Hintikka set

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Definition

Let $\mathscr{L}$ be a first order language and suppose $H$ is the Herbrand domain of $\mathscr{L}$. We say that $\Omega$ is the Hintikka set with respect to $H$ if $\Omega$ is a set of formulas that satisfy the following $7$ conditions:

1. If $A$ is an atomic formula then either $A\in\Omega$ or $\neg A\in\Omega$ but not both Caution:XOR is speculation on my part
   • With regards to the equality symbol, we require $t\doteq t\in\Omega$, where $t\in H$ Caution:Check, could be any term?
2. If $A\in\Omega$ then $\neg\neg A\in\Omega$
3. If ($A\in\Omega$ or $B\in\Omega$) then $A\vee B\in\Omega$, additionally:
   • If ($\neg A\in\Omega$ and $\neg B\in\Omega$) then $\neg(A\vee B)\in\Omega$
4. If ($A\in\Omega$ and $B\in\Omega$) then $A\wedge B\in\Omega$, additionally:
   • If ($\neg A\in\Omega$ or $\neg B\in\Omega$) then $\neg(A\wedge B)\in\Omega$
5. If ($\neg A\in\Omega$ or $B\in\Omega$) then $(A\rightarrow B)\in\Omega$, additionally:
   • If ($A\in\Omega$ and $\neg B\in\Omega$) the $\neg(A\rightarrow B)\in\Omega$
6. SAVING WORK