Given a logic over a language $L$, which has a consequence relation $\vdash$. This logic is Tarskian if for every $\Gamma \cup \Delta \cup {\alpha} \subseteq L$:
- If $\alpha \in \Gamma$, then $\Gamma \vdash \alpha$
- If $\Gamma \vdash \alpha$ and for every $\beta \in \Gamma$, it is known that $\Delta \vdash \beta$, then $\Delta \vdash \alpha$
- Monotonicity: If $\Gamma \vdash \alpha$ and $\Gamma \subseteq \Delta$, then $\Delta \vdash \alpha$
This definition is provided in Carnielli and Coniglio's book on Paraconsistent Logic. It seems to me that the monotonicity property is implied by the prior two.
Here's a proof sketch:
Assume we have $\Gamma, \Delta, \alpha$ such that $\Gamma \vdash \alpha$ and $\Gamma \subseteq \Delta$. Assume any $\beta \in \Gamma$. Since $\Gamma \subseteq \Delta$, we can show that $\beta \in \Delta$. Then by property 1 we know that $\Delta \vdash \beta$. Thus, by property 2, we get $\Delta \vdash \alpha$.
Is there anything wrong in my proof sketch? Why did the authors specifically add the monotonic property, if it is implied by the other two?
