Paradoxes of Logic

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One of the most important aspects of the $\lambda \Pi$ logical framework is that it avoids any kind of paradox. For background on these paradoxes, we discuss a paradox or logic (the liar paradox), set theory (Rusell’s paradox), proof theory (Godel’s Paradox), and type theory (Girard’s paradox).

Paradoxes of Classical Logic

The liar paradox is well known. While on its surface unintersting, it starts on the important path of understanding why logic must be restricted. For those unaware, the liar paradox is a logical paradox of the unrestricted classical logic, and is as follows:

This statement is true if it is false.

In other words, we have $(\Delta \to \bot) \to \Delta$. Then, we note that if we have $\Delta$, then we must have $\Delta \to \bot$, and if there is none, then per elimanation we also have $\Delta \to \bot$. Then, we again perform case anaylis on $\Delta$ to that that if $\Delta$ is true, then $\bot$, if it is false, then $\Delta$ and then $\bot$. Therefore, we have $\bot$.

Obviously, this is a little bit contrived. However, there are a larger class of problems it fits into, those of self reference.

Paradoxes of Set Theory

Rusell’s paradox is interesting, as it deals with a slightly differetn, but equivalent, problem that arises from unrestricted set theory. The problem goes as follows: