Metatheory for a Logic: Philosophy Forums

Philosophy Forums » Logic and Philosophy of Math » Metatheory for a Logic » Metatheory for a Logic

Metatheory for a Logic
I'm trying to construct the theorems for a Fitch style calculus.

Metatheory for a Logic

ronlongo
Initiate

Usergroup: Members
Joined: Oct 22, 2009

Total Topics: 1
Total Posts: 4

Posted 10/22/09 - 07:01 PM:
Subject: Metatheory for a Logic

Hello,

Great web site and forum. Hope to learn a lot here.

I've pretty much always been interested in logic (formal & informal, deductive
& inductive). I really love it. However, until recently I've pretty much
avoided metatheory. Well that's changed. I've worked my way through several
texts of varying levels of difficulty. I'm really loving the coherence and
consistency of the system -- it's quite beautiful.

The first calculus I learned is still my favorite, a variant of Fitch presented
in Schaum's Outline of Logic. So, I thought it would be an interesting
exercise to see if I can derive the metatheorems myself. I'm not really
working blindly. I'm attempting to create a set of theorems parallel to those
in the Stanford Encyclopedia article on Classical Logic
(http://plato.stanford.edu/entries/logic-classical/). To simplify the task
I've dropped the biconditional and am working only with the truth-functional
calculus (I'll work on the First-Order calc later). Before asking my questions
I'll present what I have so far.

----- begin edit
Sorry. I'm having great difficulty getting the formatting right in this forum editor so I've chosen to attach my work instead.
----- end edit

Here are my questions:

1. Is my definition of the deducibility relation (G |- p) correct? The use of
nested proofs seems to make it awkward. The only thing I was able to come up
with was to introduce a concept of 'open' and 'closed' formulas.

2. Additionally, I wasn't sure how to word the final clause (11), I needed some
way to state that a proof is only complete when all nested proofs are
completed (closed). Have I done this correctly?

3. I'm not all that used to applying mathematical induction. Assuming that I
even setup my inference rules correctly, did I correctly construct the proof
for the rule of weakening?

Thanks in advance for any help.

Edited by ronlongo on 10/23/09 - 03:19 AM

Attached Files:

logicPost.txt
(4 KB, 3 downloads)

Sorry, you don't have permission to post. Log in, or register if you haven't yet.