“Degenerate Case” Dialetheism
Motivation: trouble with even the most sophisticated and beautiful gappy approaches e.g. Kripke - the ‘not true’ and samesaying. Priest’s view really is better in a way. A resting place. But! Don’t get over-excited now! It’s not as if we should start to think that reality is full of contradictions or something, and look for other cases, and think of this as some kind of instructive extra special mindblowing truth - contradictory truth. As if, if you could understand this fully, that’d be the best shit. Rather, paradoxical statements are a degenerate case -
Immediate difficulty arises here: risk. The statements aren’t themselves degenerate.
Kripke’s lesson about empirical Liars is very important. (Is he the originator of the point? I wonder. It’s not as if this is even widely isolated as a potential contribution afaick.)
What is really the difference between this view and that of Priest et. al.? Perhaps it lies largely in a difference of opinion about how much this should influence logical theory, how front-and-centre it should be in our conception of logic. I think in many ways it is perfectly good to continue with bivalent logic even if accepting dialetheism about Liar-like paradoxical statements. Why does it seem such a threat? Well, it has to do with the universality pretensions - aspirations, I should say - of logic. It obviously is no good for logic mainly to issue in truths of the form ‘usually, …’ or something like that. But that is not the only way to view it - we can still say that bivalent logic is absolutely correct for all non-paradoxical statements. And yes, when we ascribe truth we run a risk of paradox, as Kripke emphasises.
Even though we do risk dialetheia, and
I talk to my students about it and make it natural to motivate LP: If you’re a dialetheist you are pushed to do nonclassical logic, since
I agree that Priest’s logic is more generally correct because it handles paradoxical cases - i.e. we do not have to stop applying it and forming our beliefs in accord with it in response to paradox, it’s idiot-proof w.r.t. paradox.
But modus ponens fails! Etc.
Look, it’s technically correct that LP is more general, and that is fascinating- some logical principles are paradox-proof and others aren’t. But at the end of the day, there is a lot to be said for paying attention to classical logic. Because paradox so rarely actually fucks us up.
((Try to think about how one might actually be led astray “applying classical logic” when in fact there’s a paradox. How does the risk show up exactly.))
((Could there be a kind of interesting contingency to this view I’m developing? Paradox rarely fucks us. Could we imagine a hypothetical scenario in which the risk is a big deal and for natural reasons we often fall into paradox? That could be very interesting.)
OK MAYBE HAVE SOMETHIGN
OK,
But, if you apply LP to your beliefs and that includes paradoxical ones, you will still have more false (but also true!) beliefs by inferring from paradoxical propositions. You’re sort of still fucked, in the sense that you might have paradoxical beliefs. But full credit, it won’t give you plain false (i.e. not also true) ones unless you have plain false ones going in to the inference.
(What is K3 like?)
Relate to notions of ‘the correct logic’.
Let’s get down to brass tacks.
Do we not even tell people about modus ponens because that’s proven wrong by paradoxical examples? Of course not.
It’’s not some embarrassing oversight that bivalent classical logic is central in the logic curriculum.
What am I really arguing here? Well, looks like a bunch of empirical normative stuff, what am I doing? But really i’m doing the conceptual work of showing how accepting certain things i.e. dialetheism by the letter does not mean ‘abandoning classical logic’ in the full sense of chucking it out or regarding it as discredited. It’s just realising a peripheral limitation.
Can A Dialetheist Endorse Classical Logic Without Being Pushed to Believe Everything?
RNs
Ripley & someone review of Spandrels in Mind
A spandrel is an unintended by-product of a design choice.
But I don’t want to lose the framing of: Liar propositions are there to be asserted, whatever we do.
This part I think I want to go along with but will have to look into and think critically about the ‘merely semantic’ bit.
Beall holds dialetheia to be ‘merely semantic’ (p. 16). In other words, expressions not involving the truth predicate belong to a part of our language that is fully classical. The only dialetheia countenanced by the view are side effects of the introduction of semantic vocabulary: paradoxes and other ‘ungrounded’ sentences. For this reason, Beall regards his view as a modest brand of dialetheism.
Abf
Degenerate Case Dialetheism - Beall’s view may be, or be close to, an instance of this. But we can get to the dialetheism not by Beall’s particular route in detail - his ttruth setup
The idea of ‘spandrels’ seems to motivate the choice of a logic as close to classical as possible while holding to non-triviality; and while it is unknown just how close is as close as possible, we can get considerably closer than BXTT. (All of the above-mentioned axioms and rules are classically valid.) Spandrels would be improved by a discussion of the known non-triviality-preserving principles that can be added to BXTT, and arguments as to why they should not be added, if indeed they should not.
This idea of getting as close as possible to classical logic is very interesting. And I think my emerging suspicion about the Liar and my view of what it means makes me think this is wrong-headed. It would be interesting to try to work this through.
What I mean is that I have a hunch of the following kind: LP presented semantically is faithful to the phenomenon and there’s no reason to think it’s wrong except wanting more classical stuff to come out valid without getting triviality/explosion.
One way to strengthen this would be to explain how the model theory and designated value choice gets things right.
((Beginning to think about this just now. I do wonder about the idea that if one disjunct of a disjunction is just-true but the other is both true and false, the disjunction is just true. But I think it may be good.))
Another way is to look for cases where some dialetheist-endorsed logic goes beyond LP and try to explain why, from the dialetheist perspective, it shouldn’t.
I’m just thinking now: it seems natural to think that in many cases where we want to apply logic, we want just-true conclusions. But who cares if we needed some dialethia to get to them? The perspective is: arguments are just-truth machines, but if we can fuel them with a mix of just-true and both premises as well as all just-true premises, all to the good, no reason not to use this degenerate fuel if it works.
(‘But the classical machine has more validities!’ Yes but sometimes we might want our deduction to be paradox risk proof, in the sense that our conclusion is still just-true even if some of our true premises were also false.)
The logics I’m aware of that we get by having 3-valued semantics and designated values all make double mention of the designated value(s): when talking about the premises, and when talking about the conclusion. So we can make a different choice each time. And that is the natural thing to do on this picture:
An argument is valid iff every model M which makes each premise either 1 or X makes the conclusion 1.
***
It is remarkable how quickly we tend to be ready to move from dialetheism to the revision of logic. Perhaps part of this is that we are so aware of it being a radical move, that radicalness is so salient, that revising logic no longer shows up as a big thing because it is side by side now with dialetheism. All bets are off in a sense.
The decisive move in the conjuring trick was the one that escaped notice.
Then one really gets used to degenerate-case dialetheism about liar-like paradox and starts to embrace it and then one is forced to reckon.
One funny thing is that many dialetheists nowadays may agree with me that it’s fine and perhaps even good to start logic instruction with bivalent logic. I am in danger of seeming to have no distinctive position. The difference is subtle and is about what is emphasised, what is treated as an unimportant practical matter, etc.
***
One thing is very important: logical laws that fail in LP but hold classically do not automatically become only maintainable in a ‘usually’ form. We have a concept of nonparadoxical sentences and propositions and we can maintain that the laws hold of absolutely all of them.
(We must remember that logical laws are in the first place thought of as applying to sentences or schemas meant in a particular way or reckoned as part of a language—or alternatively to propositions. It’s not as if they are about absolutely everything including say oranges (at least, not in the relevant sense). This is a hard point to bring out but it feels important.
Approached from another angle: if logic is just about a particular kind of thing—declarative sentences, propositions—why does it have such an aura of generality? Because this particular kind of thing plays an enormous and widely ramified role in our lives! Especially if we’re “intellectual types” and especially if we’re a certain kind thereof! Seen this way, excluding dialethia from standard logic’s scope is no real cause for embarrassment or disappointment or disenchantment. There is this odd kind of case, there are interesting proposals about how to include it, and that’s it. It just is what it is.
I need to say more about the two broad approaches of dialetheists - LP or something like that with the model theory thought of as principled and tracking what’s really going on vs. this kind of industry I’m dimly aware of of trying to get back as much classical stuff as possible while not exploding. I should perhaps talk to Ripley about this.
This all connects up with my perspicuous representation stuff too. Because wanting cool stuff to do is now one reason why classical logic gets short schrift or denigrated merely for its bivalence and explosion. Seeing that there’s plenty of other cool stuff to do without messing with bivalence and explosion makes it clearer that there are multiple directions for logic to grow in and we can rightly regard classical logic with standard modern notation as the centrepiece. (For us, at this time in history, and given our nature.)
I don’t want to be stupid about the role of paradox in modern logic. It is there. It is undoubtedly very important. It notably plays a vital role in particular (here I mean ‘vital’ quite specifically, wanting the association with life and all that, it’s not merely functioning as a synonym of ‘important’ here).
“Look, we wanted to know which argument forms were valid. And in light of the liar and dialetheism being the truth about it, we have to accept that lots of the ones we thought were just aren’t! Don’t try to weasel out of that”
There’s a lot in that. One interesting thing emerging for me here is that it’s not merely the ‘well you don’t believe everything do you, so you have to reject explosion’ thought which leads to rejecting explosion. You can also just track through the definition of validity and show invalidity ‘Look, it is possible after all for A ∧ ¬A to be true after all (and hence possible for it to be true while some conclusion B is false). What we overlooked is that A is in some cases both true and false, and so therefore is the conjunction of A with its negation.
Another good thing I’ve just noticed—it feels like good news for my point of view: I think we can maintain the generality of classical logic in another way (in contrast to saying it’s about the nonparadoxical sentences/propositions). Namely, by revising what we mean by ‘valid’ usually—namely, being clear that we’re talking about ruling out the case of the premises all being just true (i.e. not also false) and the conclusion not being so. (Alternative worth comparing: changing the last bit to ‘the conclusion being just false’.) (The latter could be implemented technically with a notion of “anti-designated” value.)
