What is the Absolute Theorem in Logic?

[powp]

Hello

Does anybody know what the Absolute Theorem is in logic?? My text box uses it in proofs but I cannot find it anywhere else.

Thanks

P

 

[HallsofIvy]

This website:
https://www.physicsforums.com/showthread.php?t=111324
that I found by googling "absolute theorem" and "logic" defines an absolute theorem as one whose true false value is alway TRUE for all values of its variables- what I would call a "tautology".

 

[powp]

Thanks for you response.

What I have is this example that show the following

|- true ≡ A ≡ A

(1) true ≡ false ≡ false <axiom>
(2) false ≡ false ≡ A ≡ A <absolute theorem>
(3) true ≡ A ≡ A <Trans + (1, 2)>

My question is where does line 2 come from? Looks like it is coming from a combination of the formula I am trying to prove and line 1.

 

[honestrosewater]

Are you leaving out grouping symbols? Can you replace them or give the rules for replacing them? What does

true ≡ A

mean?

 

[powp]

no I am not leaving out grouping symbols. This is how this is in our text/course notes.

as for what true ≡ A mean. Offically I do not know. They want us to learn the rules before we learn what True and False mean.

I believe A would evalute to equal true. So so lost.

Thanks

 

[honestrosewater]

Ouch. Do those notes happen to be available online?

Well, if equivalence is a binary operation, there must be grouping symbols or rules for grouping. I guess they leave them out since ((A ≡ B) ≡ C) -|- (A ≡ (B ≡ C)), but I imagine it might make a difference in which rules you can apply and how. Plus, they're just different formulas! Ack.

It looks like they just did this:

(1) true ≡ (false ≡ false) <axiom>
(2) (false ≡ false) ≡ (A ≡ A) <absolute theorem>
(3) true ≡ (A ≡ A) <Trans + (1, 2)>

Is that what Trans does -- allow you to substitute equivalent formulas? Can you just copy the Trans rule? Is it

A ≡ B, B ≡ C |- A ≡ C

Is Absolute Theorem a theorem or a rule? Is the line exactly the same in every example proof? What is A called? Formula, sentence, proposition? What are true and false called? The same thing, something-values?

 

[powp]

I tried to upload them but it is two large.

Think you can get the notes here.

http://www.cs.yorku.ca/~gt/papers/1090-notes-2005-I.pdf

Does order of operations matter when proving?? We can remove barkets based on the rules of which connectives have a higher priority.

I cannot find what the absolute theorom is. it is not listed at all.

Thanks for you help

 

[honestrosewater]

I tried to upload them but it is two large.

Think you can get the notes here.

http://www.cs.yorku.ca/~gt/papers/1090-notes-2005-I.pdf

Yeah, I found those and thought they might be it. I'm reading them now.

Does order of operations matter when proving?? We can remove barkets based on the rules of which connectives have a higher priority.

Yeah, I'm just now looking for that info so I can restore other brackets.

I cannot find what the absolute theorom is. it is not listed at all.

From the looks of things so far, I think it might be a Γ-theorem when Γ is empty. Oh, rock on:

0.4.5 Definition. (Theorems) Any formula A that appears in a -proof is called a -theorem.
We write ⊢ A to indicate this. If is empty ( = ∅) —i.e., we have no special assumptions—
then we simply write ⊢ A and call A just “a theorem”.
Caution! We may also do this out of laziness and call a -theorem just “a theorem”, if the
context makes clear which 6= ∅ we have in mind.
We say that A is an absolute, or logical theorem whenever is empty.

 

[powp]

Thanks for you help.

When I saw absolute theorem in the annotation I thought it be defined in the notes. But I searched and read and could not find it.

Thanks