I'm still crawling my way there. This morning I didn't have time to do my little bit of philosophy to warm up the mind. Hopefully with two of us we can guess our way through.

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.

I am a moth jumping from light to light, but I usually come back around.

This morning I find myself going back. In particular as I proceeded I started to pick up on a pattern in the writing: between theorems and conclusions.

Going to Chapter 4: Theorems are general patterns which can be seen through formal considerations of the Initials. Also axioms are used. In going back to get a better feel for the distinctions between these terms I'm also picking up on that Canon is never formally defined -- it's like a Catholic Canon in its function. Also I'm picking up on why identity is the 5th theorem -- if the calculus was inconsistent then you could come up with x =/= x. And, going back over, I'm starting to see the significance of theorem 7 -- it's what let's us build a calculus through substitution, which theorem's 8 and 9 provide the initials for that calculus in chapter 5.

This is all inspired by the paragraph immediately where I left off:

We may take the evident degree of this indeterminacy to
classify the equation in which such expressions are equated.
Equations of expressions with no re-entry, and thus with no
unresolvable indeterminacy, will be called equations of the
first degree, those of expressions with one re-entry will be called
of the second degree, and so on.
It is evident that Jl and J2 hold for all equations, whatever
their degree. It is thus possible to use the ordinary procedure
of demonstration (outlined in Chapter 6) to verify an equation
of degree > 1. But we are denied the procedure (outlined
in Chapter 8) of referring to the arithmetic to confirm a demonstration
of any such equation, since the excursion to infinity
undertaken to produce it has denied us our former access to a
complete knowledge of where we are in the form. Hence it
was necessary to extract, before departing, the rule of demonstration,
for this now becomes, with the rule of dominance,
a guiding principle by which we can still find our way.

And reviewing back up to Chapter 5 is about as far as I got this morning. I'm attempting to disentangle the procedures of Chapter 6 from Chapter 8 to give myself a better understanding of what's missing and needed to understand the next bits in Chapter 11.  

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.

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)

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?

I've been trying to think of a good response @Janus but have been unable, so perhaps this will do better. I believe this expression may be close to what you've been getting at?

You see, in both cases, the fundamental issue isn't resolved. Answering "habit" doesn't create rule-following facts for us. As with the problem of induction, we still have the gaping hole where we expected empirical data to support our assertions. Obviously, since Hume's problem attracted Kant's approach, we might expect that Kripke's problem would do something similar. Meaning isn't based on objective rule following, so maybe there's something innate about it. Maybe this innateness is a touchstone that meets each episode of communication, including this one. — frank

"innate" with respect to meaning is something I wouldn't deny as true, but only as unsatisfactory. It may be the case that innateness of meaning is the touchstone that allows you and I to communicate. When it comes to poetry, especially, that's where I gravitate towards -- asking for more words to explain words.

However we'd like to know more about something than "this is just what it means". This is getting back to a question I don't know how to answer: what do I want from a theory of meaning? To disappoint, I don't know what I want from a theory of meaning. Somehow I just ended up here with these questions, probably because I like to ask after seemingly silly things ;)

I think I'm tempted to simply accept the conclusion: there are no rule-following facts. Same with Hume and causation, though I really do admire Kant's attempt to overcome Hume's skepticism towards causation.

Fair point about the target audience. I'm asking about the speaker of the essay. If Kripke isn't offering a resolution, then I'm asking: what about these resolution-like looking paragraphs at the end? Who is offering them? That's what I mean. Who is the speaker?

The way that makes sense to me is to read the essay as presenting Kripkenstein's views, rather than Kripke's. Is that how you're reading it? (which, to be fair, that's how he starts out the essay -- saying it's an essay not on his view as much as an impression of his while reading Witti)

So we still don't have any basis for determining that S followed a particular rule. We just treat certain circumstances as if she did. — frank

True.

If I'm understanding the argument: in place of truth-conditions Kripke resolves the sceptical problem with the sceptical solution that the community provides assertability-conditions. There's no fact which justifies the assertability-conditions, though.

Finally, the point just made in the last paragraph, that Wittgenstein's theory is one of assertability conditions, deserves emphasis. Wittgenstein's theory should not be confused with a theory that, for any m and n, the value of the function we mean by 'plus', is (by definition) the value that (nearly) all the linguistic community would give as the answer. Such a theory would be a theory of the truth conditions ofsuch assertions as "By 'plus' we mean such-andsuch a function," or "By 'plus' we mean a function, which, when applied to 68 and 57 as arguments,. yields 125 as value."

...

Wittgenstein thinks that these observations about sufficent conditions for justified assertion are enough to illuminate the role and utility in our lives of assertion about meaning and determination of new answers. What follows from these assertability conditions is not that the answer everyone gives to an addition problem is, by definition, the correct one, but rather the platitude that, if everyone agrees upon a certain answer, then no one will feel justified in calling the answer wrong.

But what would you look for in an extraterrestrial signal if you were assessing for rationality? You'd probably want to see intention, right? What tells you that an action was intentional?

Some would say we want to see some signs of judgement. For instance if we would take a sequence of constants as a sign of intelligence, that would tell us that the aliens consciously chose those numbers. Choice entails normativity. They picked this number over that one.

All of this is wrapped up in rule following, which is normativity at its most basic. To follow a rule means to choose the right action over the wrong ones.

If it turns out that there's no detectable rule following in the world, normativity starts to unravel and meaning along with it. Is that how you were assessing the stakes here? — frank

I want to post Kripke's summation of his own argument. On page 107-109:

Let me, then, summarize the 'private language argument' as it is presented in this essay. (I) We all suppose that our language expresses concepts - 'pain', 'plus', 'red' - in such a way that, once I 'grasp' the concept, all future applications of it are determined (in the sense of being uniquely justified by the
concept grasped). In fact, it seems that no matter what is in my mind at a given time, I am free in the future to interpret it in different ways - for example, I could follow the sceptic and interpret 'plus' as 'quus'. In particular, this point applies if I direct my attention to a sensation and name it; nothing I have done determines future applications (in the justificatory sense above). Wittgenstein's scepticism about the determination of future usage by the past contents of my mind is analogous to Hume's scepticism about the determination of the future by the past (causally and inferentially). (2) The paradox can be resolved only by a 'sceptical solution of these doubts', in Hume's classic sense. This means that we must give up the attempt to find any fact about me in virtue of which I mean 'plus' rather than 'quus', and must then go on in a certain way. Instead we must consider how we actually use: (i) the categorical assertion that an individual is following a given rule (that he means addition by 'plus'); (ii) the conditional assertion that "if an individual follows such-and-such a rule, he must do so-and-so on a given occasion" (e.g., "if he means addition by '+', his answer to '6S+ 57' should be '125"'). That is to say, we must look at the circumstances under which these assertions are introduced into discourse, and their role and utility in our lives. (3) As long as we consider a single individual in isolation, all we can say is this: An individual often does have the experience of being confident that he has 'got' a certain rule (sometimes that he has grasped it 'in a flash'). It is an empirical fact that, after that experience, individuals often are disposed to give responses in concrete cases with complete confidence that proceeding this way is 'what was intended'. We cannot, however, get any further in explaining on this basis the use of the conditionals in (ii) above. Of course, dispositionally speaking, the subject is indeed determined to respond in a certain way, say, to a given addition problem. Such a disposition, together with the appropriate 'feeling of confidence', could be present, however, even if he were not really following a rule at all, or even if he
were doing the 'wrong' thing. The justificatory element of our use of conditionals such as (ii) is unexplained. (4) If we take into account the fact that the individual is in a community, the picture changes and the role of (i) and (ii) above becomes apparent. When the community accepts a particular conditional (ii), it accepts its contraposed form: the failure of an individual to come up with the particular responses the community regards as right leads the community to suppose that he is not following the rule. On the other hand, if an individual passes enough tests, the community (endorsing assertions of the form (i)) accepts him as a rule follower, thus enabling him to engage in certain types of interactions with them that depend on their reliance on his responses. Note that this solution explains how the assertions in (i) and (ii) are introduced into language; it does not give conditions for these statements to be true. (5) The success of the practices in (J) depends on the brute empirical fact that we agree with each other in our responses. Given the sceptical argument in (I), this success cannot be explained by 'the fact that we all grasp the same concepts'. (6) Just as Hume thought he had demonstrated that the causal relation between two events is unintelligible unless they are subsumed under a regularity, so Wittgenstein thought that the considerations in (2) and (3) above showed that all talk of an individual following rules has reference to him as a member of a community, as in (J). In particular, for the conditionals of type (ii) to make sense, the community must be able to judge whether an individual is
indeed following a given rule in particular applications, i.e. whether his responses agree with their own. In the case of avowals of sensations, the way the community makes this judgement is by observing the individual's behavior and surrounding circumstances.

Because it makes sense of your questions :D -- when I first read your questions I realized I just needed to do some of the homework. So far I've been arguing only that there is a skeptical problem or skeptical question that I see from your OP, and haven't gone so far as to offer a solution or response or even to draw out implications.

And I'm glad I did some of the homework. Kripke's mind is wild to ride along with. Look at all these incredible connections he's able to draw out, and look at how he's able to distinguish so many possible beliefs at once while maintaining a single thread of thought! It's impressive.

I think what I'd say is that there are ways of detecting if someone is following a rule, it's only that these ways are not a state of affairs in the world. Rather it's an acceptance by a community. At least this is the solution I see Kripkenstein offering. The conditions of assertability aren't in truth-conditions, but there are still conditions of assertability. You just have to learn what they are.

What Kripkenstein's skeptic points out is that our common belief that "1 + 1 = 2" doesn't have truth-conditions, but rather conditions of assertability, and in comparison with Hume's skepticism we learn the conditions of assertability through repetition and acceptance by a community of rule-followers: the force of habit reinforced by communal acceptance.

So not quite an undermining of all normativity, but possibly a re-adjustment on philosophical interpretations of meaning.

I don't believe arithmetic to be merely rule following, but I think it is something we get intuitively on account of its being naturally implicit in cognition. Some animals can do rudimentary counting, which means they must be aware of number.

So, it begins with recognition of difference and similarity, then gestalted objects, then counting of objects, and this basis is elaborated in the functions of addition, subtraction, multiplication and division. Mathematical symbols and the formulation of arithmetical rules then open up the possibility of endless elaboration and complexification. — Janus

How do you respond here to @Ludwig V's point?

Counting makes sense as a genesis of arithmetic. But is doesn't escape from the sceptical question. There is no fact of the matter that determines whether I have counted correctly - except the fact that others will agree with me. This reinforces me in my practice of counting, as my agreement with others about their counts reinforces their practice of counting. — Ludwig V

Here there's a few bases from which we could confuse one another: arithmetic as a practice, arithmetic as a part of our rational intuition, arithmetic as rule-following, arithmetic as it was in its genesis, and arithmetic as it is. It might depend on which we're thinking about in our assertions how we evaluate the skeptical position.

I hope that makes it clear how I see it. I'm happy for others to disagree, provided they disagree with things I actually think, and not some imagined position based on their misunderstanding. — Janus

Hard to attain, at times. All we can do is re-state, try again, and all that. I read you as taking an intuitionist stance, as in mathematics is a part of our natural intuition that's even shared with other creatures, and so the skeptic has no basis because the skeptic is framing arithmetic in terms of rule-following when there's more to arithmetic than rule-following, such as the intuitive use of mathematics, whereas the skeptic's use is derivative of that (and so is an illegitimate basis of their skepticism, considering that the skeptic is undermining their own position in the process)

Let me know if that's close or not.

The challenge is about rule following, specifically about rule following activity that's now in the past. It's not an epistemic problem. It's not about what a person knows about which rule they followed. It's that there's no fact (a situation existing in the world) even in terms of mental states that satisfies Kripke's criteria for a rule-following-fact.

The idea of quadition was just to convey the problem. Kripke wasn't trying to do philosophy of math, although there have apparently been philosophers of math who were interested in it. — frank

I'm not trying to do philosophy asof math. I don't think I'd reduce rationality to rule-following either.

I think what @Janus's position amounts to is that there is a kind of fact, namely the familiar rules of arithmetic, which is the natural way to believe a person to be thinking about the question "how many?"

I'm taking the position that this is not an adequate reply, and attempting to give examples, like modular arithmatic, that are actually used where the procedure is the similar to the philosopher's toy of quaddition. Just because quusing is a philosopher's toy in comparison to addition that doesn't mean we have a fact to the matter about which rule is being followed -- there are other, more "practical" operations of arithmetic which can serve the same function as quaddition in the set-up. So the familiar reply to the skeptic -- to ask the skeptic to justify their skepticism -- can be overcome because there are practical (natural) examples that look identical to addition that are not philosopher's toys.

My thoughts on it (so far) is that it fits pretty well with my belief that we aren't as rational in practice as we tend to think we are. I think some people would assume that means I end up a behaviorist, but I'd say they're making the same mistake again. They think their post hoc rationalizations are the way the world really is. It's not. — frank

Heh. I don't think I'm that deep. I see a question, but I don't see a resolution.

Why a simple and seemingly private individualist mental life in the form of altered state of consciousness... has been represented and actualized in society to be aligned or opposed to a proper ethical way of life? — kudos

Some religions use psychoactive substances. Some religions condemn them. I'd say that if we're talking about an ethical impulse towards drugs then we're dealing with religious desires.

So the opposition to drugs is religious. In the Big sense the religious acceptance of drugs might also be religious (it's not like the religions which use drugs are recreational, any-time hedonists) -- but in the usual sense of most religious believers it's because of a religious impulse: it's just bad, and that's that.

Personally, I don't find walking enjoyable or conducive to thought and I am virtually indifferent to nature. I walk a lot in the city and when I visit other cities and towns. If I can be distracted by interesting people and architecture, I don't notice that I am walking. — Tom Storm

I love walking in cities and towns (most of my walking is there) but it's not to be distracted. I look about but there's something to the rhythm of it all that is conducive to thinking. It's just as much a part of nature as the woods, and at times our "home" is more terrifying and wonderous.

My point in indicating that everyone in the tread accepts it is to say that this burden is on you. To everyone in the thread it is accepted that we know our own actions in a more immediate way than we know others' actions, and if you disagree then you will have to provide an argument. The commonsensical idea is that when I see someone else flip a coin my knowledge is mediated by sense data; but when I flip a coin my knowledge that I am acting is in no way limited to sense data. Because I am the one effecting the act, therefore I know that the act is being effected. The mediation of the former is not present in the latter. — Leontiskos

Hrm! I don't know that I'd accept "we know our own actions in a more immediate way than we know others' actions" as a true sentence, but it'd be for boring reasons: I simply wouldn't use the predicate "...immediate" with respect to knowledge in general.

You could come up with a million absurd and arbitrary rules like quusing, and all I can say is "so what?". The logic of counting is inherent in cognition; even animals can do basic counting. And I see no reason not to think that basic arithmetic finds its genesis in counting. Give me a good reason not to think that and I will reconsider. — Janus

I'd say that basic arithmetic's genesis is in abstraction more than counting. But whether that's a good reason or not is up to you.

Mathematics is strange because there are no physical instantiations of it, really, and yet it's still true. It's always abstraction. With 0 you have to recognize something that isn't there. With 2 you have to look over the differences in physical objects to see what's the same between them. With 1 you have to individuate from the rest of the world: "this is an object distinct from the world as a whole. here we have a part"

I'd put mathematics on par with language as a whole rather than counting. Counting is an operation whereby we find the number. We don't even need things, as you've stated. You just go to the next number.

But what is the next number?

With modular arithmetic the number after 12 could be 1, or if we use military time the number after 24 is 1. Since we're in the domain of time this makes perfect physical sense. It's just a way of marking the day rather than the total time. Sometimes that's more important than a count "from the beginning of time".

Quaddition is certainly an arbitrary rule. (one might be tempted to say to the external world skeptic the same thing) It's a toy.

But the rub here is that addition is too -- it's just more useful than quaddition because of the world we happen to be referencing. But if we were referencing clocks then a different, modular arithmetic might be better suited.

So maybe a more plain-language way of putting the question @frank opened with (though I haven't read the text he's supplied, so I could be wrong): the skeptic might be asking how do you know the answer is not "the time is about 10:25" given that 125 divides into 12 10 times with a rough estimate of 25 minutes.

If you have four piles of four objects then you have sixteen objects, three piles of three objects then you have nine, two piles of two objects you have four. This obviously cannot work with two objects, so I'm not seeing the relevance to deciding whether addition, subtraction, multiplication and division are basically derivable from counting operations. — Janus

Well, if they're not derivable from counting then your argument against quusing isn't really talking about the same kind of thing since you've outlined a procedure for deciding if someone is quusing by pointing out that we can count beyond the quuser. But if it's not counting then that doesn't really demonstrate that a person is adding or quusing. The operations are distinct, rather than reducible to counting.

Of course I agree that arithmetic is more complicated than counting, all I've been saying is that it is basically counting. It is the symbolic language of mathematics that allows for the elaborations (complications) of basic principles. — Janus

I disagree that arithmetic is basically counting for the reasons I've stated: there are some numbers you cannot count to which you can get to within the arithmetic operations. This is an ancient problem, so I'm not sure how much the symbolic language matters. The symbols simplify and make things easier for us, but this is a problem that's not derived from the symbology: link on incommensurability (which should show why I keep harping on the square root of 2)

And I would also argue that it all finds its basis, its genesis, in dealing with actual objects, Thinking in terms of fractions, for example, probably started with materials that could be divided.

Probably, yes. But as the influential codger said:

There can be no doubt that all our cognition begins with experience...But even though all our cognitions starts with experience, that does not mean that all of it arises from experience — Kant

You're making the very simple, complicated. these are not equations, but evaluations of f for all the possible values of a & b. the right hand side in each case is the result of simplifying the left — unenlightened

Ahhhhh.... OK I think it clicked now. I figured out my mistake. I was treating "m" in (3) as embedding the crosses to its left, but in fact it's alongside the crosses rather than embedded and so I was just doing the evaluation incorrectly. Then upon finding the wrong answer I tried to come up with various other possible ways to evaluate, which I've already shared at length, and now I see how they simplify.

So due to Theorem 2 I can easily simplify (1) and (3) -- an empty cross next to any combination of expressions simplifies to the empty cross, and m is an empty cross. (within the cross, so cross-cross = the unmarked state)

That leaves (2) and (4).

For (2):

$\left. {\overline {\, \left. {\overline {\, fm \,}}\! \right|n \,}}\! \right|$

Let f = m, and mm = m, therefore we are left with the marked state.
Let f = n, and nm = m, and we have the same.

But for (4) when we do this the evaluation comes out m or n.

Thanks for the help. I made a small mistake along the way and it resulted in a lot of confusion.

I think the changing scenery, the rhythm of breathing and walking, and the relative lack of things to pay attention to in particular is what makes walking good for thinking. I share the habit! I've noticed that I tend to have more focused thoughts when walking.

Wait, I think i see what you are doing - treating each line as an equation, and then substituting the right back in for f. — unenlightened

Yup. The first line states what f is and that's how I was treating it.

You don't want to do that! Each line is a result for a combination of an and b. There is no working shown, and almost none to do. so for (2):– — unenlightened

$\left. {\overline {\, fm \,}}\! \right|$ = $\left. {\overline {\, m \,}}\! \right|$

and the re-entered f can be ignored.

Cool. So with this solution the trouble I have is with (4). n-cross-m-cross is three crosses and so should equal m, but (4) equals n.

Right before these lines GSB states:

We can now find, by the rule of dominance, the values which f may take in each possible case of a,b.

And the rule of dominance doesn't care about the infinite depth it cares about S-sub-0, the pervading space. For solution 2 I was treating "m" as the marked space and putting an "m" then "n" alternatively as I filled out the expression. For solution 4 I'm ignoring m and n and simply marking S-sub-0 with the next letter that follows. That works for (1) through (4), but (5) it would simply be equal to "m" rather than m or n by this procedure.

But this is me explaining my failed evaluations trying to figure out how to get to a successful one. (I've actually typed out a few of these puzzlers before only to find the solution at the end and delete the puzzler in favor of the solution... but this time I was still stuck at the end of my post)

And if I follow GSB's outline for (5) and apply it to (4) I'd say we have two expressions that evaluate to m or n -- because if f equals m, then you get mn-cross-m-cross, which is m-cross-m-cross, which is an even number of embedded crosses that gives you n, but if f equals n then you get nn-cross-m-cross, which is an odd number of embedded crosses so you get m.

The other three work out that way.

Basically I'm treating all 5 as test cases for understanding how to evaluate an expression's value which has re-entry and each time I try to use one of these solutions one of the equations comes out differently from what's written in the book.

So these are my failed evaluations which I'm sharing because this time by typing it out I haven't figured out how to do it right.

Cool. I think that's enough of the concrete side of things for me to feel like I have a footing again. Thank you again for your explanations @wonderer1.

I'm really just inching along in this chapter. Every page presents a problem. Now I feel I have a handle on how we get to infinite expansions but I'm stuck on Re-entry on page 56-57 (I realize now before I was citing the pdf pages rather than the book pages. These are the book pages)

So I looked back at the rule of dominance because that's how we're meant to determine the value of infinitely expanding functions for the various values of a or b being either m or n. I went back to the rule of dominance because it seemed like basic substitution didn't work. But as I apply the rule of dominance I'm getting different values at each step of E1 in chapter 11.

$\left. {\overline {\, \left. {\overline {\, fa \,}}\! \right|b \,}}\! \right|$ = $f$

$\left. {\overline {\, \left. {\overline {\, fm \,}}\! \right|m \,}}\! \right|$ = $n$ (1)

$\left. {\overline {\, \left. {\overline {\, fm \,}}\! \right|n \,}}\! \right|$ = $m$ (2)

$\left. {\overline {\, \left. {\overline {\, fn \,}}\! \right|m \,}}\! \right|$ = $n$ (3)

$\left. {\overline {\, \left. {\overline {\, fn \,}}\! \right|n \,}}\! \right|$ = $m$ or $n$ (4)

So I've tried three different things in trying to get all the equations to equal what they state here:

1) substituting the marked state for m and the unmarked state for n while expanding each instance of "f" with one more iteration so you'd have, in the case of (1): m-cross-m-cross-m-cross-m-cross. Since you have an even number of crosses all embedded within one another you get n -- the unmarked state. But then if I try this on (2): m-cross-n-cross-m-cross-n-cross: we have an even number of crosses embedded within one another so it should reduce to the unmarked state, but (2) reduces to the marked state.

2). The rule of dominance. You begin at the deepest depth which would be "f" in each case and alternate putting m or n next to the next depth-level. So starting with (1):

$\left. {\overline {\, \left. {\overline {\, fm \,}}\! \right|m \,}}\! \right|$ (1)
$\left. {\overline {\, \left. {\overline {\, fmm \,}}\! \right|m \,}}\! \right|$ (1.1)
$\left. {\overline {\, \left. {\overline {\, fmmn \,}}\! \right|m \,}}\! \right|$ (1.2)
$\left. {\overline {\, \left. {\overline {\, fmmn \,}}\! \right|mm \,}}\! \right|$ (1.3)
$\left. {\overline {\, \left. {\overline {\, fmmn \,}}\! \right|mmn \,}}\! \right|$ (1.4)
$\left. {\overline {\, \left. {\overline {\, fmmn \,}}\! \right|mmn \,}}\! \right|m$ (1.5)

and by the rule of dominance I get m because that's what sits in the pervading space.

But that's not the right way to apply the rule, then, since we must get n from the procedure for (1). Which brings me to:

3) Substitution from the Sixth Cannon where :

$\left. {\overline {\, m \,}}\! \right|$ = $n$
$\left. {\overline {\, n \,}}\! \right|$ = $m$

But then for (3) I get m, because n-cross equals m and m-cross equals n and that leaves m after substituting.

While writing this out I came up with a 4th possibility: just mark the next m or n, as you'd do with the rule of dominance, and take the value in the outer space. But then (5) is equal to m, and not m or n, except in a fancy way of interpreting "or" which I don't think is what's going on.

So, as I said, I'm inching along and every page presents a problem. :D This is as far as I got this morning. (EDIT: Changed the number-names of each step to conform with the thread)

To be fair I was just trying to parse the argument as is rather than trying to interpret it in terms of Kant.

It's always easier to moralize than it is to be moral. — Pantagruel

Yeah, but....

1
"Do not judge, or you too will be judged.
2
For in the same way you judge others, you will be judged, and with the measure you use, it will be measured to you.
3
"Why do you look at the speck of sawdust in your brother's eye and pay no attention to the plank in your own eye?

I'd say it's Heidegger that has the plank in his eye. He didn't fail to embody his own ideals. What he ultimately deemed as authentic living was just a bad ideal that he successfully lived up to.

But, as @Joshs has already pointed out -- Levinas and Derrida have more right to condemn Heidegger than me, and they both use his philosophy. So, for better or worse, I think he's worth reading. (as a Marxist, that's damning praise, but praise all the same)

For Laws of Form Relays may be the better bit of technology to look at, though they function the same as the transistor. Looking through wikipedia at all this stuff I noticed the paragraph at the end sounded pretty similar to the basic operation in Laws of Form:

Latching relays require only a single pulse of control power to operate the switch persistently. Another pulse applied to a second set of control terminals, or a pulse with opposite polarity, resets the switch, while repeated pulses of the same kind have no effects. Magnetic latching relays are useful in applications when interrupted power should not affect the circuits that the relay is controlling. — wiki

The SR f!ip-flop circuit is symmetrical, so it is somewhat arbitrary which output is chosen to be Q and ~Q. However, the Set pin is defined as the input that can cause Q to produce a 1 (5V) output. So one could swap Q and ~Q, but to be consistent with the conventions for SR flip-flops one would also need to swap which input is labeled S and which R. So like the stoplight it is a matter of convention. — wonderer1

OK, cool. That was what I was thinking, but realized I didn't know. Given the topic of the book -- a kind of proto-logic prior to logic, or from which logic emerges (with a practical basis in sorting out electrical work and inventing a logic for that) -- it seemed important to me.

Also, flip-flops themselves don't perform logical operations. They just serve as memories that can be used to provide inputs to logic gates (or combinations thereof), and store outputs from logic gates.

Got it. This is a memory, and not an operating circuit. So it holds a 1 or a 0, and it's by convention that a side of the flip-flop is treated as a 1 and the other as 0, and it behooves us to be consistent because then we can start doing cool things like reducing our number system to binary and having circuits perform operations faster than we can.

If you wanted to count a hundred objects you could put them in a pile, and move them one by one to another pile, making a mark for each move. Then if you wanted to add another pile of, say, thirty-seven objects you just move those onto the pile of one hundred objects, again marking each move. And then simply count all the objects or marks. — Janus

Being able to count "1" is significant, as is being able to recognize when you have 0 of something. Then the journey from 1 to 2 is the act of grouping -- absence, presence, and sameness. A nothing, a something, and a set. After you have a set then I think the successor function makes perfect sense -- keep doing the thing you already did, add 1 and go to the next spot. I'm not so sure before that.

Also: division is what allows us to start asking things like "What about the numbers in between 1 and 2?" -- before that we'll just be dealing with wholes. Then we start adding all kinds of numbers to what appeared to be nothing but counting and moving stones. But that we can divide sets into equal portions, or set up ratios between numbers, I'd say is distinctly not counting as much as comparing, because some of the numbers in between 1 and 2 cannot be represented with a ratio of stones. The square root of two cannot be represented by a ratio of stones in the numerator and denominator, so it can't be counted to by counting two ratios, but it's still a number between 1 and 2. We only get there through operating on the numbers, rather than counting. But it's still arithmetic because we're just dealing with constants and what they equal.

Basically I'd say that arithmetic is more complicated than counting.

This is great. Thanks again for taking the time to write out these explanations.

At this point it is pragmatic to jump up a level in abstraction and think in terms of logic gates instead of transistor circuits. — wonderer1

Of course, though, this is what I want :D

I think what I'm wanting to settle, for myself, is whether or not the circuits are in turn being interpreted by us, or if they are performing logical operations. What makes Q and ~Q different other than one is on the left side, and the other on the right side? Do we just arbitrarily choose one side to be zero and the other side to be 1? Or do the logical circuits which have a threshhold for counting do it differently?

To my mind the circuit still doesn't really have a logical structure anymore than a stop light has the logical structure of Stop/Go without an interpretation to say "red means stop, green means go". So are we saying "Q means 1, and ~Q means 0"?

That's remarkably clear.

I've struggled to put together a QM primer before only to put it aside because it's hard to even understand, and so even harder to simplify. Plus the class I took was explicitly taught in the Copenhagen interpretation, and a lot of the discussions around here try to differentiate between the interpretations and, at least as I learned it, there wasn't really a way to differentiate between the interpretations. (though maybe that's changing now)

For chemists you're just expected to pick up the math along with the class :D. And then, to top it off, I went biochem since it seemed more employable -- and sometimes QM matters there, but not often. A rough hand-wavey understanding, or some of the simpler technically not true theories, of bonding are adequate for the predictions there.

Addition, subtraction, multiplication and division are all, as far as i can see, basically counting, and counting is basically naming different quantities. Think about the abacus. — Janus

The abacus might be a bad example for me because it would emphasize what I've said: I can certainly count the beads on an abacus, but I don't know how to do the arithmetic operations with an abacus. I never learned how to use it.

Similarly we can count marks, or we might know the the arabic numerals, but we may not know how to solve an addition problem without some sort of knowledge of figuring sums. That ability might even be relative to the numeral symbols we use -- thinking here about the trick of stacking numbers on top of one another and adding them by column from the right. Seems like that'd be difficult to do with Roman numerals.

Hrm! That helps me understand the feedback part very well -- so thank you again for taking the time. When Set is grounded the voltage from R3 no longer gives the voltage necessary for the transistor to be in the "on" state, but the parallel circuit through R2 does so the circuit flips over to Tr2. Since Tr1 is now off that means 5V goes to Q as the path of least resistance. The same holds for reset and the blue state.

And that helps me understand how it has a memory -- when you come back to it it'll be in one state or the other, so there are two possible states for the circuit to be in when at equilibrium.

And I can now see how they are switches thanks to your explanation, which was a bit of a mystery to me before.

Thanks! I'm going to type out what I understand from your explanation and the diagram and guess work, and I looked at this website too.

The story of a hole in a state of flow with an innumerable number of other holes towards ~Q: We start at 5 V and move through R1 to TR1 because the voltage at Q is lower than the voltage at ~Q (assuming we're already in a steady state), then we go through the unmarked resistor on the other side of the transistor, up through R3 and out ~Q. If you touch "Set" to the zero volts line than you ground the flow causing the voltage to switch over to R4-T2-R2-Q.

Based on the website I linked it looks like Q and ~Q are out of phase with one another. So the memory comes from being able to output an electrical current at inverse phases of one another? How do we get from these circuits to a logic? And the phase shift is perhaps caused by subtle manipulations of the transistor?

I have not understood how essences as definitions differs in salient ways from essences in terms of necessary properties. Isn't a definition a set of necessary and sufficient properties? — Banno

Me either, which is what I was trying to get at by asking for a criterion of quiddity.

As I'm reading Fine a definition is necessary, because Fine accepts the argument that if something is not necessary then it is not essential, but necessity is not sufficient.

Or, if we're going by way of Aristotelian essence, then I'm not sure "sufficiency" is the conceptual mark we should be using at all (hence my divergence into Aristotelian causes for determining whether something named has an essence at all)

But I believe you, @Leontiskos, have started to give an answer here:

One way to cash this out is to say that risibility or the ability to learn grammar supervene on rationality, and it is rationality that belongs to the essence because it is explanatorily fundamental. Thus a human being is not defined as "A risible animal" or "An animal capable of learning grammar," but rather, "A rational animal." This contains and explains the others.

Aquinas claims that, in a similar way, delight supervenes on happiness, for happiness is essentially the possession of a fitting good and not the possession of delight, and yet delight always follows upon and attends happiness such that they appear indistinguishable.

I should point out yet again that it is one thing to disagree with some real definition and another to disagree with essentialism itself. The latter is much more contentious and difficult, and would seem to involve the claim that no properties are explanatorily prior or posterior. — Leontiskos

A definition is a true description of an essence, which is a property which is explanatorily prior to other properties, including the necessary ones (like the Singleton Socrates). "Prior" is unspecificed at this point, but that's the beginning of something: definitions are meant to explain something, and the explanation is one of a priority of properties.

@Leontiskos do you accept the argument that if some predicate is not necessary of a name that then that same predicate is not an essence of the name? (only asking because then we could add to this list to say that essences are necessary, though there are necessary predicates which are not essential)

Are you looking at the 9th canon where he constructs an ever deepening series of nested a's and b's? Page 55 in my version?
If so, you just take the whole right hand expression of a & b as = r. and use J2 in reverse. — unenlightened

Yup, that's the one! Thanks.

That worked. I already became stuck on the next step. :D -- but I figured it out by going back to the demonstration of C4 and using its steps rather than the demonstrated equality between the expressions.

Then it's pretty easy to see the pattern after that: it's the same pattern as before, only being iterated upon a part of the expression in order to continue the expansion.

I can say I'm stuck with your last reported place that you're at. At least, this morning I am.

Wow, if someone implemented something like that we could have computers and an internet!

Sorry, couldn't resist. — wonderer1

This is part of my interest here -- something I've always struggled with is understanding the connection between circuits and symbols. I'm sure I don't understand how a circuit has a memory, still.

What is marriage to you? — NOS4A2

To me personally? The most current, but clearly inadequate, means for our society to answer the question "By Whom and how are these children going to be taken care of?" -- it's a legal entity for the economics of the family home which gets interpreted in various ways in their particular instances.

For purposes of this discussion I'd just focus in on the legal aspects, which will vary depending on where we're at, but the specifics shouldn't matter here. In focusing in on what can be seen, and on individuals, your account is very vague when it comes to one of the most important social realities we live with: property. Even individuals own property, but only by law -- which cannot be seen, is not flesh-and-blood, is seemingly abstract and yet seems causal in that the reason people act in such-and-such ways is because of how property is treated within the legal system.

This is what I meant by "history", I think, the culmination of our interactions with one another. That is the extent of our relationship.

Only these types of interactions, in combination with the accounts of those involved, can determine what kind of relationship — NOS4A2

Hrm -- I'm reading you as going back to your original post, then, whereas before I was reading you as allowing that marriage is real.

So a marriage is the entire list of interactions between two people, and these interactions in combination with accounts from the two people involved in the marriage determines that they are interacting in marriage-wise ways, but they have no bond or connection with one another (except for the children who had an umbilical cord when they are born) and the marriage itself is not real. "Marriage" is just shorthand for a long list of interactions between two people.

By this then even ownership can't be real, I'd say: that was the point of my example, along with the obvious bit that they're a collective. When a marriage dissolves what's at stake is who owns what. Before a marriage is dissolved you have an example of collective ownership. But if the only real things are interactions between individuals, then I'd say there's no such thing as property because it's only acted upon by individuals. The house is not flesh-and-blood, after all, and property rights are only established by law (which doesn't exist in your accounting).

Or, at least, this is where my mind goes with what you've said so far. I'm not sure if it's something you'd avoid or accept.

I'm a bit pressed for time today, but for Aristotle the fundamental issue is that a kangaroo has an essence whereas a hammer does not, and this is because only the first is a cohesive thing (substance) with its own proper mode of being and acting (and this also includes teleological considerations). A hammer is an aggregate of substances thrown together for a human purpose.

A simpler example would be a horse-and-rider. A horse-and-rider is not a substance, and it has no essence. Instead it is a composite of two substances (a horse and a human rider). We can talk about the essence of a horse-and-rider in an analogical way, as if it were just a single thing, but technically this is not quite right.

I am not opposed to talking about the "essence" of a hammer or the "essence" of a named individual, just so long as we do not forget that for Aristotle there are no such essences. More broadly, it makes sense for the Aristotelian to say that the human has being in a more primary sense than the hammer does; or that the name attached to a perceptual 'description' is more primary than the name attached to the conceptual 'description' (and that the latter should take its cue from the former). Such a distinction may seem quite odd to the modern mind, but it may also be at the root of some of these issues. — Leontiskos

I think that it's worth asking, in that case, what is the criterion of quiddity, what-it-is-ness, such that we can have names with and names without an essence? What's the criteria by which you judge an individual to have an essence? I take it your beliefs are Aristotelian-inspired, but since you're also saying "for Aristotle" it seems you may also be thinking about your account as different.

From what I read of you it comes down to whether something is a substance or is composed of substances. I'm not sure exactly how to parse that -- I put teleology as a criteria because I understand activity to be central to the essence of beings in Aristotle, and teleology is the kind of cause which is self-caused towards something -- so an olive is a seed with a teleology of becoming-tree, where its material cause is its is its plant-like embryonic structure, efficient cause would be water, soil, and nutrients, and formal cause would be the owner who planted the olives (or, in the wild, the form within the mind of God who thinks the seed-to-tree into being)

I'm going back to the four causes because it seems to me that hammers have a definition, and so I would have said that a hammer has an essence on that basis from my understanding of Aristotle's notion of essence. But you're saying that it doesn't, so I'm starting to rely upon my understanding of Aristotle's physics as a basis for differentiating what truly has a substance from that which is merely composed of substances. It seems to me that the lack of a teleological cause might be a basis for making this claim -- basically anything which is a natural kind would participate in all four causes. (the strange thing here being that the basic materials participate in teleology by having a proper place to be in the stack... which clearly goes against how we understand matter to operate today)

Moliere began a discussion of essences with the example of hammers. This is a strange move from the perspective of an Aristotelian, because hammers have no real essence. A hammer is a derivative being, a human artifact. Hammers should always be studied in relation to humans, because their existence is dependent upon humans. — Leontiskos

The discussion has moved to other considerations, but I have some thoughts here.

Aristotle is conceptually rich, so this is very much a guess in the dark:

Perhaps individual human activity, like hammering, is strange in Aristotle because he offers an ontology that has a kind of cause that accounts for the change of individuals over time such that they're still the same object while undergoing change because of the kind of object they are --- teleology. The hammer doesn't fit very well because it's not a biological entity or a natural kind -- its teleology is directed by an individual, and so its purpose is relative to the ends of not even a species but of an individual of the species. All tools are such that they are always relative to some other being's usage, and so they don't have a teleology at all -- they don't have an activity that their kind strives towards which makes them what they are.

It seems your account must have named objects without real essences alongside what has essences -- and maybe what you say here has something to do with why Heidegger used the example of hammering in putting the question of the meaning of being back on the table for philosophical investigation.

If hammers don't have essences, then what does? And on what basis are we to exclude tools from having being (or, perhaps they have being, but no essence?)?

Isn't counting adding 1 to the previous number? — RogueAI

I think "counting" is almost a primitive. It's such a simple operation or concept that we'd have a hard time defining it rigorously. But I'd put "counting" as more primitive than addition, because addition holds for more domains than counting -- such as fractional numbers that fall in-between the counting numbers.

Without defining the domain counting is strange. You can't count to the square root of 2 on the natural numbers, for instance. Counting will never get you to the real number line. And if we allow division, at least, it's pretty easy to operate on the natural numbers such that we need more numbers than what we count. One might say a difference between quusing and adding is that adding is a part of all arithmetic, and so we have access to division, where quusing is the same as addition up to a certain point but what makes it different are the rules and the domain.

Quaddition is clearly a philosophical toy, but modular arithmetic works similarly in that there is no number beyond a certain point within the mod space. Quaddition just defines, arbitrarily, what happens after you reach the end.