1

In page 19 of Mathematical Logic by Chiswell and Hodges, they provided a derivation of the sequent $\vdash (\phi\to(\phi\to\phi))$:

enter image description here

where $(\to I)$ is:

enter image description here

I understand that if we discharge $\phi$ in 1 this will work since, as $\phi$ has already been discharged, we discharge it again in 2. I don't understand the case where ? is 2, i.e, we discharge $\phi$ at that step. Since we're leaving $\phi$ undischarged until 2, how can we use $(\to I)$ in 1 without discharging $\phi$?

Thank you in advance!

Scientifica
  • 8,781
  • Hint: Introduce the initial premise twice in succession. Sounds weird, but it will work. – Dan Christensen Jun 06 '16 at 02:19
  • Thank you very much for your comment. That's a good idea. I'm not sure if this is correct but I guess we can introduce $\phi$ a second time right at the left of $(\phi\to\phi )$ above the circled $2$. Then we use $(\to I)$ a first time just as shown (circled $1$) and a second time to remove the second $\phi$, right? – Scientifica Jun 06 '16 at 06:43
  • I'm not sure of your particular rules of inference, so this may not work for you. In ordinary natural deduction it would go something like this: 1. $P$ (Premise), 2. $P$ (Premise), 3. $P \implies P$ (Conclusion, 2), 4. $P \implies [P \implies P]$ (Conclusion, 1) – Dan Christensen Jun 06 '16 at 12:16
  • @DanChristensen Thank you for your answer. The rule we work with is the one described in the image in my question by $(\to I)$. We've never used the world permise before, but I guess you mean by it an assumption, right? I'd like to make sure I understood your answer: when you say (Conclusion, 2) you mean "The implication we get from Permise 1", right? If that's it then I believe I understood your point and you're totally right. I like that approach. – Scientifica Jun 06 '16 at 14:50
  • @DanChristensen A question: Why in 3. you used Premise 2 instead of 1? Are you obliged to do so? It seems the book doesn't cover Premises since I couldn't find that word in the Index. – Scientifica Jun 06 '16 at 14:52
  • "Premise" is another word for assumption or hypothesis, or a statement that is provisionally assumed to be true for the sake of argument -- a what-if statement. When you discharge a premise, it applies to the last premise that has not already been discharged. – Dan Christensen Jun 06 '16 at 16:06
  • @DanChristensen Ok thank you. I understood a bit. It's a quite different way and I just started the book (at page 23 now) – Scientifica Jun 06 '16 at 16:22

1 Answers1

2

By Sequent Rule (Axiom Rule), for $\Gamma = \{\}$, we have the sequent $\Gamma \cup \{\phi\} \vdash \phi$, so by Sequent Rule ($\rightarrow \textrm{I}$), we have $\{\} \vdash (\phi \rightarrow \phi)$. That is, we get the tautology $\phi \rightarrow \phi$ "for free" ($\phi$ having been introduced by assumption on the previous line).

Recall the text (p. 17, immediately following the definition of Sequent Rule ($\rightarrow \textrm{I}$)) "the assumption $\phi$ in the first sequent of the sequent rule ($\rightarrow \textrm{I}$) is allowed to drop out of the assumptions of the second sequent". Note "allowed" $\neq$ "required". This is discussed further in Remark 2.4.2.

Recall also the text (p. 17, prior to the definition of Natural Deduction Rule ($\rightarrow \textrm{I}$) "in forming the derivation we are allowed to discharge any occurrence of the assumptions $\phi$ written in $D$. The rule is still correctly applied if we do not discharge all of them; in fact the rule is correctly applied ... [if] there is nothing to discharge." This indicates (1) an assumption may appear more than once in a derivation, (2) applying the rule allows you to discharge all, any, or none of them, and (3) the rule may be applied even when all assumptions are discharged.

This third case is somewhat similar to the first paragraph above. If $\psi$ is something we can deduce and $\phi$ is any statement whatever, we can deduce $\phi \rightarrow \psi$ (because $\psi$ is already known to be true).

Eric Towers
  • 67,037
  • 1
    Thank you very much for your answer! My bad, even if I read the book carefully I didn't realise that "allowed" $\neq$ "required" until you explained it to me. Thank you so much for your efforts and for using your time to write such a good explanation! – Scientifica Jun 05 '16 at 06:02