Tag: Fitch

Orthologic and epistemic modals (3): Knowability

Bas van FraassenLeave a comment

[Revised on September 29, 2022] [Remarks on atomless lattices added October 6, 2022]

It is incumbent on any treatment of epistemic modals to show what is wrong with such a statement as “It is raining but it might not be”.  To prove the relevant theorem H&M introduce a special condition, Knowability.  

This term, as well as intuitions about what the condition implies, immediately recalls Fitch’s Paradox of Knowability.  Fitch argued that if every truth is knowable then every truth is known.  That conclusion is startling because we are sure there are many propositions which are true but not known to be true, and we are inclined to think that what ever is the case could be known to be the case.  But the argument is straightforward:  for any proposition A consider A* = (A and it is not known that A).  There is no possible world in which it is true that it is known that A*.  So our ostensible certainty is refuted.

If we look for an analogue in epistemic modals, replacing “it is known that” by “it must be that” we can see immediately that Fitch is evaded in a way that he could not be evaded in classical theories of modality.  For such a statement as “It is raining but it is not the case that it must be raining”, equivalent to ““It is raining but it might not be raining” is never true, not true in any possibility.  So Fitch’s argument does not get off the ground.

But we also see in Holliday and Mandelkern’s theory that the possibility that every truth is known can be realized, and that this can actually play a role in illuminating epistemic modals.

Before making this precise, my thought quickly stated is that  in the classical reading there are indeed many examples of true propositions to the effect that A and that it is not known that A, but not in the reading where “not” is the orthocomplement.  

Propositions and the i-function

Recall here that we are dealing with the complete orthocomplemented lattice of propositions, which is formed by a closure operation on a set of possibilities. The zero and unit element are, respectively, the empty set and the set of all possibilities. For each possibility x there is a possibility i(x) such that “It must be that A” is true exactly if i(x) is in A.  The first condition on the i-function is

            Facticity.  x is a refinement of i(x).  Symbolically:  x ⊑ i(x)

The second condition is 

            Knowability.  for every possibility x there is y such that i(y) ⊑ x

Given Facticity that means that y ⊑ i(y) ⊑ x:  all propositions true at x are true at i(y) and all propositions true there are true at y.  What does it mean for the lattice of propositions for this condition to hold?

The simpler case is that of a complete atomistic lattice: A is an atom iff only the 0 element and A itself imply A.   Recall here the earlier notation for the ‘span’ or ‘support’ of a possibility [x] = {y: y ⊑ x}.  Let’s call a possibility x atomic exactly if z ⊑ x implies that [x] = [z].  So A is an atom iff A = [z] for some atomic possibility z.  

If x is atomic and [x] has more than one member, those members are for all purposes in this theory the same, indiscernible, a harmless redundancy.  In the example of a Euclidean space, where x is a vector, [x] = {y: y = kx for some number k}, and vectors which are multiples of each other belong to all the same subspaces and in physics do not represent different states.  But the redundancy is easily removed too, so without loss of generality, I’ll add here:

            Atom-uniqueness.  If x is atomic then [x] has only one member.

Atomistic and atomless lattices

A lattice is atomistic (or atomic)exactly if each element is the join of a set of atoms.  Specifically, in that case, then each possibility has a refinement which is atomic.

Lemma.  If the lattice of propositions is atomistic then Knowability holds if and only if w = i(w) for each atomic possibility w.

Clearly if each possibility x has an atomic refinement w such that w ⊑ x and w = i(w) then Knowability holds.  Conversely, if Knowability holds, and w is atomic then if y ⊑ i(y) ⊑ w then [y] = [w] = [i(y)], and so by Atom-uniqueness, w = i(w).

What if the lattice of propositions is not atomistic?  Then any element x may have an infinite chain of refinements, and the condition has to be that for at least one element y in that chain, i(y) is also in that chain.  But the same would apply to this y, and so we see an infinitely descending  subchain of the form … y(j) ⊑ i(y(j) ⊑ y(k) ⊑ ….  x.

If the lattice is atomless it is certainly possible for some element y to be such that y = i(y). In that case Knowability holds for all the elements x such that y refines x. But then, nevertheless, there is an element z that refines y, and a further element w such that w  ⊑ i(w)  ⊑ y, and hence also w  ⊑ i(w)  ⊑ x, So, if the lattice is atomless we can conclude that for each element y such that y  ⊑ i(y)  ⊑ x there is a distinct element w that refines y and w  ⊑ i(w)  ⊑ x. There is no bottom to it ….

So now what happens to Fitch’s paradox?  

Suppose the lattice is atomistic, Knowability holds, x is in A, but also in ~□A.  Then there is an atomic refinement w such that w  ⊑ x, and all of the following are true at w: A, ~□A, □A, □~ □A.  That is impossible.  So there is no possibility x in which (A ∩ ~□A) is true.  (Similarly, even if less transparently, if the lattice is not atomistic and Knowability holds.)

And yet of course it is the case that everything that is true in x is known at some other possibility, namely at the atomic possibility w which refines x, since i(w) ⊑ x.  

Notice that argument I just gave does not go through if we just suppose that x is in A and x is not in □A.  For our “not” in the metatheory is not just an orthocomplement, it is classical.  In the case in which x is in A and i(x) is neither in A nor in ~A, which is not ruled out a priori. As pointed out in the previous post, the condition of i-regularity is required even to establish that {x: i(x) is in A} is a proposition.    (And we must note that in H&M’s proof of 4.21 both Knowability and i-regularity are invoked).

Note.  In view of the lemma it would seem that the i-function is not easily identifiable.  In an atomistic lattice each element is the join of the atoms which refine it.  Supposing that z is such that [z] = [x] ⊕ [y], where x and y are atomic so that x = i(x) and y = i(y), there cannot be in general a simple relation between i(z) and the pair i(x) and i(y).   For the value of the i-function must in general be a ‘less informative’ possibility of which its argument is a non-trivial refinement.

NOTE. Reference is to Holliday and Mandelkern article, at https://arxiv.org/abs/2203.02872v3

Fitch, too, had a Paradox: (1) Possibility

Bas van FraassenLeave a comment

Are any truths unknowable? When I first saw this, my impulse certainly was to say No — at least not in principle! We poor finite beings have our limitations, limited intellectual storage capacity and low processing speed, admittedly. But those limitations are contingent, and malleable. So, leaving those aside, surely if anything is true it could be known?

In 1963, just as I was entering graduate school, Frederic Fitch published his ‘Knowability Paradox’ (though he did not call it that!), his proof that

[Fitch] if every truth is knowable then every truth is known.

The argument is simple. Suppose A is true and no one knows that A is true. Then the following statement is true:

(B) A and it is not known that A

Now suppose, per absurdum, that all truths are knowable. Then it is possible that someone knows that B. In that eventuality, a fortiori two things are also known:

(C) It is known that A

(D) It is known that it is not known that A

But if something is known then it is true. So we have to infer from (D) that it is not known that A. This is in contradiction with (C).

That’s it, our supposition that there was a truth, A, which was not known (and assuming that indeed all truths are knowable) has been reduced to absurdity .

There is a reason why magicians never explain how their tricks are done. The reason is that, if they do, you will feel very disappointed — the magic is gone, it was all too simple, there was really nothing to it.

When we see how simple Fitch’s argument really was, we have to feel disappointed, in view of the initial air of deep, profound paradox. But it is also clear from the history of this paradox and the many reactions philosophers had (it’s all in a book published in 2008, see below) that the disappointment is not definitive. We still feel uneasy: is it really impossible to know that B is true?

PLAN: first, here, I will delve into Fitch’s argument to see what it requires by way of logic. Then, in the next post, I will explain the resolution that leaves me feeling at peace with the paradox.

We begin, following Hintikka and many writers since, by regarding “It is known that” as the sort of propositional attitude assimilated to modal logic. That include classical sentential logic, which we will take for granted. Let “It is known that” be abbreviated as K, and let us take it to be governed by two postulates.

First of all, the verb ‘know’ is factive. That means:

[Factivity] For every statement X, if it is known that X then X is true.

If X is not true then you may think or believe that X, but you cannot know that X, falsehoods are not the kind of thing that can be known. And secondly we postulate that to know that two things are true is to know that each of them is true.

  1. KA ⊃ A (‘facticity’)
  2. K(A & B) ⊃ [KA & KB] (‘&-distribution’)

Now we add two suppositions. The first is that there is some true proposition that is not known, and then, to start a Reductio ad Absurdum, that someone knows that.

3. A & ~KA …. supposition, assumed factually true

4. K(A & ~ KA) …… supposition, to start Reductio argument

Then we argue:

5. KA & K ~ KA from 4. by &-distribution

6. ~KA from 5, second conjunct, by facticity

7. ~K(A & ~KA) from supposition 4. and the contradiction between 5. and 6.

Is there anything dubitable in the logic employed? Certainly, and I’ll note all of it here, but it is not going to do away with the force of the argument, as you’ll see.

The first is that 4. is too strong a supposition for this purpose. We are only allowed to suppose that 3. is knowable, not that it is known! But the retort is that if we can get a contradiction from supposing something, X, then it is also the case that a contradiction follows from the very possibility that X, as well. (This point we will revisit below, when we look at the argument through a ‘possible world semantics’ lens.)

Secondly, Reductio is disputed as a logical principle by the Intuitionists, and it is not even valid classically once supervaluations are admitted. But the form of Reductio here employed is just the half of the principle that the Intuitionists do admit! And there is no sign anywhere that supervaluations would have any relevant point of entry.

Thirdly, a bit more telling, perhaps, is that postulate 2., &-distributivity, is not as innocent as it looks. If A implies B then A is logically equivalent to (A & B). So from 2. we get the corollary that anything and everything logically implied by what is known is also known. But we are not perfect logicians, and often do not see the logical implications of what we know.

There is a familiar retort, since this objection has often come up in discussions of knowledge and its logic. I already signaled it at the outset: we say “in principle” or “implicitly”, bracketing the shortcomings due to our mortal finitude.

Now what? There is one more thing we can do: we can look at the semantic analysis of modal logic, in the possible world form, to see how the suppositions are to be imagined as realized. And we should do this, to clear up the objection that supposition 4. was too strong.

A model structure <W, R> consists of a non-empty set W (the ‘worlds’) and a binary relation R (‘knowledge-accessibility’, ‘relative knowledge possibility’). Intuitively, if x and y are worlds then xRy if and only if all that is known in world x is true in world y. Put another way, what is known in x is precisely and only all what is true in all the worlds to which x bears relation R.

We can put this more precisely, by entering some notation: let R(x) be the set {y: xRy}, and let |A| be the set of worlds in which A is true. Then the foregoing amounts to:

KA is true in world x if and only if R(x) is part of |A|

Relation R is reflexive, so that if KA is true in world x then (since xRx) A is true in x. So our postulate 1. is true in all worlds. And &-distributivity holds just because, if R(x) is part of |A & B| then it is part of each of |A| and |B|, by the usual truth-table rules for &. So postulate 2. is also true in all worlds.

We should add that “It is possible that A” is true in a world exactly if A is true in any world at all. (We could add notation for this, but it would not make the argument clearer.)

Now we focus on a particular world, call it α, ‘our world’, ‘the actual world’. We suppose that a certain statement A is true in α (that is, α is a member of |A|), but it is not known to be true in α (that is, A is false in some world in R(α)). It follows that (A & ~KA) is true in α.

Enter now the principle that every truth is knowable. That implies that it is possible that (A & ~KA) is known — which, in this analysis, means that there is some world z such that:

K(A & ~KA) is true in z.

Accordingly also:

KA is true in z

K(~KA) is true in z,

and therefore also, by facticity,

~KA is true in z

and here we have our contradiction.

(A quick note aside: Notice how this world z came into play. It has nothing to do with our world α, nor with the set R(α), it just came in because the knowability principle brought in simple possibility. There is a noticeable disconnect here that I hope to exploit.)

So either we give up the knowability principle, or we give up the idea that some statement A could be true in our world even though we do not know it to be true. To put is more grandiosely, either the very possibility of knowledge is limited or we are omniscient.

This is the end of the scrutiny of the argument. You could be content at this point. Maybe you were taken in by the opening question, but now you know better. How could any but a silly, un-self-critical philosopher ever have believed that there aren’t any unknowable truths?

But I’m not content at this point. I’m going explore what looks to me like a loophole: that there may be a difference between something being possibly known and being possibly actually known.

SOURCES

 “A Logical Analysis of Some Value Concepts”. Frederic B. Fitch. The Journal of Symbolic Logic 28 (2) (Jun., 1963), pp. 135-142.

Joe Salerno,  “Knowability Noir: 1945-1963” in J. Salerno, New Essays on the Knowability Paradox, Oxford 2008.

 

A memory of Frederick Fitch (3) God 1948

Bas van FraassenLeave a comment

Almost twenty years before I met Fitch, he had published what he described as an empirical, a posteriori, proof of the existence of God:

“On God and Immortality”. Philos. and Phenom. Research 8 (1948), 688-693.

In that proof God had turned out to be a theory.

At first blush the argument looks, besides having a conclusion entirely irrelevant to religion, not just simplistic, but dated, and most likely trivial. However, I became quite intrigued with the argument when I started asking myself what unstated assumptions Fitch was bringing to bear. Most especially, what must the language be like in which such an argument could be formulated?

I began to realize that it was a shot fired in the war Curry and Fitch were waging with a conventional wisdom derivative from Tarski and Carnap.

The crucial claim in this paper, argued to be well confirmed, indeed as well-confirmed as any empirical theory could be, is that every class of facts in the universe has an explanation.

Reading his preliminaries we are at once taken aback by how explanation and confirmation are defined. Explanation is equated with implication by a consistent theory, and confirmation with positive instances. But this was a time when the agenda for those notions was set by Hempel, and they were acccepted in the literature just as Fitch presents them.

His one departure from the orthodoxy of the time was this: a fact disconfirms a theory X iff it confirms some theory inconsistent with X. Thus confirmation and disconfirmation may come at the same time! Many facts confirmed both the Ptolemaic and Copernican theory, and thus disconfirmed both of them as well.

The crucial claim looked trivial. It is critique rather than applause for Freudian theory when people say it explains everything. Also, I’m used to reading “theory” in the way, introduced by Tarski in his ‘calculus of systems’, as “set of sentences closed under deduction, i.e. containing all its own consequences”. Then if Z is any set of true sentences the deductive closure of Z is a theory, which implies — and so, by the above definition, explains — all of Z. Trivial! But Fitch has a stronger, more demanding concept of theory, requiring a character familiar in philosophy of science:

“A theory is a proposition (that is, an hypothesis) which involves no mention of any particular thing, event, or moment of time, though propositions mentioning particular things, events, and moments of time may be deducible from it.” 

So read the crucial claim is not trivial, but looks obviously false, as long as explanation equated to implication. For a theory of this sort will not imply the sort of singular facts that scientific theories take as data or boundary conditions. To be charitable, let’s just say that Fitch needs a considerably more sophisticated concept of explanation, to run his argument. A theory of that sort might unify a whole realm of particular facts in a way that could count as explanatory, for example. But the details in Fitch’s deductive steps in the overall argument do not hinge on this, anyway, so, charitably, let’s continue.

Now Fitch formulates a theory that will bear the brunt of his reasoning, the theory S:

“Let us designate by S the proposition that every fact or class of facts has at least one explanation. The proposition S is a theory because it does not mention any particular thing or any particular event or any particular moment of time. The proposition S is also clearly consistent; hence it is an explanation of any fact that it implies, and any such fact is a con- firmation of it.”

S is an empirical theory, and is very well confirmed, he argues. For it is confirmed every time any theory is confirmed, any time when any class of facts is explained. And even if some theory is disconfirmed, which is a disconfirmation of S as well, by definition some other theory was confirmed, and that is an additional confirmation of S. Hence S is as well confirmed as any theory is, was, or ever will be! This conclusion is going to have remarkable consequences.

But stop, for a moment: what is the language like in which this argument is conducted?

S is a theory about theories, for it says that fore every class of facts there is a theory that explains that. To introduce this is not just to assume that theories can be formulated in the language in use, but that they can be talked about in that language. And indeed, S itself purports to be one of those explaining theories, about the class of facts which involve relations between facts and theories, so it is also about itself. This is where we see the contra-Tarski theme of many of his writings: our language is one in which much of the semantics of that very language can be formulated.

Theories are sets of sentences in this language, but here it is presumed possible to talk about theories in general in this language, about implication in the language, about relation to facts, about truth and falsity. So to a significant extent, it is possible in this language to talk about that language itself, about a great deal of what we would classify as belonging to its semantics.

Thus Fitch is relying on something he argued for , notably in “Universal metalanguages for philosophy” and “Self-reference in philosophy”. (I’ll have to devote another post to this!) It is part of the controversy in which Curry and Fitch were on one side, and the only side now generally accepted or even known at all, was Tarski’s.

Well, let’s quickly finish up with the startling consequences of theory S, including the existence of God:

“A corollary of S is that the class of all facts has an explanation.”

Is it possible to talk about the class of all facts, in his language in use, given that he equates facts with propositions, and propositions with what his sentences express? Paradoxes threaten, but they are just the paradoxes that he takes himself to have solved or dissolved or otherwise avoided, in his conception of language.

Let us call an explanation of the class of all facts an ultimate explanation . That there is such an explanation, Fitch argues, “means that the universe as a whole is a system”.

He derives at once that an ultimate theory, by definition consistent, must be true, for if it implied the contradictory of any fact it would still have to imply the fact itself (since it implies all facts), and so would be inconsistent. Furthermore, it is unique:

“there can be only one true explanation of all the facts in the universe, for if there were two different explanations, the fact that one of the explanations was true would be a fact to be accounted for by the other explanation, and so the two explanations would imply each other and hence be equivalent.”

And this ultimate explanation, that we must all agree to call God. For it is the “guarantee that no event in the universe is altogether without connection and relevance to the over-all scheme of things” — what else, then, could it be, other than God?

Of course he sees the obvious objection. Whatever the conception of God might be, it is not a conception of a mere theory! But, Fitch says, this is not a mere theory! For it is true and it is the ultimate explanation of everything.

To his commentator Charles Baylis at the APA, who did not see anything worthy of worship in even such a theory, Fitch retorts:

“I grant that my conception of God as the ultimate system in the universe differs considerably from the anthropomorphic conception of deity held by many people in both primitive and modern times. God seems to me indeed worthy of worship, but my account of the nature of worship cannot easily be summarized here.” (same journal and issue, page 698) 

I kept an eye out for the secret smile that I suspected behind much of what Fitch used to say, in his characteristically gentle, courteous voice. But sometimes it is hard to see.