4

The principle of explosion is this:

$$ X \land \lnot X \rightarrow Y $$

And similarly, the rule of weakening/monotonicity of entailment is this:

$$ (X \rightarrow Y) \rightarrow (X \land A \rightarrow Y) $$

If in a given logic, the rule of weakening holds, you could substitute $X = P$, $Y = Q$, and $A = \lnot P$ (or $P = \lnot P$ and $A = P$) in the rule of weakening to get something that gets close assuming the law of the excluded middle holds (see line 4) - it matches the principle of explosion at least when $Q = X = \top$:

$$ (P \rightarrow Q) \rightarrow (P \land \lnot P \rightarrow Q) \tag{1} $$ $$ (\lnot P \rightarrow Q) \rightarrow (P \land \lnot P \rightarrow Q) \tag{2} $$ $$ ((P \lor \lnot P) \rightarrow Q) \rightarrow ((P \land \lnot P) \rightarrow Q) \tag{3} $$ $$ Q \rightarrow (P \land \lnot P \rightarrow Q) \tag{4} $$

Similarly, most logics I've found that reject the law of the excluded middle also reject the principle of explosion, either implicitly like minimal logic or explicitly like relevant logic. This pervasive pattern led me to question why that pattern exists in the first place - if nobody is making their logic work with it when they drop the rule of weakening, is it even possible at all (despite the broad opinionatedness)?

My question is two-fold:

  1. Does the principle of explosion depend on the rule of weakening in any case? Or in more formal terms, does proving the rule of weakening to hold also prove the principle of explosion to hold or vice versa? If not generally, then under what constraints? (The above proof sets two constraints, but I'm not fully convinced either one of them are absolute.)

  2. Are there any known non-monotonic logics where the principle of explosion does hold? (This would serve as a counterexample to my hypothesis.)

Claudia
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    The best known logic that rejects excluded middle is intuitionistic logic, and it does recognize explosion. Intuitionistic linear logic rejects both excluded middle and weakening, but still retains explosion. – hmakholm left over Monica May 22 '18 at 12:15
  • Oh, and what kind of thing is (4) even? Usually $\vdash$ wants one or more formulas on its right-hand side but does not itself produce a formula, so (4) does not look like it's a well-formed bit of symbolism at all. – hmakholm left over Monica May 22 '18 at 12:42
  • I'm using sequent calculus syntax, but I can edit that to be a bit more traditional instead. – Claudia May 22 '18 at 12:46
  • @IsiahMeadows: There is nothing like $\cdots\vdash(\cdots\vdash\cdots)$ in sequent calculus. – hmakholm left over Monica May 22 '18 at 12:54
  • To clarify, I'm talking about predicates and logical consequence in the abstract here. Each line in 1-6 is supposed to amount to a tautology. I fixed it to be a little more traditional instead. – Claudia May 22 '18 at 12:55
  • In which possible logic is $A\to(B\to A) \Leftrightarrow B$ valid? Certainly not classical or intuitionistic propositional logic (or for that matter minimal logic), which both hold that $A\to(B\to A)$ is a tautology. – hmakholm left over Monica May 22 '18 at 12:58
  • Good catch, no clue what I was thinking when recasting that part. (I did recall that while typing the question initially, but somehow went on a tangent and took an incorrect mental leap.) I just stripped it entirely and kept it a little more accurate. – Claudia May 22 '18 at 13:04
  • @MauroALLEGRANZA I killed that section, if it helps. – Claudia May 22 '18 at 13:05
  • @MauroALLEGRANZA I specifically noted it almost worked - I stated it "gets close", not that it's a proper proof. This time around, I corrected it to be correct. (And BTW, step 4 never represented explosion.) – Claudia May 22 '18 at 13:09
  • Maybe useful Jean-Yves Béziau & Mihir Chakraborty & Soma Dutta (editors), New Directions in Paraconsistent Logic, Springer. – Mauro ALLEGRANZA May 22 '18 at 15:08
  • I think it's misleading to say that minimal logic rejects the principle of explosion. I think it's more accurate to say that minimal logic is "What if we work in positive logic, but introduce a specific propositional variable $L$ (of which we make absolutely no assumption) and give special meaning to implications of the form $A \to L$?" While it seems usual to write $\bot$ instead of $L$, I don't see a compelling reason to think that $L$ is the same underlying idea as the usual notion of a bottom element. –  May 22 '18 at 16:47
  • I fixed it to note that minimal logic doesn't explicitly reject it. However, the Wikipedia page does specifically state that it "is a variant of intuitionistic logic that rejects not only the classical law of excluded middle (as intuitionistic logic does), but also the principle of explosion (ex falso quodlibet)." I was going off of that. However, relevant logic openly rejects it, and that particular community has made explicit statements on that topic, so I kept the reference there. – Claudia May 23 '18 at 16:21
  • I know this is an old post, but the inference from 2 to 3 is invalid. Note the equivalence $((A \to B) \lor (C \to B)) \iff ((A \land C) \to B)$ – PW_246 Sep 17 '23 at 15:30

1 Answers1

-2

I find the following form of the principle of explosion more useful in proofs:

$$A\implies [\neg A \implies B]$$

I don't how much flexibility you have in choosing your axioms, but here is a proof using natural deduction (more straightforward than substitutions):

1) $A$ (assume)

2) $\neg A$ (assume)

3) $\neg B$ (assume)

4) $A \land \neg A$ (intro $\land$, 1, 2)

5) $\neg \neg B$ (discharge, 3)

6) $B$ (elim $\neg \neg$, 5)

7) $\neg A \implies B$ (discharge, 2)

8) $A\implies [\neg A \implies B]$ (discharge, 1)

See my formal proof in HTML format here.

Dan Christensen
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