If $\Phi, \phi, \neg\psi$ is inconsistent, show that $\Phi \vdash \phi \to \psi$.

Question

If $\Phi, \phi, \neg\psi$ is inconsistent, show that $\Phi \vdash \phi \to \psi$.

What I have so far:

If $\Phi, \phi, \neg\psi$ is inconsistent, then $\Phi, \phi \vdash \psi$. Then by deduction, $\Phi \vdash \phi \to \psi$.

I am not sure what this means though. Where did $\Phi, \phi \vdash \psi$ come from? Does this also mean that $\Phi, \neg\psi \vdash \neg\phi$?

Answer 1

I am not sure what this means though. Where did $Φ,ϕ⊢ψ$ come from?

Saying that the statements $\Phi, \phi,\neg \psi$ are inconsistent, means that they cannot be mutually satisfied (there is no valuation that allows the three statements to all be true).   Therefore if they are inconsistent, then any valuation that can satisfy both $\Phi$ and $\phi$, cannot also satisfy $\neg\psi$ .   In bivalued logic this means the evaluation must satisfy $\psi$, so we say $\Phi$ and $\phi$ entail $\psi$.   Ie: $\Phi, \phi\vDash \psi$ .

Does this also mean that $Φ,¬ψ\vdash ¬ϕ$?

Yes.   For the same reasons.   We can likewise use deduction to infer that $\Phi\vDash \neg\psi\to\neg\phi$ .

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(Also note that we should be using $\vDash$ for the entailment because we are discussing semantics when talking about consistency.)