Gillian Russell: Barriers to entailment

Banno

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.

Hume’s 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. I’ll 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, can’t 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 here’s 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 Hume’s 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. Russell’s approach seems to be more intuitive.

Perhaps some will find this approach interesting.

Banno

So what is Russell doing with the mooted counterexamples?

In the Prior’s 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)

↪Banno

Very interesting. I'll read Russell's paper.

Banno

↪J

Cheers. I'm not expecting this thread to garner much interest, no more than a few comments, but will write on, taking my own advice. Setting out the arguments myself is a useful way to make sure I've understood it.

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.

↪Banno

I've read the first section of Russell's paper. Do you find the putative counter-examples persuasive? They seem fishy to me, but I don't know how to give them a strictly logical refutation. Presumably Russell will go on to do this. In particular -- and this has come up in several previous threads on TPF -- we have the idea that, because p, it is necessarily the case that p. It evidently requires a temporal qualifier, though: It is necessarily the case now that p. P was not necessary until it became actual. Furthermore -- and this is the part I'm really dubious about -- it invokes an idea of necessity that seems at odds with how we think about necessity overall. I'm not saying that "Because p, therefore not not-p" is wrong. That is indeed a kind of necessity. But this "necessity of actuality," to coin a phrase, doesn't address the questions about what constitutes non-temporal, definitional or lawlike necessity. It's more of a weak sister, a glancing acknowledgment that yes, once something happens, it can't unhappen. Do we need to worry about this as a counter-example to the thesis that "you can't get claims about how the world must be from the claims that merely state how it is"?

frank

↪Banno

So she's getting rigorous about the problem of induction?

Banno

↪frank

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.

Banno

↪J

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.

there is a difference between the modal instance and the temporal instance. They are not the same. — Banno

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.

Banno

"Because p, it is necessarily the case that p", expanded, means "It is necessarily the case now that p". — J

I don't see why.

↪Banno

Maybe I'm not explaining it well. I guess it hinges on two different senses of "necessary." If I say "The squirrel is in the tree, therefore it must be the case that the squirrel is in the tree," surely that's wrong? It happens to be the case, and now that it is the case, it can't not be the case, but we want necessity to capture something else, don't we? Something more like "The squirrel is in the tree, and it is, and was, necessary that the squirrel be in the tree" -- which I take to be the same idea as "it must be the case that . . ." Neither of these formulations are true, or so it seems to me. That's all I meant.

Banno

↪J

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.

Banno

So to some detail on Section Two. Russell cites "you can’t get a universal sentence from particular ones" as the paradigmatic case, and sets about defining "Universal" and "Particular" with the counterexamples in mind. The question is, can these two terms be defined in such a way that the counterexamples are shown wanting int the way described above?

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.

Banno

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, Γ. 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 Γ.

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.

How's that? I'll look for a good analogue as well. — Banno

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:

Banno

↪J

It'd been in the greenhouse for a few years, with some success, but it proved ungainly and unmanageable, so I've found it a place on a north wall. I doubt it will do as well, it's in a bit of a wind trap.

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.

↪Banno

Ah, OK, much clearer, and now I understand why the 1st diagram seemed counter-intuitive. I hadn't understood that only the single, designated F was a. So of course the addition of something that is ¬F can't change anything with respect to a.

Carry on.

Banno

One analogue might be knitting. John starts with green wool, so the first row is green, and so is the second, and the third. So far all the rows are green. At some point John may decide to change wool to yellow, and continue adding rows. Suppose we don't know what John decides to do.

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.

Banno

A review of the book in Philosophy Now:

Barriers to Entailment by Gillian Russell

This book’s proof of the Strong General Barrier Theorem is a landmark achievement in twenty-first century philosophy. Not since Ludwig Wittgenstein’s Tractatus (1921) has such an important contribution been made to philosophical logic. — Christopher John Searle

Banno

Some features of these definitions are worth remarking on — 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 ∃₁, 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 ∃₁ 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.

The knitting analogy is a bit clunky, — Banno

No, I found it helpful.

Prior's Dilemma — 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?

Banno

↪J

Good question. Thanks for following along.

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.

↪Banno

I understand Russell's and Pryor's interpretations. I'm still not clear on how something can be neither particular nor universal. Also, why "not universal" isn't the same as "particular" -- this is perhaps just another way of phrasing the first unclarity. "Fa v ∀xGx is not universal" . . . and yet, as you show, {Fa v ∀xGx, ¬Fa} is not particular. This is hard to understand. It makes sense using the "rows" illustration, but not conceptually or intuitively; it seems like a paradox. Probably I should wait for your "more on this later."

Banno

↪J

The intuition sometimes has to give way to the logic; but with practice the intuition can change to match the logic.

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.

Banno

It might not be obvious where this break of symmetry originates. It is built in.

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".

Banno

The article now turns to applying this schema to other examples, starting with temporal logic - the grammar of time.

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.

Banno

Now some considerations of the model - how the logic is to be interpreted.

There is a sequence of times, t₁, t₂, t₃... forming a set T, with one of them nominated as "now", n. There is a binary relation "<", understood as "t₁<t₂" means t₁ occurred prior to t₂.

"<" is

Transitive: if t₁<t₂ and t₂<t₃ then t₁<t₃.
Dense: there is a time between any other two times.
Extendable: Any time has a time before it, and a time after it.
Total: given two times, one is before the other.

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 t₁, and 0 (false) to p at t₂, and so on. I(p, t) = 0 means 'p is false at time t.

Banno

In knitting terms, the set up hasn't changed much. Each row is a time t, and has a colour, p, and the row you are presently knitting is row n, but the garment - a scarf? - goes forever in both directions.

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.

↪Banno

This is particularly helpful, as is the whole idea of thinking of it in terms of an asymmetry. I think that's where my "intuition" wants to go astray; perhaps we naturally conceptualize in symmetries until shown otherwise?

↪Banno

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.

Banno

↪J

The barriers to entailment here concern logical implication. →, ⊃, ⊨, ⊢ and that sort of thing.

We can make use of Bayesian methods.

But tomorrow might well resemble today in ways other than the sun rising.

Banno

Russell moves from the specific cases so far examined to the general case.

Start with a formal language L, containing a sentence Φ. 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.

Gillian Russell: Barriers to entailment

Welcome to The Philosophy Forum!

Categories

More Discussions