Suppose we have a logical system $s$. Now, the monotonicity property tells us that: $\Gamma \vdash A$ and $\Gamma \subseteq \Delta$ implies that $\Delta \vdash A$. I see this definition somewhat problematic: What if $\Gamma = \{A\}$ where $A = \text{"Socrates likes chocolate"}$. and $B = \text {"Socrates hates chocolate"}$. Now, it doesn't true that $\Gamma \cup \{B\} \vdash A$. Is it? I mean, $\Gamma \cup \{B\}$ is always a false set of statements.
What am I missing in this definition of monotonicity?
This property is correct. In fact, if $\Gamma$ is a set of sentences that leads to a contradiction, then $\Gamma\vdash A$ for every sentence $A$. In plain words, from a false statement we can deduce anything.
Proof: Let's say that $\Gamma\vdash B\wedge\neg B$. To prove $A$, let's suppose $\neg A$. Now we deduce (because we can) $B\wedge\neg B$ from the sentences in $\Gamma$. This is a contradiction, so $\neg A$ is false and hence $A$ is true.