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With a truth table its easy to see that the two formulae $A\land B \to C$ and $A \to B \to C$ are not equivalent, for example, if $A = B = C = 0$, than the first evaluates to $1$ and the second to $0$ (because $A \to B$ is truth, and then $(A\to B) \to C$) is false).

But here

How do I memorize axioms of a Hilbert system?

it is referred to this transformation as currying, and there

on page 9 it is stated that

  • Curry and Uncurry are proofs of $$\forall P,Q,R. (P \land Q) \to R \leftrightarrow (P \to Q \to R)$$

So i am confused, when are these expressions equivalent, and if not how can I use them for "uncurrying"?

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Right arrows like $\to$ or $\implies$, etc., are usually right-associative, that is $A \to B \to C$ means $A \to (B \to C)$, not $(A \to B) \to C$. – dtldarek Jan 25 '13 at 13:16
Huh? C is not equivalent to (A->B). – Doug Spoonwood Jan 25 '13 at 13:23
up vote 7 down vote accepted

The devils is hidden in details. Be careful with parenthesis (and conventions about them).

Indeed, it's true that

  • Curry and Uncurry are proofs of $$\forall P,Q,R. (P \land Q) \to R \leftrightarrow (P \to (Q \to R))$$
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I think $P \to Q \to R$ means $P \to (Q \to R)$ but not $(P \to Q) \to R$.

$(A \land B) \to C \leftrightarrow A\to (B\to C)$ is true.

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