That Frodo depends on words isn't that "Frodo" refers to words. — Michael

It might be. We would have to be careful not to confuse use and mention in fleshing it out. But I get that you don't think you are headed in that direction.

But that leads you to insist on,

"Frodo" refers to a hobbit, — Michael

Well, in the idiom of Lord of the Rings talk, yes. "Frodo" is how one particular hobbit is called by his peers. Perhaps it is presumptuous to expect to avoid that idiom.

and hobbits exist only in a fictional piece of writing. — Michael

Do you mean, in a fictive piece of writing? But then we are back to referring (indirectly, not confusing use and mention) to hobbit-pictures and hobbit-descriptions. You were adamant that you didn't want to go there.

So perhaps you mean fictional, but a fictional world? In which case, why say writing? I think it's because that (words and pictures) is indeed where you are headed, as you quite rightly try to transcend the fictive idiom, and talk literal sense.

Frodo is a hobbit, "Frodo" is a word. Clearly there are two different referents. — Michael

That may seem clear during the phase of the game where you are confident in asserting,

When I use the name "Frodo" I am referring to the hobbit, not to the word "Frodo" or my idea of Frodo. — Michael

Later, perhaps trying to square this with the fact there are no hobbits, you must start to explore ways of restating things. Hence,

Does this entail realism regarding Frodo? Of course not. Frodo is not ontologically-independent of our language and our ideas. — Michael

The existence of hobbits is traded for the existence of, and indirect reference to, certain varieties of words and pictures and other symbols.

There are all these paradoxes as to how can we talk about things that don't exist? Pegasus, Zeus, etc. — Manuel

Only one, really, and it soon straightens out. Usually, the speaker equivocated between denying that the reference was (directly or indirectly) to words, pictures or other symbols,

When I use the name "Frodo" I am referring to the hobbit, not to the word "Frodo" or my idea of Frodo. — Michael

... and admitting as much,

Does this entail realism regarding Frodo? Of course not. Frodo is not ontologically-independent of our language and our ideas. — Michael

(6 years old, but recently exhibited.)

What are the "Ordinary Language Philosophy" solutions to common philosophical problems? — Chaz

At best: sublime readability, and a mission to dig as deep as possible (though not deeper).

At worst: bluff, imperiousness, charlatanism, guruism, preistliness, laziness, sophistry, prejudice, mysticism, ism-ism, tribalism.

:rofl:

"... an' they catch 'im... an' they say e's mental!!"

I think that final gem is the culmination of the speech by the boy pictured (used earlier in the track).

Hopefully you can see the passing relevance.

If you clicked the first link, what did you hear?

Perhaps the stimulus was too noisy, literally.

Anyway, just sharing some lovely (perhaps inauthentic) social history.

No aspersions or barbs.

Carry on.

What would you replace that power with. Criminals all get treated the same regardless of their mental health? — Isaac

There's a YouTube link in my psyche. Can't insert it here as media so that it starts at the right place, but

https://youtu.be/rLmMchi2aAQ?t=220

The source is even more genius than the reappropriation. Extract only, but the audio will be familiar to some (losers).

So, you looked over the post — Cheshire

No, over the thread. Just pointing out that absolutism has a non-cosmic variety, from which point of view correctness is absolutely achieved, and your notion of 'inherent error' is unnecessarily cosmic.

nor the relevance of your comment. — Banno

Everything as it should be, then.

In error, cheers.

Good absolutism is recognising that within a language game there is often no choice between this and not-this. For example, the puzzle,

[1] Tell me, do you think that a single grain of wheat is a heap? — bongo fury

requires a benignly absolutist grasp that answering in the affirmative isn't playing the game of English usage of 'heap'.

The good absolutist (and you would think, any competent speaker) will say,

[2] No, absolutely not. — bongo fury

Likewise with,

• Tell me, do you agree that this is not a hand?

or

• Tell me, do you agree that I may move my bishop straight forwards?

... although these don't naturally or easily create sorites sequences.

Good relativism is recognising that within a language game there is often a choice between this and not-this. For example,

[1] And tell me, do you think that adding a single grain could ever turn a non-heap into a heap? — bongo fury

deserves something like

[2] Yes, that kind of flexibility does seem intrinsic to usage of the word.

Good relativism is also about recognising that the absolutism only holds relative to the game, which can co-exist happily with other games.

Games can merge, of course, and then the relativity becomes complicated and might require loss of absolutism here and there.

Science is all about merging, and reconciling and reformulating, so while it's natural to think that

Every measurement that has ever been taken since the beginning of measuring things has inherent error. — Cheshire

it may make better sense to see the process as one of dropping or replacing or reforming whole systems of measurement that were perfectly (absolutely) stable games in their own terms, and with their own margins for error.

No, you can't be bothered, and why should you.

My bad. Carry on.

There's a double negative in what you're saying. — TonesInDeepFreeze

Of course.

RAA premise would not need to deny ~P. Rather, in this case, the premise is P. — TonesInDeepFreeze

Exactly, if for some reason you want to label the RAA line "P" rather "~P".

In a line properly signposted as RAA, and in a discussion in which someone had bothered to say

We don't need to suppose toward contradiction that there is a surjection. — TonesInDeepFreeze

it could make sense to display under that signpost (P or ~P depending on signposting preferences, or a form of words such as I chose so that the question didn't arise) the denial of what is to be shown in the argument. This denial will be the supposition toward a contradiction.

What is to be shown is that S can't, without contradiction, be in the range of f.

So the denial, the suitable RAA line, the supposition toward contradiction, if you or anyone did want to belabor the point, or understand the point about "not needing to suppose toward contradiction..." is indeed

"S is in the range of f",

and it might be interesting that this is taking the place of "f is surjective", in a proof by contradiction.

Anyway, that's not the beginning of my proof. — TonesInDeepFreeze

Agreed.

Maybe they were undecided whether it was meant to be read as a proof by contradiction or not. — bongo fury

Not a big deal — fishfry

How true.

And, of course, I wouldn't even think of denying the claim that S is not in the range of f. — TonesInDeepFreeze

Except in a line properly signposted as RAA.

The proofs prove the exact same result - nothing more nothing less, — TonesInDeepFreeze

Sure, but yours begins (read as a proof by contradiction) by denying a more specific claim of failure of surjectivity: the claim that such sets as, in particular, S will fail to be in the range of f. Was my point. Obviously that more specific claim of failure implies the more general. But the denial of it is weaker than the denial of the more general. I thought this might be the correct way to interpret

We don't need to suppose toward contradiction that there is a surjection. — TonesInDeepFreeze

(That you were saying we can suppose less.)

That's backwards. — TonesInDeepFreeze

Sure. I was prepared to guess at quantifier introduction on the backwards journey, but the "I guess" probably sounded sarcastic. Without the sarcasm it probably doesn't improve much.

In that way, fishfry's RAA is deferred in my proof to later. — TonesInDeepFreeze

Yes. It's still a proof by contradiction, just not so upfront.

I didn't write EyeX f(y) = S as a separate line, since I didn't belabor certain obvious steps; it's not a fully formal proof. — TonesInDeepFreeze

Sure. And it's still a proof by contradiction.

RAA and modus tollens are basically the same. — TonesInDeepFreeze

By RAA here I take it you mean the whole argument, while earlier it was a tag for the line that you

suppose toward contradiction — TonesInDeepFreeze

? Cool.

RAA and modus tollens are basically the same. — TonesInDeepFreeze

Another reason not to expect an important contrast in your reworking.

For some mathematicians its a stylistic preference. — TonesInDeepFreeze

signpost[ing] — bongo fury

Yes you are correct, it's cleaner to not use proof by contradiction. — fishfry

Is that a thing? Ok.

But @TonesInDeepFreeze doesn't appear to be eschewing proof by contradiction, instead merely proving (still by contradiction) a stronger denial of surjectivity than mere failure of surjectivity. Hence his supposing ("toward contradiction") that the denial is false amounts to supposing less than necessarily complete surjectivity, which would entail the whole power set being in the range of f, and amounts instead to supposing merely the presence of S in the range of f.

Thus showing, that the naturals can't be used to keep count of their own groupings/combinations/sub-sets if any of them (naturals) are needed to index (I mean count, map to) groupings they aren't in. Because that would create the set S, which would need to but couldn't without contradiction be in the range of f.

Hope I've got that right. And if I have, then the suggestion to prove only the more specific failure of surjectivity has helped me, at least.

Then again, Tones hasn't exactly signposted the supposition, that S is in the range of f (the supposition being, I thought, in order to show that it leads to contradiction), and he doesn't even explicitly state it. It just (as line 4 perhaps alludes but doesn't actually say) follows from

EyeX f(y) = S — TonesInDeepFreeze

which (I guess?) follows from

Let f(y) = S — TonesInDeepFreeze

So, I don't know. Maybe they were undecided whether it was meant to be read as a proof by contradiction or not.

Sorry.

From oasis to cess pool, in one tweak. :roll:

(R v A) -> (~R -> A)
R v A
therefore we have reason to believe ~R -> A

is not [modus ponens]. — TonesInDeepFreeze

Not if you hear it, for no reasons that are obvious to me, as talking about psychology. I hear it, for reasons of charity and extensionalism, as dialect for "therefore we have deductive reason to assert"... i.e. "therefore".

But the puzzle includes an intensional operator "believe'. — TonesInDeepFreeze

Please not. You're inviting the enthusiasts for modal logic to show off, and end up perpetuating the silly libel of a logical error subtlety. — bongo fury

Assume, assert, affirm, hold, "believe"... whatever.

an instance of modus ponens. — TonesInDeepFreeze

Indeed. And perfectly valid.

If you can't stand by all 3 lines at once, don't. They can't be a good expression of what you're trying to say. Don't necessarily involve any logic in expressing yourself, but don't think you need a better one. (I don't mean you.)

Yet — Banno

Weasel. Yes they did. If they "believed, with good reason" both [1] and [2], then they had deductive reason to believe [3].

Deductive not good enough? Sure. Deductive not always good enough. Too strict at times. Then go inductive.

"Strictly", though... deductively... modus ponens valid. Here, as anywhere.

Well... equals?

Ah, so not some one among several with the same domain.

So worlds are not in general to be identified by their domain?

That is just a nice thing about your example?

So, W1 = some world (among others) whose domain is {egg, bacon}?

W1={egg, bacon} — fdrake

A world = a sub-domain?

According to non-Bayesian statistics if the value is continuous — Cheshire

But unnecessary presuppositions aside...

there isn't one. — Cheshire

So, finally,

Unbounded precisely, i.e. not graph 4; or unbounded ever i.e. graph 2? Or unbounded how? — bongo fury

So graph 2, i.e. ditch P1, because after all, "everything's relative", or "on a spectrum".

obviously it's a puzzle if we accept also the premise that calling a single grain a heap is absurd. If calling it a heap is tolerable then, as I keep saying, no puzzle.

[1] Tell me, do you think that a single grain of wheat is a heap?
[2] Well, certainly, it's the very smallest size of heap.

Game over. — bongo fury

Which is cool, if you find [2] an acceptable description of usage, and would say that black is minimally white, bald is minimally hairy, off is minimally on etc.

the limit of observed cubits — Cheshire

... is a questionable notion, but key to my

antonym-based constructive solution — bongo fury

Which of course is of no interest if there is no problem.

Here (and unfortunately only here) is where I assume we more or less understand each other:

I imagine there's a distribution of arm lengths and as a result a very, nearly exact distribution of cubits. — Cheshire

I don't quite get the 'very nearly exact' but never mind that. The puzzle (for an enthusiast of the heap puzzle who recognises here a classic case) is: exactly where (along a reasonably long line of arms positioned in ascending or descending order of length) does the distribution of cubits end, and the distribution of non-cubits begin?

As long as everyone measured with their arm they are technically right as I understand it. — Cheshire

Well, I do hope neither of us is about to reach for Wikipedia. My point is that any such primitive measuring system is as good an example as any of the potential quandary. Line up the population of the village in order of height. Now, whose arm is the first valid cubit stick? Hence,

And then the puzzle is to specify the smallest (or largest) number of microns that is no longer a cubit. — bongo fury

Which is merely to envisage a radically longer line-up. Because you said microns. Which is fine in principle.

You don't add or subtract length to your arm to meet a standard, so this is incoherent. — Cheshire

How about now, any clearer?

It's the length of your forearm to middle finger. — Cheshire

What is? Presumably not the variance, whose unboundedness I was inquiring into.

So, people don't use tight tolerances for measures with unbounded variances. — Cheshire

Unbounded precisely, i.e. not graph 4; or unbounded ever i.e. graph 2? Or unbounded how? — bongo fury

Did you just mean, people don't use narrow tolerances to measure wide tolerances? But of course engineers do just that, as you seem aware.

It's a novelty — Cheshire

Sure. A 3000 year old novelty.

We'd all produce a different cubit if measured to the micron. — Cheshire

And then the puzzle is to specify the smallest (or largest) number of microns that is no longer a cubit.

We'd all be right relative to our arms and wrong relative to the others. — Cheshire

No, some of us would be obviously right relative to the cubit system, some of us obviously wrong, and some of us neither.

So, people don't use tight tolerances — Cheshire

Narrow tolerances or precise tolerances?

for measures with unbounded variances. — Cheshire

Unbounded precisely, i.e. not graph 4; or unbounded ever i.e. graph 2? Or unbounded how?

Okydoke.

So if epistemicism neither captures people's metasemantic awareness of their own language, — Snakes Alive

because it fails to endorse P2 or offer a substitute...

nor does it seem to describe anything 'objective' in the practice itself, — Snakes Alive

hence the faint mystical glow of the Warburton quote, which I've probably unfairly represented, I must check...

what is its utility as a hypothesis? Are you defending it in any capacity, — Snakes Alive

No.

or just using it as a springboard to talk about the difficulties with vagueness? — Snakes Alive

Yes, which it is good for from my point of view, because it at least endorses P1. It doesn't offer the kind of argument for P1 that would stop people from so carelessly abandoning it, is the shame. On the other hand, we are at least talking in terms of trying to draw a suitable graph of usage.

I could see the proposal to act like it's true, — Snakes Alive

No no no, I never proposed anything of the kind, and I absolutely propose that you carry on doing the opposite :up:

What you seem to be saying now, however, is that epistemicism isn't really true in any sense — Snakes Alive

Except in its support of P1.

it just helps us highlight some features about vague language that are puzzling to us — Snakes Alive

Yes, how to draw a convincing graph of usage.

I think vague language is vague, [...] but that doesn't make it puzzling, — Snakes Alive

It does if you accept any responsibility for the care of P1 as well as P2, and try to apply logic.

Now then...

[1] Tell me, do you construe the word 'heap' as being correctly applied to a single grain?

Surely, though, pretended things aren't so? — Snakes Alive

Is this suddenly a problem?

we all know precisely well what we mean by saying they do or don't exist, and no one is confused. — Snakes Alive

Is your position that we ought to pretend there is a single correct use of a term, — Snakes Alive

No, but in playing or describing the game we ought to respect the cases of correct and incorrect that are clear. We ought not pretend that we are playing chess by moving the bishop non-diagonally, nor that we are speaking English literally by construing the word 'heap' as being correctly applied to a single grain. We ought to pretend instead that the word 'non-heap' applies to or points to or is otherwise connected to a single grain.

, and in the case of vague language, pretend to be epistemicists? — Snakes Alive

No, I think vague language can be described as a game of pretend that tolerates disagreement about what is pretended. The puzzle is how to square the tolerance in some places (P2) with the clarity in others (P1). Epistemicism (a minority view, as its proponents admit) thinks all disagreement is a symptom of error. It says that a proper description of a vague game, if it were possible, would weed out the errors and leave a precise and perfectly consistent game.

But here, as we discuss this now, we presumably aren't pretending — Snakes Alive

We're doing our best to agree enough (pretended) reference to have viable discourse about our actual linguistic behaviour.

so shouldn't we say epistemicism is false? — Snakes Alive

Sure. Not because we aren't pretending (we are) but because the game is better described as tolerating dissent.

But then, I have to admit I fail to see the value in acting like vague language determines precise boundaries. — Snakes Alive

Quite, but the puzzle is to explain how it can have blurred boundaries. How to get from P1 and P2 to P3.

The problem is this is not true. — Snakes Alive

Oh well, if you put it as emphatically as that, with italics and all...

You seem to be hung up on the false idea that a magical barrier exists preventing people from using words in certain ways. — Snakes Alive

I honestly don't know how it seems like that, when I keep mentioning how reference is a game of pretend.

they can even move the bishop non-diagonally – try it yourself... — Snakes Alive

I have. People say, "ok, if you like, but you know it won't be chess?" And this is me, "what makes you think there's any fact of the matter whether what we do is chess?". And they're like, "sorry mate, we only play chess not philosophy."

I accept P1 because I wouldn't apply 'heap' to a single grain. — Snakes Alive

Ok, I'm grateful if someone at least avows P1. We might have a game. Now, bearing in mind,

it is a matter of arbitrary decision whether we choose to apply the word 'heap' or not, and so construe it as correctly applied or not — Snakes Alive

... Let's try.

[1] Tell me, do you construe the word 'heap' as being correctly applied to a single grain?

You seem to think that because 'heap' has some property preventing it from being applied to a single grain, therefore P1 is true because people 'can't' apply it to a single grain. — Snakes Alive

Well, whether or not I ever lapse into magical thinking, and I'm only human after all, I would indeed tend to offer scare quotes around can't, and be ready to clarify, as I keep doing, that either of us doing what they 'can't' merely prevents us from agreeing that the game is indeed 'chess', or 'spoken English' or whatever. Which is usually a game-stopper. (Which may or may not be good for the quality of the game as generally played.)

But you've got it backwards. — Snakes Alive

Surely there's no back or front? If we're not thinking magically or essentially?

It's because people don't use 'heap' for a single grain that P1 is true. — Snakes Alive

Ok, if you prefer. Although I preferred your more symmetrical "material equivalence", earlier. What I really like here is "people"...

We could turn around and decide to start applying it to a single grain, if we wanted to, and declare P1 false as a result. — Snakes Alive

My emphasis, for the same reason, that you are on the verge of recognising a general rather than personal proscription against the application, such that reversing it in a collective endeavour might create a new and different game with the same word. (Where P1 was indeed false.)

I just wouldn't want to, — Snakes Alive

Ah, rats.

Still...

[1] Tell me, do you construe the word 'heap' as being correctly applied to a single grain?

Of course, I would not reject P1, because I think using 'heap' in such a non-standard way is pointless and confusing. — Snakes Alive

Well of course, I would not reject a statement to the effect that a bishop can't move directly forwards, because I think using it in such a non-standard way is pointless and confusing. However...

All these things are a matter of adjudication. You could choose to use a word in a highly nonstandard way, and people could go along with it – but they often won't, and they'll be more unwilling to, the farther you move away from an established usage. But if you decide to use 'heap' to refer to a single grain too, then sure, go ahead, that's also a pattern of usage that could be established. It would be 'incorrect' in virtue of some prior pattern of established usage, but so what? Patterns of usage can be re-negotiated as well. — Snakes Alive

But I'm guessing you don't offer the same advice in regards to the chess move? Because it wouldn't be chess? Well I, like any dictionary compiler or competent speaker, take the same view of the single grain, that it's well outside of the range of correct application of 'heap', in ordinary English as spoken literally. (As opposed to metaphorically.)

This is a matter of how to apply the word, not an interesting inquiry either into the nature of language, or the nature of sand and piles of it. — Snakes Alive

But how to apply the word is interesting and puzzling and inquires into the nature of language because, unlike in chess, the rules are flexible at the same time as they are strict. They can be bent, but not too far, and obviously how far is the puzzle.

So, you do reject P1 with respect to general usage, in English, of the word 'heap'. I accept that you accept P1 with respect to your own usage. Your personal threshold is perhaps much further along than one. But you appear happy to acknowledge that usage as a whole allows for literal application of the term to a single grain. A linguist or dictionary compiler may beg to differ. They would offer a single grain as an obvious example of incorrect usage, or opposite meaning.

The epistemicist, in appealing to a strict notion of 'correct usage,' is invoking a kind of magical view of language. — Snakes Alive

Wasn't it clear we agree about this?

to say a word has a meaning is no more and no less than to say the word has certain causal powers in virtue of a community of speakers coordinating to use it in a certain way. — Snakes Alive

But isn't that verging on a kind of magical thinking? You'll never cash out those causal powers at the level of linguistic analysis. (Chomsky's famous ridicule of "the probability of a sentence".) Better to describe the (pretended, sure) relations and rules and moves of the game.

When someone says a certain usage is correct, they might either mean: (i) as a descriptive matter, this is how people tend to use the term, as summed up by some statistical measure (based on prior usage or an inference about disposition to future usage, or whatever), — Snakes Alive

Graphs 1 and 2.

or (ii) as a normative matter, that some use is to be singled out as to how the word is to be used. — Snakes Alive

Graphs 3 and 4.

But neither of these are descriptive facts about words having meaning as if that were something else beside how people use a word. — Snakes Alive

Yes, and we don't want them to be, but, we do want the first (1&2) to better acknowledge where a line of acceptable usage (however blurred) has been crossed, and the second (3&4) to better show how the line is both created and blurred by use.

I.e., we want P1, and we want P2.

The grain doesn't transform a non-heap into a heap. An assertion without negation does. — Cheshire

I kind of agree. Does it matter who asserts and who negates? Are you equating 'heap' with 'allegedly a heap' or with 'unanimously a heap'? (Or both or neither, or something else.)

And do I take it that you disagree with the epistemicist position, that if we each recognise said threshold at different places then fewer than two of us will be correct? — bongo fury

The epistemicist has the 'atomic number' model of metasemantics, — Snakes Alive

Agreed. It's partly because they insist on a distinction between

whether we choose to apply the word 'heap' or not, and so construe it as correctly applied or not — Snakes Alive

and whether the word is correctly applied or not.

The first question suggests a possible poll of personal thresholds likely to exhibit a bell curve when plotting popularity of threshold against grain-number:



Or, roughly equivalently, an 'ogive' or half bell curve rising from next to nothing at a single grain and levelling out to a plateau at about, maybe, who knows, a few thousand. This might represent the distribution of actual applications of the word in ordinary discourse.



Whereas, the second question is envisaged by epistemicism as an underlying fact of the matter, albeit the linguistic matter, such that an appropriate graph would extend horizontally at a height of zero, then step suddenly up to 1 at the correct threshold, and continue horizontally.






Which I think we both reject, but is what is being defended here:

Like the children we make, the meanings we make can have secrets from us. — Nigel Warburton, aeon article

Happily, probably everyone agrees with

something is a heap iff the word 'heap' is correctly applied to it, 'iff' being read as material equivalence). — Snakes Alive

The controversy is over whether to distinguish between 'applied' (and so construed as correctly applying) and 'correctly applied'. Whether there are any incorrect applications (and construals).

Now, I wonder if you will approve of any of these suggested clarifications? I should have thought you might have reservations.

I certainly do. Ridiculous as I find the 'hidden step', I think that ordinary usage deserves some kind of recognition of its ability to distinguish between correct and incorrect, in some way that doesn't fizzle out to 'relatively correct'. Usage can sometimes be a matter for negotiation, and adjudication, but sometimes not. We know that anything black is an obvious counter-example to white, and is therefore anything but minimally white, and similarly for off and on, bald and hairy, etc.

Hence my readiness to restart, and invite you to consider an absolutist position on a single grain. E.g.,

[1] Tell me, do you think that whether a single grain can be correctly called a heap in common English is a matter for negotiation or adjudication in context?

I appreciate fully that you may well see no need at all to deny that proposition. (I'll have to bluster that you don't speak English, but never mind!) But if that's because you have embraced anything like the half bell curve as a picture of usage (or of fuzzy truth), then notice that you are, after all, ditching P1 and not P2.

one is often at liberty to say that the addition of a single grain creates a heap where there was none before. — Snakes Alive

And do I take it that you disagree with the epistemicist position, that if we each recognise said threshold at different places then fewer than two of us will be correct?

Behind you!!

Ok, I didn't realize this was the format. — Cheshire

Of course it could just as easily start from P2 and P3, asking how you can possibly go bald one hair at a time, etc.

If you tell me heaps exist then you can prove the existence of a heap through some criteria. — Cheshire

I wasn't trying to prove anything. Only to look for examples we can agree on. I don't see the relevance of criteria. Unless you want to say, being a billion grain collection is a criterion, or a sufficient condition. Fine. Bring it on board. How does it help?

Where do I send the invoice? — Cheshire

Reminds me of when my bank operated a no-charge policy for "small" overdrafts...

If premise 3 is true it implies criteria for a heap exists. — Cheshire

You lost me. What exactly do we need to agree is implied by

P3. heaps exist — bongo fury

?

I take it to mean, simply, that there are some heaps.

Not that we need to straight away consider examples, but I'd offer, say, any billion-grain collection. Premise 1, on the other hand, does refer to an alleged counter-example.

But, none of this addresses a paradox. — Cheshire

The puzzle, it should be clear, is how to reach P3, or avoid denying it, while accepting both P1 and P2.