\section{Introduction}

The \emph{meaning} of a first-order formula $\phi$ in a structure $\A$ is just the set of valuations that make the formula true in $\A$. That is,
\[
\phi^\A = \setof{\vec a \in \^NA}{\A \models \phi[\vec a]},
\]
where $A$ is the universe of $\A$, and $N$ is the number of variables in $\phi$. Given a structure $\A$ and any two first-order formulas $\phi$ and $\psi$, 
\[
\phi^\A = \psi^\A 
\]
if and only if \(\A \models \phi \iff \psi\). Thus first-order logic is able to express the concept of ``if and only if.''

Independence-friendly logic (IF logic) is a conservative extension of first-order logic that has the same expressive power as existential second-order logic \cite{Hintikka:1989, Hintikka:1996}. In IF logic the truth of a sentence is defined via a game between two players, \abelard\ ($\forall$) and \eloise\ ($\exists$). The additional expressivity is obtained by modifying the quantifiers and connectives of an ordinary first-order sentence in order to restrict the information available to the existential player, \eloise, in the associated semantic game.

In IF logic, only the information available to \eloise\ is restricted, which means existential quantifiers are not dual to universal quantifiers. To compensate, negation symbols are only allowed before atomic formulas. Generalized independence-friendly logic (IFG logic) is a variant of independence-friendly logic in which the information available to both players can be restricted, making existential quantifiers dual to universal quantifiers and allowing any formula to be negated \cite{Dechesne:2005}.

Since there are IFG-sentences that are neither true nor false, it is unclear whether IFG logic can express the concept of ``if and only if.'' For instance, one can define \(\phi \hiff{J} \psi\) as an abbreviation for the formula
\[
({\hneg\phi} \hor{J} \psi) \hand{J} (\phi \hor{J} {\hneg\psi})
\]
(the subscripts indicate what information is unavailable to the players at each move), but does this formula assert that $\phi$ and $\psi$ are logically equivalent? Is \(\phi \hiff{J} \psi\) true in a structure exactly when $\phi$ and $\psi$ have the same meaning in that structure? If not, is there some other syntactical combination of $\phi$ and $\psi$ that is true exactly when $\phi$ and $\psi$ have the same meaning? The answer to all of these questions is no.
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