okay, I'm rather confused by the discussion in the section of my book 'The Logic Book' about the section The Completeness of SD and SD+.
I will write a portion of the discussion verbatim here; if it is not enough information to help you explain to me then I could try to write some of the earlier or later discussion verbatim  but I hope this will be enough to help you to explain it to me.
"6.4.7 If Γ is inconsistent in SD, then every superset of Γ is inconsistent in SD.
Proof: Assume that Γ is inconsistent in SD. Then for some sentence P there is a derivation of P in which all the primary assumptions are members of Γ, also a derivation of ~P in which all the primary assumptions are members of Γ. The primary assumptions of both derivations are members of every superset of Γ, so P and ~P are both derivable from every superset of Γ. Therefore every superset of Γ is inconsistent in SD.
But we have already proved by mathematical induction that every set in the infinite sequence is consistent in SD. So Γ(j+1) cannot be inconsistent in SD, and our supposition that led to this conclusion is wrongwe may conclude that Γ* is consistent in SD."
(Γ* was defined in earlier discussion as the union of all the sets in the series and is defined to contain every sentence that is a member of at least one set in the series and no other sentences.)
Okay, I hope that's enough to clarify my question which is: If the discussion was to develop a Proof, the one I just wrote, then they proceed to tell me that the supposition ( is that the Assumption?) is wrong  then how does that comprize a Proof?!?
need help understanding when a Proof is a Proof

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 Posts: 20
 Joined: Mon Jul 31, 2017 5:43 pm
Re: need help understanding when a Proof is a Proof
Let me try to rephrase the question ..... When you see a 'Proof' in a book about Logic  does it mean more that it is more of an argument with as many possible aspects discussed, defined, and answered with a possible outcome? Or is a 'Proof' what most people would consider it  evidence for an assertion with no other possible interpretation possible so that the assertion and its' underlying implicit ideas are certain and exactly as described?
Re: need help understanding when a Proof is a Proof
I think this a case of reductio ad absurdum,i.e. assume the negation, derive a contradiction, and then deduce the conclusion. We have seen that before in the negation introduction rule. The supposition/assumption is wrong because it introduces a contradiction. Check to see if you can understand the material now. Otherwise give me that: "But we have already proved by mathematical induction that every set in the infinite sequence is consistent in SD."ProfAlex wrote:Okay, I hope that's enough to clarify my question which is: If the discussion was to develop a Proof, the one I just wrote, then they proceed to tell me that the supposition ( is that the Assumption?) is wrong  then how does that comprize a Proof?!?

 Posts: 20
 Joined: Mon Jul 31, 2017 5:43 pm
Re: need help understanding when a Proof is a Proof
thanks! I have a new question but I think I must create a new thread  give me a few minutes and I'll post it! thanks again for your willingness to help!
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