1

I am reading through Mathematical Logic by Ian Chiswell and Wilfrid Hodges and am trying trying to finalize my understanding of the RAA rule introduced in section 2.6. The book states the rule in the following way:

Suppose we have a derivation whose conclusion is $\bot$. Then there is a derivation

$$\require{cancel}(\cancel{~\lnot\phi~}) \\\\\ D \\\\\ \quad \quad \quad \dfrac{\bot}{\phi}(RAA)$$

In my head I am interpreting this to mean, in English, "If you have a derivation of absurdity, then you can conclude any statement whatsoever (call it $\phi$). Additionally, if $(\lnot \phi)$ is used as an assumption in $D$, you are allowed to discharge any occurrences of that assumption (and remember that it may not be an assumption of $D$ at all)"

I feel like this is a good understanding of what the rule states, and I was able to do a couple of the exercises with this understanding, but I still have some confusion about how this rule is justified. The main question I'm asking in this post is, "What allows us to conclude any statement whatsoever from absurdity"? However, the following are related sub-questions that I'm also thinking about to try to answer this main question:

  1. Since RAA says we can conclude any statement from absurdity, RAA seems to include the principle of explosion a.k.a ex falso quodlibet. Is this correct?
  2. If this IS correct, then I suppose my "main question" becomes "What justifies the principle of explosion?" In this book, it seems that the principle of explosion is to be taken as primitive (we can immediately conclude any statement $\phi$ from absurdity). However, outside of this book, I have seen that the principle of explosion is something to be derived. Arguments using $\lor$ have not been introduced in this book yet, so I'm being loose with the rules and notation here, but that derivation goes something like this:

$$ P \\\\ P \lor Q \\\\ \lnot P \\\\ \therefore Q $$ $\quad \quad$ So, is the POE/EFQ something primitive or is it something to be derived?

  1. Lastly, again, if RAA is somehow introducing the POE/EFQ, then why don't the authors state explicitly that they are introducing the POE/EFQ? (Based on the index, I don't think they mention it anywhere else in the book either.) I figured this would be a mainstream term worth mentioning in the book.
Artyom Elessar
  • 109
  • It's not exactly the same question, but I believe the second half of my answer here does explain what justifies the principle of explosion. It might not be useful to you in its current form, given that you're just starting out in logic, but I don't have time to rewrite it in a more elementary context right now. Still, it's worth giving it a read to see what you get out of it. – Z. A. K. Sep 24 '23 at 06:39
  • Not exactly: the Ex Falso rule $\bot \vdash \varphi$ allows us to conclude with a formula whatever from a contradiction. This is the basic rule governing $\bot$. – Mauro ALLEGRANZA Sep 24 '23 at 08:13
  • 1
    This is PoE and is common to most logical systems: a contradiction is catastrophic. But see Paraconsistent Logic where contradictions are "managed". – Mauro ALLEGRANZA Sep 24 '23 at 08:31
  • 2
    RAA above is equivalent to Double Negation and is peculiar of Classical Logic: if we derive a contradiction from the negatio of a claim, we are allowed to assert the claim. – Mauro ALLEGRANZA Sep 24 '23 at 08:33
  • The argument you quote simply shows that the explosion principle / ex falso quodlibet can be derived from disjunctive syllogism (and other deduction rules that are even easier to accept/understand). The point maybe being that disjunctive syllogism could be considered as "less surprising" to a newcomer to logic than the explosion principle. – Daniel Schepler Sep 25 '23 at 16:33

0 Answers0