LP: A Logic in Which Contradictions Can Be True

The law of non-contradiction states that it can’t be the case that a sentence and its negation are both true.  In other words, no sentence can be both true and false.  This principle is regarded by many (perhaps most famously Aristotle) to be the most fundamental logical law, and, throughout the history of philosophy and logic, very few have questioned it.   

One might think that logic itself forbids contradictions.  It is, of course, true that classical logic does not permit contradictions without triviality.  Classical logic contains the principle of “ex falso quodlibet,” or “Explosion,” which says that, from a contradiction, anything follows.  However, it is perfectly straightforward to define a formal logic that permits contradictions without triviality. 
​
In Chapter 11 of my Introduction to Logic textbook, I present the Logic of Paradox (“LP” for short), a formal logic developed most notably by Graham Priest, in which a sentence can be both true and false.  I present LP in terms that are completely accessible to anyone who’s had just a basic introduction to logic (which you can get, among many other places, by skimming through the first five chapters of the book). 

There are several reasons why one might want to use a logic in which contradictions can be true.  In the chapter, I outline three of them.  Very briefly, they are the following:

First, even if one thinks that there can’t possibly be any true contradictions, one might want to reason about impossible scenarios in which there are.  For instance, Priest tells the story of Sylvan’s Box, which contains (and does not contain) an impossible object.  A box with such contradictory contents is surely impossible.  Nevertheless, we can reason about what is the case and isn’t the case in the story, and, insofar as we can reason about this story in which an impossibility obtains, it’s reasonable to want to formally codify how we ought to reason about such a story.  LP is capable of doing that job.

Second, several philosophers and logicians have been inclined to think that there are at least some sentences for which both truth and falsity is a reasonable candidate truth value.  Most famously, consider the following sentence:

              The Liar Sentence (L): L is false.

Is L true or false?  Well, if it’s true, then what it says is true, but what it says is that it’s false, and so, if that’s true, then it’s false.  So, if it’s true, then it’s false.  On the other hand, if it’s false, then what it says is false, but what it says is that it’s false, and so, if that’s false, then it’s true.  So, if it’s false, then it’s true.  It seems, then, that we can’t maintain that it is either true or false without maintaining that it is both true and false.   Why, then, not say that it is both?  That seems to be as intuitive of a thing to say here as any.  LP enables us to say it.

Finally, there have been several philosophers in the history of philosophy who at least have seemed to have contradictory views, views that they’ve seemed to express with contradictory sentences.  For instance, the 2nd century Indian Buddhist philosopher Nāgārjuna holds that nothing at all has intrinsic nature.  That, it seems, is the intrinsic nature of reality on Nagarjuna’s view.  Does, then, reality have an intrinsic nature?  Nāgārjuna ​seems to say, “Yes and No.”  He writes, “All things have one nature, that is, no nature.”  Jean Paul Sartre’s view of the self seems to take on a similarly contradictory status in the context of his philosophy, with Sartre apparently explicitly endorsing this contradiction, maintaining “I am not what I am.”  So, it seems that there is at least one plausible interpretive line one might be inclined to take in reading such philosophers: their views really are contradictory and their contradictory statements really express their contradictory views.   LP enables us to take this line.

In the chapter of the book, I spell out each of these three motivations in some detail, and I then go on to officially lay out the logic of LP, providing its semantics, its definition of validity, and providing a sound and complete natural deduction system for it.  Whether or not there really are contradictions in reality, LP shows that, at least from a logical perspective, it is perfectly coherent to think that there are contradictions, and after working through the chapter, you'll be able to use LP to reason coherently about them.  Check it out!

---