Talk:Deduction theorem

message: I'm having trouble understanding how the deduction theorem differs from the conditional proof inference rule in the propositional calculus. We have:
"if φ |-- ψ, then |-- φ -> ψ" (Deduction_theorem)
vs
"If ψ can be derived while assuming the hypothesis φ, we may infer ( φ → ψ )." Propositional calculus
I can't help but think these are synonymous. Maybe the problem is that the former is given in much more formal notation than the latter. Would it help me, do people suppose, if someone would translate the latter into a more symbolic form? (Can that even be done?)
In any case, there must be something subtle (at least given my background) going on here. Help? --Ryguasu 19:31, 17 May 2005 (UTC)
replies: text: The deduction theorem applies to axiomatic systems, and the rule of conditional proof to natural deduction systems. They're analogous, but different. The deduction theorem is not a rule of the formal system; it is a property of the system's deducibility relation abstractly construed. Nortexoid 05:33, 2 April 2007 (UTC)

text: Actually the deduction theorem is deeply relevant for axiomatic systems, as it justifies the modus ponens rule : the deduction theorem "says" that the arrow is deeply linked to the notion of derivation form assumptions (which is a notion missing from axiomatic systems). So, by saying that you have the deduction theorem in an axiomatic system you're justifying the modus ponens as a rule because you showed that the arrow keeps the track of a derivation. And then, by composition of derivations, you know that if you have a proof of, say, A you have a proof of B (assuming that you have a proof of A --> B). This is why it's relevant in axiomatic systems. But if you consider natural deduction systems for example, it's not relevant anymore as you already have the notion of a derivation from assumptions. Then, the deduction theorem is "internalized" within the rules for the arrow.  Ianshil (talk) 14:49, 13 February 2017 (UTC)
replies: text: What's the difference between "deeply relevant" and just plain "relevant"?  — Preceding unsigned comment added by 184.179.86.135 (talk) 14:21, 3 July 2018 (UTC)
thread_title:  difference? 

message: In the middle part of the proof:
* Q→((Q→R)→R) 8. deduction from 5 to 7 QED
Would it not be sufficient to use 5 and 7 instead 5, 6 and 7, because 6 does not show up in this line, to get to this last line? Further 6 is already used the line above. Is this not enough? Indeed very clear your article and I understand here more than in Deduktionstheorem Cheers.--de:Benutzer:Roomsixhu —The preceding unsigned comment was added by 87.187.55.111 (talk) 02:20, 2 April 2007 (UTC)
replies: text: I said "deduction from 5 to 7" rather than say "deduction 5 and 7" because all of the steps from 5 to 7, i.e. steps 5, 6, and 7 depend on the hypothesis 5. And they are used in deducing the conclusion 7. Step 7 depends on step 6 which depends on step 5. In any case, the form of this virtual rule of inference is fixed and cannot be varied depending on the peculiarities of a special case. JRSpriggs 09:43, 2 April 2007 (UTC)
replies: text: Thank you again: I only noticed that, because 6 depends on 5 an the axiom 1 (line  4) and 8 depends on 5, 6 and 7, but 6 depends on 5 and 4, and 4 is the axiom. In my calculus you have basic formulas and basic rules, and can leave the basic formulas out, that seems to be different with your calculus which has axioms. But I did not find it in our german wikipadia described that clearly. I still will do some practice, to get more secure. Point 3.2 is a simple example how to get further "meta"-formulas from the basic lattice. --de:Benutzer:Roomsixhu —The preceding unsigned comment was added by 87.187.55.111 (talk) 13:24, 2 April 2007 (UTC)
thread_title: Proof

message: I have heard this before but I am suspicious because no-one ever gives evidence. Where is Herbrand's proof and for what system? Where is Tarski's proof and for what system? And how is it known that neither got it from the other, directly or indirectly. And how is it known that no-one did it earlier?Tyro13 (talk) 18:46, 2 November 2014 (UTC)
thread_title:  References to Herbrand's and Tarski's proofs, etc.