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I managed to prove that

  1. a set $\Gamma $ is inconsistent if and only if $\Gamma \vdash a$ where $a$ is an arbitrary wff
  2. a set $\Gamma $ is inconsistent if and only if $\Gamma \vDash b$ where $b$ is an arbitrary wff

Is it appropriate make a conclusion that the selected axiomatization of propositional logic is strongly sound and strongly complete (e.g., $\Gamma \vdash c$ if and only if $\Gamma \vDash c$) based solely on the two statements above?

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  • $\begingroup$ You can see this post as well as this one. $\endgroup$ Commented Feb 15, 2022 at 13:06
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    $\begingroup$ the language "where $a$ is an arbitrary wff" seems open to misinterpretation; better to say simply "for every wff $a$". This makes it clear why (1) and (2) do not imply soundness and competeness; given just $\Gamma\vdash a$ for one particular $a$ (1) does not say anything. $\endgroup$ Commented Feb 15, 2022 at 13:11

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