Maybe no "metaphysical" notion is needed and we just speak in terms of "plausibility" and "usefulness" but these seem to easily become even murkier notions. The two most common versions of pluralism (Beall and Restall and Shapiro) cited have very different notions of which logics should "count" for instance. — Count Timothy von Icarus

My intuition is that the rules which bind coming up with mathematical formalisms are the same as those which govern writing fiction. They're in general loose, murky, descriptive, but you can tell a good description from a bad one. I'd also want to liken the relationship of formalisms to their intended objects, or intended conceptual content, to the relationship some writers have with their characters. They don't always know what the character wants, how the character would react, but they'd be able to work through how they'd feel and act if they put them in a scenario. That lets them write in a manner true to the character. I think formalisms have a similar "true to the character" expressive flavour, and the concept of an interpretation lets you come up with "scenarios" and "story beats" to flesh out the understanding of the concepts and what's written about them. Interpreting your own symbols in that extensional sense is a way of finding the meaning of what you've written. And just like writing fiction, you can find the conceptual content very resistive to your expression. A theorem may escape you just like how to put a scene.

My intuition is also that there are other principles that set up relations between the practice of mathematics and logic and how stuff (including mathematics) works, which is where the metaphysics and epistemology comes in. But I would be very suspicious if someone started from a basis of metaphysics in order to inform the conceptual content of their formalisms, and then started deciding which logics are good or bad on that basis. That seems like losing your keys in a dark street and only looking for them under street lamps.

And so it is up to monists to show what it is that all logical systems have in common. I don't see that it can be done. — Banno

It could be thought of as a regulative principle -- here we have multiple logics, but we would like them to cohere: the monist would then be more of a project than a position, the attempt to build a logic which contains all logics. (one could presumably derive the LNC from this meta-logic, for instance -- but it's just an idea)

Trying to get out of this thread, but...

a stipulated logical monist of a certain sort, that there is only one entailment relation which all of these logics ape. — fdrake

I called the pluralism/monism debate an internecine debate between Analytics because they are all univocalists. Your word "ape" here is doing a lot of work, but it seems that for both pluralists and monists such an entailment relation will be purely univocally predicable. This is why the whole game is so boring. The interesting question is an adjudication between two different paradigms, and folks like Banno and probably G. Russell are eternally stuck in a single paradigm, interpreting the other paradigm in their own terms.

My definition of logic via the Meno is something like, "That which creates discursive knowledge" (or perhaps just knowledge). Now is knowledge or discursive knowledge a univocal concept? I don't think so, and therefore there can be no univocal "entailment" relation that holds for all knowledge. For the univocalist this means that each kind of knowledge and each accompanying entailment relation are hermetically separate from every other kind, and that is precisely what analogical predication denies. This is probably something like Wittgenstein's "family resemblances," although I am not overly familiar with Wittgenstein. (And note again how drastically this univocal analysis deviates from natural language use.)

In the realm of circles we are asking about the relation between the pretheoretical grasp or notion of a circle and the formalization. We could say that the formalizations "ape" that pretheoretical notion, but the scrutiny here is entirely on the manner of aping. Yet in the case of knowledge there is something more concrete and even practical at stake in the question.

No True Scotsman doesn't admit of an easy formalisation in terms of predicate logic — fdrake

Yep.

I imagine monists are generally going to just deny this, because monism is about logical consequence relative to some non-arbitrary context — Count Timothy von Icarus

I think this is part of it too.

I'd also want to liken the relationship of formalisms to their intended objects, or intended conceptual content — fdrake

That is the big equivocation for me. Is it a relation to a non-mental reality or merely a conceptual content? Timothy's point about non-arbitrary contexts hinges on the answer.

My intuition is also that there are other principles that set up relations between the practice of mathematics and logic and how stuff (including mathematics) works, which is where the metaphysics and epistemology comes in. But I would be very suspicious if someone started from a basis of metaphysics in order to inform the conceptual content of their formalisms, and then started deciding which logics are good or bad on that basis. That seems like losing your keys in a dark street and only looking for them under street lamps. — fdrake

Sure, and I don't think this is controversial. But I don't think you've given a straight answer to the other side of the coin: are some formalisms truer than others? Is there better and worse metaphysical fan fiction? That's the nub. (And some grandchild of logical positivism is operative here, because the formalists are liable to say, "This question is not formally adjudicable, therefore there is no better or worse metaphysical fan fiction.")

(This central topic has been sidestepped in all sorts of ways. Wanting to talk about modeling or "correctly assertible" rather than truth is one of those ways. If some metaphysical fan fiction is better than others, then it is truer than others, and there is (non-deflationary) truth to be had.)

@Banno
I think Leontiskos thinks logic is the Anima Mundi. Very medieval.

My intuition is that the rules which bind coming up with mathematical formalisms are the same as those which govern writing fiction. They're in general loose, murky, descriptive, but you can tell a good description from a bad one.

Yes, I would agree with this.

If I understand correctly, you're using "deflationary" to mean restricting the interpretations of a theory to all and only the ones which are syntactically appropriate and clearly within the logic's intended subject matter. Like propositional logic and non-self referential statements. Effectively removing everything that could be seen as contentious from the "ground" of those systems. Which would then ensure the match of their conceptual content with whatever objects they seek to model, (seemingly/allegedly) regardless of the principles used to form them. Which 'deflates' truth into unanalysable, but jury rigged, coincidence.

Well, as you say:

The paper gives lots of strategies for coming up with schematic counter examples to many, many things. You can come up with scenarios where even elementary things like "A & B... lets you derive A" don't hold. So much would need to be jettisoned, thus, if The Logic Of All and Only Common Principles was taken exactly at its word, in the sense of intersecting the theorems proved by different logics.

And that's kind of a knock down argument, when you consider X is true in system Y extensionally at any rate (which is AFAIK the standard thing to do)

It is a knock down argument, but it seems to miss what monists are claiming (at least from what I've seen). Or even what the pluralists say; Beall and Restall only endorse classical logic and a few sub-classical logics.

And I agree in terms of the standard, at least that seems to be a very common way to look at it in the discipline. But I am not sure it is a useful standard in this context since it seems to allow for refuting the dominant position(s) in terms in which its advocates wouldn't recognize it.

For instance, G&P frame the position they want to argue against as: "we define logical pluralism more precisely as the claim that at least two logics provide extensionally different but equally acceptable accounts of consequence between meaningful statements."

:smile:

Well,

folks like Banno and probably G. Russell are eternally stuck in a single paradigm, interpreting the other paradigm in their own terms. — Leontiskos

made me laugh out loud.

My definition of logic via the Meno is something like, "That which creates discursive knowledge" — Leontiskos

People create knowledge. I'm not following what his claims are here. Is he suggesting that we remember logic from our previous lives?

Your chat with him puts me in mind of Kripke's lecture on the surprise test paradox, such that he might reason as follows:

If I know that Monism is true, I know that any evidence against Monism is evidence against something that is true; I know that such evidence is misleading. But I should disregard evidence that I know is misleading. So, once I know that Monism is true, I am in a position to disregard any future evidence that seems to tell against Monism.

Or

If I know that Euclidean space is true, I know that any evidence against Euclidean space is evidence against something that is true; I know that such evidence is misleading. But I should disregard evidence that I know is misleading. So, once I know that Euclidean space is true, I am in a position to disregard any future evidence that seems to tell against Euclidean space.

Or

If I know that LNC is true, I know that any evidence against LNC is evidence against something that is true; I know that such evidence is misleading. But I should disregard evidence that I know is misleading. So, once I know that LNC is true, I am in a position to disregard any future evidence that seems to tell against LNC .

All quite sound reasoning.

But I would be very suspicious if someone started from a basis of metaphysics in order to inform the conceptual content of their formalisms, and then started deciding which logics are good or bad on that basis. That seems like losing your keys in a dark street and only looking for them under street lamps. — fdrake

Yep.

Excellent post.

square circles — Leontiskos

I think Lambda Calculus had this feel to it originally. It's reputedly the simplest language in which anything computable can be... computed.

For instance, G&P frame the position they want to argue against as: "we define logical pluralism more precisely as the claim that at least two logics provide extensionally different but equally acceptable accounts of consequence between meaningful statements." — Count Timothy von Icarus

Can you link me this paper please? If it hasn't been done already.

My definition of logic via the Meno is something like, "That which creates discursive knowledge"
— Leontiskos
People create knowledge. I'm not following what his claims are here. Is he suggesting that we remember logic from our previous lives? — Banno

Could be. Meno is part of Neoplatonic project building which wouldn't get much more than a blank stare from AP.

I'd thought of Meno's "paradox" as a precursor to bits of Wittgenstein- that there are ways of understanding (knowing) that are not the result of ratiocination. These include such things as "seeing as" instead of "seeing that", "knowing how..." instead of "knowing that..." and my favourite, PI §201, that there must be a way of understanding a rule that is shown in implementing it rather than in stating it.

It's a book, so sadly not wholly available from what I can see. Google books sometimes has a decent number of pages. There is a review by Erik Stei though and his recent book would be another example for how monists frame their own case.

https://ndpr.nd.edu/reviews/one-true-logic-a-monist-manifesto/

I have to say, I love the cheekiness of the cover.





It's:

If you know something, there is no need to inquire about it because you already know it.

If you don’t know something, you wouldn’t know what to inquire about or how recognize the answer when you find it.

And how will you enquire, Socrates, into that which you do not know? What will you put forth as the subject of enquiry? And if you find what you want, how will you ever know that this is the thing which you did not know? (Plato, Meno, 80d1-4)

It sometimes gets brought up in discussions of systematic search.

It is sort of related to P = NP as well. You might be able to tell if you have a correct answer easily if you are provided with one, but finding that answer can be effectively impossible, even if you have a description of what you are looking for that is a ridged designator.

I have to say, I love the cheekiness of the cover. — Count Timothy von Icarus

Thanks. It looks a little bit like a Chuck Tingle cover.

I have to say, I love the cheekiness of the cover. — Count Timothy von Icarus

Medhurst's Moses

https://commons.wikimedia.org/wiki/File:The_Phillip_Medhurst_Picture_Torah_457._Moses_breaks_the_tables_of_the_Law._Exodus_cap_32_v_19._after_Parmagiano.jpg

I've been unable to find a substantive account of the L∞G∞S Hypothesis. It's not apparent how it might deal with paraconsistent or non-binary logics, which have been the main concern here. It must be hidden in “OTL must contain logical constants for all the isomorphism-invariant relations over its models”

"L∞G∞S" looks more like the brandname for an aftershave than a worthy hypothesis.

Like probably everyone on TPF, I have read about paraconsistent logic as I read about animals in a far off land, but I have never worked with it or made use of it. — Leontiskos

My brush with dialethiesm, and thereby paraconsistent logic, came from my studies of the liar's paradox. So for me it's the result of reading arguments about the liar's and thinking dialetheism provided the most satisfactory answer. And actually this might be related since I read you here:

Are you asking me whether I think that accepting both paraconsistent and explosive logic results in the robust kind of logical pluralism? My guess is that I would answer 'no.' Paraconsistency does not entail Dialetheism. And paraconsistent logic is often used informally in everyday life (if that counts). — Leontiskos

First to answer the question, yes that's what I'm after: attempting to define what would count as a robust kind of logical pluralism. Here it seems you indicate that, supposing a defense of dialetheism holds, logical pluralism would count? Rather than paraconsistent logic, just the notion of true contradictions would at least ask for a different kind of logic, even if not paraconsistent, and so we'd be justified in saying there is at least two kinds of logic: the ones which reject contradictions, and the ones which utilize them in some way.

Also I don't mean to say I'm an expert by any means. Just an interested reader who thinks about these things.

I also haven't seen anyone in this thread who favors logical pluralism embrace Dialetheism - other than yourself, of course. They seem to be mostly Augustinians, "Lord, give me logical pluralism, but not yet!"

I also have ulterior reasons for taking dialetheism seriously, namely Marx and Hegel. Marx's notion of contradiction I have a good feel for (but because it's more extensional it's easier to untangle Marx's notion of contradiction from the logical one by dividing wholes into parts that differ), but Hegel's continues to mystify me.

And then one day I came across Priest in reading through the Liar's sentence and as odd as it is on its face it kind of slowly grew on me. I'm not sure of extensions of the dialetheism beyond the liar's, though Priest lists several (also including some Eastern philosophy too), but I think I like dialetheism as a solution the to the liar's paradox because it's a queer conclusion that comes from the plainest understanding of the liar's: no fancy logic is really needed. I can understand thinking the liar's is incoherent -- once upon a time I thought that because it's hard to imagine an empirical use for it-- but since this concerns logic alone, and may provide some inroads to other interests I have, I find it worthwhile in trying to comprehend and use. (Also, I think it could be a promising theory to develop in fleshing out the absurd, which is where I began originally -- taking the absurd as a metaphysics seems to indicate that logic cannot contain reality, especially the absurd parts -- logic's whole thing is making sense!)

The other response to the liar's I held was that the liar's sentence is simply false. It's telling you exactly what it is on its face. there is no evaluation necessary.

But the strengthened liar's sentence persuaded me that there is at least an interesting formal concern.

Now I sit and wonder what it takes the to contain explosion, if anything rational could be proposed in empirical (rather than conceptual) cases.

To address your concern about knowledge and logic's relation to it: I think this exercise demonstrates that we can't contain the world with logic, but rather we invent the logic to fit the world. It works because we've seen this or that enough times and so we follow the habits which reward us and call it truth*.

What's interesting about this line of thinking is that it's not denying even a metaphysical truth. But rather is showing how knowledge is produced: Guess and check. There is no method that guarantees knowledge. You just have to work things out the best you can.

So it's not entirely a dry academic consideration, to me. I see lots of interesting inroads with these ideas to other things I'm interested in, and the creative nature of it all gets along with what I think knowledge generation takes: making up new things and seeing if they work.

EDIT: An afterthought -- in a way the pluralist is actually more anti-nihilist than the monist. The monist has to hold that contradictory statements cannot be logically comprehended which is, in a way, a baby nihilism: Here is the field of inquiry where no logical rules hold. The pluralist says "Well, so far, perhaps... but what if we...."

*EDIT2: That looks dangerously close to a pragmatic theory of truth. It's off topic but I'm not a pragmatist, in spite of these sayings which would easily cohere with pragmatist theories of truth.

Almost like I read philosophy to figure things out that I still wonder about ;)

I also have ulterior reasons for taking dialetheism seriously, namely Marx and Hegel. Marx's notion of contradiction I have a good feel for (but because it's more extensional it's easier to untangle Marx's notion of contradiction from the logical one by dividing wholes into parts that differ), but Hegel's continues to mystify me.

Hegel's contradiction is pretty far from most paraconsistent logics, given the unity and "development" of opposites. If you're interested though, formalization attempts have run through category theory and Lawvere is the big name here.

https://www.google.com/url?sa=t&source=web&rct=j&opi=89978449&url=https://philarchive.org/archive/CORMAA-3v1&ved=2ahUKEwjrxdPIz6CJAxURlIkEHUmyEkcQFnoECCEQAQ&usg=AOvVaw3XxnDtBEih45jE5c2zfW2d

Nlab has some stuff on this too.

I have read many commentaries on the Logic at the point. Houlgate and Wallace are my favorites (Wallace isn't quite a commentary, but he does focus on the Logic), but Taylor was useful too. Despite this and now years of effort, I find the essence chapter largely impenetrable lol. But better minds then mine might have more success.

Hegel's contradiction is pretty far from most paraconsistent logics, given the unity and "development" of opposites. — Count Timothy von Icarus

I agree. I came to the same conclusion, and was disappointed. "Further research needed" :D

I enjoy the phenomenology, but only got 1/2 through the logic and couldn't say I understand it. I could tell it was not time to climb that mountain.

If you're interested though, formalization attempts have run through category theory and Lawvere is the big name here.

https://www.google.com/url?sa=t&source=web&rct=j&opi=89978449&url=https://philarchive.org/archive/CORMAA-3v1&ved=2ahUKEwjrxdPIz6CJAxURlIkEHUmyEkcQFnoECCEQAQ&usg=AOvVaw3XxnDtBEih45jE5c2zfW2d

Nlab has some stuff on this too.

I have read many commentaries on the Logic at the point. Houlgate and Wallace are my favorites (Wallace isn't quite a commentary, but he does focus on the Logic), but Taylor was useful too. Despite this and now years of effort, I find the essence chapter largely impenetrable lol. — Count Timothy von Icarus

Thank you! Next time I feel like trying the Kilimanjaro of philosophy I'll be referencing these ahead of time to prepare.

What if in place of Kant’s Transcendental categories we substituted normative social practices? Doesn’t that stay true to Kant’s insight concerning the inseparable role of subjectivity in the construction of meaning while avoiding a solipsistic idealism? Don’t we need to think in terms of normative social practices in order to make sense of science? — Joshs

That’s what pragmatist-hermeneutical and poststructural models of practice are for. For Hegel and Marx the dialectic totalizes historical becoming. In these latter models cultural becoming is contextually situated and non-totalizable. — Joshs

Yours has been the hardest to respond to for me. Hence my tardiness.

If we substitute normative social practices for Kant's Transcendental categories, what does that look like? In a very literal sense, which I don't think you mean but this is why I'm asking for clarification, I could substitute a model of practice for quality, quantity, relation, and modality -- substitution seems to need some relation of sameness, if not strict equality, and I'm not sure how practices would work within Kant's categorical frame.

I'd reach more for the ethics, but it becomes even more confusing there lol. So I'm reaching for what's making sense to me right now to respond in kind.

It is normativity all the way down.

How does this claim escape the charge of totalizing?

That's a bit beyond me. How does it fail?

so as I understand it it has much the same power as first order logic. Not a failing, quite curious actually. But difficult to work with.

:up:

A professor I had told me about a reading of Kant more in line with an Averroist "material intellect" shared by all men. That's another solution for the slide towards solipsism I suppose. And I believe it was also somewhat normative too, the constructive mind is the "mind of Europe," in which all participate and which has been so marked by Newton and modern science

Unfortunately, I don't recall the name of the person advancing it.

But I am not sure it is a useful standard in this context since it seems to allow for refuting the dominant position(s) in terms in which its advocates wouldn't recognize it. — Count Timothy von Icarus

Sider calls this "hostile translation." From the QV/Sider thread:

This is what Sider refers to as a "hostile translation" on page 14. It is interpreting or translating someone's utterance in a way that they themselves reject. — Leontiskos

@fdrake wants to talk about "good counterexamples," and he relies on notions of "verbatim" or "taking someone exactly at their word" (even in a way that they themselves reject). The problem is that if these are still hostile translations then they haven't managed to do what they are supposed to be doing: they haven't managed to produce good counterexamples.

It is interpreting or translating someone's utterance in a way that they themselves reject. — Leontiskos

I disagree that that is what is going on. When someone stipulates a definition, they are committed to that definition insofar as it relates to the intended concept. Rejecting a criticism of a definition on the grounds that the criticism doesn't portray your intents is a fine thing, so long as it isn't pointing out something which your stated commitments entail. Isn't this a basic idea in reasoning itself, playing out in how people codify ideas?

Indeed, you offered an alternative informal definition of logic:

"That which creates discursive knowledge" (or perhaps just knowledge) — Leontiskos

Which could equally mean "mind", "minds", "people", "institutions", "thought processes", "scientific experiments", "scientific theory", "perceptions", "deductive reasoning", "deductive reasoning using formalisms" and so on. Which are perhaps in the intended scope, and perhaps not.

But something like a research institution creates knowledge in a sense, and I doubt that is in the scope. And we could play the same game as we played with the formalisms out in natural language. What would make a research institution fail to be logic?

Is there better and worse metaphysical fan fiction? That's the nub. — Leontiskos

Yes. I thought it went without saying. Some things people think of are more appropriate than others in some contexts, and strictly better by some metrics. Some fiction is more valuable than others. If a thingy works better than another thingy on every relevant facet, the first thingy is better than the second thingy.

How would you judge that for a given context? Well I suppose you'd look for examples, see what pans out, provide definitions of things to see if they capture the relevant phenomena... Maybe you'd refine your criteria for what counts as a good thing in a given context based on the what you've seen and what's been created, too.

I still have the impression that you think of this is as an Objectively Correct vs Subjective-Relativist sense, and I don't want to accept the Subjective-Relativist role in the discussion since the proofs and refutations inspired epistemology of mathematics isn't relativist in the slightest, because its emphasis is on communities of people agreeing on what follows from what by following coordinating norms and demarcating those norms' contexts of application. Minimally then, it's intersubjective, and communities create knowledge about collectively understood subject matters.

If you read through Proofs and Refutations, which is an amazing book, the most clear cut resolution and associated proof of the book's central topic is offered using an entirely separate formalism than what had been considered up until that point. It was a substantial theoretical innovation and reframing that cleared away the old problems, but was nascent within them. Lakatos' approach has a dialectical flavour in that regard.

I'd thought of Meno's "paradox" as a precursor to bits of Wittgenstein- that there are ways of understanding (knowing) that are not the result of ratiocination. These include such things as "seeing as" instead of "seeing that", "knowing how..." instead of "knowing that..." and my favourite, PI §201, that there must be a way of understanding a rule that is shown in implementing it rather than in stating it. — Banno

I think Meno's paradox shows that some knowledge is innate. The story we surround that with probably reflects worldview. For Plato, it meant transmission from another level of reality. We might be mysterian about it and call it hinge propositions, or we can decide it must have come from evolution.

@Leontiskos, as this thread draws to a close, has still not addressed, and apparently is yet to read, the article from which this topic derives, nor even viewed the Lecture.

He entered this thread with an attack not on the topic but on on me: , and maintained that personal abuse. He sets out to frame the topic in strict Aristotelian terms, not talking of formal logic as it is now understood, indeed showing a neglect of that topic.

In the discussion of mathematics with @fdrake and others he repeatedly refused to consider the alternative maths on offer, insisting on framing each part of the discussion in Euclidean terms.

And now he talks of "hostile translation".

Might leave it at that. What more can one do but laugh.