Alright, forget New York because we're just talking past each other. There is no disagreement there and clearly the example is not making what I intended clear

And there is a formalistic definition of truth, a statement is true in a theory when that statement holds in every model of that theory. Like "swans are birds" is true because there are no swans which are not birds, but "swans are white" is false because there are swans which are not white.

And you don't think assuming that this definition is what is meant by "truth preserving," is question begging? Don't logical monists generally claim that their position is true tout court?

I would posit that axioms can be considered to be correct when they entail the intended theorems about the object you've conceived.

I don't see how these two together don't presuppose a deflationary theory of truth. We could debate the merits of deflation, but its presupposition seems to be very relevant.

Whether you have true premises is a different issue. When you stipulate axioms, you treat them as true. Are they true? Upon what basis can they be considered as such?

An excellent question for a field that revolves around truth, no?

I have no qualms with setting aside metaphysical considerations of truth for formal analysis. And this is perhaps rightly the norm for cases. But it seems inappropriate in this case.

How would you compare Peano Arithmetic and Robinson Arithmetic, for example? Which one is true? Is one "more true" than another? What about propositional logic and predicate calculus? These aren't rhetorical questions btw.

I'm not sure we have to choose between these. We're talking about truth relative to some stipulated sign system. There are multiple stories about what happened to Luke Skywalker after the original films. Are any of these more true than any other? However, it seems to be something quite different to claim that allclaims are true only relative to stipulated systems and that none are more true than any other.

However, it seems to be something quite different to claim that all[/ claims are true only relative to stipulated systems and that none are more true than any other. — Count Timothy von Icarus

Are any of these more true than any other? — Count Timothy von Icarus

Indeed. It doesn't seem meaningful to claim that the axiomatic systems are true or false in toto. But nevertheless, if there is a single unifying, bivalent truth concept, and two systems have incompatible theories, we should be able to say which is true and which is false. If we did not need to, we'd have to suspend that some claims are not evaluable as true or false in principle - and thus jettison bivalence by destroying the assignment mechanism of statements to truth values. And if we did need to, we'd have to claim that some systems are... false.... somehow, even when they seem to adequately represent concepts in precisely the same manner as others, just different concepts.

claims are true only relative to stipulated systems and that none are more true than any other. — Count Timothy von Icarus

I don't think logical pluralists are committed to that. Everyone agrees what follows from what stipulations. So it's true to say that "not every group is abelian". You can think of stipulations as disambiguations - which is what lemma incorporation works like.

The underlying issue seems to be that everyone can agree that eg groups have certain properties, but if you stipulate the definitions differently you change the properties. But you don't change the properties of the intended object when that object is the group, you perhaps change what the intended object is tout court.

I think that is in the direction of the intended thread topic. Because the ability to stipulate lemmas that make an axiomatic system better track an intended object's properties thereby lets you make more universal judgements about more precisely demarcated structures. Everyone will agree that Euclid's definition of circle captures plane circles, but not all pre-theoretically intuited circles are plane circles.

In effect this is a way of massaging the "complete generality" predicate in the OP's argument. You can restore a sense of "complete generality" by using lemmas, by speaking about something ultra specific and formalised you can guarantee that it works in that way for that system, the latter applies without exception. Applies without exception in the sense that "fdrake is sitting drinking tea now" is true at time of writing, and thus applies at that time without exception forever. Only "now" for those refined systems is a new lemma, allowing them to better specify their intended conceptual content.

I don't think logical pluralists are committed to that.

Not necessarily, as I noted before, many "weak" versions of logical pluralism start to look indistinguishable from weak forms of monism (something Russell discusses as well). And I would imagine most don't want to be committed to this view. It's a different question whether is this essentially presupposed as a background assumption though.

I mean, in your response to the question of: "in virtue of what are logics to be considered correct," you presented a textbook deflationary account of truth. Now I understand that you might not advocate that view as absolute. But if we "roll with it for the purposes of analysis," it seems like it will play a key role in seemingly deciding the issue.

Everyone agrees what follows from what stipulations.

Do they? Isn't one of the questions at issue whether anything follows from anything else?

To quote Russell:


arguments are often said to be neither true nor false, but
rather valid or invalid. This is correct as far as it goes, but a principle containing a turnstile as its main predicate can be regarded as a sentence making claim about the relevant argument. Such a claim will be true if the argument is valid, false if it is not. Hence the nihilist can be said to believe that there are no true atomic claims attributing logical consequence.

In effect this is a way of massaging the "complete generality" predicate in the OP's argument. You can restore a sense of "complete generality" by using lemmas, by speaking about something ultra specific and formalised you can guarantee that it works in that way for that system, the latter applies without exception. Applies without exception in the sense that "fdrake is sitting drinking tea now" is true at time of writing, and thus applies at that time without exception forever. Only "now" for those refined systems is a new lemma, allowing them to better specify their intended conceptual content.

An interesting practical approach, to be sure.

Do they? Isn't the question one of the questions at issue whether anything follows from anything else? — Count Timothy von Icarus

I don't think that's an issue at stake at all. If no principle holds in complete generality, they may still hold in certain well understood and well demarcated cases and contexts. Such as modus ponens in propositional logic.

The idea that (nothing follows from anything else in virtue of a valid argument) if (there are no principles which hold in complete generality) is ultimately not a premise in the OP's argument. It could be, and I believe Gillian Russel lectures as if, there are valid arguments even if there are no principles which hold in complete generality. Because she specifies what context she's speaking in. It then remains to be seen if a sense of complete generality can be restored by supplanting restricted statements - like Euclid's definition of a circle - with disambiguating phrases - like "in plane geometry".

Do they? Isn't the question one of the questions at issue whether anything follows from anything else? — Count Timothy von Icarus

Does everyone who understands a system and a proof in it believe the conclusion if the proof is correct and understood? Yes. Everyone agrees that P & P=>Q allows you to derive Q in propositional calculus. That's less about there being rules which cover everything, and more about there being followable rules. That's a followable, derivable rule.

I think this is the simplest version of what I was thinking.

Given a sphere centered about A,
pick any three points in the sphere,
those three points determine a unique plane,
the intersection of that plane and the sphere is a circle.

We're just taking a section of the sphere, without any further reference to the point A, which has already done everything needed to guarantee that its coplanar subsets are circles. In particular, we did not need to project A onto the plane that sections the sphere. (We can project it onto that plane, using the obvious orthogonal projection, or anything we like.)

Am I getting something wrong here?

Am I getting something wrong here? — Srap Tasmaner

Nah it looks fine. I'm just confused, it's doing away with the centre by providing an equivalent construction of the centre. Which is also fine, I just want to see what you're seeing in it.

those three points determine a unique plane, — Srap Tasmaner

Are those points in the interior of the sphere or on its surface?

Sphere, not ball. The surface. The 2-manifold.

Sphere, not ball. The surface. The 2-manifold. — Srap Tasmaner

With you. Yeah.

Way back when we started, what interested me was decoupling the point with reference to which the circle is constructed from the plane within which it is constructed.

Then I noticed you can decouple the point used to construct the circle from the (in-plane) center of the circle, because that's a projection, but it's a projection you don't need to do to construct the circle. Which means you can project the circle's originating point anywhere in the circle's plane.

I guess it would be better, and simpler, to say we can decouple the projection onto the plane of the originating point from the center of the circle.

And I thought there might be something interesting there, just in the geometry, but then realized the model I was creating was suggestive of stuff I've been thinking about a lot. That happens to me all the time.

That makes sense. Equivalence classes of pre-images of projections under some relation seems like a cool idea.

We're just taking a section of the sphere, without any further reference to the point A, which has already done everything needed to guarantee that its coplanar subsets are circles. In particular, we did not need to project A onto the plane that sections the sphere. (We can project it onto that plane, using the obvious orthogonal projection, or anything we like.) — Srap Tasmaner

Nice. That cleared something that I was puzzling over. A Great Circle is defined by only two points on the surface. It can do this becasue it is a straight line. So as on a plane, a line can be defined by two points and a circle by three.

I don't think that's an issue at stake at all.

IDK, that's how I've often seen nihilism defined. Per Russell it is "the claim that there are no laws of logic, i.e., no pairs of premise sets and conclusions such that premises logically entail the conclusion."

It could be, and I believe Gillian Russel lectures as if, there are valid arguments even if there are no principles which hold in complete generality. Because she specifies what context she's speaking in

Yes, and this makes sense if deflation vis-á-vis truth is presupposed. You can have nihilism and truth preservation via entailment because truth is just defined in terms of the formal context.

And it might make sense in other contexts as well. Just thinking back to philosophical history, there is certainly a long history of concepts of vertical reality—some things being "more real," or "more true." True might be predicated analogously like being and might not be fully captured by language and discursive human reason (e.g. Plato's Seventh Letter).

I'd have to think about it more but my intuition it would play havoc with other theories of truth. For example, in simple correspondence theories, X is true just in case X actually is the case. Now I'm not sure what it means for "truth preservation" if it is possible to have valid arguments that persevere truth while variously affirming and denying that "X is actually the case." I suppose people might counter that logic is now properly the study of formalism, not truth qua truth, or even natural language, to which I would disagree, the former will always sneak in the back door if left unacknowledged.

Per Russell it is "the claim that there are no laws of logic, i.e., no pairs of premise sets and conclusions such that premises logically entail the conclusion." — Count Timothy von Icarus

But... P & P => Q entails Q in propositional logic, who is denying this? It does not seem Russell is:

like thinning, cut, and the sequent forms of conjunction elimination. The
reason is this: a natural interpretation of the claim that there is no logic is that
the extension of the relation of logical consequence is empty; there is no pairing
of premises and conclusion such that the second is a logical consequence of the
first. This would make any claim of the form Γ |= φ false, but it would not
prevent there from being correct conditional principles.10

And footnote ten:

A note about vocabulary: arguments are often said to be neither true nor false, but
rather valid or invalid. This is correct as far as it goes, but a principle containing a turnstile
as its main predicate can be regarded as a sentence making claim about the relevant argument.
Such a claim will be true if the argument is valid, false if it is not. Hence the nihilist can be
said to believe that there are no true atomic claims attributing logical consequence.

The logical consequence relation is preserved, even if the intended objects it's supposed to refer to can be taken as counter models. Like "This sentence is false" might be taken as a countermodel for the law of excluded middle, or the great circle might be taken as a countermodel for Euclid's definition of a circle.

Consider Russell's proof and refinement of LEM:

Either φ is true in a model M, or it is false. In the first case, φ∨¬φ is true in M because of the truth-clauses for ∨. In the second case, ¬φ is true in M because of the truth-clause for negation, and
so again φ ∨ ¬φ is true in M. So either way it is true in the model, and—since M was arbitrary—it is true in all models. So φ ∨ ¬φ is a logical truth...

So we examine our simple proof and realise that our assumption that the sentence could only be true or false is violated by the monster*. Hence our culprit is the assumption that sentences can
only be true and false. Still, perhaps there are some sentences which can only be true or false—sentences in the language of arithmetic might be like—and our result would hold for these. Our new theorem reads: for any φ which can only be true or false, φ ∨ ¬φ is a logical truth. Just as the geometry teacher dubs polyhedra which satisfy the stretchability lemma simple, so we could give a name to sentences which meet our assumption. Perhaps bivalent would be suitable. Then we can retain the proof above as a proof of:

For all bivalent φ, φ=>φv~φ

I underlined "bivalent" in the final bit, since you produced a similar repair to the argument:

1 ) Gillian is in Banf
2) Therefore, I am in Banf.

by understanding "I" as "Gillian", then adding this as a specification in the argument:

1 ) Gillian is in Banf
2 ) I am Gillian.
3 ) Therefore, I am in Banf.

Your repair could well have read "For all I-s who are Gillian", just like Russell's repair of LEM reads "for all bivalent φ".

It's also worth noting that Russell's countermodels, monsters and context specifying information (eg "for all bivalent") aren't necessarily in the object language in question. EG propositional logic just
assumes bivalent φ, so LEM applies, so you couldn't formulate a "neither" valued statement in its standard operation.

And since her countermodel of a statement which evaluates to "neither" does not have an interpretation in terms of standard propositional logic, she expands what ought to clearly be the scope of any logic of propositions to include that statement, goes "bleh, any logic worth its salt should account for this...". marks down on the page "eh, propositional logic as is works fine for bivalent φ" and then moves onto new pastures of polyvalent φ.

Russell's approach is largely telling logical nihilists not to throw the baby out with the bathwater, just because they expect logical laws to behave like The One Law To Rule Them All, a kind of context invariant divine providence.... and when they don't, why not just say they work when they work and find out where they work?

So a logical nihilist might say "Aha, "this sentence is false" disproves LEM!, we cannot use propositional logic". and Russell invites us to say: "I'm going to use propositional logic only for sentences we know satisfy LEM". The latter constrains the range of stuff you can sensibly throw into the collection of models of the logic, and so you end up filling up the semantic entailment relation again in the system by artfully removing the counterexamples.

In effect the nihilist doubt machine gets going by noticing that there's arbitrary degrees of contextual variation, and throws every available piece of crap against the expectations of logical form a universalist has (like @Leontiskos and I's discussion earlier), when ultimately only the universalist need read the nihilist doubt machine as nihilist - it's just a doubt machine, you can tell it to sod off by specifying the exact mess you're in.

A Great Circle is defined by only two points on the surface. It can do this becasue it is a straight line. So as on a plane, a line can be defined by two points and a circle by three. — Banno

Something's not right here, which is just sloppiness and rustiness on my part.

In general, three non-colinear points in 3-space determine a unique plane, a unique triangle, and a unique circle. And then it takes a fourth point, not in the plane of the first three, to pick out a unique sphere.

When I was talking about sectioning a sphere ― after I realized that using a non-coplanar point to determine a circle could be thought of this way ― I reached for three points to pick out the sectioning plane out of habit, thinking that the section is guaranteed to be a circle because it is (a) planar and (b) a subset of a sphere.

Which is super super dumb. What the sphere guarantees is that the points selected are non-colinear, which hadn't even occurred to me.

All this sphere business ran roughshod over my original thinking, which was very cone oriented, as the drawings show.

Sheesh.

But... P & P => Q entails Q in propositional logic, who is denying this?

No one. But logical nihilism is not a position about "what is true in propositional logic." It seems like you're still presupposing deflation here, truth has to be "truth relative to this formalism."

Thanks, . I might add the following, more or less by relating your comments to the article.

It perhaps comes down to what is meant by "truth-preserving". A sentence in a classical extensional logic is consistent if there is at least one interpretation in which it is satisfied. If there is no such interpretation, then it is contradictory. If it is satisfied under every interpretation, then it is valid. Here, "truth-preserving" is replaced by satisfaction.

A given sentence is neither true nor false until given an interpretation. "Γ ⊨ φ" is understood as "Γ satisfies φ". So since Tarski, truth and validity are defined in terms of satisfaction.

Logical nihilism, is the view that "there are no laws of logic, where a law of logic takes the form "Γ ⊨ φ"(p.4). That is, logical nihilism is the view that there are no cases in which Γ satisfies φ.

Russell lists three approaches, as follows:

The cases approach allows us to say more about what logical nihilism amounts to: it is the view that for any set of premises Γ and conclusion φ whatsoever, there is a case in which every member of Γ is true, but φ is not. — p.4

On the interpretations conception then, logical nihilism is the view that for every argument, Γ φ, there are interpretations of the non-logical expressions in Γ and φ which would make every member of Γ true, but φ not true. — p.5

On (the universalist) approach, logical nihilism would be the view that for any argument, there is an assignment which makes all the premises true without making the conclusion true. — p.5

She adopts the interpretations approach, but for simplicity. She gives the impression that her argument might be made using the other two approaches. She proceeds to show how P →Q,Q ⊨P is truth-preserving if the interpretation includes only T; but not if it includes both T and F. That is, it is a logical law under one interpretation, but not under another. She then shows how the law of excluded middle is a logical law in the interpretation (T,F), but not in (T,F,N).

Now what this shows is that truth-preservation is a function of the interpretation. So yes, in your rough terms, truth and validity do depend on the system being used, since that system includes the interpretation.

Now I am not at all sure what you mean by 'deflation". But I am confident that all of the above could, at least for extensional cases, be put in terms of satisfaction, without mention of truth-preservation. If that for you is deflation, than so be it.

I'm not sure where that leaves our chat.

In effect the nihilist doubt machine gets going by noticing that there's arbitrary degrees of contextual variation — fdrake

I think the univocalist extreme of splicing everything apart and analyzing it separately is representative of sophistry (or nihilism?). Namely, the methodology precludes reasoning and knowledge. If one does not admit analogical predication in one form or another then they can deny but they can never affirm. They have created a method that can only deny; a skepticism machine.

For example:

1) Gillian is in Banf.
2) Therefore, I am in Banf.

to

1) Gillian is in Banf
2) I am Gillian
3) Therefore, I am in Banf — fdrake

Has it been fixed? The "sophist" would say no, and can quibble endlessly. They might ask you to specify what exactly "I am Gillian" means; what 'I' means; what a name is; what the predication of amness means (all difficult questions). They might splice (1) and (2) into different contexts, pointing out that (1) is a third-person description and (2) is a first-person description, and that it is not clear that these two discrete contexts can produce a conclusion that bridges them. "Shit-testing" seems to have no limits and no measure.

There is an interesting question about the great circle, but the method which outright denies that the great circle is a circle can outright deny anything it likes. It is the floodgate to infinite skepticism. I think we need to be a bit more careful about the skeptical tools we are using. They backfire much more easily than one is led to suppose.

Edit:

you can tell it to sod off by specifying the exact mess you're in — fdrake

Can you? There is an idea that floats around, according to which one can give quibble-proof arguments. I don't think this is right. I'd say the idea that there is some quibble-proof level of exactness won't cash out.

Cheers. I've got you off your topic - apologies - becasue what I was considering is what a circle might be defined intrinsically on the surface of a sphere. I might need to draw a diagram...

Anyway, the relevance was the difference between the maths of a sphere in $\mathbb {R}^3$ and intrinsic spherical geometry.

Logical nihilism is not a claim about what is true in classical extensional logic. It is presumably a claim about all truth preserving arguments.

Likewise, if truth can be defined arbitrarily, if we follow Carnap in the claim that: "in logic there are no morals. Everyone is at liberty to build his own logic, i.e. his own language, as he wishes. All that is required of him is that, if he wishes to discuss it, he must state his methods clearly, and give syntactical rules instead of philosophical arguments," it seems logical nihilism is trivial, but the question is effectively begged.

As for deflation: https://plato.stanford.edu/entries/truth-deflationary/

Has it been fixed? The "sophist" would say no, and can quibble endlessly. They might ask you to specify what exactly "I am Gillian" means; what 'I' means; what a name is; what the predication of amness means (all difficult questions). They might splice (1) and (2) into different contexts, pointing out that (1) is a third-person description and (2) is a first-person description, and that it is not clear that these two discrete contexts can produce a conclusion that bridges them. "Shit-testing" seems to have no limits and no measure. — Leontiskos

Those are quite different I believe. There's no attempt to change the verbatim meanings of argument terms in @Count Timothy von Icarus's repair, in fact there's an insistence on representing the conceptual content of what's said in spite of the means of its representation (predicate logic vs "I"). In effect, Timothy's takes the truth of the argument for granted and treats the inability of the verbatim machinery of propositional logic to reflect that truth as a failing of the logic... thus repairing the argument by explicitly spelling out the context sensitivity of "I".

Whereas your examples do not insist on taking the conceptual content of what's said for granted, indeed they're attempting to distort it. Allegorically, the logic of shit testing is that of a particularly sadistic genie - taking someone at their word but exactly at their word, using whatever pretheoretical concepts they have. The logic of your sophist is closer to doubting the presuppositions which are necessary for the original problem to be stated to begin with.

Our dispute was similar to the former - we both have the same pretheoretical intuitions about what a circle is. Agreeing on Euclid's and on the great circle's satisfaction of it. And we'd probably agree on the weird examples containing deleted points too, they would not be circles even though if you drew them they'd look exactly like circles. The issue we were having is that Euclid's definition clearly did not accurately represent our (mostly) shared pretheoretical intuition regarding what a circle was - what it looked like -, and I kept asking you to repair it.

Remember even Euclid saw fit to define a circle axiomatically. And his works exactly as planned in the plane. Just circles also live outside the plane, and thus are not bound by Euclid's plane figure definition of them verbatim.

"For all circles in the plane... (Euclid's theorems follow)" - another example which could've been in Russell's paper.

Logical nihilism is not a claim about what is true in classical extensional logic. — Count Timothy von Icarus

Sure. I drew attention to that. But that's were it starts. We can move on to formal intensional logics, if you like, and their algorithmic interpretations. Probably should leave that until we have a bit more agreement, though. It's important to understand that this is an area of development, and not all questions have been answered. For intensional logics, use is made of Kripke's theory of truth, but I certainly don't have the details.

I don't understand why you are talking about truth being defined arbitrarily. Tarski's definition is far from arbitrary.

And yes, I have a rough idea of what deflation is with regard to truth. I'm just not sure what part you take it to play here. For extensional languages we can define truth in terms of satisfaction. I gather you understand that as deflationary? Fine. What's the problem? Is it that you object to such an approach setting up truth in terms of interpretation? But it works.

Can you? There is an idea that floats around, according to which one can give quibble-proof arguments. I don't think this is right. I'd say the idea that there is some quibble-proof level of exactness won't cash out. — Leontiskos

Sure. Here is a quibble proof argument.

Let x belong to the field of real numbers.
Stipulate that x+1=2
therefore x=2-1
therefore x=1

Where's the issue?

There is an interesting question about the great circle, but the method which outright denies that the great circle is a circle can outright deny anything it likes. It is the floodgate to infinite skepticism. I think we need to be a bit more careful about the skeptical tools we are using. They backfire much more easily than one is led to suppose. — Leontiskos

To be clear you would have been compelled to deny the great circle was a circle by only using Euclid's definition of it verbatim, I would not have!

Russell's approach is largely telling logical nihilists not to throw the baby out with the bathwater — fdrake

This is what always seems to happen with these shiny new theories. It is motte and bailey. The controversial claims that stimulated attention dissipate upon closer examination.

I'm not really sure what you are arguing, fdrake. It doesn't sound like you hold to logical nihilism or logical pluralism in any strong or interesting sense. Am I wrong in that?

You talk a lot about the great circle:

the great circle might be taken as a countermodel for Euclid's definition of a circle — fdrake

Let's suppose it is a countermodel. How does the logical pluralism arise? I can only see it arising if we say that a "circle" means both Euclid's definition and the great circle countermodel, and that these two models are incompatible. Is that what you hold?

-

Our dispute was similar to the former - we both have the same pretheoretical intuitions about what a circle is. Agreeing on Euclid's and on the great circle's satisfaction of it. And we'd probably agree on the weird examples containing deleted points too, they would not be circles even though if you drew them they'd look exactly like circles. — fdrake

Given that I disagree with all of this, does it follow that you were the sophist and not the sadistic genie?

and I kept asking you to repair it. — fdrake

I kept asking you to offer a reason why it needs to be repaired, because it "clearly" was fine. You are begging the question in your own favor with words like "clearly."

Why are we to believe that a three-dimensional abstraction (i.e. the great circle) does not contain a two-dimensional abstraction (i.e. a circle)? In any case, the easier disagreement here is over the question of whether one can delete a point.

Whereas your examples do not insist on taking the conceptual content of what's said for granted, indeed they're attempting to distort it. Allegorically, the logic of shit testing is that of a particularly sadistic genie - taking someone at their word but exactly at their word, using whatever pretheoretical concepts they have. The logic of your sophist is closer to doubting the presuppositions which are necessary for the original problem to be stated to begin with. — fdrake

This is helpful, but I'm not convinced it is cogent. The sadistic genie is not taking them at their word by being overly pedantic, he is just being a sophist. I see the distinction you are making, but I would say that the sadistic genie is a sophist, even if not every sophist is a sadistic genie.

I saw my cousin who has Asperger's, "Your hair is long, how long has it been growing?" "Since I was born!" He is fun, and this is an example of the sadistic genie, but it is not a non-example of a sophist. Taking someone "exactly at their word" is a good way not to take them at their word.

Where's the issue? — fdrake

To take a few, you haven't defined the operations, commutativity relations, numbers, variables, etc.

To be clear you would have been compelled to deny the great circle was a circle by only using Euclid's definition of it verbatim, I would not have! — fdrake

I don't follow, but you seem to think "verbatim" is a fix; a quibble-proof solution; a univocal meaning. I don't think the buck stops there or anywhere else. Literal meaning is a puzzle as much as anything else. To use the word "verbatim" and assume you have won the argument will not do.

Good posts, though. I have to run but I hope to come back to this soon.

To take a few, you haven't defined the operations, commutativity relations, numbers, variables, etc. — Leontiskos

Understand them as you usually would. + and times are spelled out in the field axioms (see classical definitions). Add that subtraction of a is equivalent to adding -a. IE x-a=x+(-a)

@fdrake, what is the confusion here, do you think? Is it to do with the commensurability of differing logical systems? If logical monism is the view that all logical systems are commensurable, then there is presumably some notion of translation that works between them all. I find that difficult to picture. Perhaps all logics might be found to be variations on Lambda Calculus or some other "foundational" logic, in which case there would be one true logic, begging for some wit to find a logic that is not based on that foundation.

fdrake, what is the confusion here, do you think? Is it to do with the commensurability of differing logical systems? If logical monism is the view that all logical systems are commensurable, then there is presumably some notion of translation that works between them all. I find that difficult to picture. Perhaps all logics might be found to be variations on Lambda Calculus or some other "foundational" logic, in which case there would be one true logic, begging for some wit to find a logic that is not based on that foundation. — Banno

I think it's a confusion regarding the connection of meaning to truth, and about truth. It might not be a confusion, it could be an insistence on a unified metalanguage having a single truth concept in it which sublanguages, formal or informal, necessarily ape.

It's quite suspicious that you can talk about "for all bivalent phi" in Russell's paper but also "for all phi which are true, false or neither" in natural language, and the reader will understand some birthing of new context and propagate their understanding into that context. As if there's some big Understanding Truth Machine that gazes through the eyes as soon as you see someone write down a new system of axioms.

All the while you know there's a wealth of intended objects for the symbols to capture.

begging for some wit to find a logic that is not based on that foundation. — Banno

They're always going to need semantics, too. I've no idea how to specify the connection between a syntax and a semantics without using some informal metalanguage, so there will always be some unformalised remainder I think!

I suppose the question is whether you read the necessity of that unformalised remainder as a sign that all systems should be thought of univocally, or whether you can erect little fortresses of axioms and interpretations amid the sea of chaos whose waves are one voice.

Thinking of all systems as univocal would appear to be putting unnecessary restrictions on the development of logic.

(Quine's rejection of modal logic)

When else are you called a Nihilist for not accepting something is perfect? If there was a one logic it would still be people using it.There might be one way things are and many ways to understand it without being 'truth adverse' or whatever that system would look like; mostly free association with sprinkles?