What's meaning of the inverted Greek letter iota “ι” in Principia Mathematica I* 14?

It is not an assignment. $\phi$ is a predicate. $(\iota x)\phi x$ is the one and only object $x$ for which the predicate $\phi$ is true.

For example, if $\phi x$ means "$x$ is the current monarch of the United Kingdom" then $(\iota x)\phi x$ denotes Queen Elizabeth. As another example, $(\iota x). x\ge 0\mathrel . x^2=3$ denotes the number $\sqrt 3$.

This is explained in the front matter of the Principia, including what the meaning is when there is no unique $x$ (either because there is none or because there is more than one). If the denoted predicate is not unique, then the truth value of the formula is undefined.


This is the descriptor operator. $(\iota x)\varphi x $ is the unique $x $ with the property specified by $\varphi $ (should it be the case that, indeed, there is precisely one such $x $). The Wikipedia entry on Principia has a very decent explanation of their notation.

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