# 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? – A Rogue Ant. Nov 7 '18 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. – Gregory Nisbet Nov 7 '18 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? – elliot svensson Nov 8 '18 at 0:09
• Carnielli and Marcos discuss intermediate between non-explosive and explosive logics (finitely trivializable) in Paraconsistency, p.22. – Conifold Nov 8 '18 at 1:24
• @ElliotSvensson yes that’s correct. – Gregory Nisbet Nov 8 '18 at 1:51

## 2 Answers

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.