# Is it possible to deny that affirming a logical contradiction implies every possible proposition?

I find myself aligned with the motivation behind paraconsistent logic, they seem to me reason enough to warrant an attempt to structure logical systems which deny logical explosion. It does seem very odd to me that classical logic automatically trivialities the entire string of propositions in a proof by proving every possible proposition whenever contradictory propositions are held.

I am having trouble evaluating the many possible ways to deny logical explosion, and there's just too many implications to some of these systems. I am only able to evaluate systems like non-adjunctive logic, and three-valued logic where it's quite simple to see the impact these system has has on what is a valid inference, I am liking relevant logic right now, but I can't quite pin down why to believe this over other solutions, given the immense implications. Just how do I evaluate the various ways to deny {A , ¬A} ⊨ B?

Also, I still evaluate everything with classical logic, how does one evaluate the sorts of things we evaluate everyday with some of these paraconsistant logical systems?

• I am rolling your title back so as to make the answers below less confusing. Please feel free to ask followup questions and link back to this one if you wish. Commented Aug 27, 2011 at 15:02

One way to evaluate a new different logic is to see what new things you can prove in it and what old things you can't prove; and also whether you want these new changes or not. That is, start with the restriction (e.g. relevance logic only allows proofs where propositions have antecedents that appear in the consequents (i.e. hypotheses have relevance to inferences)) and then see if you get something you don't like (like p -> (q -> p)).

Another way to decide is how -long- your proofs are; removing rules of inference or restricting their action can make some things not provable, but sometimes it leaves them provable but just with much longer proofs (see for example cut-elimination)

• That's one of the things I am finding hard to do... evaluating these alternative systems based on what's provable in them, and testing them against my intuitions. I am liking relevant logic because I've never liked proofs like p -> (q -> p). But which route do I go. Do I bring in modality? should Commented Aug 20, 2011 at 17:39
• Also related en.wikipedia.org/wiki/Vacuous_truth and perhaps en.wikipedia.org/wiki/Dialetheism Commented Aug 28, 2011 at 0:32

The SEP article on Paraconsistent Logic describes a number of systems that deny logical explosion; perhaps you can begin there.

In terms of everyday life, most paraconsistent systems function just as you'd expect-- you operate identically to classical logic until you reach a contradiction; if you do reach a contradiction, you note it and move on, with nothing else necessarily affected. In this way, paraconsistent logics are much more adaptable to the real world than classical logic, where upon meeting a contradiction, one would be forced to accept any and all claims whatsoever, and forfeit any hope of rational argument.

• I am not sure if all paraconsistent logics operate in the same way as classical logic until a contradiction happens... Non-classical logic usually affect what you can prove. Relevant logic for instance was made to avoid the paradoxes of material and strict implication which are tautologies in classical logic. For instance, p → (q ∨¬q) is not veridical in relevant logic. I did look at the options in the SEP article, I just have problem evaluating the options. How do you make sense of 5 value logic for instance, is that a better way than fuzzy logic? Can't pick one with confidence. Commented Aug 20, 2011 at 17:17
• I don't find there are any problems with using classical logic in the real world. When confronted with a contradiction, this is just the punchline in an absurdio which indicates that I made a poor assumption someplace. Commented Sep 6, 2011 at 14:30
• @Niel: Not necessarily-- to a follower of Dialetheism (plato.stanford.edu/entries/dialetheism) there are true contradictions on the world. Commented Sep 6, 2011 at 14:38
• @Michael Dorfman: it does not follow, from the fact that "Dialethiasts" believe that there are true contradictions, that someone who uses classical logic will agree, or find their preferred mode of reasoning in any way inadequate. Commented Sep 6, 2011 at 14:43
• @Niel: Sorry, I wasn't clear enough. I wasn't disagreeing that one could use classical logic in the real world. I'm just suggesting that when one is confronted with a contradiction, it is not necessarily an indication that one made a poor assumption someplace-- it could be the case that the contradiction is real. In other words: when confronted with a contradiction, one could remain committed to Classical logic, and find a incorrect assumption, or, in the absence of one being found, one could embrace Dialetheism and accept the contradiction as valid. Commented Sep 6, 2011 at 14:51

It's not quite correct to think that deriving a contradiction implies all propositions. What deriving a contradiction does is prove a modal realm to have arbitrary truth values. It says that, in order for your argument to be valid, you have to live in an alternate, crazy, universe, where anything is possible.

If an argument leads you to ~A==A, it doesn't break the world. It breaks your argument. The solution is to fix your argument. This is the whole point of Paraconsistent Logic: to force people to abandon flawed arguments.

There are a great many questions in philosophy that are considered important, but which rely on apparent contradictions: "Do I exist?", "Can god create a stone heavier than he can lift", "This sentence is not provable". It's an interesting feature of human linguistic ability that we can create seemingly meaningful sentences like those. HOWEVER, to then spend hundreds (thousands) of years arguing about them is viewed as counterproductive by many philosophers. We could afford to sit by the fire and sip sherry and indulge ourselves back when philosophy included all science. Now? Not so much.

• I thought in paraconsistant logic contradictions don't break your arguments. Solution is to fix what's inferable from contradictions. e.g. Dialetheias or true contradictions can be cashed out as a 3rd value in 3 valued logic (true, false, and both/maybe/neither). But Just what on earth is a 'true contradiction?' How to make sense of that, let along evaluate that. This is why it's so puzzling for me. Even the most straight forward solution of simply treating propositions like 'this proposition is false' as a function to the 3rd truth value is hard to understand with real world in mind. Commented Aug 20, 2011 at 17:31