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I have one or more second-order logic formulas that correct for some finity model. What operation can I perform on them, so that resault is also correct for some finity model?

Just conjunction is impossible, example: $x$ has finity model, $\lnot x$ has finity model, but $x \wedge \lnot x$ has not finity model. Are there any restriction?

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Out of the standard logical operations ($\lor$, $\land$, $\neg$, $\implies$), the only ones that meet your conditions are $\lor$ and $\implies$.

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  • $\begingroup$ Thank you. And how do you define this? Is there a rule? $\endgroup$ – VsSekorin May 29 '18 at 11:16
  • $\begingroup$ What non-standard operations exist that meet this condition? $\endgroup$ – VsSekorin May 29 '18 at 11:18

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