I am trying to prove the equivalence of these two ways of stating the Soundness Theorem for FOL.
I am proceeding by reductio for each direction. So on the left to right direction I reason as follows:
- If $ \Gamma\vdash\alpha$ then $\Gamma\vDash\alpha$
- Suppose it is not the case that if $\Gamma$ is consistent, then $\Gamma$ is satisfiable.
- From 2, $\Gamma$ is consistent.
- From 2 also, $\Gamma$ is unsatisfiable.
- From 4, there is no structure $\mathfrak{A}$ and variable assignment $s$ such that for every $\gamma\in\Gamma\vDash_\mathfrak{A}\gamma[s]$.
- From 3, there is no wff $\alpha$ such that $\Gamma\vdash\alpha$ and $\Gamma\vdash\lnot\alpha$
- For every $\gamma\in\Gamma,\Gamma\vdash\gamma$.
- So by 1, $\Gamma\vDash\gamma$ for every $\gamma\in\Gamma$
- By 8, there is a structure $\mathfrak{A}$ and variable assigment $s$ such that for every $\gamma\in\Gamma\vDash_\mathfrak{A}\gamma[s]$.
- 9 contradicts 5. So not 4.
- If, if $ \Gamma\vdash\alpha$ then $\Gamma\vDash\alpha$, then if $\Gamma$ is consistent, then $\Gamma$ is satisfiable.
On the right to left direction I started to reason as follows:
- Suppose if $\Gamma$ is consistent, then $\Gamma$ is satisfiable.
- Suppose it is not the case that if $ \Gamma\vdash\alpha$ then $\Gamma\vDash\alpha$.
- From 2, $\Gamma\vdash\alpha$
- From 2, $\Gamma\nvDash\alpha$
- From 4, for some structure $\mathfrak{A}$ and variable assignment $s$, for all $\gamma\in\Gamma,\vDash_\mathfrak{A}\gamma[s]$ and $\nvDash_\mathfrak{A}\alpha[s]$
But now I am a bit stuck. My main questions are this:
Does the left to right direction work? In particular, I justify step 7 by way of the fact that every wff deduces itself. Is that legit?
With the right to left direction, I'm stuck. Maybe the reductio's the wrong strategy here. Thanks for any and all help. I really appreciate it.