Does leaving universal quantifiers up to context lead to ambiguity?

Question

An example of this notation would be extensionality: $$ \forall x \forall y [\forall z (z \in x \Leftrightarrow z \in y) \Rightarrow \forall w (x \in w \Leftrightarrow y \in w)].$$

If we write $$ (z \in x \Leftrightarrow z \in y )\Rightarrow (x \in w \Leftrightarrow y \in w ),$$ then does this make things ambiguous?

Answer 1

Yes, it does generate ambiguity. Let $E$ and $O$ be predicates for "is even" and "is odd," respectively, and consider the difference between $$\forall x(E(x)\rightarrow O(x))$$ and $$(\forall x(E(x)))\rightarrow (\forall x(O(x))).$$ In your approach each would be represented as "$E(x)\rightarrow O(x)$," but they're clearly different statements: in the usual context of the natural numbers, the first is obviously false and the second is obviously (and vacuously) true.

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You might be tempted to try to fix this issue by prohibiting variable reuse, but that doesn't help: consider $\forall x\forall y(E(x)\rightarrow O(y))$ versus $(\forall x (E(x))\rightarrow (\forall y(O(y)))$. Again, each would be represented in your approach as "$E(x)\rightarrow O(y)$," but the first is false and the second is vacuously true.

Answer 2

$ \forall x \forall y [\forall z (z \in x \Leftrightarrow z \in y) \Rightarrow \forall w (x \in w \Leftrightarrow y \in w)]\tag1$

This statement means $$ ∀w∀x∀y\color\red{\exists z}\;\Big( (z \in x \Leftrightarrow z \in y )\Rightarrow (x \in w \Leftrightarrow y \in w )\Big).$$

$ (z \in x \Leftrightarrow z \in y )\Rightarrow (x \in w \Leftrightarrow y \in w )\tag2$

This strongly suggests the statement $$∀w∀x∀y\color\red{∀z}\;\Big( (z \in x \Leftrightarrow z \in y )\Rightarrow (x \in w \Leftrightarrow y \in w )\Big).\tag3$$

Does leaving universal quantifiers up to context lead to ambiguity?

So, leaving the universal quantifiers in example $(1)$ "up to context" is actually misleading rather than ambiguous. (Similarly, in each of the two pairs in Noah's answer, omitting the universal quantifiers unambiguously suggests the first statement and not the second statement.)