Philosophy Now Forum

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 wrong-we 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?!?

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?

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?!?

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."

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!