Thursday 14 March 2024

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.

Friday 9 September 2022

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.

Tuesday 28 June 2022

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.

Friday 3 December 2021

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.

Saturday 20 November 2021

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:


Friday 5 November 2021

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 biography 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.

Thursday 21 October 2021

Reversing Logical Nihilism - Forthcoming in Synthese

Final draft available on PhilPapers.

Abstract: Gillian Russell has recently proposed counterexamples to such elementary argument forms as Conjunction Introduction (e.g. ‘Snow is white. Grass is green. Therefore, snow is white and grass is green’) and Identity (e.g. ‘Snow is white. Therefore, snow is white’). These purported counterexamples involve expressions that are sensitive to linguistic context—for example, a sentence which is true when it appears alone but false when embedded in a larger sentence. If they are genuine counterexamples, it looks as though logical nihilism—the view that there are no valid argument forms—might be true. In this paper, I argue that the purported counterexamples are not genuine, on the grounds that they equivocate. Having defused the threat of logical nihilism, I argue that the kind of linguistic context sensitivity at work in Russell’s purported counterexamples, if taken seriously, far from leading to logical nihilism, reveals new, previously undreamt-of valid forms. By way of proof of concept I present a simple logic, Solo-Only Propositional Logic (SOPL), designed to capture some of them. Along the way, some interesting subtleties about the fallacy of equivocation are revealed.