# Are there "partially explosive" logics?

Roughly speaking, I'm wondering if it's possible to meaningfully grade different systems on how explosion-tolerant they are.

In classical sentential logic and intuitionistic sentential logic, a single contradiction `P ∧ ¬P` lets you conclude any well-formed formula.

Let `T` be the set of theorems and `L` be the set of the well-formed formulas. Let `φ` be a particular contradictory well-formed formula that is not a theorem in `T`.

If we take the closure of `T ∪ {φ}` under the inference rules in an explosive system, we get back `L` .

If we take a system with no inference rules at all, then the closure of `T ∪ {φ}` is just `T ∪ {φ}` . We would get no spurious theorems besides the contradiction. So the system with no inference rules is "minimally explosive".

In general, can we distinguish different non-explosive logics from each other by characterizing what the consequences are of accepting a contradictory premise / temporarily adopting a contradictory axiom? Is this a useful way of thinking about different paraconsistent logics?

As far as I can tell, an explosion-tolerant logic is one that does not admit the following the following inference rule.

``````P ∧ ¬P
------
Q
``````

So then, by the contrapositive of the deduction theorem, it suffices to show that `(P ∧ ¬P) → Q` is not a tautology.

There's a simple three-valued logic given here defines the connectives in terms of truth tables:

``````neg           P → Q     Q         P ∨ Q      Q       P ∧ Q    Q
P ¬P                  1 ? 0                1 ? 0            1 ? 0
----                 +-----               +-----           +-----
1  0               1 |1 ? 0             1 |1 1 1         1 |1 ? 0
?  ?             P ? |1 ? 0           P ? |1 ? ?       P ? |? ? 0
0  1               0 |1 1 1             0 |1 ? 0         0 |0 0 0
``````

For the purposes of identifying tautologies, the two designated truth values are `T`/`1` and `?`

In order to show that `(P ∧ ¬P) → Q` is not a tautology, we consult the truth table and work backwards, as shown below.

``````1. (P ∧ ¬P) → Q  falsifiable
2. Q false   and   P ∧ ¬P non-false
3. if P is "?", then P ∧ ¬P is non-false
4. {P="?", Q=⊥} witnesses the falsifiability of (P ∧ ¬P) → Q
``````

This example does a good job of showing us why the asymmetry in the definition of `→` is there. A premise whose truth value is `?` is treated just like a true premise.

It certainly seems like the set of theorems doesn't grow much in this system if a contradictory premise is assumed. I think that with the assumption of `P ∧ ¬P`, you only get `P` and `¬P` through conjunction elimination, but I'm not sure how to prove that.

Also, there might be other paraconsistent logics that are less tolerant of contradictory premises than this one.

• Is this is a candidate for being moved to math.stackexchange? Commented Nov 7, 2018 at 23:17
• Yes, if the community thinks that’s more appropriate. I’m not sure what the guidelines are for where to put a logic question. Commented Nov 7, 2018 at 23:26
• I added the expression "-tolerant" above the first grayed-out box... this is correct, isn't it? Not admitting explosion means it's tolerant, right? Commented Nov 8, 2018 at 0:09
• Carnielli and Marcos discuss intermediate between non-explosive and explosive logics (finitely trivializable) in Paraconsistency, p.22. Commented Nov 8, 2018 at 1:24
• Have you already checked a general review about Systems of Paraconsistent Logic ? Commented Nov 8, 2018 at 10:34

An example is da Costa's hierarchy of propositional paraconsistent calculi

The three-valued logic of Lukasiewicz can be viewed as a paraconsistent logic, since ¬(P ∧ ¬P) is not a universal law that applies to all statements, but a contingent statement, applicable to some statements but not others. If P has the middle truth value, so does ¬P. A statement and its negation are thus not necessarily contradictory, and (P ∧ ¬P) is not explosive.

However, it is possible to formulate other expressions that are explosive, using the operators Mp and Lp.

The logic that is closest to Classical Logic that is only partially explosive is Minimal Logic, which can be achieved in intuitionistic natural deduction by removing explosion from the set of inference rules. However, Negation Intro. is retained such that (P&~P)→~Q remains valid.