A paradox is a proposition that seems contradictory. The following statement is a short version of the lier’s paradox:
“This statement is false”
If the statement is the truth (True), then the sentence will have us accept that the statement is False. If the sentence is False, it follows that the statement is True. And so we go round and round in a logic circle.
Paradoxes come in many forms. Bertrand Russel created a paradox in set theory (Mathematics). The paradox utilizes the concept of “sets that do not contain themselves”. I will try to explain what such sets look like:
- A set of books in a household does not contain itself, as a set of books is not a book. A set (or collection) of stamps does not contain itself as the stamp collection is not a stamp in itself.
- The set of all “ideas” could be said to contain itself, as “a collection of all ideas” could be thought to be an idea in itself.
So “sets not containing themselves” are normal everyday sets. The paradox is constructed by defining a set, namely the set of all “sets not containing themselves”.
Does the set of all “sets not containing themselves” contain itself? Lets examine that:
- If the set of all “sets not containing themselves” is in its own set, then it “contains itself”, and so by the definition it cannot be in the set, which is a contradiction.
- If the set of all “sets not containing themselves” is NOT in its own set, then it “does not contain itself”, and then by definition it has to be in the set, which is a contradiction.
Hence the paradox.
So how are paradoxes constructed?
Truths are either derived, or assumed (a truth is either an axiom, or it is proved). If you assume and prove something at the same time, you invite the possibility of inconsistency. Paradoxes contain (a set of) rules or statements that refer to themselves , or stated differently, paradoxes are self-referential. The Liers’ paradox and also the (set/mathematical) definition uses itself whilst defining it’s own characteristic. The above paradoxes use negation to discredit the self-reference (I am not myself).
The moral of the story is to be careful with recursive logic (For example, do not get a person to vouch for him/herself).
On an aside, Gödel’s incompleteness theorem is proven using similar self-reference as the Lier’s paradox!