Definition
Let $\gamma\in \text{Form}$. A proof of $\gamma$ is a sequence of formulas $\phi_1,\phi_2,…,\phi_n=\gamma$ where each $\phi_i$ is an instance of an axiom or was obtained by modus ponens from two previous formulas.
The axioms we use are
Axiom 1: $ (\alpha\implies (\beta \implies \alpha)).$
Axiom 2: $((\alpha \implies (\beta \implies \gamma ))\implies (( \alpha \implies \beta) \implies (\alpha \implies \gamma))).$
Axiom 3: $((\neg \alpha \implies \neg \beta) \implies ((\neg \alpha \implies \beta) \implies \alpha)).$
Definition
We say that $\Gamma\vdash \gamma$ if there exists a finite proof of $\gamma$ from $\Gamma$ (that is, each formula of $\Gamma$ can be treated as an axiom).
Definition
We say a set $\Gamma$ is inconsistent if there exists a formula $\alpha$ such that $\Gamma \vdash \alpha$ and $\Gamma\vdash \neg \alpha$.
Now I want to show that if $\Gamma\cup \{\gamma\}$ is inconsistent, then $\Gamma\vdash \neg\gamma$, but I'm pretty lost.
We know there exists some $\varphi$ such that $\Gamma\cup \{\gamma\}\vdash \varphi,\neg\varphi$, but I don't see how this helps.
I also proved that a $\Gamma$ set is inconsistent if and only if $\Gamma\vdash \beta$ for every formula $\beta$, thinking this would help, but didn't get far.
Could someone help me out?
Also, a side question, does the system I described here have a common name? I could find very little information about it (I'm trying to prove the completeness theorem using this system, and need this for Lindembaum's lemma).
Best Answer
Hint
For Mendelson's system, see :
We need :
Lemma 1.8 [ page 27 ] : $\vdash \varphi \to \varphi$.
With it, (Ax.1) and (Ax.2), we can prove Prop.1.9 (Deduction Th) [ page 28 ] and some useful results [ page 29 ]:
Corollary 1.10(a) : $\varphi \to \psi, \psi \to \tau \vdash \varphi \to \tau$
and :
Lemma 1.11(b) : $\vdash \lnot \lnot \varphi \to \varphi$.
First, we can prove the "easy" version :
a) if $\Gamma \cup \{ \lnot \gamma \}$ is inconsistent, then $\Gamma ⊢ \gamma$.
Proof
1) $\Gamma \cup \{ \lnot \gamma \}$ is inconsistent, i.e. $\Gamma \cup \{ \lnot \gamma \} \vdash \varphi$ and $\Gamma \cup \{ \lnot \gamma \} \vdash \lnot \varphi$, for some formula $\varphi$
Thus :
2) $\Gamma \vdash \lnot \gamma \to \varphi$ --- from 1) by Ded.Th
3) $\Gamma \vdash \lnot \gamma \to \lnot \varphi$ --- from 1) by Ded.Th
4) $\vdash (\lnot \gamma \to \lnot \varphi) \to ((\lnot \gamma \to \varphi) \to \gamma)$ --- (Ax.3)
Finally, for the sought result :
we have to apply Noah's suggestion.
As in case a) above, we have :
1) $\Gamma \vdash \gamma \to \varphi$
2) $\Gamma \vdash \gamma \to \lnot \varphi$
3) $\vdash \lnot \lnot \gamma \to \gamma$ --- by Lemma 1.11(a)
4) $\Gamma \vdash \lnot \lnot \gamma \to \lnot \varphi$ --- from 2), 3) and Corollary 1.10(a)
5) $\Gamma \vdash \lnot \lnot \gamma \to \varphi$ --- from 1), 3) and Corollary 1.10
6) $\vdash (\lnot \lnot \gamma \to \lnot \varphi) \to ((\lnot \lnot \gamma \to \varphi) \to \lnot \gamma)$ --- (Ax.3)