Hintikka set

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Definition

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

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