The Paradox Paradox

One of the greatest paradoxes I have come across is the question of the existence of paradoxes. Think of it this way:
  • A paradox is a statement that is both true and false.
  • No statement can be both true and false.
So which is it?

Dictionaries neatly sidestep this issue by defining paradoxes to be statements that are seemingly or apparently self-contradictory. This would imply that we will discover, upon closer examination, that a paradox will not be determined to be self-contradictory. 

Naturally, if that were true, then the statement would no longer appear or seem to be self-contradictory. It would not be a paradox. 

If paradoxes cannot exist by their own definition, then can it be said that they exist at all?

The Paradox of the Heap and the Theseus' Ship paradox are two versions of the same fundamental paradox. The idea in both cases is that there is essentially an arbitrary difference between a collective concept (such as a heap, a ship, or a forest) and the individual components that define it (such as a grain of sand, the plank of a ship, or a single tree). 

There is no universal number of sand grains that make up a heap, nor a critical number of "original planks" that define a single ship. Instead, we choose to perceive something as a heap, or a ship, or a forest more or less on an individual level.

We also call them "mashed potatoes," even if the total number of potatoes used to make the dish is one.

Paradoxes always come down to either a problem of definition or a problem of perception. Either the language used in our definitions is not precise or accurate enough to encapsulate what we're trying to describe, or we are attempting to describe something universally, when its definition is entirely arbitrary. 

Mashed potatoes are a problem of definition because we have defined them to be plural no matter what the reality of the situation is. Forests are a problem of perception because there exists no specific number of trees such that that number of trees makes up a forest, but one fewer tree does not.

The interesting thing about paradoxes is that they are both a problem of definition and of perception. The definition can never be true, and their existence is in fact only a matter of perception.