The child does not learn a rule or preparation for counting, she learns how to count. If she learned correctly she not only affirms that it is true that there are 4 apples on the table, but by counting beans, sticks, fingers and other things she can affirm that it is true that 2 units + 2 units, or, in short, that 2+2=4. — Fooloso4

I don’t disagree with this, except to say that the expression “2+2=4” is not necessarily counting anything.

They are conscious, but they are exclusively operating in the domain of your neural systems, not on paper in logical formality. — Garrett Travers

I don’t know what this means. They are conscious but they are not conscious?

Hinge-propositions are not that at all. There propositions that emerge in thought, and are either accepted by us individually, or rejected. The acceptance or rejection of those propositions inform our behavior, and behavior is function. — Garrett Travers

Not sure that I agree with this. Hinge "propositions" are not conscious judgments, so we do not accept/reject them in any rational or considered manner.

Thus, if I'm to move to open a door to leave my home, I must have accepted the hinge proposition that the door will open in the first place to let me out. Furthermore, if I am leaving the house to go to whole-foods, the action is predicated on the belief of such a places existence. Does that make sense? — Garrett Travers

Yes, this sounds more like it.

Given Wittgenstein's assumption that mathematics is a human construct, it follows that 1+1=2 is neither true nor false in that it not an empirical statement and so is outside the concept of truth as correspondence.

But is he going against his own admonition to just look? — Fooloso4

I believe W's view is that "1+1=2" is not counting, but is instead a rule or a preparation for counting, much like learning the meaning of a word is not actually using the word, but is instead a rule or preparation for the use of that word in a language-game. This also helps to explain why W considers it neither true nor false that the Paris metre is one metre long - because it is a rule or a preparation for making metric measurements and is not itself a measurement.

(And I still owe you a response to your previous post, which I am still considering)

That's because "hinge-propositions" are a neurological phenomenon. Meaning, the acceptance of a truth value, much like logical validity itself, does not imply the accurate conclusion of truth value one way or the other. But, that such an acceptance must take place before action, and thereby function, can be initiated. — Garrett Travers

Sorry, but I don't really know what you're trying to say. Could you express it more plainly?

according to Wittgenstein, the mathematical equation is nonpropositional
— Luke

Where does he say this? — Banno

From Moyal-Sharrock's book again:

For Wittgenstein, to be a proposition is to be bipolar; that is, to be susceptible of truth and falsity. From the first, Wittgenstein’s technical concept of the proposition is internally related to bipolarity: ‘In order for a proposition [Satz] to be capable of being true it must also be capable of being false’ (NB 55); ‘Any proposition [Satz] can be negated’ (NB 21); ‘A proposition [Satz] must restrict reality to two alternatives’ (4.023) [...] In the thirties, Wittgenstein still upholds bipolarity: ‘In logic we talk of a proposition as that which is true or false, or as that which can be negated’ (AWL 101); ‘ “A proposition [Satz] is whatever can be true or false” means the same as “a proposition [Satz] is whatever can be denied” ’ (PG 123); ‘it is a part of the nature of what we call propositions [Satz] that they must be capable of being negated’ (PG 376). The ruling out of the possibility of falsity amounts to the ruling out of propositionality: ‘There is no such proposition as “Red is darker than pink”, because there is no proposition that negates it’ (AWL 208; my emphasis). In other words, so-called analytic and synthetic a priori propositions are not propositions.
The claim that propositions are essentially bipolar cannot be consistent with accommodating rules, tautologies or anything else which is necessarily true within the propositional fold.

[...]

G.E. Moore reports Wittgenstein as asserting both that a proposition ‘has a rainbow of meanings’ (MWL 107) and, of the ‘kind of “proposition” ’ that has traditionally been called ‘ “necessary”, as opposed to “contingent” ’, such as ‘mathematical propositions’,

...he sometimes said that they are not propositions at all...They are propositions of which the negation would be said to be, not merely false, but ‘impossible’, ‘unimaginable’, ‘unthinkable’ (expressions which [Wittgenstein] himself often used in speaking of them). They include not only the propositions of pure Mathematics, but also those of Deductive Logic, certain propositions which would usually be said to be propositions about colours, and an immense number of others. (MWL 60)

However Moore may have taken Wittgenstein’s ‘puzzling assertion that 3 + 3 = 6 (and all rules of deduction, similarly) is neither true nor false’ (MWL 80), there is no ambiguity about Wittgenstein’s ‘declaration’ and ‘insistence’ that mathematical ‘propositions’ are ‘rules’, indeed ‘rules of grammar’ (MWL 79) and that these ‘rules’ are ‘neither true nor false’ (MWL 62, 73). And this cannot be dismissed as ‘early Wittgenstein’. He is still making the same claim in the Remarks on the Foundations of Mathematics:

There must be something wrong in our idea of the truth and falsity of our arithmetical propositions [Sätze]. (RFM, p. 90)

It is important to underline that Wittgenstein does not only attribute nonpropositionality to mathematical ‘propositions’ but, as he makes clear in the AWL passage above, and it is worth repeating, to any ‘proposition’ of which the negation would be said to be, not merely false, but ‘impossible’, ‘unimaginable’, ‘unthinkable’ and these

...include not only the propositions of pure Mathematics, but also those of Deductive Logic, certain propositions which would usually be said to be propositions about colours, and an immense number of others. (MWL 60) — Understanding Wittgenstein's On Certainty, pp. 35-38

So your argument, if I understand it, concludes that 12 x 12 =144 is not true.

Hence why not reject your argument by reductio? — Banno

Perhaps you could if it implied that 12 x 12 = 144 was false. But, according to Wittgenstein, the mathematical equation is nonpropositional, so it is neither true nor false.

But feel free to present an argument.

Are you claiming that a hinge is not susceptible to being false? Or are you making a claim about mathematical hinges? — Fooloso4

My immediate point was that the equation 12x12=144, and similar fundamental mathematical statements more generally, are not susceptible to being false. That this can also be extended to some empirical statements was W’s concern in OC.

Is the concept (I am trying to avoid the term proposition and the confusion it may cause, independent of OC) of the earth revolving around the sun a hinge? Is it susceptible of being false? At one time the sun revolving around the earth was a hinge. — Fooloso4

Yep, hinges can become propositions and propositions can become hinges. I think that today it is incontrovertible that the earth revolves around the sun, so it would be a hinge, unless or until some new scientific discovery were to change that.

Could a mathematical proposition that is true be false? No. Could a mathematical proposition be false? Yes. — Fooloso4

What I meant, wrt my post to Seppo regarding the bipolarity of propositions, was: is 12x12=144 susceptible of being false? If not, then it is not a proposition (in Wittgenstein’s view), and neither is it susceptible of being true.

I do not think it is the case that hinges are unknowable and lack truth value:

655. The mathematical proposition has, as it were officially, been given the stamp of
incontestability. I.e.: "Dispute about other things; this is immovable - it is a hinge on which your
dispute can turn."

Certainly we know that 12x12=144 and that this is true. — Fooloso4

Could it be false? (see my post to Seppo above regarding bipolarity)

It's a fair point, but I don't consider mathematical propositions to be the sort of hinge propositions that Wittgenstein is concerned with in the text. What motivates W's concern in OC is the "peculiar logical role" of Moore's statement "Here is a hand", which has the form of an empirical (i.e. bipolar) proposition, but which functions more like a mathematical or logical proposition:

136. When Moore says he knows such and such, he is really enumerating a lot of empirical propositions which we affirm without special testing; propositions, that is, which have a peculiar logical role in the system of our empirical propositions.

308. [...] we are interested in the fact that about certain empirical propositions no doubt can exist if making judgments is to be possible at all. Or again: I am inclined to believe that not everything that has the form of an empirical proposition is one.

401. I want to say: propositions of the form of empirical propositions, and not only propositions of logic, form the foundation of all operating with thoughts (with language).[...] — Wittgenstein, OC

I believe W would categorise mathematical propositions in the same or a similar class as logical propositions. Wittgenstein draws the distinction and compares mathematical/logical propositions (i.e. rules) with empirical propositions, for instance:

350. "I know that that's a tree" is something a philosopher might say to demonstrate to himself or to someone else that he knows something that is not a mathematical or logical truth.[...] [W's italics, indicating that the philosopher does not technically (JTB) know this] — Wittgenstein, OC

Moyal-Sharrock does not use the phrase "hinge proposition".

He talks of "hinges" — Banno

She.

Having a truth-value is an essential, characteristic trait of propositions. Just as having three sides is an essential, characteristic trait of triangles. Different types of triangles can and do differ from one another... just not in having three sides, since if they don't have three sides they aren't a triangle. And in exactly the same fashion, different types of propositions- hinge propositions, for instance- may differ from one another in various ways, but not in having a truth-value or not. — Seppo

You've made a category error. Equilateral triangles and triangles (in general) are not two different types of triangle. Triangles (in general) are simply triangles. Likewise, propositions (in general) are not a type of proposition, and cats (in general) are not a type of cat. Therefore, hinge propositions and propositions (in general) are not two different types of proposition. This is why I said "it remains to be demonstrated that hinge propositions are of the same type as propositions in general."

An isosceles triangle may be a different type to a scalene, but a particular type of triangle cannot be different from triangles in general. Triangles in general are simply triangles, so if a particular "type" of triangle is different from triangles (in general) then it is not a triangle. Likewise, if hinge propositions are different from propositions in general then they are not propositions, and if siamese cats were different from cats in general then they would not be cats. You have acknowledged that hinge propositions are different from propositions in general, which implies your acknowledgement that hinge propositions are not propositions.

Even if this was not implied, you would still need to specify in what respects hinge propositions and propositions (in general) are the same, despite being different (as you have acknowledged), without merely presupposing that hinge propositions are propositions.

If hinge propositions lack a truth-value, then they are not propositions, just as a triangle that didn't have three sides wouldn't be a triangle. — Seppo

This is not a problem for me, because I agree that hinge propositions are not propositions. While you and Banno have been patting each other on the back so loudly that you cannot hear anyone else, this has been my point the entire time. It's sad that this isn't even an exaggeration or caricature.

This is why this is frustrating, neither I nor anyone else should have to explicitly make such an argument. — Seppo

You really shouldn't have. Not because it's so obvious, but because you've been oblivious to the possibility that "hinge proposition" could be a misnomer. Your repeated argument of "otherwise it wouldn't be a proposition" has been both futile and tone deaf.

Not a very rewarding discussion from my perspective. — Seppo

Join the club.

Right, he never uses the phrase "hinge propositions"... but, as I have already pointed out, and you either ignored and forgot, he does refer to them as "propositions". — Seppo

As Moyal-Sharrock points out in her book, Understanding Wittgenstein's On Certainty, Wittgenstein was "still in the process of determining whether a certain kind of statement is a proposition or not (e.g. OC 167)" while writing OC. Furthermore, it is "his translators who are more often than not responsible for its [the word "proposition"'s] appearance in his works."

Which makes me wonder what motivates this denial that they are truth-apt, particularly since no one seems to be able to give a coherent argument for why we should doubt or deny that they have a truth-value, while simultaneously characterizing them as the sorts of things that are truth-apt (propositions, certainties, beliefs). I mean, where did this notion even come from? — Seppo

As Moyal-Sharrock says in the same book, "For Wittgenstein, to be a proposition is to be bipolar; that is, to be susceptible of truth and falsity." Given that a hinge (qua hinge) cannot be false, then it is not susceptible of both truth and falsity, so it cannot be a proposition and neither can it be true. The same applies to grammatical and mathematical "propositions" (rules). As the author notes more generally: "nonpropositionality is attributed to any string of words that constitutes a rule or a norm".

It remains to be demonstrated that siamese cats are of the same type as cats in general. — Banno

More like: It remains to be demonstrated that koala bears are of the same type as bears in general.

The argument being offered is akin to: All bears hibernate in the winter, therefore koala bears must hibernate in the winter. Otherwise, they wouldn't be called bears. ...Solid argument.

Siamese are different from other cats, lets' say.

You want to conclude that Siamese are therefore not cats. — Banno

It remains to be demonstrated that hinge propositions are of the same type as propositions in general.

Where's Sam26? — Banno

Yeah, where is he? He was also arguing that hinge propositions do not have a truth-value.

If hinge propositions are different from "propositions in general", then hinge propositions need not bear a truth-value.
— Luke

That's not valid reasoning, and it's not cogent. — Banno

It is valid reasoning. Seppo's argument is:

All propositions (in general) are bearers of truth-value
Hinge propositions are propositions (in general)
Therefore, hinge propositions are bearers of truth-value

If hinge propositions are different from propositions (in general), then hinge propositions are not propositions (in general). I'm not sure about you, but Seppo has acknowledged a difference between hinge propositions and propositions (in general).

If hinge propositions and propositions (in general) are in some respects the same, then it needs to be explained in what respects they are the same, without assuming that they must be the same because they are both propositions (in general). They are not both propositions (in general).

If hinge propositions were sufficiently different to other propositions so as not to be truth bearing, they would arguably no longer be propositions. — Banno

Yeah, as I said in my last post, and according to the article I linked to in my last post, hinge propositions are not propositions.

I'm not sure one can have a rule that is not a proposition. — Banno

This is a very anti-Wittgensteinian sentiment imo.

A rule presumably says how things should be, and how they should be is a possible state of affairs, and hence a proposition. — Banno

What "possible state of affairs" or "[way] things should be" is given by a rule of grammar, logic, or chess?

As Moyal-Sharrock (again) notes:

When we learn rules, we do not learn content, but a technique, a skill, a mastery – how to proceed. To follow a rule is not to make a judgement, but to make a move. — Daniele Moyal-Sharrock, Wittgenstein's Hinge Certainty

Well, no, they need to differ in some way. But if they are propositions, then having a truth-value is not where they can differ, because having a truth-value is what propositions do. — Seppo

That would depend on how they differ. You cannot just assume that they are the same in the respect of being the same.

If hinge propositions don't have a truth-value, then they are not propositions. — Seppo

Ya, that's my point. You've simply reiterated things that I already addressed in my last post as if you didn't read what I said, so this may be a lost cause. Did you read the article? Hinge propositions are not propositions, but rules of grammar.

Hinge propositions not being propositions is self-contradictory. — Seppo

Wittgenstein never called them "hinge propositions". Apparently you have no interest in discussing whether or not hinge propositions are propositions, or in discussing Wittgenstein's work.

Hinge propositions are set apart from other propositions not in virtue of lacking a truth-value, but in their inability to be justified, — Seppo

You keep repeating this without any apparent regard for Wittgenstein's work, but based solely on the designation of "hinge proposition".

Its sort of a trainwreck of non-sequiturs, and in this post you've simply reiterated things that I already addressed in my last post, as if you didn't read what I said. — Seppo

You're right, I didn't read closely enough. You said in your last post:

But hinge propositions are indubitable (in a sense), unjustifiable, and in virtue of being unjustifiable, unknowable... because they form part of the background against which we doubt, justify, or come to know propositions in general (and hence themselves being subject to those processes would involve circularity). — Seppo

You make a distinction here between hinge propositions and "propositions in general". This is the same distinction that I have been trying to get you and @Banno to acknowledge. If hinge propositions are different from "propositions in general", then hinge propositions need not bear a truth-value. Or, at least, you cannot use the definition of a "proposition in general" to support any claims about a hinge proposition.

As Daniele Moyal-Sharrock points out (here), a hinge proposition is not a proposition at all, but a rule of grammar:

Hinges are really logical bounds of sense or rules of grammar; they form ‘the foundation of all operating with thoughts (with language)’ (OC 401). For Michael Williams (2001, 97) and many epistemologists, to ask for the ground of a belief is to ask for yet another belief, for only propositional beliefs (and other intentional states) can stand in logical relation to other propositional beliefs. What Wittgenstein makes clear is that this is an invalid assumption; for grammatical rules can stand in logical relation to propositional beliefs – as, indeed, they must: as the necessary enablers or determinants of sense. I cannot come to the belief that ‘It is indeed a hand I see on a blurry photograph’ unless I am ‘hinged’ on the grammatical rule that ‘This is what we call “a hand”. — Daniele Moyal-Sharrock, Wittgenstein's Hinge Certainty

Yeah, reading Luke's latest post, I'm afraid this is a bit of a lost cause — Seppo

What's wrong with the argument?

A reminder of the definition that you introduced into the discussion:

The term ‘proposition’ has a broad use in contemporary philosophy. It is used to refer to some or all of the following: the primary bearers of truth-value, the objects of belief and other “propositional attitudes” (i.e., what is believed, doubted, etc.[1]), the referents of that-clauses, and the meanings of sentences. — SEP article on Propositions

You have argued that a hinge proposition is a bearer of truth-value because all propositions are bearers of truth-value.

The same argument could be made that a hinge proposition is an object of belief and other "propositional attitudes" (including doubt) because all propositions are objects of belief and other "propositional attitudes" (including doubt).

However, a hinge proposition is not an object of belief and other "propositional attitudes" (including doubt), because a hinge proposition excludes doubt:

341. That is to say, the questions that we raise and our doubts depend on the fact that some propositions are exempt from doubt, are as it were like hinges on which those turn.

342. That is to say, it belongs to the logic of our scientific investigations that certain things are in deed not doubted. — OC

Therefore, a hinge proposition need not be an object of belief and other propositional attitudes (including doubt).
Therefore, a hinge proposition need not have a truth-value.

ETA:

What interests us now is not being sure but knowledge. That is, we are interested in the fact that about certain empirical propositions no doubt can exist if making judgments is to be possible at all. Or again: I am inclined to believe that not everything that has the form of an empirical proposition is one. — OC 308

Indeed, your comments concerning propositions show some deep misconceptions. — Banno

Such as?

Well, no, not exactly; doubt is a propositional attitude, and being the object of propositional attitudes is part of how a proposition is typically defined (it sort of follows from the fact that they are bearers of truth-value, since propositional attitudes just are the different positions we may take wrt the truth or falsity of a proposition) — Seppo

I would say that the only “different positions” we can take wrt the truth or falsity of a proposition is that a proposition is either true or false. I don’t see how being an object of a propositional attitude, such as doubt, follows from the fact that propositions are bearers of truth-value. That a proposition is an object of doubt, and that we have propositional attitudes more generally, is something in addition to a proposition simply being a bearer of truth-value.

... but that doesn't necessarily mean that every proposition can coherently be the object of every propositional attitude at all times. — Seppo

Nowhere in the SEP definition of a proposition does it refer to “every propositional attitude at all times”. According to the SEP article you provided, a proposition is defined in contemporary philosophy as being not only a bearer of truth-value, but also as being an object of doubt (and other propositional attitudes).

Your argument was that a hinge proposition must have a truth-value by definition (SEP). By the same logic, a hinge proposition must also be an object of doubt (and other propositional attitudes) by definition (SEP). However, a hinge proposition cannot be an object of doubt because a hinge proposition is indubitable. Therefore, if a hinge proposition does not meet the SEP definition of a proposition wrt being an object of doubt, then why must a hinge proposition meet the SEP definition of a proposition wrt to being a bearer of truth-value? Hinge propositions are not the same as the (ordinary) propositions defined in the SEP article, so you cannot justify that a hinge proposition must bear a truth-value based on the definition of an ordinary proposition.

Think of Descartes and his cogito, "I exist" is a proposition, but it cannot coherently be doubted (since doubting something entails that you exist to do the doubting). — Seppo

“I exist” might be the ultimate example of a hinge proposition, and the fact that a hinge proposition cannot be an object of doubt is precisely the point, because it contradicts the SEP definition of a proposition. Conforming to the definition of a proposition is the only argument you have offered for why a hinge proposition must be the bearer of a truth-value. If hinge propositions do not conform to the SEP definition of a proposition wrt being an object of doubt, then why must hinge propositions conform to the SEP definition of a proposition wrt being a bearer of truth-value? That is, if the definition can be contradicted by not being an object of doubt, then it can also be contradicted by not being a bearer of truth-value. Hinge propositions are not ordinary propositions.

Your argument is that hinge propositions must be bearers of truth-value because they are propositions (e.g. as defined by the SEP). Using the same logic, hinge propositions must also be objects of doubt because they are propositions (e.g. as defined by the SEP). However, hinge propositions are not objects of doubt (as W demonstrates), so your argument fails. Hinge propositions needn't have a truth-value merely because they are propositions (e.g. as defined by the SEP). In fact, hinge propositions are not propositions (e.g. as defined by the SEP).

There's nothing that says a proposition has to be able to be known or justified, and as above a proposition needn't necessarily be able to be doubted either. — Seppo

Now you are contradicting the SEP article that you cited. If you are to insist that a proposition must be the bearer of a truth-value due to the definition, then you must equally insist that a proposition must be an object of (propositional attitudes such as) doubt due to the definition. After all, you provided the definition.

Moore doesn't know whether "here is a hand", not because "here is a hand" is neither true nor false (how could it be neither true nor false? What is a proposition without a truth-value, other than a contradiction in terms?), but because "here is a hand" is, to use your previous analogy, one of the rules of the game: that here is a hand is one of the hinges upon which our evaluation of other propositions swings. — Seppo

93. The propositions presenting what Moore 'knows' are all of such a kind that it is difficult to imagine why anyone should believe the contrary. E.g. the proposition that Moore has spent his whole life in close proximity to the earth.— Once more I can speak of myself here instead of speaking of Moore. What could induce me to believe the opposite? Either a memory, or having been told.— Everything that I have seen or heard gives me the conviction that no man has ever been far from the earth. Nothing in my picture of the world speaks in favour of the opposite.

94. But I did not get my picture of the world by satisfying myself of its correctness: nor do I have it because I am satisfied of its correctness. No: it is the inherited background against which I distinguish between true and false.

95 . The propositions describing this world-picture might be part of a kind of mythology. And their role is like that of rules of a game; and the game can be learned purely practically, without learning any explicit rules. — Witt, On Certainty

93 shows that Wittgenstein is talking about Moorean (hinge) propositions at 94. 94 shows that hinge propositions are the background against which we distinguish between true and false (and therefore lack a truth value themselves). 95 shows that hinge propositions needn't be articulated (and 87 does the same).

All of this adds weight to the suggestion that hinge propositions are unlike ordinary propositions in that hinge propositions are indubitable, unknowable, unjustifiable and lack a truth value.

I can imagine Wittgenstein drawing an analogy and questioning whether we can attribute a truth value to (e.g.) the law of identity, or to the rule of a game, while leading us in the direction of a negative answer. (And that these examples are equally not the objects of doubt, knowledge, or justification.)

Do you acknowledge that ordinary propositions are not the same as hinge propositions?
— Luke

Well, the way you set this up, no. — Banno

If you cannot acknowledge that ordinary propositions are not the same as hinge propositions, then it appears that you don't know what a hinge proposition is.

And if a proposition is to be taken as undoubted - and that seems to be the case - then by that very fact it is true. — Banno

Hinge propositions are not merely "undoubted"; they are indubitable. More accurately, they cannot be an object of doubt, and (hence) neither can they be an object of knowledge.

A proposition - stated or no - is the sort of thing that can have a truth value... — Banno

Then that true proposition is (or should be) knowable. But hinge propositions are unknowable.

No, being justifiable, dubitable, or capable of being know are not part of the standard definition of a proposition. A proposition, in contemporary philosophy, is something which has a truth-value, a bearer of truth/falsity. — Seppo

This may be of some use here: https://plato.stanford.edu/entries/propositions/ — Seppo

Thanks for the link. Its opening paragraph states:

The term ‘proposition’ has a broad use in contemporary philosophy. It is used to refer to some or all of the following: the primary bearers of truth-value, the objects of belief and other “propositional attitudes” (i.e., what is believed, doubted, etc.[1]), the referents of that-clauses, and the meanings of sentences.

So the article you cited in support of your claims does not limit the definition of a proposition to being only “a bearer of truth/falsity”. The article you cited explicitly states that a proposition is also the object of belief, doubt and other “propositional attitudes”. So it appears that the capacity to be doubted is also part of the definition of a proposition in contemporary philosophy. It’s not a huge leap to infer that the capacity to be known and justified could also be included.

As the article rightly states:

The best way to proceed, when dealing with quasi-technical words like ‘proposition’, may be to stipulate a definition and proceed with caution, making sure not to close off any substantive issues by definitional fiat.

If we can agree that the definition of a proposition includes being the bearer of truth/falsity and having the capacity to be doubted, known and justified, then the question remains why hinge propositions should differ from ordinary propositions in one (or three) respect(s) but not the other.

No, the argument is that if its a proposition, then must have a truth-value, because that just is what a proposition is (i.e. the sort of thing that has a truth-value). — Seppo

By the same logic, if it is a proposition then it must be justifiable, dubitable and capable of being known, because that is just what a proposition is. Yet hinge propositions are none of these things.

The distinction that myself, Banno, and Jamalrob have urged between ordinary propositions and hinge propositions is that the difference lies in the latter's inability to be justified (and that because of the role hinge propositions play in language, particularly in the process of justification). — Seppo

I’m aware. I’m “urging” the further distinction that they do not have a truth value either.

And a proposition without a truth-value would be a contradiction in terms. — Seppo

But a proposition that cannot be justified, known or doubted isn’t a contradiction in terms?

Well, most obviously, because hinge propositions are propositions. — Seppo

But that is also in question here. Again, W does not refer to “hinge propositions” in OC. Also, if they cannot be doubted or known, then they are unlike (ordinary) propositions in at least some other ways.

What is an "ordinary" proposition here? — Banno

Unless you know of any other types, ordinary propositions are those which are not hinge propositions. Do you acknowledge that ordinary propositions are not the same as hinge propositions?

Those things can presumably be stated.

Some folk call such statements hinge propositions.

Hence hinge propositions are true. Hinge propositions are undoubted. Hinge propositions are unjustified. — Banno

Your argument appears to be that if a proposition can be stated then it must have a truth value. But this is just to ignore the distinction between ordinary propositions and hinge propositions and does not explain why hinge propositions must have a truth value, especially given your acceptance of the other differences between hinge propositions and ordinary propositions that have been noted.

Hinge propositions are indubitable, and hence true. — Banno

Hence we know them?

It is not that they cannot be known because they do not have a truth value... — Banno

I haven’t made any such argument. I’m asking why they must have a truth value when they are not ordinary propositions and they are distinguished from ordinary propositions in other ways.

One gets to hinge propositions by then inferring that one could state what it is that is indubitable — Banno

I don’t follow why you would accept that hinge propositions are not like ordinary propositions in the sense that hinge propositions are indubitable (and therefore unknowable) whereas ordinary propositions are not. Yet you insist that hinge propositions must be like ordinary propositions in the sense of having a truth value.

I can't make sense of the idea of a proposition that does not have a truth value - not a proposition for which we don't know if it is true or false, but a proposition which is not eligible for truth or falsehood. Sam26 was entertaining that idea here. — Banno

What reason is there to assume that hinge propositions are no different to ordinary propositions? It may be worth noting that Wittgenstein doesn't use the phrase "hinge proposition" in OC. If hinge propositions are just ordinary propositions then why does W appear to indicate that they cannot be doubted or known? We can doubt and know ordinary propositions. Or do you think he is talking about some other (third) type of proposition in this regard?

And yes, what does "both" mean here? We have a horse moving in 1878. We can pick out pairs of frames, but that horse isn't "here" in 2022 according to you. 1878 was a long time ago. — InPitzotl

As the horse is now dead, I would say that the horse no longer exists. Alternatively, an eternalist would say that the horse still exists in 1878, and that dinosaurs still exist at the time they existed and that the future now exists in its entirety, too. Given that you claim the object, O, exists at both t=1 and at t=2, then you seem to be leaning towards eternalism. "Both" simply means that the object exists at t=1 and at t=2.

Maybe this will help:

There are different ways to oppose presentism—that is, to defend the view that at least some non-present objects exist. One version of non-presentism is eternalism, which says that objects from both the past and the future exist. According to eternalism, non-present objects like Socrates and future Martian outposts exist now, even though they are not currently present. We may not be able to see them at the moment, on this view, and they may not be in the same space-time vicinity that we find ourselves in right now, but they should nevertheless be on the list of all existing things.

It might be objected that there is something odd about attributing to a non-presentist the claim that Socrates exists now, since there is a sense in which that claim is clearly false. In order to forestall this objection, let us distinguish between two senses of “x exists now”. In one sense, which we can call the temporal location sense, this expression is synonymous with “x is present”. The non-presentist will admit that, in the temporal location sense of “x exists now”, it is true that no non-present objects exist now. But in the other sense of “x exists now”, which we can call the ontological sense, to say that “x exists now” is just to say that x is now in the domain of our most unrestricted quantifiers. Using the ontological sense of “exists”, we can talk about something existing in a perfectly general sense, without presupposing anything about its temporal location. When we attribute to non-presentists the claim that non-present objects like Socrates exist right now, we commit non-presentists only to the claim that these non-present objects exist now in the ontological sense (the one involving the most unrestricted quantifiers).

The present is the year 2022, at the time of this writing. All frames in this video are in 1878. That horse is long gone in 2022. — InPitzotl

I could quibble that all frames in this video were photographed in 1878. We clearly still have those frames (now) in 2022, making the YouTube video possible.

Anyway, according to you, does the horse still exist (in the ontological sense)?

I'm advocating past facts don't change. You're by contrast advocating that they both do and do not: "O was at (1,1) at t=1" but "O is not at t=1". O changes time, from past to present, but still has a past location at past time t=1. — InPitzotl

You appear to be accusing me of a contradiction, but those statements are entirely consistent: The object was at t=1 and is not (presently) at t=1. The latter statement ("O is not at t=1") need not be read tenselessly.

O changes time, from past to present, but still has a past location at past time t=1. — InPitzotl

I reject the claim that O "still has a past location at past time t=1" at any time after t=1.

But t=1 to t=2 is still a change in time. It's over that change in time that a moving object changes position. — InPitzotl

I wouldn't consider it a moving object unless it changes its spatiotemporal location. Given that the object always exists in all of its spatial locations (including x=1 and x=2), and at all of its temporal locations (including t=1 and t=2), then how does it change its spatiotemporal location? In what sense does it move?

You're injecting O's being being dragged along through time into your reading of the phrase, but that's not what the phrase means. — InPitzotl

And you're rejecting O's "being dragged along through time" from your reading of the phrase. O's "being dragged along through time" is precisely what motion is. What else could it be? Motion is a type of displacement. No object is ever displaced on the four-dimensionalist view because all objects forever remain at every place and time of their existence. Therefore, no object is ever in motion.

No, the present changes.
— Luke

It's not just that Luke. You're not just talking about time changing. You're talking about O's being at a particular time changing... you're specifically arguing about O "changing time", presumably changing into the present. — InPitzotl

According to presentism, the only time that exists is the present time and the only objects that exist are those that exist at the present time.

I believe that our present evidence, theories and statements are the truth bearers of facts.
— Luke

That's not enough. How can an object move if it can't be in some place at all in the past, and how can it be in some place in the past if all objects are only in the present? And that's just the starters... wait a blink, and that very question gets re-asked about the past. — InPitzotl

I think a presentist can say that there were objects in the past, even if there are no objects [in the ontological sense] in the past now. A presentist can say that an object was in the past and that it changed from t=1 to t=2. I don't believe an eternalist can say the same and remain consistent with their eternalism.

...whether the horse "exists at both times" (whatever "both" means) or not. — InPitzotl

You don't know what "both" means?

Let's be clear: are you advocating a four-dimensionalist, eternalist view of time where time is a space-like dimension, and where all past, present and future times exist, or not?

The change in time is from t=1 to t=2. — InPitzotl

I don't see how you reconcile this with your assertion that "O's being at (1,1,1)" and "O's being at (2,1,2)" are both true. If the change in time is from t=1 to t=2, then is the statement of "O's being at (1,1,1)" true when O is at t=1 but false by the time O changes to t=2? If not, then how can you say that O changes in time from t=1 to t=2? Isn't your claim that O exists at both t=1 and at t=2?

You're making the claim that the past changes — InPitzotl

No, the present changes.

The word "was" doesn't help; "was" is just the word normal people use to talk about something in the past. — InPitzotl

"Was" is a word people use to talk about something which is presently in the past. You also used the word "was" in the exchange that I quoted and replied to.

So what then is the truth bearer of facts about the past, if there isn't anything in the past? — InPitzotl

I believe that our present evidence, theories and statements are the truth bearers of facts.

Otherwise, there is also the growing block theory of time, which posits that both past and present exist and that new things come into existence as the present moves forward in time.

§94 is about one's picture of the world, not propositions. That picture is the background against which propositions can be seen to be true or false. That picture shows hinge propositions to be true. Or better, as becomes clear in other sections, our actions mkae the truth of the proposition. — Banno

As Moyal-Sharrock writes, from the article that @Sam26 posted here:

Yet the clear message of On Certainty is precisely that knowledge does not have to be at the basis of knowledge. For Wittgenstein, underpinning knowledge are not default justified propositions that must be susceptible of justification on demand but nonpropositional certainties – certainties 'in action' or ways of acting – that can be verbally rendered for heuristic purposes and whose conceptual analysis uncovers their function as unjustifiable rules of grammar. So that basic certainties stand to nonbasic beliefs, not as propositional beliefs stand to other propositional beliefs, but as rules of grammar stand to propositional beliefs. Hence the absence of propositionality as regards them.

This also demonstrates that not all empirical statements can be hinge propositions. Hinge propositions are only those that function as "unjustifiable rules of grammar". A statement like "my truck weighs 2800 pounds" is something that is verifiable but not something that everyone acts like they know with certainty. Maybe a statement like "most people cannot lift a truck over their head" would be closer to a hinge proposition, as it goes without saying.

93. The propositions presenting what Moore 'knows' are all of such a kind that it is difficult to imagine why anyone should believe the contrary. E.g. the proposition that Moore has spent his whole life in close proximity to the earth.—Once more I can speak of myself here instead of speaking of Moore. What could induce me to believe the opposite? Either a memory, or having been told.—Everything that I have seen or heard gives me the conviction that no man has ever been far from the earth. Nothing in my picture of the world speaks in favour of the opposite.

94. But I did not get my picture of the world by satisfying myself of its correctness: nor do I have it because I am satisfied of its correctness. No: it is the inherited background against which I distinguish between true and false. — Witt, On Certainty

My take, which will probably just be a terrible rehash of Moyal-Sharrock:

I think it needs to be kept in mind that Wittgenstein is talking about empirical propositions, which are traditionally considered to be contingently true (or false). Hinge propositions, however, have the special status of being empirical statements that are quasi-necessarily true. W likens them to mathematical statements (e.g. see §340). Hinge propositions are beyond doubt, beyond truth (see §94 above), beyond justification, and non-epistemic.

I say "quasi-necessarily true", because they are treated as necessarily true and beyond true (beyond doubt) only when they form part of the background assumptions that we do not usually consider consciously and that we use (consciously or not) "as a rule of testing" (§98). When these same empirical propositions are instead consciously considered and used as "something to test by experience" (§98), then they revert to being normal, contingent, empirical statements that lie within the scope of epistemology, knowledge, doubt, truth and justification.

87. Can't an assertoric sentence, which was capable of functioning as an hypothesis, also be used as a foundation for research and action? I.e. can't it simply be isolated from doubt, though not according to any explicit rule? It simply gets assumed as a truism, never called in question, perhaps not even ever formulated.

96. It might be imagined that some propositions, of the form of empirical propositions, were hardened and functioned as channels for such empirical propositions as were not hardened but fluid; and that this relation altered with time, in that fluid propositions hardened, and hard ones became fluid.

98. But if someone were to say "So logic too is an empirical science" he would be wrong. Yet; this is right: the same proposition may get treated at one time as something to test by experience, at another as a rule of testing.

100. The truths which Moore says he knows, are such as, roughly speaking, all of us know, if he knows them. — Witt, On Certainty

The object being at t=1 and then at t=2 is also a change.
— Luke

In what sense? The object is at (1,1) at t=1; therefore the object is at t=1. The object is at (2,1) at t=2; therefore the object is at t=2. So O is both at t=1 and at t=2. Where's the change? — InPitzotl

Ask yourself that question. You have made clear your belief that O does not change in time, presumably due to its four-dimensional existence at both t=1 and t=2. However, this change in time is required by the definition of motion: "change in position over change in time". So where is the change in time that is required in order for you to say that O moves? (Also, where does O move from/to?)

In my opinion, this is easy to answer: The change in time is that O is first at t=1 and then subsequently at t=2. However, since you hold that O exists at both t=1 and t=2, and that O therefore does not change in time from t=1 to t=2, then how do you account for a change in time?

Nope. The object's position changes over time; that's exactly what the definition requires. — InPitzotl

The object changes from being at one time to being at another; that's exactly what the definition also requires. Change in position over change in time.

I'm not the one saying that O's being at (1,1,1) doesn't change. It does change.
— Luke

And yet it apparently doesn't:

If O's being at (1,1,1) were to change, then by what means do you think you get to say O was at (1,1)?
— InPitzotl

Yes, it was at (1,1) at t=1. That's what is denoted by O being at (1,1,1).
— Luke — InPitzotl

I think you overlooked the word "was".

I guess by change, you mean that "O's being at (1,1,1)" changes from true to false and back again based on when Luke thinks he needs to say O's not at (1,1,1) and when Luke thinks he needs to say O is at (1,1,1). — InPitzotl

"O's being at (1,1,1)" is true iff O is at (1,1,1). If "O's being at (1,1,1)" and "O's being at (2,1,2)" are both true, then how can O possibly change from being at (1,1,1) to being at (2,1,2)? What does "change" mean in that case?

Being at one position at one time doesn't preclude being at a different position at a different time. — InPitzotl

No, it doesn't, but a difference is not necessarily a change. Existing at both times precludes changing from one time to another.

The change from (1,1) to (2,1) is a change over time. O's at (1,1) at t=1; it's at (2,1) at t=2. Those different positions are at different times. — InPitzotl

How does O change from one time to another if it always exists at both t=1 and t=2?

You're just parsing the English wrong. Here, it's "(from (1,1)) (at t=1)", not "(from ((1,1) at t=1))". The "at t=1" describes when it was "from (1,1)", not where it's moving from. Similarly for the other phrase: "(to (2,1)) (at t=2)", not "(to ((2,1) at t=2))". — InPitzotl

Okay then:

You say that there is motion from [(1,1)] [at t=1] to [(2,1)] [at t=2]; and
You say that there is not motion from (1,1,1) to (2,1,2).

How is [(1,1)] [at t=1] different from (1,1,1)?
How is [(2,1)] [at t=2] different from (2,1,2)?

It's motion from (1,1) to (2,1); not motion from (1,1,1) to (2,1,2). The former is a change; when it's at (2,1), it's not at (1,1) any more. That's a change over time; it's at (2,1) at time t=2; it's at (1,1) at time t=1. — InPitzotl

To emphasise my point:

You say that there is motion from (1,1) at t=1 to (2,1) at t=2; and
You say that there is not motion from (1,1,1) to (2,1,2).

Please explain:

How is (1,1) at t=1 different from (1,1,1)?
How is (2,1) at t=2 different from (2,1,2)?

Change refers to an alteration in the state of a thing. — Banno

The “state” of a thing is its condition at a given time.

Of course it is motion from (1,1,1) to (2,1,2). Why is it not?
— Luke

Because it's still at (1,1,1). That didn't change. — InPitzotl

Then what has changed?

Sure. It's motion from (1,1) to (2,1). That is a change. — InPitzotl

The object being at t=1 and then at t=2 is also a change. How can you allow for the object to change position if you do not allow for the object to change time?

Right. It's motion from (1,1) to (2,1); not motion from (1,1,1) to (2,1,2). The former is a change; when it's at (2,1), it's not at (1,1) any more. That's a change over time; it's at (2,1) at time t=2; it's at (1,1) at time t=1. The latter is not a change; when O is at (2,1,2), it's still at (1,1,1). That's what that underlined phrase represents, right?: — InPitzotl

You seem to be saying that the object does not change time. But isn't that a requirement of motion, per the definition?

You're saying that if O's being at (1,1,1) were to change, then you can still say O was at (1,1) at t=1, because O's at (1,1,1). At once, O's being at (1,1,1) changes, and it doesn't change? — InPitzotl

I'm not the one saying that O's being at (1,1,1) doesn't change. It does change. What you are accusing me of with respect to time is what you are guilty of with respect to position. You are saying that the object both changes and does not change position. You say that the object is always at (1,1,1) but you also allow for it to change from (1,1) to (2,1). You are the only one of us saying that O doesn't change in some respect. I'm saying that it changes with respect to both position and time. Therefore, the contradiction is yours.

You introduced changeless facts into the discussion. If, according to your logic, changeless facts imply that change of any sort is a contradiction, then that's on you.

You have still not answered the question of what an increase in the x-coordinate represents with respect to Banno's image or the hill.

I'm claiming it's not motion from (1,1,1) to (2,1,2). The change here is from O's being at (1,1) to O's being at (2,1); not from O's being at (1,1,1) to O's being at (2,1,2). — InPitzotl

Of course it is motion from (1,1,1) to (2,1,2). Why is it not? It denotes a change in position over a change in time, which is the definition of motion that you provided earlier.

Think about it; (1,1,1) and (2,1,2) are different points-in-time, sure, but so are (1,1,1) and (1,1,2), and the latter is just called staying still. But it gets worse than this... — InPitzotl

If the latter is called staying still, then the former must be called moving. So are you saying that the object does move from (1,1,1) to (2,1,2)? But you have just said "I'm claiming it's not motion from (1,1,1) to (2,1,2)". It can't be both motion and not motion.

The change here is from O's being at (1,1) to O's being at (2,1); not from O's being at (1,1,1) to O's being at (2,1,2). — InPitzotl

I think what you are trying to say is that the object does not move from t1 to t2 (per the definition of motion), or that a change in time (only) is insufficient for motion, but that's not what you have said. Obviously, the object does change from being at t1 to being at t2. Moreover, it is also obvious that the object cannot change position (i.e. move) unless it also changes time.

If O's being at (1,1,1) were to change, then by what means do you think you get to say O was at (1,1)? — InPitzotl

Because it was at (1,1) before it changed. Right? That's what change is.

When would it have even been there... at t=1? — InPitzotl

Yes, it was at (1,1) at t=1. That's what is denoted by O being at (1,1,1).

[As an aside, you were earlier using notation of (x,y,z,t), but in this post you have changed to (x,y,t). I am just trying to follow your notation.]

Nope; that's no good... that's the very thing you'd be claiming changed... that O was at (1,1) at t=1. — InPitzotl

O was at (1,1) at t=1 and then it changed (e.g. to being at (2,1) at t=2). I don't see the issue.

So if you can't say that O is at (1,1) at t=1, given you're going to claim that its being there changes, then how can you claim it was ever not at (2,1)? — InPitzotl

It is at (2,1) at t=2. At least, I presume that's what you've been saying. I don't see why I can't say that O was at (1,1) at t=1 and then it changes/moves to (2,1) at t=2. You may need to spell out further what the problem is here.

That is the contradiction, and it's on your end. — InPitzotl

What is the contradiction on my end? That objects can change position and time?

If you're going to claim that facts about where an object is in place-and-time change, then you cannot get motion off the ground in the first place. — InPitzotl

I never said anything about changes in facts. I have only spoken about the changes in the positions and times of an object.

You have switched from talking about changes in positions and times of an object to talking about changes in facts about the positions and times of an object. Are you implying that changeless facts about the positions and times of an object implies changeless positions and changeless times for that object? Then motion is impossible.

To repeat, if changeless facts (about the positions and times of an object) implies changeless positions and changeless times (for that object), then motion is impossible. But I have not said anything about changeless facts. However, you have said that O does change position, and you said this as recently as your latest post:

The change here is from O's being at (1,1) to O's being at (2,1) — InPitzotl

Then again, you have also said:

O does not move from A to D; O is always at A and always at D. — InPitzotl

These contradictions are your own.

What the heck is an a-pixel? And what do you mean "replace x-coordinates"? a and b here are numbers; (a,b) expresses an x,y coordinate with x=a and y=b. — InPitzotl

Right, x=a. My point was that a-pixels of a given value are no different to their associated x-coordinates of a given value, because they are both associated with the same colour. That is, the a-pixels of a given value form a vertical line of the same colour and they all cross the x-coordinate at the same point. Or, again, x=a. I asked you what the increase in the x-coordinate represented and you explained it in terms of a-pixels. My criticism was that this did not explain what the increase in the x-coordinate (or an increase in the a-pixels: from a to a+1) represented, and that you had merely expressed the same thing in a different way.

Exhibit B:

You've said that nothing moves or changes position in various ways recently:

There's no such thing as a thing that moves from (1,1,1) at t=1 to (2,1,1) at t=2
— InPitzotl
O does not move from A to D; O is always at A and always at D.
— InPitzotl
B2 doesn't change places to B1; B2 and B1 are merely different places on the same image.
— InPitzotl
— Luke

None of those things say "nothing moves"; none of them say "nothing changes". Incidentally, isn't this you?:
Exhibit B2:
— InPitzotl

Nothing about it has changed
— Luke

...so I would like to know, Luke, if you're going to prefer to be consistent and claim that you are saying nothing changes, or honest and admit that you are just building straw men. — InPitzotl

Yes, that's me. My position is that nothing changes without time. I thought you held the opposing view? But it seems you hold both views and think it's unproblematic.

Surely the first statement of yours quoted above (in bold) can be read as saying that nothing moves, or that there is no such thing as a thing that moves from one position at time t1 to another position at time t2? Do you also assert that there is something that moves from one position at t1 to another position at t2? Maintaining both statements is a contradiction.

...exactly what you would expect, if the increase is mathematical. That 5 is an increase from 3 ipso facto makes it an increase in value because it is that value being described by increase. — InPitzotl

I simply thought that the mathematics somehow linked back to the examples of Banno's image or the hill. I have asked numerous times what the increase in the x-coordinate represents but your only response has been that it represents the increase in the x-coordinate itself, not that it represents anything about Banno's image or the hill.

In Banno's image, x is a number and "f(x)" is a color. — InPitzotl

What does an increase in x represent in Banno's image (in terms of the image, but not in terms of the function/colour)?

It might be clearer if you could explain what the increase represents in terms of the hill. What does the increase in the x-coordinate represent there?
— Luke

Coordinates are grammar school material, Luke. You shouldn't be confused in the first place. — InPitzotl

I'm not confused; that's not why I asked.

The value of your opinion is proportional to the justification. You're not only lacking that; you're apparently so allergic to opposition, you invent straw men even on points you agree with (exhibit B). — InPitzotl

It's not a straw man to point out your contradictions.

So let's talk about that word change. That is an English word; used by English speakers. Applied to change-over-place, we can examine how people in the wild use that word. Here are some samples: — InPitzotl

I think that when the word "change" is used "in the wild", then it isn't typically assumed that time is absent, like we are assuming in this philosophical discussion. This is evident from your quoted examples. In the absence of time, what you call a "change", I would call only a "difference". There is no going or getting from left-to-right, top-to-bottom, or movement in any other direction. This is why I keep asking what the increase in the x-coordinate represents.

Do you agree that D1 and E1 do not animate? — InPitzotl

Yes, I agree.

Ostensively speaking, D1 and E1, especially opposed to B1/C1 and F1/G1 and friends, are changes that are not changes over time. — InPitzotl

What "road narrows" usually indicates is that the road gets narrower as you travel down the road. Absent of time, there can be no travel down the road.

This doesn't tell me what the increase in the x-coordinate represents. I imagine it represents something in space?
— Luke

The coordinates are labeled x and y thusly: (x,y). For any a and b, (a+1,b) is one pixel right of (a,b). (a,b+1) is one pixel down from (a,b). (0,0) is the coordinate of the leftmost topmost pixel. — InPitzotl

This does nothing but replace x-coordinates with a-pixels. None of this explains what the increase (a+1 or x+1) represents or why there is an increase. What is represented by the increase from a to a+1? You have used mathematics to demonstrate that there is an increase, but you then explain this increase as mathematical. I don't agree that there is any change over space without time. Where does this change come from? What, in spatial terms, does this change represent? Your explanation is circular, referring to the increase itself, not to what it represents.

It might be clearer if you could explain what the increase represents in terms of the hill. What does the increase in the x-coordinate represent there? Does it represent a change in position on the hill? If so, what has changed position on the hill? (Please do not say that the x-coordinate has changed position on the hill.)

The only special thing about increase in x coordinates here is that Banno posted an image and made a claim about a change of color from left to right. — InPitzotl

The claim is that there can be a change in colour over a change in space, absent of time. I am of the opinion that, absent of time, nothing changes in space. Therefore, I think you need to account for the assumed change in space (absent of time) and/or what such change represents or corresponds to. If your argument is that things can change in space (absent of time), then you can't simply assume that things do change in space (absent of time) in your argument. You want to say "as the x-coordinate increases, then there is a change in colour/height". But you first need to account for how the x-coordinate increases (absent of time), and what that increase represents. Otherwise, I think you are begging the question.

Nope. I never said nothing changes or moves position; you said I said that. — InPitzotl

You've said that nothing moves or changes position in various ways recently:

There's no such thing as a thing that moves from (1,1,1) at t=1 to (2,1,1) at t=2 — InPitzotl

O does not move from A to D; O is always at A and always at D. — InPitzotl

B2 doesn't change places to B1; B2 and B1 are merely different places on the same image. — InPitzotl

I was chocking this up to a mistake earlier and ignoring it, but you repeated it here. The increasing x-coordinates are not approaching the color on the right; — InPitzotl

Isn't each colour associated with an x-coordinate? If the colours are approaching the colour on the right, as you say, then I don't see how each x-coordinate isn't also approaching the colour on the right. You said that the x-coordinates were increasing, yes?

The increasing x-coordinates are not approaching the color on the right; they are just increasing, as it says on the tin. — InPitzotl

Why are the x-coordinates increasing?

The colors are approaching the color on the right; but that phrase is an underspecification. Approaching is something an ordered sequence does, and we have to specify how the colors are ordered so we can meaningfully say it's approaching the color. That is what the phrase "as the x-coordinate increases" does... it imposes the order. — InPitzotl

This is still circular: the x-coordinates increase due to the ordered sequence, and the ordered sequence is ordered due to the phrase "as the x-coordinates increase".