Johannes C. Mayer's Shortform

Johannes C. Mayer

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Any n-arity function can be simulated with an an (n+1)-arity predicate. Let a and b be constants. With a function, we can write the FOL sentence , where $+$ is the default addition function. We can write the same as $\forall x, (+_{P} (a, b, x) \to x > 1)$ where $+_{P}$ is now a predicate that returns true iff $a$ added to $b$ is $x$ .

The reverse is also possible: an n-ary relation can be represented as an n-ary function which maps instances of the relation to the object "true" and non-instances to the object "false".

So which is better?

Enter Oppenheimer & Zalta: Relations vs functions at the foundations of logic: type-theoretic considerations:

Though Frege was interested primarily in reducing mathematics to logic, he succeeded in reducing an important part of logic to mathematics by defining relations in terms of functions. By contrast, Whitehead & Russell reduced an important

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