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I have a doubt about the completeness proof of propositional logic that appears in Chiswell and Hodges´ book. In page 93, case 2, i) I don´t understand why I can replace χ_1 by χ_1 and χ_2, because I don´t know if the rules for and introduction are followed in derivation D. I also don't know what I accomplish by making such a replacement, because χ_1 and χ_2 are in D and clearly I can put after χ_1 and χ_2. Also the book says the result is a derivation of ⊥ whose undischarged assumptions all lie in Γk , but I think that these undischarged assumptions all lie in in Γk+1, since I put in D χ_1 and also χ_2.

I am not a mathematician, I am sorry if the question is very easy.

Pablo Ib
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  • Great question! It may be helpful to those trying to answer your question if you try to format it in a way that it is easier to read. – Burt Apr 28 '20 at 16:06
  • I assume that you have understood how the proof is working... In order to build the set $\Delta$ starting from $\Gamma_0 = \Gamma$, the proof "travels" down the derivation tree and at step $i$ we add formulas to $\Gamma_i$ according to the inference rule used at that step. – Mauro ALLEGRANZA Apr 28 '20 at 16:22
  • You did not delete the off-topic post on SE meta:( – Martin James Apr 28 '20 at 16:24
  • In case 2i we have a conjunction $\chi_1 \land \chi_2$; thus, we apply rule $\beta$ (that regarding $\land$) and we add both conjuncts to $\Gamma_i$. – Mauro ALLEGRANZA Apr 28 '20 at 16:25
  • See page 92: "The idea of the construction is to take the requirements for a Hintikka set, one by one; as we come to each requirement, we build the next $Γ_i$ so that the requirement is met." – Mauro ALLEGRANZA Apr 28 '20 at 16:34
  • And page 91: "Definition 3.10.4 We say that a set $Γ$ of formulas is a Hintikka set if it has the following properties: (1) If a formula (φ ∧ ψ) is in Γ then φ is in Γ and ψ is in Γ. [...]" – Mauro ALLEGRANZA Apr 28 '20 at 16:35
  • In conclusion, I suggest you to re-read the chapter trying to ask a more focused question :-) – Mauro ALLEGRANZA Apr 28 '20 at 17:28
  • Another question intimately related:why is it necessary to check(in page 94) that Δ meets the requirements of Hintikka sets,taking into account that we have already done the construction of Δ with such requirements(in pages 92 and 93)?The book says that formula (φ ∧ ψ) could be some φ_j for j > or = i, but the construction of each Γi seems to assume tha i=0,1,,,n, so it would be unnecessary to examine the case j. Could someone explain this? – Pablo Ib May 12 '20 at 10:29

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