Maureen Eckert (left) and Graham Priest (right) on deviant logic.

According to classical systems of logic, anything follows from a contradiction: the relation of logical consequence is explosive. But recent decades have seen growing interest in “deviant,” paraconsistent systems that include non-explosive relations of logical consequence. Further, some deviant logicians, such as Priest, assert the existence of dialetheias (true contradictions). In this conversation, Eckert and Priest discuss whether and how deviant logic should be studied in the undergraduate classroom. Then (starting at 29:40) they look for dialetheias in the areas of emotions, legal norms, and contradictory fictions.

Related works

by Eckert:

Edited (with Steven Cahn): David Foster Wallace, __Fate, Time, and Language: An Essay on Free Will__ (2009)

Edited (with Robert Talisse): __A Teacher’s Life: Essays for Steven M. Cahn__ (2009)

by Priest:

__Doubt Truth to Be a Liar__ (2008)

__An Introduction to Non-Classical Logic__ (2001)

“Sylvan’s Box: A Short Story and Ten Morals” (1997)

I am quite happy to have watched this video. I am an undergrad studying philosophy in the US, and the only logic offered is classical. In fact, anything other than classical is considered “scary”. I hope to study some paraconsistent logic after modal logic, but that said, I find the idea of both P and ~P being true to be quite philosophically perverse. We shall see.

Hi Tyler,

I’m so glad that you were happy to watch the video. In my own background that I allude to, non-classical logic also seemed dark and scary. The chink in the C/NC divide for me began with modal logic. I doubt that you can do without some working knowledge of modal logic in your studies at some point, and there’s not really much to find scary about it. As Graham indicates, modal logic can be considered an extension of classical logic—a means to increase its expressive power. It doesn’t represent a slippery slope towards dialethism, so no worries. What got me interested in modal logic were philosophical debates about the status of possible worlds—the metaphysical issues were fascinating. Were possible worlds real, causally closed entities over which we could quantify, as Lewis argued? Or could they be utilized with much lighter metaphysical commitments (Fictionalism, Ersatzism)? It was pretty interesting stuff to me and, moreover, there were numerous contemporary debates in other areas of philosophy—Philosophical Zombies in philosophy of mind, for example—that made me confident that it was worthwhile studying modal logic.

I’ve mentioned this to encourage you to look into modal logic. And I can assure you that things get even more interesting. While studying classical logic, you may have come across the paradoxes of the material conditional and strict implication, or you might have wondered about the way certain inferences that are valid in classical logic do not depend on the relevance between premises and conclusion. This is where research in Relevance Logics comes in—you can find a whole book by Stephen Read online: http://www.st-andrews.ac.uk/~slr/Relevant_Logic.pdf.

When it comes to paraconsistent and dialethic logics, you still have options with different commitments about the status of contradictions, as we tried to elucidate. I think it’s really good to be skeptical about just what one wants to commit to in this case. I like that what it takes to be able to decide for oneself—say, choosing between paraconsistent and dialethic logics or between Jc Beall’s dialethism and Graham Priest’s—are sophisticated questions. For instance, as we briefly mention, applications of paraconsistent logics are interesting and important if one is asking questions about what one is to do with inconsistent data sets. How information is grouped and treated matters in artificial systems—and even natural systems (like minds) that are modeled artificially. While research in logic seems quite “pure,” this isn’t quite the whole story. You may find in the future that, as you study more philosophy, your resources for understanding differences between, say, Jc Beall and Graham Priest, increase as well. I learned about the deflationary theory of truth in graduate school, but nothing about dialethic logics. Later on when I approached this area of logic, I already was familiar with some of the issues. “Light dawns gradually on the whole,” and I don’t think that an undergraduate student needs to decide right away about where they stand. I do think that knowing that there’s so much more to the field of logic is pretty good information to have.

Very best wishes,

Maureen

great episode! I haven’t study logic much so I have nothing really relevant to say.

Although, something less-than-really relevant, is my interest in russell’s paradox, which is interesting to me mostly because I do not see the paradox in it! If a contradiction arises from supposing a set may be a member of itself, then I see this as obvious evidence that somehow the basic nature of a set as a group of things is that it cannot be one of its own members. Besides, it seems fairly intuitive that a thing cannot be in both places at once. Does this make any sense? I can elaborate if anyone is interested.

Hi Maureen,

I really enjoyed this. Great to see Priest in extended conversation with someone thinking intelligently about these issues. Given Priest’s views on true contradictions being the very stuff of many metaphysical limits (Beyond the Limits of Thought) I was surprised that, when you were probing this kind of area, he did not mention such limits as real contradictions that exist ‘in the real world’. Perhaps he does not think they are quite real enough? What do you think?

Your discussion of emotion was very interesting and I would like more exploration in this area. Priest was right to point out that hate is not quite the same as not-love and therefore there may be no contradiction in hating and loving someone at the same time; but it does seem to me that to hold some contradictions in our heads (or, perhaps, better, to FEEL some contradictions) has something of the essence of being human (as opposed, say, to being monkey or computer).

It was interesting that you focussed on the pain, arising in grief, of feeling both presence and absence. The negativity arising this particular contradiction, I would argue, is the exception rather than the rule. The awareness of true contradictions often leads to feelings of unity and completeness which, far from being unpleasant, are the very stuff of revelation as attested by religious writers as diverse as Meister Eckhart and the great sufi mystic Hazrat Inayat Khan. There is more than a whiff of Hegel here in this notion of resolving things, of course, and perhaps there is no harm in saying than there is some sort of cognitive synthesis of contradictory emotions which leads to these deeply positive feelings. It is not the holding/feeling of such contadictions per se which is important but the fact that doing so successfully points beyond such contradictions to some unified state in which no conflict exists.

This synthesis is often not immediate however. Often it is hard won. Cf Augustine: ‘O Lord, grant me continence [that I may not ruin myself], but not just now!’ This synthesis is some kind of ‘cognitive breakthrough’, but I don’t know enough about this area to say any more on the psychological aspects.

I think one can build a broad historical case, however, to show that those who live happily with such emotional contradictions often give up on what we might call strictly ‘rational’ enterprise – they become poets or musicians or religious – whereas others who, either by chance or by emotional design never confront this issue, do not take true contradictions to be any part of a world view. Such an idea simply never enters their consciousness or, if it does, they banish it as too destructive to normal thinking.

What is so remarkable about Priest’s work is that it extends logic towards these insights from other sorts of thinkers. It moves towards joining rational, logical traditions and mystical poetic ones – though I’m not at all sure he would see it like that!

Nathan,

By the Axiom Schema of Comprehension: If P is a property, then there exists a set Y= {x : P(x)}; there exists a set of elements with that property. This definition allows for Russell’s paradox. So, consider the property, P:= Is not a member of itself. As you point out, this is an intuitive enough property. But, if Y is the set of *all* sets such that the set has property P then Y is paradoxical and a contradiction follows. For, assume Y is a member of Y. Then Y has property P – i.e. it is not a member of itself. On the other hand, if Y does not have property P – i.e. it is a member of itself – then Y is not a member of Y. So we have both that Y is a member of Y and that Y is not a member of Y. I rather like the cute version: If a barber shaves those and only those who do not shave themselves, then who shaves the barber?

You also state that, “I see this as obvious evidence that somehow the basic nature of a set as a group of things is that it cannot be one of its own members.” Many would agree; it makes no sense to have a set that can have an element of the same rank. By the Schema of Separation: If P is a property, then for any X there exists a Y={x ∈ X: P(x)}. Now there is a hierarchy of sets: Either Y=X or Y is a subset of X. However, the paradox, and solution, posed a problem for Frege, whose program of trying to represent the laws of arithmetic simply as a branch of logic was now crippled since it relied on naïve set theory. The interesting question is, in light of the wonderful discussion between Dr. Eckert and Dr. Priest, can naïve set theory be salvaged in light of the possibility of dialetheia?

Please forgive any conceptual blunders I have made. All of this is new to me, but quite fascinating. Nonetheless, I am quite prone to error, *especially* when dealing with such rich and complicated material.

Ted Locke

Undergraduate at the University of North Florida

Hi Ted,

Are you saying you see it as I do, or that there’s more paradox than that?

It seems to me that assuming sets cannot be members of themselves solves all the paradox. I believe this is something like Russell and Whitehead’s “theory of types” from Principia Mathematica, but that, so I hear (for I dare not read it myself), was taken down in the end by the paradox. I don’t see why there approach wouldn’t have worked – mine is similar.

Take “X is not a member of itself,” the root of the paradox. It’s not as obviously fixed (by assuming sets cannot be members of themselves) as “Y is a member of itself,” but fixable still. If no sets are members of themselves, then X is the set of everything. But as this includes other sets, X, like Y, does not exist.

Best,

Nathan (I’m an undergrad at UC Santa Cruz :D )