Hopefully we can translate the structure of the proof into knitting, line by line.

"Suppose Γ is a satisfiable set of R-preserved sentences and $\delta$ is R-fragile."
Γ is the rows of some scarf that have already been knitted, while $\delta$ tells us about some arbitrary set of any rows at all.

"Let M be a model which satisfies Γ"
Let M be any scarf with the rows Γ already knitted.

"Either $\delta$ is true in M or it isn’t."
Either the rows described by $\delta$ will be added to M, or they won't.

"If it isn’t, then M is a counterexample showing that Γ⊭ $\delta$"
If the rows $\delta$ are not added to M, the the rows Γ could not have led us to conclude that they would be added.

"But if $\delta$ is true in M, then since $\delta$ is R-fragile there is some M' such that R(M,M') and $\delta$ is not true in M'."
But if the rows $\delta$ are added to the scarf, then since they might not have been added (they are fragile), there is some other scarf M' such that the rows $\delta$ were not added.

"Since each member of Γ was particular, each member of Γ is also true in M'."
Since the rows Γ have already been knitted, they are the same in both scarves. M' also has the rows Γ

"Hence M' is our counterexample, and Γ⊭ $\delta$."
In which case, the other scarf M' has the rows Γ but not the rows $\delta$, and so again, the rows Γ could not have led us to conclude that the rows described by $\delta$ would be added.


Clear as mud? There was a bit of trouble with the parsing, such that I had to use mathjax for the delta but not the Gamma. Odd. Let me know if it doesn't parse well.

The next section shows the structural similarity between Prior's objection and Pp ⊨ FPp

That some sentence was true in past implies that in the future it that sentence will be true in the past. Prima facie, a derivation about the future from a premise about the past. But FPp is on Russell's account neither past nor future, and so Pp ⊨ FPp does not derive a sentence about the future from a sentence about the past.

The logic sets out the incoherence of the intuition.

There'd be a naive objection along the lines that all Russell had done is avoid the issue by re-defining "future"; that the sentence "In the future, p will be in the past" is about the future - it's right there in the syntax.

Trouble is, this is just to give an alternate formal definition of "future" and "past", as if a sentence were "future" when the outermost tense operator is F. Russell's semantic definition gives us a general case that applies also to the Prior dichotomy, while also giving logical support to the intuition that what was true int he past need not be true in the future.

the syntactic version does not generalise, and does not explain why certain inferences do not work. And it is no surprise ot find that the surface syntax can mislead us as to the logical character of a sentence.

Russell now moves on to considering another application of the barrier to entailment, that of the impossibility of deriving an ought from an is. The main problem faced here is the lack of a widely accepted formalisation of deontic logic. We don't have a settled semantics for "ought".

(It might be worth pointing out here that deontic here is not much related to the "deontic" that is so often contrasted with utilitarianism and virtue ethics, as concerning absolute moral rules. "Deontic" here means to do with ought, not necessarily to do with moral rules. So it includes utilitarianism, virtue ethics and other ethical systems)

Russell noted right back in the introductory paragraph that she does not actually present a section on Hume's Law, but rather is building towards it.

What is needed in order to apply Russell's account is a set of sentences that are fixed, and a set of sentences that switch. The obvious candidates here are for the fixed statements, those that concern what is the case, and for the switching statements, those that concern what ought be the case. Descriptive sentences would be preserved under normative switching, normative sentences would be fragile. Which is just to say that there are different normative approaches to any fixed description of how things are.

But there is a sense in which this is already to assume Hume's law. To define what we ought do as fragile is to presume that it is distinct from what is the case, that we can clearly seperate normative sentences from descriptive sentences.

The danger is that Russell presumes rather than demonstrates Hume's law. In which case she will have provided a powerful way for us to talk about deontic logic but not have settled the issue.

Somehow I managed to miss all of these over the last several days. Sorry for the crickets -- I will catch up.

Thanks, but don't feel obligated. This is as much. or more, me writing my own notes as it is seeking comment. I want a clear idea of how the logic relates to Hume's Law, so I'm working through the article far too meticulously for most folk. I'm not at all surprised this hasn't garnered much attention.

Of course, comments and criticism is welcome.

this is just to give an alternate formal definition of "future" and "past", as if a sentence were "future" when the outermost tense operator is F. — Banno

This, if I understand it, is an important point. Could I paraphrase it this way?: A sentence with a tense operator does not automatically become about that temporal location. Pp ⊨ FPp is about p, not the future (or the past).

But there is a sense in which this is already to assume Hume's law. To define what we ought do as fragile is to presume that it is distinct from what is the case, that we can clearly seperate normative sentences from descriptive sentences.

The danger is that Russell presumes rather than demonstrates Hume's law. In which case she will have provided a powerful way for us to talk about deontic logic but not have settled the issue. — Banno

Yes. And your reservations about how to formalize "ought" are also significant. So there've been attempts to create a semantics for "ought," but they haven't succeeded? That's interesting. Similar attempts to standardize ordinary-language uses of "ought" also have failed, as far as I know.

A sentence with a tense operator does not automatically become about that temporal location. Pp ⊨ FPp is about p, not the future (or the past). — J

Not quite since, FPp might tell us something that will be true in the future - that the past will not have changed; so it is also about the past. It's this lack of being definitely about the future or definitely about the past that Russell brings out. It's a bit of both, fragile in some models, preserved in others.

Similar attempts to standardize ordinary-language uses of "ought" also have failed, as far as I know. — J

There are various ways to formalise ought. The simplest is just to adopt an operator "Oρ", roughly "we ought ρ". Whether they fail or not depends on what one is doing with them. The advantage of formalising language is that the consistency of what we say is made clear. There is more than one way to formalise "ought", each perhaps brining to the fore a different aspect. I wouldn't count this as a "failure". The task for Russell is to find an account that can avoid question begging.

In pondering the next part of the essay, I've come across How to Prove Hume’s Law, a more recent paper.

That paper was downloaded 300 time last month. Seems topical.

Down the rabbit hole.

And so to Vrana's objection.

One of the ways of setting out a obligation in first order logic os to simply incorporate an opperator, O. Op is then just "One ought p" This has the advantage of simplicity, Humes rule being the brief remark that there can be no valid inference form φ to Oψ, φ being some statement concerning what is the case, Oψ some statement as to what ought be the case.

Following Russell's strategy, we'd be looking to perhaps show that φ was preserved, while Oψ was fragile, and hence no entailment relation can hold between them. φ → Oψ, then, is mixed, and so in the scheme of things, neither descriptive nor normative.

φ → Oψ is neither preserved nor fragile.

Now the General Barrier Theorem says roughly that no set of satisfiable sentences , each of which is preserved, entails a sentence which is fragile. It is about sentences that are either preserved or fragile.

Vrana's objection is that since φ → Oψ is neither preserved nor fragile, the General Barrier Theorem says nothing about it. So on this account, the Barrier Theorem tells us nothing about Hume's Law... but that's what we wanted!

And the second horn of the dilemma. Suppose we go along with the criticism, and strengthen our barrier to entailment so that no "is" statement can result in φ → Oψ; then we have ~φ ⊨ φ → Oψ; but that is exactly what we do not want! If we strengthen it enough to avoid Vrana's criticism, then it's demonstrably false.

Russells response is to move from individual sentences to sets of sentences. Is that justified?

I think so. To start with, an argument is a set of sentences. Consider the example "If Alice is a first-year, then she ought to hang her coat on one of the blue hooks." On it's own, this is invalid. It needs an additional premise: All first-years ought hang their coats on the blue hooks".

Is this justified for Hume's Law? Consider the ubiquitous quote from whence it came:

...when of a sudden I am surpriz’d to find, that instead of the usual copulations of propositions, is, and is not, I meet with no proposition that is not connected with an ought, or an ought not. — Hume, Treatise, Bk. III, Pt. I, §1

It's clearly about sets of sentences.

Entailment is a characteristic not of individual sentences, but of sets of sentences. And Russell is concerned with barriers to entailment. Hence she is concerned with sets of sentences.

So I think we can grant that what might look as if it is an ad hoc reaction to a criticism is instead an adjustment that follows form the context.

Entailment is a characteristic not of individual sentences, but of sets of sentences. — Banno

This is surely true. So if the move from individuals to sets works at the formal level, then it's justified, since it captures the concept of entailment more precisely than sticking to individual sentences.

One of the ways of setting out a obligation in first order logic is to simply incorporate an operator, O. Op is then just "One ought p" — Banno

I'm still not very happy with this. Don't we need to better understand what "ought" means in ordinary language before we can be sure that "O" (and all the formal moves we can make with O) captures this? Might there not be something about how "ought" works in OL that prevents the "simple incorporation"? Just asking.

I'm still not very happy with this. — J

Nor should you be. There is certainly more going on here. But what we can do is set out some minimal requirement, and at the very least we do say for some sentences that we ought do as they say.

What we might be looking for is a way to clearly articulate what can be done consistently with at least these sentences.

But there's more here, of course. Hare pointed out that formal logic treats of statements that are true or false, but here we are dealign with imperatives sand proscriptive sentences. There are a number of options open to Russell, perhaps the simplest of which is to pars the imperative as a statement, such that "It ought to be that p"; so from "Alice ought hang her coat on a blue hook" we get "It ought be that Alice hangs her coat on a blue hook".

it is a point to consider.

@Banno

Don't we need to better understand what "ought" means in ordinary language before we can be sure that "O" (and all the formal moves we can make with O) captures this? — J

The bus ought to be here at 8:00 AM every day (according to the timetable). But it isn't always or inevitably.

Ordinary language has use for 'ought' exactly in cases where there is an ideal that may not be realised The bus has fallen from the state of grace; it has sinned against the timetable, and in such a lamentable condition, what ought to be is not what is. The bus is late, or worse, early. My thesis is that one only uses the term in cases where 'what is' departs from 'what ought to be'. I only say 'Dogs ought to have waggy tails' when I come across a wagless dog tail, or a tailless dog, or some other aberration. There's always an implied complaint that the world is out of sorts and not up to my high standards.

So I would say I don't need a formal logic to prove this, to suggest otherwise is not to understand the grammar of language. What ought to be isn't, and what is ought not be. Far too often! By definition!

I wrote this yesterday, and then thought I might be interrupting your reading. Coming back to it, I am struck by the "according to". I think philosophers should probably cite their source whenever they use the word.

According to probability theory, a fair die should roll a six on average ...
According to current physics, the smallest mass that can sustain fusion of hydrogen to helium through gravitation should be ...
According to unenlightened, philosophers ought to cite the source of any obligation they present.
According to Jesus, we should love our enemies.
According to the law of the UK, one ought not use a phone whilst driving.

What ought to be is someone's idea of things, not the way it is.

I'm not at all sure what that was about, but I enjoyed it anyway.

Natural languages will always have more to them than can bee shown in a formal language. Indeed it might be good to think of formal languages as just one part of natural languages. What formal language can do is to set out a bit more clearly how the bits of language might relate to each other.

Damn. That reply to Un deleted my next bit of exposition.

Hi Banno! Hope you are doing well. So, I'm wondering if this new critique of entailment could be applied to Gettier's two cases? That may be different or somehow opposed to your purposes/intent here, but if it's not, could you set it out? Case I in particular.

Damn. That reply to Un deleted my next bit of exposition. — Banno

Once more with feeling.

:groan: