~(A or B) <--> ~A & ~B

(1)

A B
--------(Or Intro) ----------- (Or Intro)
A or B ~(A or B) A or B ~(A or B)
----------------------- (~ Elimination) -----------------------(~E)
absurdity absurdity
------------------------(~ Intro "RAA") ---------------------- (~ Intro)
~A ~B
------------------------------------------------------------------------(& Intro)
~A & ~B
------------------------------------------------------------------------(--> Intro)
~(A or B) --> ~A & ~B

(2)
~A & ~B
----------- (&E "Simplification")
~A A or B ~A & ~B
------------------ (Disjunctive Syllogism) ------------ (&E)
B ~B
------------------------------------------------------------- (~E)
absurdity
--------------- (~I)
~(A or B)
------------------------------(-->I)
~A and ~B --> ~(A or B)

Combining derivations (1) and (2) by (<--> I)
-----------------------------------------------------
~(A or B) <--> ~A & ~B

At last!!

Many's the weary, lonely night I've sat up through the wee small hours, 'Oh, but for yet another proof of de Morgan's Law!!'

I can sleep now, as soon as I add this one to all the others.

Boru

What I got from that was...
Dancing Queen

Ala...
ABBA

As a passionate student of formal logic, it is abhorrent your use of it. Keep studying it and put aside your holy book. Only then you will be able to properly understand it :-s

ETA: On a side note, I thought theists avoided formal logic like the plague in favour of intuitionist logic (as you should know, avoids proof by RA+). Either way, you are not remotely proficient at it.

A logic master you are? I guess others are just godlike logicians by your standards.

Dude, Thats like the first lesson in Boolean algebra, first they give you the axioms of the theory of Boolean algebra then they prove the basic stuff like De-morgan's .

With all honesty, your presentation of the proof is really bad and hardly readable .

(May 25, 2014 at 1:16 pm)Marsellus Wallace Wrote: With all honesty, your presentation of the proof is really bad and hardly readable .

And not to mention boring as fuck.

(May 25, 2014 at 1:16 pm)Marsellus Wallace Wrote: With all honesty, your presentation of the proof is really bad and hardly readable .

Touché. The arrogance is strong with this one. Its not even the forum SW fault, one can do that proof without even use tags.

LastPoet; I don't use my holy book in logic, I always avoid including anything in my holy book because atheists hate even the fact that i am a Muslim, anyway we theists don't avoid formal logic, for me at least i mastered LP (Language of propositions) and a lot of subjects in mathematical logic, I sat down days and ours on complicated proofs in symbolic logic, which i mastered all. I don't take logic from a philosophic account, but from a mathematical account, in my opinion formal logic, is a subset of mathematical intuition, everything is mathematics, any proposition can be transformed into numbers using Godel's numbers.

---

And i don't reject (RAA), and the purpose is:

I don’t want to explain logic from scratch, but all the attempts to refuse the third law of Aristotle (The excluded middle) are because they didn’t like changing the mathematical constructive way of proof to using these logical tools, so some mathematical philosophers (intuitionists, contstructivists, etc.) rejected this law and this leads to rejecting some powerful mathematical & logical tools like (RAA). But a lot of these philosophies failed when Kurt Gödel came with the incompleteness theorem. Simply the mathematics that depends on arithmetic (Peano’s arithmetic) are incomplete. That means Constructivism failed because now there are true statements that are unprovable. This result even blew away Hilbert’s program.

If you don’t know about Gödel’s incompleteness theorem here is a simple example:
Let G be the proposition “G itself cannot be proved”, if G is false and we could prove G from our system of axioms, that means we proved a false statement and our system is inconsistent, but if G is true then we couldn’t Prove G, But that’s what G said so G is True. This means there exists true unprovable statements.
Sorry but this is the simplest way to show the incompleteness theorem, the proof is far more complicated, it involves Diophantine sentences and computable enumerable sentences so as Peano’s arithmetic, a complicated mathematical logic, if I want to explain everything I will have to write a book instead.
Gödel saved us from the possibility of someone writing a computer program that will solve all our mathematical problems or give all the proofs for all the theorems.

And proof by contradiction is still being used by mathematicians these day’s and it’s a very powerful tool, if you reject it then you will reject the theorem:
“There are infinitely many prime numbers”
Which is a mathematical standard. If you deny it then you will refuse number theory, and then the computer your sitting on.

So... LogicMaster, what is the faulty premise on which you base your logical religiosity?

It becomes more clear when you quote it, exposing the format, but damn dude, use an equal sign.