and later remembers that he has forgotten Lampe then he has fulfilled the order, but that would imply that he still remembers Lampe!
CASE 2: if Mr. Kant forgets Lampe and later he does not remember that he has forgotten Lampe, then he will have forgotten Lampe but he will not have fulfilled the order because he cannot remember that he has forgotten Lampe!
CASE 3: obviously if Mr. Kant has not forgotten Lampe at all, then he has neither fulfilled his order nor has he forgotten Lampe.