Prove or disprove: if Γ ⊢ α and Γ ⊆ ∆ then ∆ ⊢ α

Question

I am trying to solve this question but I am doubting about the answer. Namely: Γ ⊢ α means that there is a derivation with conclusion α and with all hypothesis in Γ. Since Γ ⊆ ∆, we can use the same derivation because all formulas in the hypothesis of Γ are also in the hypothesis of ∆. So from this it would follow that ∆ ⊢ α , i.e. de statement is true. However, what if we have a formula in ∆\Γ that contradicts a formula in Γ? Can we still use the derivation that follows from Γ ⊢ α? Or would this mean the statement is false?

Thank you in advance!

Answer 1

First order logic has three (often overlooked) structural rules -- weakening, contraction, and permutation as follows:

$$\frac{\Gamma \vdash A}{\Gamma,B\vdash A}(\text{Weakening})$$ $$\frac{\Gamma, B, B\vdash A}{\Gamma, B\vdash A}(\text{Contraction})$$ $$\frac{\Gamma_1, B,C,\Gamma_2\vdash A}{\Gamma_1,C,B,\Gamma_2\vdash A}(\text{Permutation})$$

The weakening rule lets you arbitrarily add hypotheses.

So formally, you would write $\Delta-\Gamma = \{A_1,\ldots,A_n\}$ and proceed by induction.

In practice, however, we usually ignore them by assuming that the context is a multiset instead of an ordered list.

Answer 2

To elaborate on the comments, the Principle of Explosion says:

$$P, \neg P \vdash Q$$

For any statements $P$ and $Q$, if $P$ and $\neg P$ are both true, then it logically follows that $Q$ is true.

See the Wikipedia article.