# Thread:

1. ## 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?

2. 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.
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 16, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9627494812011719, "perplexity_flag": "head"}