# derive formulas in mathematical logic

• Mar 3rd 2010, 11:26 PM
sfitz
derive formulas in mathematical logic
Hey everyone,

Here's the problem:

Show that if $\Gamma \vdash \phi$ and $\Delta,\phi \vdash \psi$, then $\Gamma,\Delta \vdash \psi$

I soooort of know how to start, it's supposed to be like,
We already have a derivation of $\psi$ from $\Gamma$, so start with:
.
.
.
(k) $\phi$
(k+1)
(k+2)
.
.
.

I have a hard time getting this naturally, because to me it feels like I should be able to assume $\Delta$ in line (k+1) and then have $\psi$... but then, that's not right because it doesn't use Modus Ponens or any of the three axioms.

Any help or hints as to how I should be thinking?
• Mar 4th 2010, 02:25 AM
PiperAlpha167
Quote:

Originally Posted by sfitz
Hey everyone,

Here's the problem:

Show that if $\Gamma \vdash \phi$ and $\Delta,\phi \vdash \psi$, then $\Gamma,\Delta \vdash \psi$

I soooort of know how to start, it's supposed to be like,
We already have a derivation of $\psi$ from $\Gamma$, so start with:
.
.
.
(k) $\phi$
(k+1)
(k+2)
.
.
.

I have a hard time getting this naturally, because to me it feels like I should be able to assume $\Delta$ in line (k+1) and then have $\psi$... but then, that's not right because it doesn't use Modus Ponens or any of the three axioms.

Any help or hints as to how I should be thinking?

From what you've described (three axioms schemes, and one rule that you've identified as MP), I'd say you might be thinking:

First, an application of the Deduction Theorem, followed by an application of MP.