Gillian Russell: Barriers to entailment
Another paper by Gillian Russell that I think worth some consideration. This one is of direct relevance, amongst other things, to discussions of induction, the is/ought barrier, temporal logic and modality.
Humes Law and other Barriers to Entailment
This is the 2013 paper that led to her book, Barriers to Entailment: Hume's Law and other Limits on Logical Consequence, 2023. Ill stick to the paper here, since it is openly available.
A barrier to entailment occurs when a set of premises that might on the face of it appear to be good reasons for some conclusion, cant be made to formally perform as required. So, for instance, it can be argued that if every time one finds a poppy, it is purple, one might feel entitled to conclude that all poppies are purple. This is the familiar problem of induction - no sequence of individual statements, Here is a purple poppy, There is a purple poppy, can deductively justify the move to All poppies are purple.
A couple of other such barriers have been mooted. Of particular interest are that no list of things that are the case can lead one to a conclusion as to what ought to be the case. And no list of things that have been the case can lead to a conclusion as to what will be the case in the future. And no sentence about what might be the case can lead to a conclusion as to what must be the case. There are others, but these will do for now.
Now, each of these carries its own baggage of some considerable literature, and each has its arguments and answers. Part of what is special about the present paper is that it attempts to analyse their common logical structure, formally. And it attempts to do so using the machinery of first-order logic.
Russell sets out her case quite formally, but finishes with a discussion of natural language examples. Informally, she identifies a group of satisfiable sentences that do not change their truth value if new objects are introduced into the conversation, and another set that can be made false by introducing new objects. So That is a purple poppy will remain true if we introduce more poppies, either purple or red; but This is the only purple poppy will change its truth value if another purple poppy is introduced. Russell calls sentences that do not change their truth value particular and those that must, Universal.
It should be noted that the term Universal is not a reference to the quantifier. These are not, for instance, just those sentences beginning with a universal quantifier. It is also worthy of note that the categories particular and universal do not cover all sentences - there are sentences that are neither.
We can form sets of particular sentences, such as {this is a purple poppy, that is a purple poppy}. These sets will themselves remain particular, in that adding more objects to the discussion - more purple poppies, or a white swan - will not change the particularity of the set.
And heres the point. From sets of particular sentences alone, we cannot derive a universal sentence. There is a barrier to entailment between particular and universal sentences such that the first cannot imply the second.
That account is of course a mere outline, the devil is in the detail of the article. The relation to induction should, I hope, be apparent. Russell offers ways of understanding Humes Law, Temporality, and other instances of barriers to entailment using this frame. There are other approaches, notably some involving relevance logic, that seek a similar result. Russells approach seems to be more intuitive.
Perhaps some will find this approach interesting.
Humes Law and other Barriers to Entailment
This is the 2013 paper that led to her book, Barriers to Entailment: Hume's Law and other Limits on Logical Consequence, 2023. Ill stick to the paper here, since it is openly available.
A barrier to entailment occurs when a set of premises that might on the face of it appear to be good reasons for some conclusion, cant be made to formally perform as required. So, for instance, it can be argued that if every time one finds a poppy, it is purple, one might feel entitled to conclude that all poppies are purple. This is the familiar problem of induction - no sequence of individual statements, Here is a purple poppy, There is a purple poppy, can deductively justify the move to All poppies are purple.
A couple of other such barriers have been mooted. Of particular interest are that no list of things that are the case can lead one to a conclusion as to what ought to be the case. And no list of things that have been the case can lead to a conclusion as to what will be the case in the future. And no sentence about what might be the case can lead to a conclusion as to what must be the case. There are others, but these will do for now.
Now, each of these carries its own baggage of some considerable literature, and each has its arguments and answers. Part of what is special about the present paper is that it attempts to analyse their common logical structure, formally. And it attempts to do so using the machinery of first-order logic.
Russell sets out her case quite formally, but finishes with a discussion of natural language examples. Informally, she identifies a group of satisfiable sentences that do not change their truth value if new objects are introduced into the conversation, and another set that can be made false by introducing new objects. So That is a purple poppy will remain true if we introduce more poppies, either purple or red; but This is the only purple poppy will change its truth value if another purple poppy is introduced. Russell calls sentences that do not change their truth value particular and those that must, Universal.
It should be noted that the term Universal is not a reference to the quantifier. These are not, for instance, just those sentences beginning with a universal quantifier. It is also worthy of note that the categories particular and universal do not cover all sentences - there are sentences that are neither.
We can form sets of particular sentences, such as {this is a purple poppy, that is a purple poppy}. These sets will themselves remain particular, in that adding more objects to the discussion - more purple poppies, or a white swan - will not change the particularity of the set.
And heres the point. From sets of particular sentences alone, we cannot derive a universal sentence. There is a barrier to entailment between particular and universal sentences such that the first cannot imply the second.
That account is of course a mere outline, the devil is in the detail of the article. The relation to induction should, I hope, be apparent. Russell offers ways of understanding Humes Law, Temporality, and other instances of barriers to entailment using this frame. There are other approaches, notably some involving relevance logic, that seek a similar result. Russells approach seems to be more intuitive.
Perhaps some will find this approach interesting.
Comments (64)
In the Priors Dilemma example, on the one horn, if we call Fa v UxGx a universal sentence then Fa ? Fa v UxGx is a counter-example; and on the other horn, if we call it not a universal sentence then Fa v UxGx ^ ~Fa ? UxGx is a counter-example, so either way we are stuck.
But that the answer, from Russell in 2.1 is that Fa v UxGx is neither universal nor particular (2.1, p. 9)
If you are tempted to get the book, note that the Kindle edition suffers the inability of Kindle to render LaTeX, so badly that some of the equations are simply absent. Get the paperback, which is only a few dollars more.
So she's getting rigorous about the problem of induction?
Much more than that. She is showing at a logical issue common to the problem of induction, the is/ough barrier, and to "nothing about what was the case tells us about what will be the case", amongst other things.
The general application is broad.
Each of the examples has it's own context, and there is a difference between the modal instance and the temporal instance. They are not the same. However, the treatment given by Russell applies to all. And it does seem to set out why the mooted counterexamples are fishy. But the detail...
I'd hope most folk would share the intuition that we can't logically get something in a conclusion that is not at least implicit in the presumptions. So if someone gets an ought from premises that contain only is, they have somewhere gone astray. Same if they get everything is thus-and-so from arguments that say only that this and this are thus-and-so, or that begin with "this is true" and end "necessarily, this is true".
There are further problems with causation, seperate to the issue being addressed here. Ubiquitously, those who make most use of causality are unable to tell us what it is. It certainly is not that if A causes B then in every possible world in which A is true, B is also true. But this is how it is often mistakenly understood - that B follows necessarily from A
The plan was to post on the proof strategy yesterday, but problems with lining a graphic had me baulk. Hopefully soon.
True. But the alleged modal counter-example has to make use of a qualifier or caveat about time, doesn't it? "Because p, it is necessarily the case that p", expanded, means "It is necessarily the case now that p". Otherwise, the modal necessity is very weak; this is the "fishy" aspect of saying of absolutely anything that obtains, that it therefore must do so.
I'll watch for your post on the proof strategy.
I don't see why.
The natural language use of "necessary" is ambiguous. And "it must be the case that..." is not quite the same as "It is necessarily the case that..."
That's one advantage of formal systems over natural languages. When necessity is defined in terms of access to possible worlds, these ambiguities dissipate.
It is not the case that the possum is in the tree in every possible world. But if we so choose, we can limit ourselves to talking only about those worlds in which the possum is in the tree - which is just a way of saying, if the possum is in the tree, then it must be the case that the possum is in the tree... That "if" is understood as stipulating that we restrict ourselves only to those worlds in which the possum is in the tree.
She uses first-order logic, like Fa, where this means that a is one of the things that is in the group "F". The "a" roughly a proper name for a; it picks out a and only a..
She also uses models. A model is just a group of things that are assigned to those proper names. The group of things is called the domain, D, and the mapping is called the interpretation, I. So the group of all the things, the domain, might be {a,b,c}, and the interpretation, that "a" stands for a, "b" stands for b, and so on. If a and b are both also members of F, then we can call {a,b} the extension of F
?xFx is read as "for all x, x is f", or "everything is f".
The novelty here is that we give consideration to what happens when the domain is extended - when more individuals and predicates are added.
So see fig 1:
Extending the domain by adding something that is also F and something that is not F cannot change the truth value of Fa.
And Fig 2:
Extending the domain by adding something that is also F and something that is not F can change the truth value of ?xFx.
I'll stop there for a bit. I keep getting distracted by other posts. And I should be moving a Passionfruit vine.
And so to the central argument.
Definition 1 sets out what is meant by extending a model. A model, again, is a bunch of individuals that have been assigned various predicates. The extension of a model adds some more individuals, and the predicates belonging to those individuals. Importantly, it does not change the individuals already in the model, nor their predicts.
Definition 2 sets out that a sentence is particular if and only if it's truth value does not change when the model is extended.
In contrast, Definition 3 sets out that a sentence is Universal if and only if its truth value can change when the model is extended.
So, speaking roughly, we have a bunch of individuals, and their predicates, and if we add more individuals and predicates without changing any of the existing ones, we have extended the model. Those sentences who's truth value does not change are particular, and those sentences who's truth value can change are universal.
Even more briefly, if a sentence is true in M and in any extension M' then it is particular. If it is true in M and false in at least one extension of M' then it is Universal.
Now comes the proof. What is to be shown is that from any true collection consisting only of particular sentences, we cannot derive a universal sentence. The proof works by considering the only two possibilities: Either the universal sentences is true, or it is false. Now if it is false, then it cannot follow from a true collection of particular sentences, since no collection of true sentences can imply a falsehood. And if it is true, then by definition 3, there is some model in which that universal sentence is false. But in that model we would again have the collection of true particular sentences implying a falsehood, which again cannot happen.
Too quick? Let's break it down. We have a collection of true particular sentences, ?. We want to show that this collection cannot imply a true universal sentence, .
Now either is true, or it is false.
If it is false, then it cannot be implied by any collection of true sentences, and so cannot be implied by ?.
And since is a universal sentence, there is an extension of our modal in which it is false. So even if in our model it is true, there must be a model in which it is false. And in that model, our particular sentences would still be true, and we would again have an instance of true sentences implying a falsehood.
So in neither case can a collection of particular sentences imply a universal sentence.
How's that? I'll look for a good analogue as well.
It should be noted that here I've skipped over the whole extensional mechanism of satisfaction, preferring to talk just of truth - on the presumption that truth is a bit more intuitive. Russell uses satisfaction, making her case both more robust and tighter.
What remains to be seen is how this account is to be generalised.
But first, a few important notes.
Pretty sure I get it, thanks. An example with English nouns and predicates would help too, I think. Or maybe this is what you mean by a good analogue. (The most counter-intuitive aspect, for me, is the very first step, in which Fa remains true even though ¬F has been added to the domain.)
Good luck with your vine. :smile:
Yes, I wondered about the diagram. It's both clear and misleading. The circle is the domain, each dot is an item, an individual in the domain. Only one is labeled - given an interpretation. That's the dot at top left. In the left circle there are two other individuals, both of which satisfy F; that is, both of which belong to the collection of things that are F. In the right circle, the domain has been extended, with one thing that satisfies F and another that does not - the one labeled "¬F". In both domains, a is F. indeed, by the rule for forming extensions, we can't construct an extension in which a is not F, because extending a model by definition does not change what is there already. So Fa is particular. But ?xFx, while true for the circle on the left, is not true for the circle on the right, since we can add an item that does not satisfy F, without changing what's there already. That is, ?xFx is universal.
Carry on.
Now regardless of what John decides, the first row remains green. "The first row is green" is true and particular. It will not change, regardless of what John does. Generally, any statement that if true does not become false when more rows are added is particular.
In contrast, "All the rows are green" is true unless John changes wool; and then false. So "All the rows are green", whether true or not, is universal. Generally, any statement that can change its truth value as rows are added is universal.
Now the proof seeks to show that no set of particular statements can logically entail a universal statement. That is, that a set of statements that cannot change when we add rows cannot logically entail one that can change it's truth value as rows are added.
Let's look at the universal statement "All the rows are green". Either all the rows are green, or they are not. The collection "Row one is green, row two is green, row three is green" is not enough to tell us that all the rows are green, since John may have decided to change to yellow at some subsequent row.
Now to be sure, "Row three is green, row four is yellow" does entitle us to conclude that not all rows are green. But "not all rows are green", if it is true, cannot be changed by adding new rows - if there are yellow rows, adding more rows will not change that. So "not all rows are green" is not a universal statement but a particular one.
No matter what we do, no set of particular statements can entail a universal one.
The knitting analogy is a bit clunky, but it might be of use when we get to temporality. It amounts to, no set of statements about what John has already knitted logically entails what he will knit next, which I hope has an intuitive appeal.
Barriers to Entailment by Gillian Russell
Yes, indeed, and are part of the prompt for this thread rather than just accepting the article.
First feature: Fa and ?x(x?a v Fx) are equivalent. They always have the same truth value. Yet Fa is particular while ?x(x?a v Fx) looks on the face of it to be universal - after all, that's a universal quantification right there at the front. The syntactic approach, that would classify these equivalent sentences differently, is qualified by the modelling approach adopted here, giving greater coherence. ?x(x?a v Fx) is particular.
Second feature, and prenex normal form. That's just a standard way of writing any first-order sentence with all the quantifiers - "?"'s and "?"'s - at the front. So ?x(P(x)??yQ(y,x)) becomes ?x?y(¬P(x)?Q(y,x)). This is used in computation because it feeds into Turing Machines easily. It's a syntactic definition, as opposed to the model definition Russell uses. So here Russell points otu that there already is a syntactic definition of the particular sentences.
Third, logical truths are particular. Pretty clear why - a sentence S is particular iff, whenever it is true in a model M, it is also true in all extensions of M. And logical truths are true in all models - that's what a logical truth is. But it is a curious result. It looks odd because tautologies such as ?x(Fx ? Fx) again look as if they are universal.
I'm puzzling over how that's compatible with the second feature - ?x(Fx ? Fx) in prenex normal form is, I think, ?x(¬Fx ? Fx); now Russell's observation is that the set of particulars is identical to ??, which are those prenex normal form sentences with only existential quantifiers; now ?x(x = x) is in prenex normal form, and is I believe the only ?? that is a candidate for a logical truth, at least in a non-empty domain. So what's going on? I think we have to take Russell quite literally here, and suppose a non-empty domain so that ?x(x = x) is a tautology, and since all tautologies are true, they all have the same truth value - that this is what she means by their being "identical". We will raise this question again when we get to section 4, as Russell notes. But I'm not overly content with this bit.
Forth, and I find this quite interesting, there are sentences which are neither universal nor particular. We noted Fa v UxGx in talking about Prior's Dilemma, above. This is her answer to Prior, in a nut shell: that Fa ? Fa v UxGx is not a counterexample, because while Fa is particular, Fa v UxGx is not universal, and so he does not give an example of a particular implying a universal.
And fifth, nor are they exclusive - p^~p, by way of an example. If it were true in M it would be true in M', as per the definition - but of course it is never true. And if it were true in M then there is at least one extension in which it is true, so it is universal... yep, but it's never true in M,
Logic is odd.
No, I found it helpful.
Quoting Banno
A question here. If we agree, as we should, that Fa v UxGx is not universal, how does that help in addressing the second version of Prior's counterexample, the one that derives UxGx? UxGx is a universal, correct? And ¬Fa is particular. So we're getting a universal conclusion from (1) a premise that is not universal [Fa v UxGx] and (2) a premise that is particular [¬Fa]. When you speak of "sentences which are neither universal nor particular," I assume that Fa v UxGx is such a sentence. But how does its not being universal mean that we haven't derived a universal from a particular? Is the idea that both premises must be particular, in order to claim to have derived a universal from a particular?
On a glance, the second horn of the dilemma is that if Fa v ?xGx is not universal, we have as an example of a derivation of a universal from a particular:
Fa v ?xGx
¬Fa
___________
?xGx
Either the first row of our knitting is yellow, or all the rows are green. The first row is not yellow, so all the rows are green. That's valid. And ?xGx is universal - that all the rows are green can change if we add a yellow row. So is it an example of deriving a universal from a particular?
¬Fa is particular - adding more rows will not change the colour of the first row. But Fa v ?xGx is not universal. Fa can't change, but ?xGx can. and given Fa v ?xG together with ¬Fa, ?xGx must be true. but ?xGx is universal.
Prior's suggestion was that {Fa v ?xGx, ¬Fa} is as a whole, particular, but ?xGx, universal. But on Russell's account, {Fa v ?xGx, ¬Fa} is not particular. Adding more rows may make ?xGx false.
How's that?
More on this later.
The definition of a particular sentence is that it can't change when we knit more rows.
The negation of the particular sentence "Row one is green" is the particular sentence "Row one is not green". However, the negation of the universal sentence "All rows are green" is the particular sentence "Not all rows are green". This break in symmetry is central to what comes next.
Contraposition
?xFx ? Fa, by contraposition gives ¬Fa ? ¬?xFx. This looks like particular ? universal... but it's not, because ¬?xFx is particular, not universal. Negating the universal in this case yields a particular.
In terms of our knitting,
?xFx ? Fa. if every row is green, then we can conclude that row one is green. A universal implying a particular.
¬Fa ? ¬?xFx. If it's not true that row one is green, then it's not true that every row is green. A particular implying another particular. Contraposition doesn't generate counterexamples to the particular-universal barrier thesis, because the barrier only blocks inferences from particulars to universals, and ¬Fa ? ¬?xFx is an inference from a particular to another particular. Note the broken symmetry.
In the knitting analogue, we only ever add rows, never deleting them. There's the broken symmetry.
Atomic Existential statements, such as Fa or "Row three is green", once made, are never taken back. This goes for all particular statements - it's the definition of "particular".
But universal statements, once given, can be made false by new particular statements. That's the definition of "universal".
Some explanations.
P - past existential, so Pp is read "p was true (at some time) in the past"
F - future existential, so Fp is read "p will be true (at some time) in the future"
G - future universal, so Gp is read "p is going to be true in (all of) the future"
H - past universal, so Hp is read as "p has historically (always) been true"
In the particular/universal first order logic case, the model was extended by adding more individuals and their predicates. Here, the model is extended by changing a future. The mooted barrier becomes "No set of premises about the present or past entails a sentence about the future".
Just as a particular fact will remain true when the model is extended, a past fact Pp will remain true into the future. And even as ?xFx can become false by adding more individuals, Fp may become false if the future turns out differently than expected.
So we have a structure similar to the previous first-order logic example, but in the place of extending the model we have what Russell calls "future switching", switching amongst alternative futures.
For the purposes of this temporal logic, sentences about the present behave in the same way as sentences about the past, so we can consider the "P" operator to also apply to them.
There is a sequence of times, t?, t?, t?... forming a set T, with one of them nominated as "now", n. There is a binary relation "<", understood as "t?
"<" is
So time flows in one direction - the breach of symmetry again. And it is continuous, goes forever into the past and the future and there are no gaps.
Yep, the set T is analogous to the real numbers. Russell chose this set up from among a number of alternatives, She might have chosen a different set, with beginnings and ends or analogous to the natural numbers rather than the reals, and achieved much the same outcome. Nothing in particular hangs on the choice of temporal formalism.
Every atomic sentence in the model is assigned either a 1 or a 0 (roughly, true or false) at each time by a function I, such the same sentence may be true at some time yet false at some other. So I may assign 1 (true) to p at t?, and 0 (false) to p at t?, and so on. I(p, t) = 0 means 'p is false at time t.
The rather scary looking Definition 5 just asks that we consider two scarves, identical right up to an including the row we are presently knitting, but differing thereafter. The one is a future-switch of the other.
And we can add the notion of preservation and fragility. A row that is already knitted is preserved - no further knitting will change it. A row that has not yet been knitted is fragile - it might change.
And extending those terms to the temporal case, a sentence is preserved if true in M, and true in all future-switches of M (Definition6). It is fragile if true in M, but there's some future-switch where it's false (Definition 7).
And a sentence that is future-switch preserved is Past, (Definition 8), while one that is future-switch fragile is Future (Definition 9)
In the first-order argument, we found that we could not derive sentences about all the individuals from any set of sentences about some of the individuals. Here, we find that we cannot derive sentences about the future from sentences about the past. We are now well-positioned to construct a general account of barriers to entailment.
Got it. My only question is a natural-language quibble: When you say "We find that we cannot derive sentences about the future from sentences about the past," that hinges on a certain strict understanding of "derive" (and maybe "about" as well). Nothing prevents us from saying either 1) "The sun is overwhelmingly likely to rise tomorrow" or 2) "It is logically certain that, if the future resembles the past [in the relevant respects], then the sun will rise tomorrow." These are surely true sentences about the future, based upon knowledge of the past, they just aren't "derived" according to a model that takes into account terms such as "likely" or "resemblance." This has no bearing on what Russell is demonstrating, of course.
We can make use of Bayesian methods.
But tomorrow might well resemble today in ways other than the sun rising.
Start with a formal language L, containing a sentence ?. All the language consist of are symbols and rules for stringing them together. For it to be of use, we give it an interpretation by assigning things to the variables. Each assignment is a model, and there are lots of different ways to make the assignments, so lots of different models - the set U.
And we can have relations between these models - R. The reaction might be having the same individuals up to a certain point, or having the same true sentences up to a certain time, or having the same rows of knitting up to a given row, and so on.
Definition 10 just defines a sentence as being R-preserved if for all models, if it is true in one then it is true in all the others that are related by R. Definition 11 just defines a sentence as R-fragile if whenever there are models in which it is true, there is an R-related model in which it is false.
Note again the lack of symmetry.
The General Barrier Theorem (Theorem 12) says that no R-fragile sentence is entailed by some satisfiable (true in some model) sentences that are all R-preserved.
Both the particular/universal barrier and the past/future barrier are special cases of this general result.
"Suppose ? is a satisfiable set of R-preserved sentences and [math]\delta[/math] is R-fragile."
? is the rows of some scarf that have already been knitted, while [math]\delta[/math] tells us about some arbitrary set of any rows at all.
"Let M be a model which satisfies ?"
Let M be any scarf with the rows ? already knitted.
"Either [math]\delta[/math] is true in M or it isnt."
Either the rows described by [math]\delta[/math] will be added to M, or they won't.
"If it isnt, then M is a counterexample showing that ?? [math]\delta[/math]"
If the rows [math]\delta[/math] are not added to M, the the rows ? could not have led us to conclude that they would be added.
"But if [math]\delta[/math] is true in M, then since [math]\delta[/math] is R-fragile there is some M' such that R(M,M') and [math]\delta[/math] is not true in M'."
But if the rows [math]\delta[/math] 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 [math]\delta[/math] were not added.
"Since each member of ? was particular, each member of ? is also true in M'."
Since the rows ? have already been knitted, they are the same in both scarves. M' also has the rows ?
"Hence M' is our counterexample, and ?? [math]\delta[/math]."
In which case, the other scarf M' has the rows ? but not the rows [math]\delta[/math], and so again, the rows ? could not have led us to conclude that the rows described by [math]\delta[/math] 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.
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.
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.
(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.
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, 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).
Quoting 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.
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.
Quoting J
There are various ways to formalise ought. The simplest is just to adopt an operator "O?", roughly "we ought ?". 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.
That paper was downloaded 300 time last month. Seems topical.
Down the rabbit hole.
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 ? to O?, ? being some statement concerning what is the case, O? some statement as to what ought be the case.
Following Russell's strategy, we'd be looking to perhaps show that ? was preserved, while O? was fragile, and hence no entailment relation can hold between them. ? ? O?, then, is mixed, and so in the scheme of things, neither descriptive nor normative.
? ? O? 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 ? ? O? 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 ? ? O?; then we have ~? ? ? ? O?; but that is exactly what we do not want! If we strengthen it enough to avoid Vrana's criticism, then it's demonstrably false.
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:
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.
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.
Quoting 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.
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.
Quoting 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.
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.
Once more with feeling.
Yours is a quite interesting question. There might be some potential to use Russell's work to at least show something about the relation between JTB and knowledge. But Russell's account is about truth (satisfaction), and the two Gettier cases are both true and valid. So I don't think it applies directly. that is, a JTB is always true, as is a piece of knowledge, and so formally a JTB always entails knowledge.
There might be something we could do in the detail, though. Ona. quick look I can't quite see how to make it work, although Claud somewhat disagree. Instead, I wonder if the upshot might be to show some of the problems with Gettier accounts more clearly.
Being true is persistent, yet both belief and justification are fragile; more information might change them. But is knowledge persistent or fragile? Or neither? Or both? And is the error then to treat knowledge as if it were persistent when it is fragile?
What you've done is shown how Russell's approach might have unexpected application.
I merely asked a question. You did the shewing. :wink:
I don't have much time nowadays, which is good, but what I had in mind fit into the truth as satisfaction aspect as well as being germane to issues with entailment. In the first case Gettier invokes the rules of entailment to move from d to e.
The satisfaction issue, it seems to me, is that those two conjunctive propositions do not mean the same thing. That is obvious because they have different truth conditions. Does Russell's approach find itself capable of addressing that?
Again, the detail is going to make or break any case here.
But "I have ten coins in my pocket and I will get the job" and "The man with ten coins in his pocket will get the job" are extensional identical. That is, they are satisfied by the exact same state.
So for the purposes of any extensional model we might use, the two propositions do meant the same thing.
The difference, if there is one, must be between "I know that p" and "I have a justified true belief that p". A case might be that P is persistent, but that there is a gap between P being justified and believed and P's being known. But that would have to be demonstrated. But where - the move form indexical to description - first to third person?
I had it backwards(again), but corrected it while you were replying. I suppose it's hard for me to accept that Smith would count the coins in Jones' pocket! :lol: Anyway...
Awww. I'm sorry.
And yet, the two have very different truth conditions Banno. "The man with ten coins in his pocket will get the job" is true regardless of which man gets the job, so long as he has ten coins in his pocket. Whereas, "Jones will get the job and has ten coins in his pocket" is true, if and only if, Jones gets the job and has ten coins in his pocket.
A change in truth conditions is a change in meaning.
If you wish to return to what you were working on before I entered, please do! I was just stopping in to see if the new approach by Russell was applicable to what I've hinted at here.
Theorem 16 applies this to particular sets of sentences. The structure of the argument is familiar. Then Definition 17 uses this to set up Premise-relative particularity.
That's a bit shorter than the previous versions, but it'll do.
And it's different to a man with ten coins in his pocket.
So again, far more detailed analysis is needed.
I'm traveling, so doing this somewhat sporadically. I'll try to get the rest of the reply to Vranas Objection down before I go further with your interesting aside. But thanks.
In this version, a consequent of some set of particular sentences inherits particularity from that set. So this new version bypasses Vranas' objection that the Barrier says nothing about the particularity of the mixed conditional - it says that they can be derived only if the consequent is particular.
In the discussion of tense fragility, the definition of stuff that was fragile and stuff that was past was the same. The difficulty with this is made evident in the section "Identities between Names" . That a=b should be true in both the past and the future, but appears to be true only in the past.
Russell makes the variations she does in order to formalise what in a natural language we might call an eternal temporal status. The structure she creates can accomodate a wider variety of tensed sentences, including those that survive both past and future switching, (eternal), those that survive past switching ("it will rain"), and those that survive future switching ("it rained").
That enables eternal truths that are not tautologies, such as a=b.