And that's a totally trivial observation that nobody ever disagreed with. — Marchesk

Are you serious? I just said that on the deflationists account there is *nothing* more to truth than the conventions that govern it's usage as a predicate. Literally every other theory of truth denies this, especially correspondence since truth is not simply a predicate with use-conventions. It articulates the instantiation of a relationship (correspondence) between do two ontologically distinct kinds of things (sentences/propositions and facts) thus yielding the substantive property of truth as held by the proposition.

Again, the T-scheme does not make reference to facts or correspondence with the use of p.

1) <p> is true
2) if and only if
3) p

Hence it's perfectly compatible with non-correspondence theories like deflationism.

I presume that you're asking what justifies the rules, and the answer is that the rules don't need to be justified, no more than the rules of chess need to be justified. The question is mostly senseless — Sam26

I think this is mistaken. I mean the chess analogy breaks down too quickly to be a useful comparison. On the outset, chess is just a leisure activity, it doesn't play the broad role that logic does.

But more to the point, we know there's a huge debate within mathematical logic about which rules we ought to adopt, which ones we are justified in taking on and which we ought to dispense with. So intuitionists believe classical logicians are mistaken in their use of the the Excluded Middle rule (and any rules that yield EM) in their proofs when placed within a universal quantifier, because they take logic to be about constructive provability. Isn't that just a case where logicians are arguing about which rules of logic are justified?

The cat on the mat is true if and only if the cat is on the mat. — Marchesk

You're goofing the syntax.

"The cat is on the mat" is true iff the cat is on the mat.

Deflationists don't argue that there's a correspondence relation that maps propositions to states of affairs. Truth, in other words, is not taken to be a substantive property that a proposition had. Rather (again, depending on the account) will mean that truth is really all and only about the linguistic conventions governing the predicate "is true".

The mistake you're making is thinking that the T-scheme (<p> is true iff p) assumes correspondence. It doesn't, that's just a nice heuristic to get one to understand the schema but it's not actually assumed in the schema. Many philosophers who aren't correspondence theorists still accept instances of the schema.

Check out Graham Priest, the most well known defender of Paraconsistent logic, and Dialetheism in particular. "Doubt Truth to be a Liar" is probably the book of his I liked the most. He does a number of talks online about it as well.

A lot of this, in my estimation, doesn't make sense under scrutiny.

I'm of the opinion an actual infinity cannot exist. I feel strongest about the impossibility of an infinite past, because that would entail a completed infinity: how could infinitely many days have passed?

Um, before every day there is another day. QED. Or to put it more directly, the cardinality of the set of days prior to day "n" can be put into a one-to-one correspondence with the members of the set of natural numbers. Ergo, the number of past days are infinite. I don't know if this is actually true, but there is no logical argument against the *possibility* of it.

However, this wasn't even really what I was suggesting. Between any two moments of time there's another moment. That's what I had in mind. And it's even clearer with the divisibility of space. It's nearly always taken to be a continuum, meaning it would be infinitely divisible.

That doesn't imply the philosophical analysis is wrong, it just means that we don't know of any particular limits

What philosophical analysis? If we are adopting perfectly standard mathematics (or even most non-standard math systems) there is no contradiction whatsoever in supposing the past days are infinite. This will play into a bigger point I make at the end.

My opinions are consistent with the dominant opinion among philosophers prior to Cantor's set theory, but that doesn't seem like a very good reason to believe an actual infinity exists in the world.

I hope it doesn't come across rude, but that just reads as "If you ignore the last 150 years of mathematics most philosophers would agree with me". Well that's... a defense anyone can make to defend their belief in whatever.

Look, my broader issue/point is this. The interplay between our beliefs about the world and the formal tools (maths, logics) is more complex than often made out (i.e. the influence goes both ways). However, generally the idea is that our physics needs math to guide it's conjectures, and our beliefs about the world ought to be in line with the dominant physical theories. If maths has explicated infinity as a coherent, precise concept - and it has - then presumably it becomes irrational to say (as I understand you to be saying) that "Yea yea, there's infinity in mathematics and in physics, but if you try to apply it to real things it entails a contradiction." I just don't get it.

Infinity is not a contradictory concept, so how is it supposed to produce a contradiction if applied to real things? Or is it supposed to be a category mistake? But how does that work? We talk about infinite collections in mathematics all the time, it's central to set theory. That doesn't mean infinite collections (or other infinite whatevers) can exist in our universe, just that you cannot rule them out as incoherent and thus fail to obtain in every possible universe.

You don't seem to have understood me. Some things may well be self identical. My statement is in fact identical with itself. But that has no bearing on *everything else* also being identical with themselves.

So for instance, take Schrodinger himself in "Science and Humanism":

"When you observe a particle of a certain type, say an electron, now and here, this is to be regarded in principle as an isolated event. Even if you observe a similar particle a very short time at a spot very near to the first, and even if you have every reason to assume a causal connection between the first and the second ob servation, there is no true, unambiguous meaning in the assertion that it is the same particle you have observed in the two cases. The circumstances may be such that they render it highly conve nient and desirable to express oneself so, but it is only an abbre viation of speech; for there are other cases where the 'sameness' becomes entirely meaningless; and there is no sharp boundary, no clear-cut distinction between them, there is a gradual transi tion over intermediate cases. And I beg to emphasize this and I beg you to believe it: It is not a question of being able to ascer tain the identity in some instances and not being able to do so in others. It is beyond doubt that the question of 'sameness', of identity, really and truly has no meaning."

Now, that's like seventy years old, but the view has received modern defenses (not dominant views, mind you) by the likes of Newton da Costa by means of distinguishing "identity" from "indistinguishability", and further by Wittgenstein in suggestion we drop identity from our logic as "it is to say nothing". Or on a more practical level, the database language SQL has violations of the Law of Identity:

SELECT * FROM tbl WHERE NULL=NULL

In the logic of SQL, this expression will never return as true.

Not that I endorse dropping identity (it's pretty obviously useful), just that it's not (as you suggested) literally impossible to do so without falling into incoherence.

However, the cat might not be on the mat, and thus the assertion could be false. What makes the statement true or false? Whether the cat being referred to is actually on a mat. And that's a situation in the world, which we might call a state of affairs or fact of the matter. It is the world which makes statements about the world true or false. But this is the correspondence theory of truth. — Marchesk

You're mixing the two. Deflationism (which has many articulations of course) argues that truth won't play an explanatory role in, say, theories of meaning. Correspondence theorists will say truth has an important role in that discussion. Similarly, for deflationists the entire theory of truth is basically the T-scheme. But for the correspondence theorist, truth can be analyzed in other terms, namely the correspondence relation between sentences (or propositions) and reality. Deflationists often criticize this correspondence relation.

In many ways the difference is just a simplified ontology given the belief we don't need all these extra additions to our metaphysics (e.g. propositions, correspondence, facts and so on).

I would say that space and time exist, and both are generally thought to be infinite.

I'm just saying even the deepest intuitions about logic don't have a sort of indubitability (what I thought you were saying).

The thing you missed here is the unspoken inference you make. The cardinality of the set of Natural Numbers is not infinity (which is defined as having no limits) as by referring to Natural Numbers you are limiting it to Natural Numbers alone. You are not including anything which is not a Natural Number, it does not include different colours, shapes, texture etc. It is a concept limited to that which is considered a natural number.

How was it unspoken if I literally said the assumption (the the natural numbers are infinite)? That aside, you aren't making sense. That the natural numbers do no, for instance, include the Real Numbers does not entail that the set of Natural Numbers is not infinity. In mathematics, infinity is not (as you claimed) defined as "having no limits". In this case that's especially obvious, because by "limit" you're already sneaking in the assumption of finitude (e.g. the natural numbers are finite, somehow, because the set doesn't include other types of numbers). This argument makes no sense.

You can say that the numbers have no end.. or could go on forever.. or go on indefinitely.. but you cannot refer to them as infinite as you contradict yourself by describing them as such. As they are limited... to Natural Numbers. I am aware that mathematicians are fond of using the word infinite, but I would argue that its an illogical thing to do. As I think I have sufficiently shown.

No because then you're not talking about the infinite any more.

Consider the following:

1. There are two infinite numbers, A and B
2. A is not B, and B is not A.
3. A is larger than B.

this isn't a description of something without limits. You are specifically saying that A is limited to A and does not include B. And that B is limited to B and does not include A. These are limits.

You can say it has no limits in one specific sense but has limits in others, but then you are not referring to the infinite or to a limitless thing anymore.

You are simply ignoring the definition of infinity that mathematicians use and thereby conclude that it's incoherent because of we assumed your definition we'd get a contradiction. QED, your definition is wrong because it leads to a contradiction. That's ridiculous.

Your argument makes an obvious assumption, namely that all infinite sets are of the same size That's quite literally rejected in mathematics. Infinite sets which are countable, like the natural numbers, have the ability to be put into a one-to-one correspondence with a proper subset of themselves, e.g. we can map all the even numbers onto the set of natural numbers. Uncountably infinite sets (e.g. the reals) cannot do this mapping with the natural numbers, entailing that such sets are larger. Your definition leaves no real ability to use infinity in mathematically useful ways, e.g. Calculus.

You are if you are saying this thing has no limits when it defined within the specific limits of Real or Natural numbers as in the examples you gave. You are therefore saying that this thing is both limited and not limited simultaneously. Which is a contradiction. It cannot be A and ~A.

Incorrect. The natural numbers are the counting numbers, so they do no include the reals. That does not entail the Natural Numbers have a finite *cardinality*, it simply means the set of natural numbers leaves out particular types of numbers. This simply means the set of natural numbers has a particular size of infinity.

There's a lot of... weird statements made in this thread.

Of course [one] is correct. Logic has to rest on something. It’s not controversial, although it never ceases to surprise how many people seem to think it’s fishy or suspect.

That said, logic is not a form of omniscience. There may indeed be many things beyond logic, or for which logical analysis is unsuitable. But insofar as it is real, then the law of the excluded middle, or the law of identity, don’t need further justification - they simply are, they’re what Frege would regard as ‘primitive elements’, like natural numbers. If you ask why one and one equals two, the response can only be: just is so. — Wayfarer

That's a nice view, happens to be not only controversial but probably false. Frege also thought the Comprehension Schema was "self -evident", but Russell proved it led to Russell's Paradox. And if you think Excluded Middle and such "just are", then you are behind on, oh, the last 80 years of technical work in mathematical logic. Intuitionistic logic, paraconsistent logic (esp. dialetheism), da Costa's work on logics without the Principle of Identity (look up "Schrodinger logic", or non-reflexive logic). This is hard technical work, and by no means is any assumption here not controversial. That's just not true, whether you agree with the views or not. Like I'm not an intuitionist, but constructive mathematics is hella useful, even in computer science where I'm more comfortable. It would be borderline stupid for me to tell an intuitionist that they're work is simply incorrect and that's all there is to it.

It's not even like this is new. Aristotle, despite defending Excluded Middle, held in his work on metaphysics that there are exceptions to the Excluded Middle, namely contingent statement about the future), or some have (contra-Russell) taken propositions such as "The present king of France is bald" to be an exception to EM.

You may consider it a working presumption, if you like, which enables all kinds of things, including our talk.

How so? That some objects may not be self-identical doesn't seem to have anything to do with me talking.


If I were to attempt to answer OP, I'd say in a sense logic doesn't need to be justified. But that assertion is rather difficult to unpack shortly and I'm a bit too lazy to do the whole thing. Because on one hand the truths of logic are necessary, true in all models, whatever. But that's trivial, because the truths of *every* logical system are true relative to the logic in question, e.g. Excluded Middle is a tautology in classical logic but not so in Constructive logics.

Rather, I think logics are justified like any other theory: By the virtues (and deficits) of the system under consideration.

You actually said "No" in answer to the proposition that metaphysics is journeyed through by personal choice (as opposed to rational necessity) — Pseudonym

I stopped reading the moment you said this, because no matter how much you insist that, I can go up and quote what I said:

I wouldn't deny that *in general* preferences guide what we believe.I was just saying axioms in particularare not, necessarily, simply picked arbitrarily on grounds of preference. — MindForged

This is a waste. You cannot seem to read, it seems trivial to show how you are misrepresenting what was said. That's very directly opposed to what you claimed I said.

It's a very convoluted interpretation that has this exchange saying what you claim. Not only do you specifically make the claim that "both metaphysical theories and logical theories have the worth assessed via how virtuous their theories are." — Pseudonym

How is that convoluted??? People argue about their theories in this way, but that does not entail that the acceptance and historical proliferation of theories is guided in this way. As I also said, but which you seem incapable of reading, is:

I wouldn't deny that *in general* preferences guide what we believe.I was just saying axioms in particular are not, necessarily, simply picked arbitrarily on grounds of preference. — MindForged

This conversation has a fairly narrow topic matter (the justification of axioms). Please actually read what was said instead of trying to overgeneralize. My point is that using a model of theory choice allows you to give justification to the axioms you choose (the example I gave being how intuitionists argue for their logic).

The arguments people use to justify their theories and the reason they chose them are not the same thing. — Pseudonym

Which I quite clearly said above. Again,.you simply do not read

If, however, you're claiming that regardless of how theories are actually chosen, they are defended using rational theory choice models, then your argument is a non sequitur, the OP is about how theories are chosen, not how they're defended. I very much doubt the OP was confused into thinking that philosophers actually admitted to his claim. — Pseudonym

Hmm, let's try reading the OP:

Differences in choice of axioms must originate with our preferences (likes and dislikes).

Therefore, philosophy is not so much about rationality as it is about our personal preferences. — TheMadFool

OP is talking about whether the axioms we pick can be given rational justification, concluding that they essentially "must originate with our preferences". I articulated how they can be chosen rationally and argued that the manner used to pick theories is rather general, and can and has been used to support even fundamental positions rationally. This does not entail me claiming something about how positions gain traction/adherents in philosophy.

Yet your claim is that it is responsible for selecting one set of axioms over another. I can't think of very much that matters more in explaining the differences in fundamental philosophical lines of thought, which is what the thread is about. — Pseudonym

No, what is responsible is simply the differences in the weights you give, not their specific value. People nearly always place a higher importance on consistency than on existence of ad hoc elements, adequacy to the data is taken to be most important, etc. The values can be arbitrary so long as these relationships are established (especially since the values can be reduced anyway).

So I repeat, please actually read my posts, and actually read the OP. Tends to make one cranky to have to get this all in order when you could have just read...

A theory with little evidence that is organized and worded in a pleasing manner may very well get more attention than a theory with better evidence but poor organization. A theory that sparks the imagination, makes people really wonder, can be completely false yet nevertheless garner significant support. — darthbarracuda

Bare possibilities aren't fun to discuss, it would be prudent, I think, to articulate real examples of this. Otherwise it doesn't seem like an interesting thesis, especially when it's tantalizingly described as flamboyant.

Supposedly, the natural world is amoral and so the further away we keep scientific inquiry from morality, the better (or so it goes). — darthbarracuda

I suppose my confusion is, then, how these examples really elucidate your point. By your own admission, they mean the same as their more direct counterparts and seemingly everyone knows why they're articulated that way. Without an example of how this actually, supposedly, causes mistakes it seems like much ado.

But "most rational" theory may still be give too much credence to the theory. Often theories are taken as true because they cohere well to other theories, and ultimately cohere with a naturalistic picture of the world. But the evidence going for it is still not very strong. — darthbarracuda

Coherence with other theories is only one criterion of theory choice. A theory which is dominant purely on that ground (as opposed to strong evidence) is a poor theory overall. It's adoption would transparently be because there was nothing else on offer with the desired traits. People insist on having a theory of some sort, whether it's correct or not, so bad theories will go mainstream from time to time, most often early in a discipline's development.

Why are they like this? Because relativism threatens the moral project. Under relativism, they can't argue for absolute moral truths. Why do religious fundamentalists hate atheists? Because atheism threatens the central pillar of Judeo-Christian-Islamic religious belief. As such theory in general is threatened by skepticism. Yet skepticism is ignored, treated as "childish". Why? — darthbarracuda

I don't dismiss relativism out of hand, but it's hardly the case (as I read you as saying) that relativism (of whatever sort) has established itself and that people just moved on instead of really refuting it. Really, it's no different than anything else in philosophy. Positions (and even while branches) come and go from being vogue. Sometimes for good reasons, other times because it's not with the spirit of the times. Logicism in classical mathematics died especially due to Gödel and Tarski; good reasons. Metaphysics came back because of the resurgence of formal discussions and developments of modality; also good. Logical positivism, despite definitely not being a view of endorse, wasn't really refuted. Sometimes it's bad, or represents a tendency that people can't let go of. Maybe moral realism is like this, maybe not. But it's not obvious either way to me.

Anyway, Hume's induction problem is interesting but I don't think it throws into doubt the notion of evidence supporting a theory (if I understood your OP correctly).

To be honest (and hopefully not come across dismissive...), I think this is much ado about nothing... mostly. Like this:

Literary flamboyance can be used to mask theoretical gaps.
(...)

They may repeat banal observations at key points in time, so as to create the illusion of complexity. They may quote a long-dead poet to add intrigue, mystique or garner an impression of being a "worldly" person who has interests in a variety of tastes, when in reality, a poem by itself does not prove anything "scientifically" - it's fluff. — darthbarracuda

Is less likely to happen among people in the same discipline, this sounds like a critique of science advocates (e.g. Neil Degrass Tyson). Flamboyance can be attempted by academics to their fellows, but assuming it's in a relatively formal setting it's not likely to function as a way of passing off nonsense.

Sometimes, a theory is taken to be true in the absence of any other contenders. — darthbarracuda

I think you're confusing accepting a theory because it's most rational theory with the unilateral assertion of its truth. Much of the time when pressed, it's the former that's being communicated. The latter is generally reserved for very well evidenced theories.

Yet once again, its usefulness does not prove it is true. It merely means it is useful, which may be completely accidental. — darthbarracuda

Whereas the possibility you're representing is both unevidenced and extremely unlikely. Reality could be all sorts of ways, but what you're basically saying is this. "Oh yeah, those beliefs are useful and allow you to,.by believing them, correctly predict and plan for reality to be such and such a way. However, in fact it just seems like it." You might as well have said "No" for all you gave us to chew on for that view. If Newtonian dynamics tells me some object will move in some way in certain conditions, and it does under continued testing, it's just useful. It's (the theory) is either true or near enough that anything else is probably irrelevant to most tasks.

I mean, I could run your argument against mathematics or logic, but most wouldn't accept that argument. (Even despite us knowing mathematics is not, as often said, some beacon of indubitable knowledge. It is, however, indispensable).

By and large the assumption seems to be that the world is a rational place and thus can be ascertained and understood rationally, when in fact there is no possible evidence that could prove or disprove this assumption. — darthbarracuda

It's not merely an assumption. The idea is that if reality can be understood, we ought to be able to predict how it will be in some scenario or how it was in some scenario. Repeatedly, such that the assumption becomes ever more likely, asymptoting nearer a probability of 1 that reality is, in fact effable, understandable. The idea that no one has ever responded to Hume's argument is odd. If you, as Kant did in responding to Hume, accept the existence of the synthetic a priori then you seem to have gotten around the argument. You can probably find many more responses on the SEP, like Bayesian probability being applied.

I don't know what model of theory choice you might be referring to, but I can't think of a single exposition to which "everyone" refers. There are perfectly well respected critics of pretty much every version from Popper to Becker. Its just wishful thinking to suggest we have some universally approved means of judging the value of theories (prior to formal proofs). — Pseudonym

You misunderstand, my point was that arguing based on theory choice is used in all fields, not that there is a single model of theory choice everyone subscribes to. In fact, the bit right after what you quoted references this:

the only difference being how highly respective parties rate particular theoretical virtues and how they conceive of them. — Me

And even if the above were the case (which I'm certain it isn't) then the weight to give to each factor would itself be axiomatic and so subject to rational choice theory, and so on.

The specific value given to each weight probably doesn't matter, all you need is to be able to give them different weights (of we have to get deep down, primitive recursive arithmetic is more or less never doubted). I already said people disagree on the particulars of this.

So the popularity of theological-based metaphysics just about the time when most people were religious, and it's subsequent waning as atheism grows is just a coincidence?
(...)
you're still trying to claim that philosophical theories are largely accepted or rejected on their virtues? — Pseudonym

Um, no. I'm talking about how to rationally pick between theories, hence why I made no claim about this playing out historically exactly according to any particular model (much more complicated). So instead of going off about it in an aggressive way, you could try a simple question, it's more civil. This can be seen above:

I wouldn't deny that *in general* preferences guide what we believe.I was just saying axioms in particular are not, necessarily, simply picked arbitrarily on grounds of preference.

Even logically, axioms shouldn't lead to contradictions. — TheMadFool

Consistency is a theoretical virtue, but it's one we do not always get the pleasure of attaining. What's interesting is that even if a contradiction pops up, one can work around it. People have inconsistent beliefs all the time, and want to get rid of that if made aware of it. But oftentimes they cannot, and so have to tolerate the inconsistency for a time.

In such cases isn't it personal preference, not rationale, that swings our vote? — TheMadFool

I wouldn't deny that *in general* preferences guide what we believe.I was just saying axioms in particular are not, necessarily, simply picked arbitrarily on grounds of preference.

Reflexivity of the accessibility relation just says that the actual world (whatever that is) is always a possible world (whatever that is). So: — fdrake

That's basically what I said. If worlds couldn't access themselves then they wouldn't be a live possibility... with respect to themselves!

You could say the same of any possible world, actuality becomes just an indexical property if its sense is equated with the reflexivity of an accessibility relation. — fdrake

I don't think I equated actuality with reflexivity, I brought that up when I was trying to think of what else you might have meant by "possibility in this world" Besides physical possibility. I think only on a modal realist's account is actuality just an indexical property. Arguably, that is a really attractive view in how to define actuality, though the rest of the theory is a bit... much.

physical possibility < metaphysical possibility < logical possibility — fdrake

I've often seen this, but I'm always struck by how odd the last two are. Much of the time, metaphysical possibility/necessity are cashed out in terms of logic. That is, X being consistent is necessary and sufficient for being metaphysically possible, being a logical truth is necessary and sufficient for metaphysical necessity, and impossibility is for the contradictions. It makes those two modalities seem to be synonyms. At the risk of bringing this thread too far of course, how do you differentiate metaphysical and logical modalities?

I tend to think of logical modalities as a sort statement about a particular logical formalism, e.g. P & ~P => Q is logically necessary in Classical Logic. Giving them too much metaphysical baggage seems to result in odd questions like, what's the metaphysical analogue of these laws? In the logic, it seems like the rules I establish as valid, and I can do all sorts of manipulations that have no metaphysical equivalent (like deploying a contradiction).

possible is internalised to the world — fdrake

I think in that case you're talking about physical possibility, whereas I was referring to metaphysical possibility. And besides, even in that example, I don't see the issue. The actual world is part of the set of possible worlds. So a rock having the potential energy to fall can be said to be possible within that world itself. After all, nearly every accessibility relation of nearly any modal logic will imbue give that relation a the reflexive property; worlds can access themselves.

I think the whole attempt to cash out possibility in terms of possible worlds is a giant mistake, and that any analytic metaphysics that takes that route is basically a new scholasticism not deserving of being taken seriously. — StreetlightX

Eh, I can't go that far. Partly, because possible worlds are, as the current history suggests, the impetus behind the resurgence of metaphysics within analytic philosophy. Kripke, that frigging genius, developed it when he was in high school (we should all feel depressed; luckily we can find solace in Kripke's hilariously awful voice).

I mean, even in computer science I basically have to use possible worlds (which in my work I will construe as possible states of a computer), so there's a real boon to using them; I can make recourse to well understood formalisms to make sense (on a logical level) how to describe what I'm doing.

Now of course, there are many semantic & metaphysical views about what possible worlds are that I can't accept or understand (I'm still not sure I understand the position known as "possibilism"), but I don't think I personally would compare them to scholasticism.

attempts to take seriously the need to account for the individuation of possibility. It does not take the possible as a 'given', simply waiting in the wings to be actualised, even if as a second-order 'non-live' possibility. In the scientific context in which the concept was elaborated, the adjacent possible is created or brought into being where it simply did not 'exist' before hand even qua possible

Well I guess I just don't see the upshot of that position. Possibilities are individuated in talk of possible worlds. It's just that some possibilities are only accessible given some other possibilities being the case (e.g. the Bruce Wayne being a redhead example I gave).

Or to put it otherwise, what I like about the adjacent possible is that it provides what I think is another, far superior, scientifically grounded way of thinking about possibility than the idealist logical toys of modern day analytic metaphysicians. — StreetlightX

What does it provide that is superior? From the Kauffman quote, he seems to be using essentially the same idea of modality as I'm familiar with:

that as enabling constraints “create” the Adjacent space of possibilities into which evolution can become. — Kauffman

This is exactly what I understand worlds being "more accessible" to other worlds to mean in Possible Worlds Semantics. Certain possibilities open up a space of other possibilities which would otherwise not be adjacent to them, to use Kauffman's terminology.

You're saying that the respective axioms must cohere with each other and that counts as reasons supporting the entire collection of axioms. — TheMadFool

Depending on what you mean by "cohere", no I'm not saying that. We can (and have had) theories which were contradictory (early calculus, Bohr model of the atom) but which were still the more rational theory at the time because they trumped their competition in how much they explained.

It, being the first, doesn't have the luxury of such a support structure because no other theories exist alongside it. — TheMadFool

I don't really understand this. The model of theory choice works for any kind of theory, including theories strictly about logic. Logicians essentially argue this way about their axioms, e.g. the intuitionist says to the classical logician, "My theory of logic is more rational because it makes less extravagant assumptions and is computationally useful."

This way of justifying theories does not depend on any particular axioms because it has basically universal support (everyone from logicians to sociologists essentially makes reference to such theories being made better or worse on these grounds), the only difference being how highly respective parties rate particular theoretical virtues and how they conceive of them.

metaphysics is a region of space which is filled with contradictory philosophies and there is no actual process by which we may verify or disconfirm theories in it. Such a place is journeyed through only per choice; no rationale. — TheMadFool

No, metaphysical theories rest on logical theories (for various reasons) and both metaphysical theories and logical theories have the worth assessed via how virtuous their theories are.

In other words, possibility itself can be thought of as indexed to the real, and is not simply 'prior' to it — StreetlightX

This doesn't make any sense to me. Within modality, to be "possible" simply means to obtain in at least one possible world. Given that "emerging possibilities" obviously reference possible ways realities, this doesn't force any changes in how we conceive of modality.

Really, what the philosopher is likely (and correctly) to say is this. There are possible world's which are closer to some world's than others because certain states for affairs are live possibilities for some worlds and not for others, e.g. "It's possible that Bruce Wayne has red hair" is only accessible from worlds in which Bruce Wayne exists. Modal logics have an accessibility relation.

This "adjacent possibility" idea is basically already part of modern discussions of modality.

Axioms, by definition, have no supporting reasons. So, can't be rational. — TheMadFool

If you apply a model of theory choice we very well get supporting reasons. If the axioms taken on result in a theory which introduces a bunch of ad hoc elements, isn't fruitful, is too inconsistent relative to another theory with the same results but sans-those defects, the latter theory is more rational.

Axioms can be supported with reason, just check out Penelope Maddy's "Believing the Axioms".

The interesting thing is that any two conflicting stances are reasoned positions. — TheMadFool

Surely you don’t mean that? The flat earther is not holding a reasoned position.

Therefore the difference between thesis and antithesis must lie with the axioms of the arguments offered in support of them. — TheMadFool

Well, yes, the differences in views come from the different underlying assumptions.

Differences in choice of axioms must originate with our preferences (likes and dislikes). — TheMadFool

Well I don't know about that. The project of picking axioms is intended to be done rationally, not "I like that assumption". So theory choice models come into play.

It's not just the wealthy that have power, it's not that simple. The government is also an entity in its own that has power over the rich people, — BlueBanana

Which it does not exercise for the obvious reason that politicians won't bite the hands that feed them (lobbying, campaign finance, etc.). The example you gave (CIA human rights violations) are predominately regarding what is done to non-Americans, so it seems disconnected from your statement about the government having power over rich people. So while the government does all sorts of awful things in the States and abroad, the most it will do at home that "hurts" the wealthy is what the Democratic Party will do, which is very little (Democratic party can't even get as far as supporting disallowing private financing of election runs so they too won't bite the hand that feeds them). I think the inheritance tax is about as much as they support in practice.

That's too deep underlying imo. It's the actions that matter more, not their reasons, especially if we're looking at the reasons for the fear of citizens. — BlueBanana

I don't think you're really addressing what the OP is talking about and unjustifiably acting as if it is broader than it is. The OP is referring to the fact that congressional legislation is nearly always (something like >90% of the time) in line with what the very wealthy wish. These include tax cuts for those very wealthy as opposed to the middle or working class, rejecting health care reforms common in developed countries (because corporations prefer the veritable free for all we had before) and ever increasing rates of both college tuition and the intentional preferences universities make for wealthier applicants in their acceptance rates. These and others demonstrably don't help "the common man" and are widely trumpeted as being good for everyone despite how clearly they are directed at those with high income.

I should note that despite the following, I wouldn't say I follow a single school of philosophy, I often find myself in agreement with incompatible views.

Essentially, it's for the sake of consistency. Consistency is a theoretical virtue, that is, most of the time consistency is a property of a theory which makes the good or better relative to a theory which is otherwise identical save a for a contradiction amongst its assumptions or entailments. The more you pick views and assumptions between schools, the more likely you are to introduce contradictions into your set of beliefs. Of course, one can still find themselves in a school of thought who's tenets are inconsistent or results in a belief system with some other unwanted feature (ad hoc-ness, poor explanatory power, lack of fruitfulness, etc.)

I aspire to one day use "pecksniffery" and not feel like an asshole for doing so, lol.

The America and Europe of yesteryear are not the same as today's. "Traditional" isn't synonymous with "correct". I mean, Republicans are fine with the massive military invading other nations at a whim (on second thought, that's pretty American), intentionally crippling the security of technology so they (and thus any government or sufficiently powerful corporation) can spy on Americans, or crush unions and protests because business and government don't help each other out, etc.

These are core to the Republican Party in particular (Democrats love it too), who champion these as good. Yet they are inarguably examples of unchecked government power.

Charming" is probably not the word I would use. If the intention all along was to start a discussion about racial differences in intelligence, then "misleading" is more appropriate. — T Clark

It was intended to be a bit tongue in cheek. As I said, OP seems circumspect about their intent here so I more or less agree with you.

What we have is something that clearly works, but we haven't got the vocabulary to express it mathematically. That's a mental state familiar to everyone who's ever had to construct a proof. We get to the point where we can SEE what's going on, but we can't mathematically SAY what's going on. That's where Newton and the mathematicians of the 18th and 19th century got stuck till they finally worked out a proper formalization. — fishfry

We need only go back to 1696 to see that yes, a formal contradiction was provable. The only way, as far as I can see, to say it was just an inability to formalize is to say that it was understood but couldn't be made explicit. From Analyse des Infiniment Petits (with some modern fixes in notation and such)

I don't know how to do that Math notation stuff on this forum, but the paper "Handling Inconsistencies in the Early Calculus" (use Sci-hub to download the paper) goes over the example I had in mind when trying to find a tangent to a curve. When calculating the differenial, in the early calculus dy had to be unequal to zero, yet once the fraction had been simplified it needed to be equal to zero. Prior to limits being formalized, this was contradictory. Dy's value was necessarily non-zero at one step and then zero afterwards.

Probably, though the phrasing of the OP was charmingly circumspect (or maybe I'm just cynical and being mean).

The Russell set is not what anybody had in mind when they first had the idea of sorting the world into classes. The Liar is not what anybody had in mind when they first had the idea of communicating a fact about the world to another person. What you're both missing is how perverse the paradoxical cases are — Srap Tasmaner

Why is that relevant? I certainly never said people intended to create such paradoxes. My point is precisely that the existence of such things are what motivates changes in the logic. They are perverse, at least in the sense of being unintended consequences of seemingly reasonable principles (hence the designation "paradox"). The layperson will not understand it if you tell them there are different sizes of infinity but we know it's true in math, but we don't take that as evidence against Cantor's work on infinity.

This is just to say that I'm not suggesting people blindly forge ahead. To make sensible use of Unrestricted Comprehension, you have your use a paraconsistent logic. Classically (and in every other logic), doing so makes the resulting mathematics trivial and therefore useless.

Basically. There's a clear difference in reasonability between the contradiction in "This sentence is false" and "It's raining and it's not raining". If one insists, diffusing the former requires at least giving up some reasonable principles, the latter is simply false and doesn't seem to require losing anything reasonable to dismiss it.

I think you are making a mistake. A direct contradiction that doesn't result from a compelling argument (e.g. some bachelor is married) is not the same as a paradox, which does show up as the conclusion of a compelling argument. Even if you reject the paradox, it is not of a kind with any old contradiction.

This means either our math is fundamentally wrong or it is incomplete. — Jeremiah

I'm somewhat confused as to what you're arguing, and I probably go further than you do on this issue. Surely the classical mathematician agrees that naive set theory was fundamentally wrong. Specifically, that it was fundamentally wrong in taking the Unrestricted Comprehension Schema as an axiom. In ZF, the paradox cannot be articulated, because the axiom of separation blocks sets from containing themselves.

Ah ... a while back you objected that I misquoted you saying that incompleteness was on point here. But in fact I believe I was originally correct. You think this is about incompleteness. It's not. In incompleteness we fix a given system of logic (first-order predicate logic in fact) and draw conclusions about sets of axioms. In paraconsistent logic we alter the logical rules to obtain different theorems. That is not the same thing at all. — fishfry

I think you are mixing up points. Earlier I referenced how Russell's Paradox led to a reformulation of set theory and I further said that in spite of this reformulation, the Incompleteness Theorems still left the door open to Paraconsistent logic. After all, paraconsistent logics can tolerate inconsistency but, as per Gödel's theorems, the resulting system will be complete (the opposite of standard maths).

Appreciate that! Of course that doesn't mean that 20 or 30 years from now we won't be teaching paraconsistent logic to the undergrads. But it doesn't have much debating force today. — fishfry

In that bit, I was referring to Paraconsistent logic lacking as many practical applications (in comparison to standard maths) that would serve to justify it's use. It still has such practical uses (models of human reasoning, databases and some digital logic stuff) and uses in mathematics where standard maths drops the ball. It's just not as inherently persuasive given how much more standard maths does across a broader spectrum of applications.

This is recency bias, not a reasonable explanation IMO. — fishfry

Not really. That primes might be finite would take a lot of work to justify believing, accepting that sets are extensions of properties seems far simpler.

Unrestricted comprehension "seems reasonable" till we prove it's not. You're privileging an incorrect intuition and saying, "Who are you going to believe, an absolute logical proof, or my vague intuitions?"
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Nice intuition, turns out to be false. No reason to privilege this intuitive error. You assume it and you derive a contradiction, so it's false.
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What do you mean by dismissing the possibility of paradoxes? — fishfry

(Answering these together)

This seems like the circular reasoning I mentioned before (and this will answer a similar questions you ask elsewhere in your post). You are in effect pre-excluding the the axiom when the question is exactly if the axiom is acceptable. If I'm entertaining the possibility of accepting Russell's Paradox then pointing out the contradiction obviously isn't the defeater for me. Yes, it leads to a contradiction but to avoid it you have to spell out *why* the principles causing the contradiction are to be rejected. We know there are more primes beyond whatever prime one thinks is the largest one. They become rarer, not nonexistent. That's evidence against the view that there's a largest one. So in attempting to reject unrestricted comprehension, you have to have something beyond pointing out the obvious fact that there's a contradiction. We want to know why something has gone wrong (if indeed it has).
And on that point:

Arggg! That's EXACTLY what I'm disputing. And even though it's still called a paradox, nobody treats it that way. We treat it as a rigorous and convincing demonstration that naive comprehension must be rejected. — fishfry

This seems a little silly. Russell's Paradox is a paradox *in naive set theory*. When I said no one disputes that, I meant simply that it was the conclusion of the rules of naive set theory, but it wasn't clear why those rules were faulty beyond "we get a contradiction". We knew there was a contradiction, what made (and makes) the Russell Set a paradox in naive set theory is that the principles that give rise to it seem reasonable, a standard mathematicians employ routinely. We can avoid it of course, but in doing so Zermelo was explicit that he was trying to avoid the paradox in a new theory, not solve it (i.e. provide an explanation for why the axiom separation is a more reasonable axiom). This plays off my earlier comparison, which I'd like you to address:

I can avoid the Liar Paradox. It's easy. I'll simply define a formal language (because natural languages are known to be inconsistent) and by some means disallow self-reference. Poof, no more Liar paradox. After all, via a simple proof by contradiction I know something has gone wrong and so some assumption must be discharged. So I dismissed a feature necessary to create the paradox and voila.


This is obviously missing the point, in exactly the way you did. Evading it is easy, giving a justification for how you did so and why your methid ought to br adopted is not easy. That's what a paradox is, that's why tossing up a proof by contradiction is silly. There are consequences to whatever method you use to dissolve such issues and dismissing self-reference has a large number of negative repercussions, far more than a contradiction does. Sometimes those black marks are acceptable, but that has to be shown and a proof by contradiction does not do that. That's my point in referencing theoretical virtues. The theory resulting from the change in axioms has to be assessed for its worth and consequences. If you address nothing else, this is the main point I'd like to see you tackle.

He got the point right away. You agree that Frege himself got the point right away. Yes? — fishfry

Sure? Think I already said so. But as I also said, Frege was reasoning with the classical logic he'd just created and had no other logical theory (besides the old Syllogistic) with which to explore alternate possibilities. Classically, Russell's Paradox is unacceptable because the set theory trivializes everything, but we know that's possible to contain with Paraconsistency, even if you don't go to dialetheism.

It would not be reasonable for a physicist to reject a method that works in practice simply because it lacks mathematical rigor.
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And for what it's worth, nobody rejected calculus; but they worked very hard for 200 years after Newton to get it straightened out. — fishfry

This is exactly the point I was making. Calculus was so useful and good a theory that in spite of known contradictions, it was more reasonable to keep it. But you didn't really answer the problem I posed. Neither of us disputes that calculus was inconsistent for a while. On your view, it seems like you'd have no recourse by to dismiss the theory on grounds of inconsistency. Pointing out that physics blazes the trail is besides the point; physicists still use logic. In effect, I can't see how your view isn't inherently flawed because a single contradiction justifies any change needed to avoid it. You certainly didn't reference any limit on what one is rationally committed to doing when faced with a contradiction. So without further explanation, you'd be committed to dispensing with it and other developments depending on it. Ditto for any inconsistent scientific theory.

Well my issue was that even if I'm deploying a formal logic that doesn't mean I will flawlessly deduce the conclusion. People make errors, so knowing every fallacy isn't going to make one immune to errors in deductive reasoning. Hell, mathematicians mess up calculations too, so they have others check over their work and use automated theorem provers to decrease the likelihood of mistakes.

It doesn't matter, this is just ridiculous. If someone doesn't know, they can ask or google. It's pointless to expect everything to be explained upfront and it's not as if I'm just spewing accusations of "fallacy this, fallacy that".