Notes on "Degenerate Case" Dialetheism

 “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.)

In what sense can classical logic be wrong?

Failing to capture stuff is not being wrong, so for e.g. indicative conditionals not being material conditionals does not mean that classical logic is wrong, only that it doesn’t by itself handle the validity or otherwise of arguments involving indicative conditionals.

But the threat from truth-theoretic considerations, gaps and gluts, is different here.

Also the threat from the more general idea that there’s a relevant sense of logical consequence whereby explosion (ex falso quodiblet) isn’t valid. 

In both cases things come to a head with: this argument is valid according to classical logic but really isn’t.

With the first threat, the problem is not in the notion of validity—that can stay. With the second, the frame of mind is that of wanting a conception of consequence/validity in which explosion simply lacks that status, isn’t sort of pseudo-valid, i.e. valid in a stronger pseudo-logic where we ignore some real possibilities. 

With the first, you can say: explosion instances are often good arguments in some sense, even if not strictly valid. They’re truth-preserving w.r.t. all cases not involving dialetheia. Not, of course, good arguments in the sense that you’d ever follow them from premise to conclusion! But we can say yep, anything “follows from” a contradiction if we ignore models in which dialtheia occur. Provided you know you have no dialethia in the mix, you have your guarantee that you’re not gonna be led from truth to falsity. 

On this first conception, i.e. the dialethic one, how does classical logic err? Where does it go wrong? 

‘You say it is not possible for P&~P to be true while Q is false. But it is possible, because, when you interpret P, sometimes your classical model which corresponds to reality, while it rightly captures the fact that P is true (false), goes wrong in not also capturing the fact that P is false (true). Well, actually, in these cases, two of your classical models will correspond to reality.’ (One fix, make the valuation function a relation—on that implementation, we can say the classical model rightly maps P to T (F) but fails to also map it to F (T). Another, add a third truth value representing the dialethic status - but that’s a different mode of presentation and so you don’t get the perspicuous sense in which the classical model just leaves something out. — In the relation mode of presentation, you can take a classical model with one letter and get two full models - the one where you do nothing, and the one where you also map it to the other value. But with the three-value mode of presentation, while a classical model is straightforwardly still a special case of one of these full-story models, it is no longer the case that you can take a classical model of a situation and make it correct by only adding something (or doing nothing)-if you have a dialethia, you have to unmap it from T and instead map it to X. So that makes it look like the classical model has said something wrong — and of course we can look at a classical model what way, if we treat T as “true and true only” or regard the model as making an implicit claim to telling the whole story about which letters have which of the two properties truth and falsity. But here the principle of charity, and general good sense, should tell us to not regard that as part of classical logic itself. So let’s put that aside.

From this point of view, classical logic knows what validity is alright, and doesn’t get any individual thing wrong semantically, but the semantics is incomplete—the models lack information sometimes (and the way the notion of model is set up precludes putting it in). And this leads to cases where counterexamples fail to show up, because the classical models miss parts of the picture without which the picture doesn’t show a counterexample.

Notes on Modal Issues Regarding the Ontology of Propositions

Cross-posted here.

A true sentence like ‘John is here in this room’, and its Twin Earth counterpart, express different propositions, since they are about distinct people. And that means that propositions sometimes constitutively involve particular external things that they are about.

What, in light of this, should we say about how, if at all, what propositions there are—what claims exist—varies across possible worlds?

One side of this issue is: could propositions like the ones expressed by a normal true use of ‘John is here in this room’ have failed to exist? Do they fail to exist in (or with respect to, perhaps?) all worlds in which John does not exist? (I set aside Williamsonian necessitarianism about what there is.)

My notion of the internal meaning of a sentence, or the way it is used, gives me a way of agreeing that there’s something right about the idea that the meanings of sentences are just there and exist necessarily. Given a normal occurrence of ‘John is here in this room’, the way the sentence is being used—which it has in common with its Twin Earth counterpart—may be regarded as a pure abstract object, like a way of dancing, which we can say is just there and could in no sense have failed to exist.

Here is another question we might ask: propositions about particular people and physical things—suppose they do exist in some possible worlds apart from the actual world. But do they themselves have different properties in worlds where the things they are about have different properties? A way of using a sentence, we might say, is just what it is and doesn’t have different intrinsic properties at any rate in different worlds—it will of course have different extrinsic properties such as ‘having been instantiated by someone wearing a blue hat’. But if a claim constitutively involves the object it is about, is the object with respect to the claim like a diamond set in a piece of jewellry, so that the piece’s properties change whenever the diamond’s do, since the diamond is part of it? I think perhaps this need not be so. We could instead use the model of something which needs to be tied to something else, and which disappears, or at least ceases to be that thing, if we cut the tie or remove the something else.

A tremendous complicating factor is that there are undoubtedly, in some sense, claims about things that do not in fact exist. We cannot here follow Kripke in Reference and Existence into the view that these sentences do not in fact express propositions, anymore than we should follow him in analyzing particular existential statements as talking about whether there is such-and-such a proposition. (That theory is I think clearly tortured but this is not the place to mount objections but see Postscript.) And recall there that even Kripke was keen to avoid the seeming absurdity of having to hold that the correct analysis of a statement can depend on whether it is true or false. However! It seems to me there is one thing in this general vicinity which we might indeed have to come to terms with. Namely, that the modal profiles and identity conditions of propositions expressed by statements involving names that happen to be empty differ from those of propositions expressed by statements which are being used in exactly the same way but where the names aren’t empty. 

Someone might want to say: just because we can’t pick out particular propositions about physical objects and people etc. that do not actually exist, doesn’t mean they don’t exist. (Anymore than the fact that non-actual people can’t pick out our propositions means that they don’t exist.) But this is only really correct given something like Lewisian modal realism.

‘What if Vulcan had existed?’—Are we to follow Kripke in his view of unicorns and apply that even to the case of names, i.e. say that there is no particular possibility in question at all here? A lot of what I am otherwise tending toward does seem to be leading me that way—but I suspect that here the shoe might really pinch, and that dwelling on this part of the issue and trying to do it justice will lead to a breakthrough—-a better view. A kind of more nuanced view which, pace recent Williamson, would not be a case of overfitting.

‘What if the claims made by some astronomers about Vulcan had been true? I don’t mean what if they had been right when they spoke. I mean, consider the claims they expressed about Vulcan. What if those claims had been true? Is there a possible world in which they are true?’

Postscript. It seems a very important objection to Kripke’s analysis of negative existentials in R&E that he is kicking the can down the road. For how does it get to be true that ‘There is no such proposition as that Vulcan exists’ expresses a true proposition? If we interpret it metalinguistically, it’s wrong as an analysis. So then how do we interpret it? The ‘no such’ has a soothing effect and as it were shrouds the occurrence of ‘Vulcan’ in a haze. But we still need to account for what it’s doing there and how we get different statements when we pop in different empty names.

This post is dedicated to the memory of the late Queen Elizabeth II.

Meaning and Metaphysical Necessity - now out with Routledge

 

My book Meaning and Metaphysical Necessity is now out with Routledge. I began seriously developing the ideas in it in 2011 when I began my PhD, which is also the year this blog started. Many of the posts here over the years were devoted to working out the views in the book.

The Threefold Root of the How-Question About Mathematical Knowledge

Platonism is the default, almost obviously correct view about mathematical objects. One of the major things that puts pressure on Platonism is the question 'How do we know about mathematical objects, then?'. What gives this question its power? I think three things conspire and that the third might be under-appreciated:

1. Real justificatory demands internal to mathematical discourse. For particular mathematical claims, there are very real 'How do we know?' questions, and they have substantive mathematical answers. The impulse to ask the question then gets generalized to mathematical knowledge in general, except that then there's no substantial answer.

2. A feeling of impossibility engendered by a causal theory of knowledge. If you only think about certain kinds of knowledge, it can seem plausible that, in general, the way we get to know about things is via their causal impacts on us. This then makes mathematical knowledge seem impossible.

3. Our deeply-ingrained habit of giving reasons. The social impulse to justify one's claims to another is hacked by a monster: the philosophical question at the heart of the epistemology of mathematics.

If it were just 1 and 2 getting tangled up with each other, the how-question would not be so persistent. With existing philosophical understanding we'd be able to see our way past it. But 3 hasn't been excavated yet and that keeps the whole thing going.

A Puzzle about Abbreviation and Self-Reference

Let us use 'ONE' as an abbreviation of 'This sentence token contains more than one word token'. Now consider whether the following is true:

    ONE

Major Inaccuracies in Misak's review of Journey to the Edge of Reason, a new Gödel Biography

In the Times Literary Supplement there is a review of a new biography of Gödel by philosopher Cheryl Misak. The review is called 'What are the limits of logic? How a groundbreaking logician lost control'. This paragraph contains two major inaccuracies:

Gödel proved that if a statement in first-order logic is well formed (that is to say, it follows the syntactic rules for the formal language correctly), then there is a formal proof of it. But his second doctorate, or Habilitation, published in 1931, showed that in any formal system that includes arithmetic, there will always be statements that are both true and unprovable. The answer to the Entscheidungsproblem was, therefore, negative.

The first one is that being well formed is like being grammatically correct - among the well formed formulas of first-order logic, there are formulas that are false no matter what (false on all models or interpretations), formulas that can go either way, and formulas that are true no matter what (these ones are often called logical truths, or logically valid formulas). What Gödel showed is that for every logical truth, there is a proof that it's a logical truth. 

The second major inaccuracy is that the answer to the decision problem (Entscheidungsproblem) is not shown to be negative by the incompleteness theorem that Misak alludes to. The negative answer became known only in 1936, when Alonzo Church and Alan Turing independently showed it.