Also, don't forget that quus is only one example of many other possible rules so actually you have been using some other strange rule since you started learning math and you have been using it fine — Apustimelogist

What other "strange rule" have I been using? Basic arithmetical procedures are simply the infinite iterability of addition and subtraction, and the fact that things can be grouped together in terms of different quantities.

I agree that many rules have been extrapolated out of these basics, but the extrapolations are not arbitrary in the kind of way quaddition is. They just show what can be done with these basic conceptual tools.

So, I don't agree that these basic procedures are "under-determined".

We apparently see things very differently; so much so that I cannot even tell where you are coming from with this.

What other "strange rule" have I been using? — Janus

other rules like quus. there are probably a multitude of them which are consistent with all of the addition you have ever done so far in your life and you can't rule them out.

Basic arithmetical procedures are simply the infinite iterability — Janus

uhhh don't you mean quu-nfinite quu-terability?

I agree that many rules have been extrapolated out of these basics, but the extrapolations are not arbitrary in the kind of way quaddition is — Janus

why should it be that just because a description is general or extrapolatable means it is any more or less true than a description which is specific. Is the fact you are using addition any less true than the more general description of using an operator? is the more general description of being a mammal somehow more true than the more specific description of being a human?

other rules like quus. there are probably a multitude of them which are consistent with all of the addition you have ever done so far in your life and you can't rule them out. — Apustimelogist

How can they be consistent if they don't yield the same results.

uhhh don't you mean quu-nfinite quu-terability? — Apustimelogist

No, I wasn't referring to gibberish.

why should it be that just because a description is general or extrapolatable means it is any more or less true than a description which is specific. Is the fact you are using addition any less true than the more general description of using an operator? is the more general description of being a mammal somehow more true than the more specific description of being a human? — Apustimelogist

All that seems irrelevant. I may be missing something to be sure, but if it is so, no one seems to be able to point it out. I've reached burn-out on this...

And as for people claiming they are following "other rules", there might be some plausibility to that if the other rules yielded the same results. — Janus

Judging from the ordinary understanding of basic arithmetic and logic I would say their results are self-evident to anyone who cares to think about it. — Janus

It's worth remembering that in geometry, it turned out that rules other than Euclid's (with all their intuitive plausbility) turned out to yield consistent systems, which, in the end, turned out to have "practical" applications.

indeed, it seems that Kripkes proof shows rules are not objectively true. — Banno

I don't think that's a particularly interesting result. Rules are instructions, so they aren't either true or false. That is, the rules of chess are not true or false; but they do yield statements that are true or false, such as "Your king is in check".

Well, neither is quite right. It's a question about meaning. What do we claim when we say "Jenny can add"? And more generally, what do we claim when we say that someone follows a rule? — Banno

Well, I would suggest that what is at stake is the refutation of a certain conception of what rules are - the idea that logic/mathematics is some kind of structure that determines the results of all possible applications in advance. Nothing can reach out to infinity. What we have is ways of dealing with situations as they come up which do not appear to have any limitations to their applicability. (That phrase could be misinterpreted. I mean just that "+1" can be recursively applied indefinitely. What we can't do is apply it indefinitely.)

Again, the point is underdetermination so its not about whether one rule is workable or not, any time you use addition it has an underdetermined characterization, and your ability to use it and practise it has little to do with that. — Apustimelogist

Yes, but that doesn't mean that we cannot have ways of responding to, and dealing with, problems as they come up - if necessary, we can invent them - as we do when we discover irrational numbers, etc. or find reasons to change the status of 0 or 1. In the case of 0, we have to modify the rules of arithmetical calculation.

How can they be consistent if they don't yield the same results. — Janus

Because I can construct a rule just like quus which is consistent with all of the additions you have done so far in your life and would yield the same results as all sums you have ever done, analogous to how someone may have never done sums with larger than 57 will have been doing an operation totally consistent with both quus and plus. Now the question is whether there is an empirical difference that differentiates the rule you have been using so far as either addition or this other rule? The answer is no, because so far every answer for addition is the same as this new rule. And remember there will be a multitude of these rules; for all your past behavior, this will be underdetermined and this will continue to be the case for t+1, t+2... t+n ad infinitum for every new sum you do.

No, I wasn't referring to gibberish. — Janus

The point is that you have defined what you mean by the fact that addition is diffetent to quudition, but the words in this description are susceptible to the same kinds of skepticism, and further attempts to elucidate what you mean will bring a regress of these definitions on which skepticism can be applied.

All that seems irrelevant. — Janus

well you seemed to be appealing to the extrapolatability or generalizability of addition as to why it is more true but I don't see why this is a fact in making a description more true or not. why should it be that a description that extends to more cases than another be somehow more true?

I don't think that's a particularly interesting result. Rules are instructions, so they aren't either true or false. That is, the rules of chess are not true or false; but they do yield statements that are true or false, such as "Your king is in check". — Ludwig V

Buy "you are following x rule" is factual. What do you think is the interesting result of this story then?

Yes, but that doesn't mean that we cannot have ways of responding to, and dealing with, problems as they come up - if necessary, we can invent them - as we do when we discover irrational numbers, etc. or find reasons to change the status of 0 or 1. In the case of 0, we have to modify the rules of arithmetical calculation. — Ludwig V

I don't think this problem has anything to do with practical problems. The quus issue has no bearing on someones ability to perform math.

other rules like quus. there are probably a multitude of them which are consistent with all of the addition you have ever done so far in your life and you can't rule them out. — Apustimelogist

Does Kripke question the extent to which consciously following a rule even applies?

For example, having worked with digital logic a fair bit, I have all the powers of 2 up to 2^13 memorized and if I see 2048 + 2048 I simply recognize that the sum is 4096 without following any step by step decimal addition rules.

What is supposed to be the significance of arriving at sums via different cognitive processes?

Keeping the analogy between Hume and Kripke's sceptic:

Hume's questioning of the place of causation doesn't yield reliably workable results. Scepticism isn't as much about reliable workable results as truth.

Quaddition's workability isn't really at issue. I think the sceptic would say "no, that's not useable for engineering. But what's the fact you can point to that lets us know the engineer is using addition?" Quaddition is there as a conceptual contrast to addition to help in understanding the question "What's the fact I can point to that justifies my belief that I'm adding?"

To make a similar function to quaddition that'd be easier to accept in light of engineering: Instead of Quaddition we could posit Googol-ition -- where the rules of arithmetic are the same up to a googol. If you find an example of an engineer whose used a number that high, then you can raise the googol to the power of a googol, and posit the googol^googol-ition. What's being asked after is a fact which demonstrates that we're performing addition, and googol-ition is there to give a conceptual contrast (and highlight that there's no factual difference, or at least make that challenge).

And the sceptic believes there is no fact at all -- there's a rule being followed rather than a truth being stated.

Does that make the question make sense?

Not sure about Kripke but Wittgenstein definitely mentions stuff l like that in philosophical investigations. rules and explicit definitions are more like signposts than prescriptions on how to behave. in fact, i think a major point in PI is that meanings and definitions in language are effectively so impoverished that it should render language un-usable, but it doesn't. A repeated theme it seems to be this underlying inscrutable, implicit underlying behavior where there is room for the kind of indeterminacy, fuzzyness or perhaps plurality about how people accomplish things.

Excerpt from PI:

"232. Let us imagine a rule intimating to me which way I am to obey it; that is, as my eye travels along the line, a voice within me says: "This way!"—What is the difference between this process of obeying a kind of inspiration and that of obeying a rule? For they are surely not the same. In the case of inspiration I await direction. I shall not be able to teach anyone else my 'technique' of following the line. Unless, indeed, I teach him some way of hearkening, some kind of receptivity. But then, of course, I cannot require him to follow the line in the same way as I do.
These are not my experiences of acting from inspiration and according to a rule; they are grammatical notes.

235. It would also be possible to imagine such a training in a sort of arithmetic. Children could calculate, each in his own way—as long as they listened to their inner voice and obeyed it. Calculating in this way would be like a sort of composing.

234. Would it not be possible for us, however, to calculate as we actually do (all agreeing, and so on), and still at every step to have a feeling of being guided by the rules as by a spell, feeling astonishment at the fact that we agreed? (We might give thanks to the Deity for our agreement.)

235. This merely shews what goes to make up what we call "obeying a rule" in everyday life.

236. Calculating prodigies who get the right answer but cannot say how. Are we to say that they do not calculate? (A family of cases.)

237. Imagine someone using a line as a rule in the following way: he holds a pair of compasses, and carries one of its points along the line that is the 'rule', while the other one draws the line that follows the rule. And while he moves along the ruling line he alters the opening of the compasses, apparently with great precision, looking at the rule the whole time as if it determined what he did. And watching him we see no kind of regularity in this opening and shutting of the compasses. We cannot learn his way of following the line from it. Here perhaps one really would say: "The original seems to intimate to him which way he is to go. But it is not a rule." "

Kripke's mistake (assuming I am recalling his position correctly), was phrasing the skepticism as a circular question to a mathematician where he asked to defend the validity of his judgements, as in

"How do you know that your present usage of "plus" is in accordance with your previous usage of "plus" ?"

That question is easily viewed as nonsensical, since it is easily interpreted as asking a person to question their own sanity. Similarly bad phrasing, leading to pointlessly circular discussion is found throughout the philosophy literature on private language arguments. — sime

Today we're talking in the meta-language about the object-language of yesterday, or right now we're talking in the meta-language about the object-language of addition. What, in the object-language, is the fact that we're adding at all? Would you say that this version of the question is easily viewed as nonsensical?

One of the things that I keep thinking on is how I tend to think of facts not as things but rather as true sentences. So in reading the essay, to make it make sense, I'd probably put it that -- rather than there is no fact to the matter -- there are no truth-conditions which make 68+57 equal 125. It's true because that's the answer we should obtain according to the conditions of assertability, but there are no truth-conditions that make it true.

In saying that much -- the question begins to make a kind of sense because mathematics is abstraction. So, in a way, there shouldn't be truth-conditions of addition. If there's a physical unit involved then there are possibly truth-conditions, but that's not the question. It's much more a question about meaning because of the abstraction. (at least, as I'm understanding it so far)

Thanks for taking the time. I'm reading PI right now, but haven't gotten that far.

Insightful stuff.

It can be really difficult to read to be fair. Its one of those books where possibly what the book says has not been as influential as what othwrs have said about the book.

For example, having worked with digital logic a fair bit, I have all the powers of 2 up to 2^13 memorized and if I see 2048 + 2048 I simply recognize that the sum is 4096 without following any step by step decimal addition rules. — wonderer1

I admire your memory! But isn't it exactly the same as we all (?) do when we memorize the standard multiplication tables and recall what 12x11 is. (It's just a convention that we stop at 12. The table for 13 is no different in principle from the table for 12.) Multiplication reduces to addition, but adding 12 11's by that procedure is long and boring. By memorizing the standard multiplication tables, we have a quicker way of dealing with some questions and of calculating bigger numbers. (Incidentally, how do you deal with 2 to the power of 35?)

rules and explicit definitions are more like signposts than prescriptions on how to behave. — Apustimelogist

I agree with that. Though Wittgenstein would ask what makes the sign-post point? Again, there's a practice of reading sign-posts, which we all somehow pick up/learn. Perhaps by recognizing a similarity between a pointing finger and the sign-post.

I don't think this problem has anything to do with practical problems. The quus issue has no bearing on someones ability to perform math. — Apustimelogist

I agree with that. It's a pointless difficulty. Like most sceptical arguments. I like Hume's response - essentially that it is not possible to refute the argument but it has no power to persuade me to believe the conclusion. But that's not how the philosophical game is played - for better or worse.

Buy "you are following x rule" is factual. — Apustimelogist

There's a nest of complications buried in that. In one way, you are just raising the original question again. However, there is a fact of the matter involved - that I gave 125 as the answer to the question. Whether I was following the rule "+1" is another question. In one way, it depends on whether I had that rule in mind when I gave the answer. In another way, it depends on whether we agree with the answer - and that may depends on the wider context (consistency and practical outcomes).

It's true because that's the answer we should obtain according to the conditions of assertability, but there are no truth-conditions that make it true. — Moliere

Forgive me, I don't really understand what "conditions of assertability" are as opposed to "truth-conditions". Are they facts? In which case, we may be no further forward.

What do you think is the interesting result of this story then? — Apustimelogist

That's an interesting question. In one way, the desired result is to defuse the question so that I don't get bothered by it - that is, don't need to take it seriously. Whether that's interesting or not depends on whether you are philosophically inclined or not.

But I am learning from this. One result is that I now know how to defuse Goodman's "grue". Another is that it seems that Kripke has made the private language argument superfluous. I need to think about that. A third - minimal - result is that Kripke has added to the stock of examples that pose Wittgenstein's problem. The fourth is that I notice that we have all appealed to the wider context, both of mathematics and of practical life to resolve it. Kripke's case is effective only if we adopt his very narrow view, The wider context makes nonsense of it. (I'm not saying that a narrow focus is always a bad thing, only that it sometimes gets us into unnecessary trouble.)

Wittgenstein says somewhere that he has got himself into trouble because he is thinking about the pure world, but what we need to do is return to the rough ground.

It (sc. Philosophical Investigationscan be really difficult to read to be fair. Its one of those books where possibly what the book says has not been as influential as what othwrs have said about the book. — Apustimelogist

I agree with that. One of the difficulties is that the text is not difficult to understand (contrast Hegel or Derrida). The difficulty is to understand what the point is. That's where the commentators can help - and sometimes hinder, so don't read them uncritically.

This book is not written, as most philosophy books are, in the belief that laying out the arguments clearly ("clearly" is complicated, of course) is the most effective way of changing someone's mind. No doubt it is, sometimes. But W thought that philosophical problems were not really susceptible to that treatment. So he provide hints and leaves you to work out what he's getting at. Some of his followers do the same thing.

That's where the commentators can help - and sometimes hinder, so don't read them uncritically. You'll need a general introduction to start with. Sadly, I'm so out of date that I don't know which are the best ones. But there'll be reviews that will help you choose and you could a lot worse that read an encyclopedia entry, which would be shorter.

Forgive me, I don't really understand what "conditions of assertability" are as opposed to "truth-conditions". Are they facts? In which case, we may be no further forward. — Ludwig V

I think it'd depend upon how we're trying to judge if someone knows something or not. With arithmetic those conditions are spelled out in books and habit and embodied within a community of arithmetic speakers. I'm thinking that it has more to do with a community's process of acceptance than facts.

So the teacher has a handful of representative problems which if the student is able to do without aid we then accept them as part of the community of arithmetic speakers.

Same goes for accepting whether a person knows the meaning of such-and-such for particular topics, or whether they know a language: the meaning isn't a fact as much as what you have to do in order to be accepted within a community of languagers.

I agree with that. Though Wittgenstein would ask what makes the sign-post point? Again, there's a practice of reading sign-posts, which we all somehow pick up/learn. Perhaps by recognizing a similarity between a pointing finger and the sign-post. — Ludwig V

Yup, and then the issue regresses as to what makes someone recognize a similarity between pointing a finger and sign-posts. This is all what I meant when I said that meanings and definitions are so impoverished that language should not be usable, yet it is.

I like Hume's response - essentially that it is not possible to refute the argument but it has no power to persuade me to believe the conclusion. — Ludwig V

I think for me, its about how such insights might reveal something about how brains and minds work.

There's a nest of complications buried in that. — Ludwig V

Well I am just saying that it is a factual statement regardless of whether there turns out to be or not be a fact of the matter. It straightforwardly makes sense as a factual statement.

In one way, it depends on whether I had that rule in mind when I gave the answer. — Ludwig V

Or perhaps even what it means to have a rule in mind.

But I am learning from this. One result is that I now know how to defuse Goodman's "grue". Another is that it seems that Kripke has made the private language argument superfluous. I need to think about that. A third - minimal - result is that Kripke has added to the stock of examples that pose Wittgenstein's problem. The fourth is that I notice that we have all appealed to the wider context, both of mathematics and of practical life to resolve it. Kripke's case is effective only if we adopt his very narrow view, The wider context makes nonsense of it. (I'm not saying that a narrow focus is always a bad thing, only that it sometimes gets us into unnecessary trouble.) — Ludwig V

Can you elaborate?

But isn't it exactly the same as we all (?) do when we memorize the standard multiplication tables and recall what 12x11 is. — Ludwig V

Yes, it just seemed relevant to me to point out that there are a variety of mathematically legitimate ways that can yield correct mathematical results and therefore it seems weird to me to focus so, on whether some particular rule was used in some specific case. So I brought it up in hopes of getting a better idea of what Kripke was trying to get at.

(Incidentally, how do you deal with 2 to the power of 35? — Ludwig V

With a calculator. :wink:

Well though the math thing you get at is related, its not exactly the same. The rule thing is about definition and description and is just meant to be a single example of a type of issue that is generic to everything. I guess the point is that semantic definitions and descriptions are not intrinsically embedded in the world; instead, we impose labels on the world at out own discretion and there are no fixed set of boundaries for those concepts or force us to impose concepts in a particular way.

This is all what I meant when I said that meanings and definitions are so impoverished that language should not be usable, yet it is. — Apustimelogist

Yes. There isn't a way of resolving that without going beyond that way of thinking. W's does that. His appeal to games, practices, forms of life etc. is an attempt to explain it. As a general thesis, it is quite unsatisfactory, (cf. God of the gaps), but as a tactic applied in specific situations, it works well (as in this case). There's an obvious catch that it may be misapplied. But that doesn't necessarily mean it is never appropriate.

therefore it seems weird to me to focus so, on whether some particular rule was used in some specific case. — wonderer1

I think the intention is to distinguish between a heuristic which may be useful in some circumstances, but not in all, and how we would settle the question whether the output of the heuristic is correct or not. The intriguing bit is why we accept one way of calculating as definitive (conclusive). Kripke's problem muddles up the two different ways of getting an answer.

we impose labels on the world at out own discretion and there are no fixed set of boundaries for those concepts or force us to impose concepts in a particular way. — Apustimelogist

That works in some ways. But the picture of the world out there, waiting to be "carved at the joints", is partial. The world reaches in and prods us, tickles us, attracts us and repels us. We do not start out as passive observers but as engaged actors in the world - which does not always behave in the way that we expect.

Can you elaborate? — Apustimelogist

H'm. My posts are quite long as it is. I'm concerned I might outstay my welcome or run up against the TL:DR syndrome. Some focus would help.

I think it'd depend upon how we're trying to judge if someone knows something or not. — Moliere

I get that distinction. Indeed, arguably an assessment whether the knower is in a position, or has the capacity, to know p is appropriate in assessing any claim to knowledge. And I can see that final truth will often be distinct from any such assessment. (The jury has a perfect right to find the prisoner guilty or not. Yet miscarriages of justice do happen - and proving that is different from proving whether the prisoner is guilty or not. (A miscarriage might have reached the right result.)) But I still feel that the distinction is quite complicated. After all, the truth would be the best assertability condition of all, wouldn't it? And the assertability conditions would themselves be facts, wouldn't they? Of course, they need not be the same facts as the truth conditions.

Yes. There isn't a way of resolving that without going beyond that way of thinking. W's does that. His appeal to games, practices, forms of life etc. is an attempt to explain it. As a general thesis, it is quite unsatisfactory, (cf. God of the gaps) — Ludwig V

That works in some ways. But the picture of the world out there, waiting to be "carved at the joints", is partial. The world reaches in and prods us, tickles us, attracts us and repels us. We do not start out as passive observers but as engaged actors in the world - which does not always behave in the way that we expect. — Ludwig V

For me, what the point of the impoverishment of language shows is that the way we use words and concepts does not trickle down prescriptively from definitions and meanings that possess some invariant, essential nature. Rather, definitions are idealizations that are constructed or inferred in a bottom up manner from the statistics and dynamics of experience. The kinds of fuzziness, ambiguities, context-dependence, indeterminacy that characterizes Wittgenstein's analyses can be explained by appealing to the nature of how brain processes perform inference, effectively extracting lower-dimensional, more coarse-grained, more generalized underlying patterns (concepts) from complicated observations. These are extremely complicated machines processing an extremely complicated world and so the processing they do does not necessarily reflect very simple, linear, straightforward transformations between observations and the resultant inferred concepts.

Essentially the missing link in Wittgenstein's observations is the fact that we have a brain, one whose processing is extremely complicated yet also totally hidden from us, generating our complicated thoughts and behavior from below on the fly, making it look like we are acting in these kinds of mysterious ways that seem somewhat messy and underdetermined by our concepts and so can only be described as "games, practises, forms of life".

"The world reaches in and prods us, tickles us, attracts us and repels us" as a product of the mechanistic message passing and hebbian timing-dependent learning between neurons that are physical enslaved by the patterns of activation at our sensory boundaries (e.g. retina, inner ear, receptors under the skin), impelling the perceptions forced upon us, complicated behavior we are capable of, the higher-order concepts that we construct, but also the metacognitive insight that such concepts could have been otherwise. The brain completes the picture.

I think the intention is to distinguish between a heuristic which may be useful in some circumstances, but not in all, and how we would settle the question whether the output of the heuristic is correct or not. — Ludwig V

I dunno; I think looking at this way, as I seem to understand what you have said, plays down everything else that Wittgenstein seems t be getting at in philosophical investigations.

Some focus would help — Ludwig V

Well you just put forward your four points without any reference to what you mean by those points. Basically, all these points are lacking a "how".

neurons that are physical enslaved — Apustimelogist

impelling the perceptions forced upon us — Apustimelogist

I hope so. It's the only way that we get reliable information - and, in great part, we do.

making it look like we are acting in these kinds of mysterious ways that seem somewhat messy and underdetermined by our concepts and so can only be described as "games, practises, forms of life". — Apustimelogist

I'm sure there's a lot of quick and dirty solutions and heuristic dodges involved. Anything remotely like formal logic would be too slow to be useful.

I dunno; I think looking at this way, as I seem to understand what you have said, plays down everything else that Wittgenstein seems to be getting at in philosophical investigations. — Apustimelogist

I was only talking about relying on a memorized table, instead of doing the basic calculations. It's an example of a quick and dirty solution.

One result is that I now know how to defuse Goodman's "grue". — Ludwig V

This "paradox" is structurally the same as Kripke's. Here's the link to Wikipedia https://en.wikipedia.org/wiki/New_riddle_of_induction, which mentions, but does not discuss, Kripke and his solution. I think that Wittgenstein's discussion of rule-following applies to both of these puzzles. Does that help? To take it much further would probably require another thread, don't you think?

Another is that it seems that Kripke has made the private language argument superfluous. I need to think about that. — Ludwig V

This point is made elsewhere. The complication is that the private language argument does rely on some of the things he says about rule-following, particularly the importance of understanding what does and does not conform to the rules about ostensive definition. But numbers are not sensations, so the cases are not exactly the same.

A third - minimal - result is that Kripke has added to the stock of examples that pose Wittgenstein's problem. — Ludwig V

W likes lots of examples. In one way, Kripke's case is just another one, although W does mention the point at PI 201 "This was our paradox: no course of action could be determined by a rule, because every course of action can be brought into accord with the rule. The answer was: if every course of action can be brought into accord with the rule, then it can also be brought into conflict with it. And so there would be neither accord nor conflict here." I had forgotten this quotation. In time, I could no doubt find what he was referring back to. It gives a short answer to both Goodman and Kripke.

The fourth is that I notice that we have all appealed to the wider context, both of mathematics and of practical life to resolve it. Kripke's case is effective only if we adopt his very narrow view, — Ludwig V

Isn't that an accurate reflection of what we've been saying about practices?

I hope that's helpful.

I'm sure there's a lot of quick and dirty solutions and heuristic dodges involved. — Ludwig V

I am not sure I would characterize them all as heuristics or "quick and dirty" solutions since they are just the same processes that underlie everything we do. Its just that the actual statistical structure of much of the world is much more complicated and non-linear than the simpler idealized concepts we like to dealing with in academics.

I think the problem with the answers that brains give though is they are finely contextualized by different personal histories, individual differences in brain structure, noise etc. What people learn and the information they store is probably different for everyone, but in places like academia we want to remove all ambiguity. The side effect of neat clean concepts is they lose all the fuzzy non-linearity which makes them exceptionally good at being used in real life.

I think the problem with the answers that brains give though is they are finely contextualized by different personal histories, individual differences in brain structure, noise etc. What people learn and the information they store is probably different for everyone, but in places like academia we want to remove all ambiguity. The side effect of neat clean concepts is they lose all the fuzzy non-linearity which makes them exceptionally good at being used in real life. — Apustimelogist

:100:

I get that distinction. Indeed, arguably an assessment whether the knower is in a position, or has the capacity, to know p is appropriate in assessing any claim to knowledge. And I can see that final truth will often be distinct from any such assessment. (The jury has a perfect right to find the prisoner guilty or not. Yet miscarriages of justice do happen - and proving that is different from proving whether the prisoner is guilty or not. (A miscarriage might have reached the right result.)) But I still feel that the distinction is quite complicated. After all, the truth would be the best assertability condition of all, wouldn't it? And the assertability conditions would themselves be facts, wouldn't they? Of course, they need not be the same facts as the truth conditions. — Ludwig V

Yeah, I'll admit it's complicated. Or at least vague. I don't know if the truth is the best assertability condition, though, because here we have truths that we arrive at because of the conditions of assertability -- at least this seems to make sense of Kripke's position as an interesting position. If it all came back to truth then what's the deal with pointing out that there's no fact to the matter?

Also I think Kripke takes us to this place in his essay, but then doesn't say much more. I'm still uncertain that I have the exact right interpretation of Kripke here, too -- this is just what comes to mind when I attempt to make sense of Kripke's arguments.

There's something queer for myself at least in holding that facts are true sentences, that mathematical sentences are true, and yet they are not true in virtue of truth-conditions. It would seem that under this interpretation that I'm committed to some way of coming to know true sentences aside from truth-conditions. Given that we're talking about meaning that seems to be where I'd have to go. And there's a historical precedent there in the analytic/synthetic distinction, but I wouldn't want to rely upon that distinction because I pretty much agree with Quine on it being fuzzy.

So, yes, to hold to my interpretation of Kripke's conclusion along with some of my other beliefs and defend them I'd have to do some work on what these conditions of assertability are.

The side effect of neat clean concepts is they lose all the fuzzy non-linearity which makes them exceptionally good at being used in real life. — Apustimelogist

Yes, that's true. I'm a bit inclined to say the W sees that "fuzzy non-linearity" as inherent in all concepts. So what do we say about logic? What makes it special? (I'm not asking because I know, or think I know, the answer.)

because here we have truths that we arrive at because of the conditions of assertability — Moliere

Maybe we should distinguish between what brings the rule into effect (I chose that word carefully because after it becomes effective it is correct to say that there is a rule that ...) Can we see conditions of assertability as comparable to the licence conditions for someone to perform a wedding? If so, laying down a rule is or at least is comparable to, a speech act. We then have to explain that in some cases, the rule is not formally laid down, but informally put into effect (as when language changes, and "wicked" comes to mean the opposite of what it meant before). Once the rule is in effect, there is a fact of the matter, as when your king is in check or 68+57=125.

Yes, that's true. I'm a bit inclined to say the W sees that "fuzzy non-linearity" as inherent in all concepts. — Ludwig V

Yes, I would agree with this and agree with it myself. I don't know of there is anything particularly special about logic and am drawn in the direction of logical nihilism or pluralism.

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Can you demonstrate how quus is dealt with by the approaches you have said? — Apustimelogist

A philosophical commonality of those approaches, is that mathematical objects are treated as being finite. So for example, plus is permitted to exist intensionally in the sense of an algorithmic specification but not in the extensional sense of a completed table (unlike with the quus function). Likewise, all sequences are treated as being necessarily finite and generally unfinshed. This apparent restriction is compensated by allowing objects to grow over time (technically, growable objects are describable by using what are now referred to as coalgebras and coinduction).

Intuitionism would call a sequence of n numbers x1,x2,...xn that were generated by iteratively applying the "plus" function as constituting a lawful sequence of n terms. But another sequence consisting of exactly the same numbers x1,x2,..xn, that wasn't assumed to be generated by some underlying function, would be considered a lawless sequence. In the case of quus, it can be considered to be a lawless sequence , since it is describable as a table of exactly 57 rows and 57 columns. So if that table is defined to mean "quus" as a matter of tautology, then skepticism as to what the function quus refers to can only concern operational assumptions regarding how the table should be evaluated - however skepticism of this sort is accommodated by intuitionism, since intuitionism doesn't consider mathematical truth to be a priori and the properties of unfinished sequences are allowed to change over time . The situation with quus is at least a partial improvement upon plus, whose table cannot even be explicitly stated. Quus is more or less a truncation of plus, and roughly speaking, intuitionism considers such "truncations" as constituting the basis of mathematical analysis.


Additionally, in Linear Logic terms and constants can only be used once. So two uses of a function demands two separate and distinct copies of that function. Linear logic includes a copy operation, (the so-called exponential rules), meaning that the logic can be used without loss of generality, but this forces the mathematician to justify and account for his use of resources in the same way as engineers who must always consider the possibility of numerical overflow and hardware failure.

Kripke allows that mathematicians can adequately specify the rules of addition. That's not being called into question.

logical nihilism or pluralism. — Apustimelogist

I prefer pluralism coupled with pragmatism. Horses for courses. Logical analysis can give a kind of clarity.

Maybe we should distinguish between what brings the rule into effect (I chose that word carefully because after it becomes effective it is correct to say that there is a rule that ...) Can we see conditions of assertability as comparable to the licence conditions for someone to perform a wedding? If so, laying down a rule is or at least is comparable to, a speech act. We then have to explain that in some cases, the rule is not formally laid down, but informally put into effect (as when language changes, and "wicked" comes to mean the opposite of what it meant before). Once the rule is in effect, there is a fact of the matter, as when your king is in check or 68+57=125. — Ludwig V

I'm inclined to say there is a fact, but that it's not the fact which justifies, say, the license conditions for someone to perform a wedding. The rule is in effect, and in some sense then it produces facts -- but the production of facts is not justified by the facts so produced. The rule itself has no factual justification, though we could only judge if a person knows how to add if we know the fact we'd obtain by performing the rule. (at least, in my way of speaking where facts are true sentences. this could very well be me bringing in an inconsistency, though, whereas Kripke wouldn't bother with this notion of facts being true sentences. I'm not sure there)

Or, at least, I can't help but think that there has to be some distance between rules and facts for Kripke in order for the position to be philosophically interesting. The skeptic has to be pointing out that we're inclined to believe there's a fact where there is none in order for the skeptic to have a point at all, or else we're more or less just stating that the skeptic does not succeed in pointing out a skeptical problem.

The rule is in effect, and in some sense then it produces facts — Moliere

There are rules for which the process that brings them into effect is quite clear. They are what we call laws, but there are other varieties. They are imperatives, not really different from the order given by the general. Other rules, like mathematical rules about how to calculate are different. There are proofs of such rules. What makes them effective? Which is to say, what justifies them? That's where the sceptical pressure (which W also applied) and his appeal to practices comes in. But that involves saying that the rule doesn't really of itself produce facts; human beings have to carry out the calculations (or psersuad machines to do it for them. Those results are facts, i.e. have the authority of facts? Only the calculation, which can't produce a wrong result. That means that if a result does not fit in to our wider lives in the way it is expected to, we look for the fault in the calculation and the calculator, not the rule.

At the bottom of this is the fact that "+1" can be applied to infinity. How can we know that? Certainly not by applying the rule to infinity. By definition, we can never exhaustively check that the infinite application of the rule will work out.

The skeptic has to be pointing out that we're inclined to believe there's a fact where there is none in order for the skeptic to have a point at all. — Moliere

If you mean a fact that justifies the rule and/or justifies how the rule is applied. I sometimes think that the quickest way to state the problem is to point out that the rule cannot be a fact, because the rule has imperative force and no fact can do that - a version of the fact/value distinction. For the same reason, no fact can, of itself, justify the rule.