Tag: modality

Wilfrid Sellars’ apocalyptic vision

Bas van Fraassen1 Comment

1.   Sellars’ fundamental project and apocalyptic vision 1

2.   Sellars’ Dialectic: Two Stages    3

3.   The Thing-Kind Framework, for the Representation Of Nature    4

4.   Sellars’ turn to pragmatics 6

5.   Sellars:  induction as rule formation   7

6.   Analyticity and Inductive Risk    8

Wilfrid Sellars was our first, and perhaps so far only, master of dialectical writing since Bradley.  His many students and admirers have written along lines learned from his writings, each in their own way.  I’ll do so here, without apology or attempt to preempt objections, in the hope that this can have a place in the continuum of ways of understanding Sellars.

I am not a disciple, but I took his courses and seminars in Pittsburgh in the sixties and have never ceased to be fascinated by him.

1.   Sellars’ fundamental project and apocalyptic vision

Sellars’ ultimate concern consists of a vision and a project.  Behind all he writes there is his vision of the end of days, of the perfection of science, at the end of the Peircean long run.  And his project is to reconcile that vision with the way we in fact conceive of ourselves.

When Peirce writes about truth he requires a concordance of a statement “with the ideal limit towards which endless investigation would tend to bring scientific belief”. Sellars is acutely conscious of how language changes in tandem with changes in the accepted scientific theories.  Our statements will not be statements in the language of a future, let alone final, science.  Indeed (as we will see better below) the most important aspect of an ideally improving science is its endlessly improving language.  But as with Peirce, we can best portray the view of how it all should be, by thinking about that ideal limit, however unrealistic.

No Reduction. Sellars was wont to say that it was he, not Feyerabend, who was the revolutionary. [Note 1] Unlike Feyerabend, Sellars would say, he was not a gradualist.  Our language in use is deficient, and it does change in response to new developments in the sciences.  But we cannot replace or reduce our language in a piecemeal fashion.  There is no reduction of language about thought and feeling to language about the body, for example. A completely adequate language could not be the end result of a series of reductions of common sense notions, not even with common sense evolving in the process.

One of my colleagues in Princeton, Mark Johnston, once remarked “The great lesson of 20th century philosophy was that nothing reduces to anything else!”  On this point Sellars was prophetic, he saw that from the beginning.

Final Descriptive Language. But the ideal end of our rational progress will nevertheless be that our descriptive language is the factual language of the final science of everything.  This language will be extensional, and entirely devoid of modalities, or of any elements of what Strawson had called person language.

How can these two seemingly conflicting views be placed in harmony?

The Twin ‘Shall’ Language. This descriptive language will not be all there is to rational discourse!  Indeed, we could not do with only that, neither in science nor in life.  The ideal descriptive language has a twin language, so to speak, for discourse in which we express our individual and communal intentions.  When we say “The desert will bloom” we are making a prediction, in factual descriptive language.  But something different happens when we say “The desert shall bloom!” (with our without exclamation mark).  With this form of words we express our intention to act, we voice a decision.  This twin language will and must always remain integral to scientific practice as well. [Note 2] 

What About Our  Current Discourse? Meanwhile, living in these pre-apocalyptic times, Sellars’ project is to do justice to our actual discourse about the universe and ourselves.  Our current everyday language, both practical and scientific, is shot through and through with modalities, it is intensional, even hyper-intensional, and it does not admit of a faithful translation into extensional descriptive language.  

His project is to explain our practice in a way that will not make it a counterexample to his vision. It would be diminishing, and grossly unfair, to just mark this with Berkeley’s “think with the learned and speak with the vulgar” [Note 3].  Sellars’ project is on grand scale:  we are to understand ourselves and our own place in the world by seeing it from the perspective of that ideal end of days, from the imagined perspective of that envisioned ideal end.  And step one will be to appreciate the radical disparity, and irreconcilability, of the envisioned us of then with the actual us of now.

Two ‘Mistakes’. There are two great mistakes which Sellars wants to persuade us not to make.  The first is to think that, because we will always need the “shall” language to express our intentions, the ultimate description of what we do will inevitably involve such concepts as intention.  No, in Sellars’ vision, the description of what we are doing, when we express intentions, will in the ideal end be purely and entirely ‘physicalist’, just like the description of anything else.  The second mistake would be to infer that the factual descriptive language will need to have in it a translation or reduction of the intensional discourse of our current practical and scientific life.  No, it will replace that discourse entirely.

Our versus Their description of ‘Shall’.  Our description today of what will happen then is that those persons at the end of time will be expressing intentions when they engage in the “shall” language.  Yes, indeed (speaking today’s language, I say) that is what they will be doing.  But that is not what those persons will say when they describe what they are doing.  “Intend” is a word in person-language, and they will not have any such word available for description.  We should also understand that what follows the “shall”, when they speak, will not have the content of what we would say today.  They will never say something that would translate into our “Romeo shall marry Juliet”, only things like “Romeo shall at all times stay in close spatial proximity to Juliet”, or the like.

Rorty And Churchland. Some, in those days, shared Sellars’ view.  It was the young Richard Rorty who defended this  position as eliminative materialism (1970) and the young Paul Churchland who sought to make it palatable:

“It is important for us to appreciate, if only dimly, the extent of the perceptual transformation here envisaged. These people do not sit on the beach and listen to the steady roar of the pounding surf. They sit on the beach and listen to the aperiodic atmospheric compression waves produced as the coherent energy of the ocean waves is audibly redistributed in the chaotic turbulence of the shallows.” (Churchland 1979: 29)

Specifically, Churchland argued that in person-language there is a sentence structure, used especially for the description of thought, volition, and feeling, that will disappear.  It is the pattern “So and so X-es that p”, a pattern that is a hallmark of intensional language.  His argument for this is that the basic sentence form in modern scientific theories is “Quantity Q, pertaining to So-and-so, has value x” which does not have the form to express a propositional attitude. 

That is not a strong argument. First, the sentence form is not a great obstacle in itself, since propositions can be represented by two-valued quantities.  In any case we cannot remove from science such statements as: the probability that a given measurement will have outcome y equals x.  Even if this can be put in the different form, for example, ‘the quantity p(m, y) = x’ it is still the case that probability is a modality. But even if Churchland’s arguments are not strong, his conclusion depicts Sellars’ vision of the end of days.

2.   Sellars’ Dialectic: Two Stages

I am not endorsing Sellars’ vision, and so, not his project.  

But there is something interesting, tantalizing even, about how Sellars went about that project, of ‘doing justice’ to what he thought would ultimately be eliminated.  

In philosophy of science this included a certain take on causal discourse, natural and physical modalities, laws of nature, and the counterfactuals that laws support.  This comes in two stages: the first stage is a realist account and in the second stage the first is aufgehoben in a way that Sellars saw as the completion of Kant’s ‘Copernican revolution’.   The realist account is presented in modal discourse, and the retrenchment in the second stage shows how realism about modalities disappears upon analysis. 

Philosophy of Science in a new key.  Sellars wrote “Counterfactuals, Dispositions, and the Causal Modalities” in a reaction against the puzzles about counterfactual conditionals that had been made salient by Nelson Goodman (1947) and Roderick Chisholm (1953).  The topic was not new, they were related to issues that Reichenbach and Carnap had raised in philosophy of science.  Sellars wanted to show that those puzzles could be dissolved or dismissed as far as science was concerned, though they pointed at more important, deeper issues there.

In a different post I have presented an analysis of Sellars’ theory of conditionals in that paper.  So I will here mention only the minimum; what is important here is how the second stage, the ‘Copernican revolution’ is implemented.  

A conditional is an “if … then” statement.  Sellars favored the old-fashioned form “If this match be struck …”, where “be” is meant to be neutral between “is” and “were”.  Traditionally, conditionals are closely linked to arguments, as exemplified in the Great Law of Implication:

A, B implies Q if and only if A implies (B → Q)

But now it had become clear that in natural language, conditionals do not behave accordingly.  

Sellars echoed the typical example: 

the assertion that if the match be struck it will light does not commit us to the assertion that if it be struck and made wet, it will light

Symbolize the disputed argument as “If S→L then (S & W) → L”.   If the Great Law of Implication held for this sort of example, this argument would be valid:

Suppose S → L, so then S implies L.  But if S implies L, so does (S & W).  Therefore (S & W) implies L, and thus, (S & W) → L.

Thing-Kind Language. At first sight, Sellars grants that these puzzles about matches and light bulbs affect the language of science in practice.  They share their form with statements of importance there.  They belong to a class of conditionals that appear within “the conceptual framework in terms of which we speak of what things do when actedupon in certain ways in certain kinds of circumstance” (1958: 225).  

The basic pattern he describes as follows:

Suppose we have reason to believe that

𝜙-ing Ks (in circumstances C) causes them to 𝜓

(where K is a kind of thing – e .g., match). Then we have reason to believe of a particular thing of kind K, call it x, which is in C, that 

x, would 𝜓, if it were 𝜙-ed.

And if it were 𝜙-ed and did 𝜓, and we were asked “Why did it 𝜓?” we would answer, “Because it was 𝜙-ed”; and if we were then asked, “Why did it 𝜓 when 𝜙-ed?” we would answer “Because it is a K.” If it were then pointed out that Ks don’t always 𝜓 when 𝜙-ed, we should counter with “They do if they are in C, as this one was.” (ibid., 248.)

But the crucial point is that the antecedent is an input (action or interaction) statement, the consequent an outputstatement, and neither input nor output statements describe circumstances (standing conditions).  As long as we are dealing realistically with conditionals as they appear naturally in this ‘thing-kind’ discourse, we do not run into the logical puzzles Chisholm and Goodman raised.

Fine, but this thing-kind discourse is intensional.  It has the modal word “causes” and allows us to make assertions not just to state the facts but to say what would have been the case if other possibilities had been realized.  This is the discourse of realism about modalities in nature.  

Looking at it from the perspective of the ideal end of days, where the language of science is extensional and that is the only form of description, how is this thing-kind discourse to be understood?

3.   The Thing-Kind Framework, for the Representation Of Nature

This is Sellars’ version of natural philosophy.  It is at the same time the counterpart in Sellars of Kant’s Inaugural Dissertation, before his Critical Period.  Like Kant, Sellars means to go on from that stage in philosophy to a point of view where it is not truths about nature but truths about our representation of nature that are at stake. Before that move can have some content, however, we have to have a clear idea of the earlier ‘realist’ stage.

General Form.  Representation of a thing-kind K.  There is a logical space (phase space, state-space) H, and family Q of quantities that characterize the states.  Conditions (boundary conditions, circumstances) are something separate.  There is a family G of possible actions. If 𝜋 is an action, it will (under given conditions) send the current state into a new state, or into a set of states with different probabilities.  Hence there is a set R of transition functions:  for each action 𝜋 a there is a function that takes any state (plus condition), into a state or set of states, possibly with an assigned probability.

This formulation allows for an indeterministic thing-kind.  For the deterministic kind, the transition functions are from state (+ condition) to state.  

Note that this representation is from the God’s eye point of view.  We may know a great deal about thing-kind K without having all the details about the transition functions.  

In his earlier paper, “Concepts as involving laws and inconceivable without them” (1948) Sellars goes further in his explication of this form of representation of nature.

A given thing x of kind K will have a history, which is a trajectory in its state-space.  What is characteristic of kind K is not only that specific state-space, but a selection from the set of possible histories:  the family of histories such a thing can possibly have, the thing’s possible histories. 

In speaking of a family of possible worlds, what are we to understand by a  “world”?  Let us begin with the following: A world is a spatio-temporal structure of atomic states of affairs which exhibits uniformities of the sort we have in mind when we speak of the laws of nature. (1948: 293)

This passage he immediately follows with the admonition to abandon the term “world”, and to speak of possible histories instead:

Our basic framework is thus a family of possible histories, one of which is the actual … history. (ibid.)

Turning then to the language in which systems of kind K are described, Sellars enters the assumption that it makes sense to speak of truth and falsity with respect to any given possible history.  And so it will make sense also to say of a given statement about things x of kind K that “x will 𝜓, if it is 𝜙-ed” is true in all possible histories.

Laws.  And this statement, though it can be a universal truth about things of kind K, and is not just about what happens in the actual world, is still not a logical truth.  The reason it is not, is precisely that kind K is characterized by a restricted family of possible histories:  the histories which alone are possible for things of kind K.  And this is the sense that Sellars can give to the idea of necessity in nature or of a law of nature:

A natural law is a universal proposition, implicative in form, which holds of all histories of a family of possible histories; as such it is distinguished from ‘accidental’ formal implications which hold of one or more of possible histories of the family, but do not hold of all. (1948: 309, italics in original)  

These laws in conditional form are not the subjunctive or counterfactual conditionals discussed at the outset.  They are universal material conditionals about the possible histories.  But what holds in all possible cases is precisely what we call necessary.  So we are in the land of C. I. Lewis’ strict conditional, where “if A then B” can be read as “Necessarily, either not A or B”.  However, though not themselves in the subjunctive mood, these laws form the background structure which provides warrant for such assertions as “x, of kind K, will (would) 𝜓 if it be 𝜙-ed”.

4.   Sellars’ turn to pragmatics

it is in pure pragmatics … that the lingering ghost of naïve realism (as a philosophical perspective) is finally exorcized, and Kant’s Copernican revolution receives its non-psychologistic fruition. (Pure Pragmatics and Epistemology page 185)

Sellars makes a sort-of-Kantian move, by which he can shift, apparently effortlessly, from the representation of nature to pragmatics, the representation of our discourse in terms of use, by us, in practical situations.  That is a shift in the philosophical discussion from ontology to methodology, to use a phrase he assimilated from Carnap.  It was a theme for Sellars from the beginning:  it was in his 1947 paper “Pure Pragmatics and Epistemology”, Sellars’ first published paper, that he described this sort of move as the way to complete Kant’s ‘Copernican revolution’.

Looking back at the main passage about conditionals that I quoted above, notice that Sellars is not concerned there with questions about the truth or falsity of the conditionals.  Instead it is all about what we would, or would not assert, what we would answer if we were asked certain questions in given circumstances.  And those circumstances are described in terms of what we would have reasons to believe. 

Thus presented, this subject of counterfactual conditionals, is placed within practical reasoning, the subject that in the end will be addressed by the “Shall” language rather than the descriptive language.  And indeed, Sellars indicates, with a clearly waved red flag, so to speak, that this will be his line.  For he proposes a resolution of the traditional controversies in which 

‘the core truth of Hume’s philosophy of causation” is combined with the “ungrudging recognition of those features of causal discourse as a mode of rational discourse on which the ‘metaphysical rationalists’ laid such stress but also mis-assimilated to describing (1958: 285, my italics).  

The use of the descriptive language is to state facts, while the use of the twin “Shall” language is to express intentions.  The grammar  functions as an indirect conveyance that the conditional 

“is accepted on inductive grounds … The statement, in short, sticks its neck out. It is this neck-sticking-out-ness . . . which finds its expression in the subjunctive mood.” (1958: 268-69.). 

Acceptance is a pragmatic, not a semantic concept.  To stick your neck out, that means to willingly and intentionally do something that is risky, to place yourself at risk.  Induction, accepting something on inductive grounds, is not an inference but a decision.

5.   Sellars:  induction as rule formation

So ”the core truth of Hume’s philosophy of causation” will appear when we appreciate that causal discourse and its associated counterfactual conditionals form a mode of rational discourse that is mis-assimilated to describing.  This discourse involves an ineliminable pragmatic element into the understanding of that mode of discourse.  

In recent literature on conditionals the line pursued by Justin Khoo is that they encode inferential dispositions.  That may be a way to say what Sellars meant.   The term “disposition” is then not apt, however. Given Sellars’ vision of the combination of factual descriptions solely with the “shall” language, in which intentions are expressed, we should say that we form, not inferential dispositions, but inferential commitments.

Laws Between Mere Fact And Analytic Truth.  What Sellars calls the rationalist understanding of this discourse, is that it is a description of modalities, and specifically of entailment, in nature.  This is much clearer in that earlier paper, “Concepts as involving laws and inconceivable without them” (1948).  He begins with a dilemma posed by C. I. Lewis.  

Lewis’ Dilemma.  Consider a putative law like that lead melts at 621oF.  We cannot construe its being a law as simply a universal material conditional, saying only that all actual or real lead samples which are heated to at least 621oF, and only those, melt.  But, says Lewis, the logically stronger statement that all possible, or thinkable, lead samples are thus, can only be true if it is analytic.  And the law that lead melts at 621oF is not analytic, it is not a logical truth.

Today one might well retort that statements about the possible can be true without being analytic, and some might even say that it is a contingent matter which possible worlds there are at all.  That would have surprised C. I. Lewis, and Sellars as well.  Both hold that what is true depends, and depends only, on what there is in the actual, real world.  

C. I.  Lewis’ solution is that there must therefore be a real modal connection in nature, an implication which is neither material nor logical.  And we arrive at knowledge of these connections by some form of induction (more or less what we would now call inference to the best explanation).

From Induction To Conceptual Truth.  Sellars counters this with a different understanding of induction, or ampliative inference in general.  In fact, these terms are already prejudicial, for they come with the picture of something that is like, even if lacking certainty, logical inference, from given information to concluding statements.  On Sellars’ view the process of going beyond the data is nothing like that, it is not that we arrive at new statements of fact but rather that we talk ourselves into new rules to reason by.  

(Nota bene: this turn of phrase, “talking ourselves into”, signals at once that there is no question of recipes or rules that could constitute an inductive logic, in the sense of a method leading from premises to rationally compelled conclusions.  Talking ourselves into something involves taking up options which have alternatives that we need to choose between, and hence, none of which are rationally compelled.)

Briefly and crudely put, we say of a piece of lead that it would melt if it were heated to 621oF, to convey that we believe the factual truth that all pieces of lead will either melt or not be at a temperature below that, and that we SHALL follow a corresponding default inference rule, ready to apply to any lead sample we may ever come across.  Our assertion “Lead melts at 621oF” has a dual character, it is both description and expression.  

It is precisely here that scientific practice involves the twin language of “shall” versus “will”, the language used to express intentions and decisions.  That we are going to follow such a default rule is a matter of decision, it is introduced by the emphatic “I shall …” or “we shall”.

Sellars’ title conveys the main point: the concepts that we have and apply in our descriptive language cannot be understood separately from these rules by which we reason with them.  Here we have come to an aspect of his view that will not have been apparent at the beginning.  The descriptive language, even if it is the language of the final science at the end of days, cannot stand alone.  For to be a language is not simply to be a symbol system, it is something defined by its use, and this use can only be by beings who cannot use it without having something more involved, something more that is not served by description.

To repeat, “inferential disposition” is not the apt term here, it must be “intention” or “commitment”.  For when we talk ourselves into adopting such a default rule, we are amending our policies for engagement with whatever will come our way, in the full awareness that in doing so we are sticking our neck out, accepting the inductive risk.

Now, however, Sellars would seem definitely to be in a quandary.  The input to factual information processing must be factual information, it must be a proposition.  But our modalized pronouncements, such as those subjunctive conditionals about vases and lead samples, while apparently carrying factual descriptive information, also encode inferential commitments.  And the subjunctive mood signals those commitments, with our consciously taken inductive risk.  

So when the input involves conditionals, what’s the story?

6.   Analyticity and Inductive Risk

The first part of the answer is already in the title “Concepts as involving laws and inconceivable without them.”  And the second part is the connection between the theory of concepts and laws there presented, on the one hand, and the pragmatics of causal and modal discourse on the other.  Sellars’ spelling out of this answer is long, complex, and subject to different readings, as well as to finicky objections.[Note 4]  It is, I think, one of the ways we can glimpse Wittgenstein in Sellars:

It is clear that our empirical propositions do not all have the same status, since one can lay down such a proposition and turn it from an empirical proposition into a norm of description.  (Wittgenstein 1969, para. 167)

I will explain it with an example.  Why doesn’t water burn?  Methyl alcohol looks like water, but it burns.  So even that water does not burn may have been a general belief formed ampliatively, ‘by induction’.  

That process of ampliative belief forming is one of talking ourselves into new rules to reason by (cf. 1958: 287-8, 291, 293): new inferential commitments.  But there is a second stage, when these concepts are revised: those rules now change into criteria that constitute the concepts in question.  The rule turns into a law, involved in the concept of water, and water becomes inconceivable without it.  The concepts evolved into our current concept of burning as oxidationand of water as H2O, a fully oxidized substance.  At that point it is no longer an empirical generality: it is now true ex vi terminorum that water does not burn.  

Similarly it is today no longer an empirical claim that lead melts when heated to 621oF, it has become an analytic statement.  After a certain time not long ago, anything that turns out not to melt when heated to 621oF is something that does not instantiate the (evolved) concept of lead.

Where Is The Inductive Risk Then?  Analytic statements are not risky.  So, what has happened to the element of inductive risk, the way or extent to which we stuck our necks out to begin?  That has not disappeared, it has been transplanted to two new locations.  (1) We have no guarantee that our conceptual framework will not break down in the face of new and previously unimagined phenomena.  (2) There is equally ineliminable risk in the judgement that this or that liquid sample is water, that this concrete thing instantiates the concept of water.  We have tests for water, and we have the general belief that any bit of liquid which passes those tests is water.  That general belief is no foundation for the specific instance judgment, for all we have are the tests so far.  We are lucky, that our environment has so many stable regularities, and passing a test once is typically enough – but that is not a logical truth.  That general empirical assertion is equally at the mercy of nature’s continuing compliance.  

Vindication And Disappointed Expectations.  Taking an inductive risk is not something that can ultimately be justified, it can only be vindicated in fulfilled expectations or, of course, fall prey to disappointed expectations.  (Hence the title of a later article, “Induction as vindication”.)  We are entirely in the realm of practical reasoning, but as we see, that practical reasoning leads to changes in the mode of description, in the evolving descriptive language.

NOTES

Note 1. “Feyerabend arrives at the ontological truth that the world is in principle what scientific theory says it is, he does so by chopping the structure of science with a cleaver rather than carving it at its conceptual and methodological joints. As I see I it, only someone who is unaware of the subtle interdependence of the various dimensions of the framework of empirical knowledge, would speak cavalierly of the piecemeal replacement of part by scientifically better part.”  (Sellars 1965: 187)

Note 2. This had in effect been argued by Hans Reichenbach in 1952.  See Reichenbach (1959: 198) and the clarification by Maria Reichenbach (ibid.: 193).  Just as a side note, these multiple functions of language, often not distinguished syntactically in ordinary language, had been a theme in the Significs movement, but I don’t know of any links to connect Significs with either Reichenbach or Sellars.

Note 3. Berkeley’s Works: Principles (W2, 51) and Alciphron (W3, I, 12, p. 53).

Note 4. As always with Sellars, there is a lot more to be said about the idea of induction.  Here I have been attending to his dispute with C.I. Lewis in (Sellars 1948) as well as the 1958 article.  The complete answer should have been in his later “Induction as vindication” (1964), and there is a lot there, but it is quite difficult to disentangle the strands in its tangled skein.

BIBLIOGRAPHY

Churchland, Paul [1979]. Scientific Realism and the Plasticity of Mind.   Cambridge, UK. 

Rorty, Richard (1970). “In Defence of Eliminative Materialism” in The Review of Metaphysics XXIV. Reprinted Rosenthal, D.M. (ed.) Materialism and the Mind-Body Problem (Englewood Cliffs: Prentice Hall 1971).

Sellars, Wilfrid (1947) “Pure Pragmatics and Epistemology”.  Philosophy of Science 14: 181-202. 

Sellars, Wilfrid (1948) “Concepts as involving laws and inconceivable without them”. Philosophy of Science 15: 287-315.

Sellars, Wilfrid (1958) “Counterfactuals, Dispositions, and the Causal Modalities”. Minnesota Studies in the Philosophy of Science II: 225-308. Open Access at https://cla.umn.edu/mcps/publications/minnesota-studies-philosophy-science

Sellars, Wilfrid (1964) “Induction as vindication”. Philosophy of Science 31: 197-231.

Sellars, Wilfrid (1965) “Scientific Realism or Irenic Instrumentalism”.  Pp. 171-204 in R. S. Cohen and M. W. Wartofsky (eds.) Boston Studies in the Philosophy of Science, Volume Two.  New York: Humanities Press. 

Van Fraassen, Bas C. (2023) “Wilfrid Sellars’ theory of conditionals” posted on  https://basvanfraassensblog.home.blog/

Wittgenstein, Ludwig (1969) On Certainty. New York: Harper.

Orthologic and epistemic modals

Bas van Fraassen1 Comment

A brief reflection on a recent paper, “The orthologic of epistemic modals” by Wesley Holliday and Matthew Mandelkern 

  1. The motivating puzzle p. 1
  2. Inspiration from quantum logic p. 1
  3. Propositions and truth p.2
  4. Could quantum logic be the logic of natural discourse? p. 3
  5. Why this, so far, is not enough for epistemic modals p. 4
  6. Pure states and mixed states p. 4
  7. An open question p. 5

1.   The motivating puzzle

Here is a puzzle for you:

(Puzzle *) We, Able and Baker, A and B for short, are two propositions.  Baker does not imply the negation of Able.  Yet our conjunction is a self-contradiction.  Who are we?

In any first or even second year logic course the right answer will be “you do not exist at all!”  For if Baker does not imply the negation of Able then their conjunction could be true.

But the literature on epistemic modals furnishes examples, to wit:

“It is raining, but it might not be” cannot be true.  Yet, “it might not be raining” does not imply “It is not raining”.

Such examples do rest on assumptions that may be challenged – for example, the assumption that the quoted sentences must all be true or false.  But let that go.  The interesting question is how such a logical situation as depicted in (Puzzle *) could be represented.  

2.   Inspiration from quantum logic

That sort of situation was studied in quantum logic, with its geometric models, where the propositions are represented by the subspaces.  

A quantum mechanics model is built on a finite-dimensional or separable Hilbert space.  In quantum logic the special properties of the infinite-dimensional, separable space do not play a role till quite late in the game. What matters is mainly that there is a well-defined orthogonality relation on this space.  So it suffices, most of the time, to think just about a finite-dimensional Hilbert space (that is, a finite-dimensional inner product vector space, aka a Euclidean space).

 For illustration think just of the ordinary 3-space of high school geometry but presented as a vector space.  Draw the X, Y, Z axes as straight lines perpendicular to each other.  The origin is their intersection.  A vector is a straight line segment starting at the origin and ending at a point t, its tip; we identify this vector by its tip.  The null vector 0 is the one with zero length.  Vectors are orthogonal iff they are perpendicular, that is, the angle between them is a right angle.

In the diagram, the vectors drawn along the axes have tips (3, 0, 0), (0,5,0), and (0,0,2).  The vector with tip (3, 5, 2) is not orthogonal to any of those.

If A is any set of vectors, its orthocomplement ~A is the set of vectors that are orthogonal to every vector in A.  The subspaces are precisely the sets A such that A = ~~A.  In this diagram the subspaces are the straight lines through the origin, and the planes through the origin, and of course the whole space.  So the orthocomplement of the X-axis is the YZ plane.  The orthocomplement of the solid arrow, with tip (3, 5, 2) is thus a plane,  the one to which it is perpendicular.

About (Puzzle *).  Our imaginative, intuitive picture of a 3-space provides an immediate illustration to solve (Puzzle *).  In quantum logic, the propositions are the subspaces of a Hilbert space.  Just let A and B be two lines through the origin that are not orthogonal to each other.  Their conjunction (intersection) is {0}, the ‘impossible state’, the contradiction. But neither is in the other’s orthocomplement.  In that sense they are compatible.

3.   Propositions and truth

That the propositions are taken to be the subspaces has a rationale, introduced by von Neumann, back in the 1930s.  The vectors represent physical states.  Each subspace can be described as the set of states in which a certain quantity has a particular value with certainty.  (That means: if that quantity is measured in that state, the outcome is that value, with probability 1.)  

Von Neumann introduced the additional interpretation that this quantity has that value if and only if the outcome of a measurement will show that value with certainty.  This became orthodoxy: here truth coincides with relevant probability = 1. 

Given this gloss, we have: 

subspace A is true in (the state represented by) vector v if and only if v is in A.  

We note here that if vector u = kv  (in our illustration, that they lie on the same straight line through the origin) then they belong to all the same subspaces. As far as truth is concerned, they are distinct but indiscernible.  (For the textbook emphasis on unit vectors see note 1.)

Since the subspaces are the closed sets for the closure operation ~~ (S = the ortho complement of the orthocomplement of S), they form a complete lattice (note 2).   

The self-contradictory proposition contains only the null-vector 0 (standardly called the origin), the one with zero length, which we count as orthogonal to all other vectors.  Conjunction (meet) is represented by intersection.  

Disjunction (join) is special.  If X is a set of vectors, let [X] be the least subspace that contains X.    The join of subspaces S and S’, denoted (S ⊕ S’), is [S ∪ S’].  It is a theorem that [S ∪ ~S] is the whole space.  That means specifically that there is an orthonormal basis for the whole space which divides neatly into a basis for S and a basis for ~S.  Thus every vector is the sum of a vector in S and a vector in ~S (one of these can be 0 of course).

One consequence is of course that, in traditional terms, the Law of Excluded Middle holds, but the Law of Bivalence fails.  For v may be in A ⊕ B while not being either in A or in B.

The term “orthologic” refers to any logic which applies to a language in which the propositions form an an orthocomplemented lattice. So orthologic is a generalization of quantum logic.

4.    Could quantum logic be the logic of natural discourse?

The idea, once advanced by Hilary Putnam, that the logic of natural language is quantum logic, was never very welcome, if only because learning quantum logic seemed just too hefty a price to pay.  

But the price need not be so high if most of our discourse remains on the level or ‘ordinary’ empirical propositions.  We can model that realm of discourse by specifying a sufficiently large Boolean sublattice of the lattice of subspaces.

For a non-trivial orthocomplemented lattice, such as the lattice of subspaces of a Hilbert space, has clearly identifiable Boolean sublattices.  Suppose for example that the empirical situations that we can discern have only familiar classical logical relations.  That means that, in effect, all the statements we make are, precise or vague, attributions to mutually compatible quantities (equivalently, there is a single maximal observable Q such that all humanly discernible quantities are functions of Q).  

Then the logic of our ‘normal’ discourse, leaving aside such subtleties as epistemic modals,  is classical, even if it is, only a (presumably large) fragment of natural language.  For the corresponding sublattice is Boolean.

5.   Why this, so far, is not enough for epistemic modals

Quantum states are variously taken to be physical states or information states. The paper by Holliday and Mandelkern (henceforth H&M) deals with information, and instead of “states” they say “possibilities” (note 3).  Crucial to their theory is the relation of refinement:

x is a refinement of y exactly if, for all propositions A, if y is in A then x is in A.

I will use x, y, z for possibilities, which in our case will be quantum states ( those, we’ll see below, are not limited to vectors).

If we do take states to be to be vectors and propositions to be subspaces in a vector space, then the refinement relation is trivial.  For if u is in every subspace that contains t then it is in [t], the least subspace to which t belongs (intuitively the line through the origin on which t lies) and that would then be the least subspace to which u belongs as well.  So then refinement is the equivalence relation:  u and t belong to the same subspaces.  As far as what they represent, whether it is a physical state or an information state, there is no difference between them.  They are distinct but indiscernible.  Hence the refinement relation restricted to vectors is trivial.

But we can go a step further with Holliday and Mandelkern by turning to a slightly more advanced quantum mechanics formalism.       

6.   Pure states and mixed states

When quantum states are interpreted as information states, the uncertainty relations come into play, and maximal possible information is no longer classically complete information.  Vectors represent pure states, and thought of in terms of information they are maximal, they are as complete as can be.  But it is possible, and required (not only for just practical reasons), to work with less than maximal information.  Mixtures, or mixed states, can be used to represent the situation that a system is in one of a set of pure states, with different probabilities.  (Caution: though this is correct it is, as I’ll indicate below, not tenable as a general interpretation of mixed states.)

To explain what mixtures are we need to shift focus to projection operators.  For each subspace S other than {0} there is the projection operator P[S]: vector u is in S if and only if P[S]u = u, P[S]u = 0 if and only if u is in ~S. This operator ‘projects’ all vectors into S.

For the representation of pure states, the job of vector u is done equally well by the projection operator P[u], which we now also refer to as a pure state.  

Mixed states are represented by statistical operators (aka density matrices) which are, so to speak, weighted averages of mutually orthogonal pure states.  For example, if u and t are orthogonal vectors then W = (1/2)P[u] + (1/2)P[t] is a mixed state. 

 Intuitively we can think of W as being the case exactly if the real state is either u or t and we don’t  know which.  (But see below.)

W is a statistical operator (or density matrix) if and only if there are mutually orthogonal vectors u(i) (other than 0) such that W = Σb(i)P[u(i)] where the numbers b(i) are positive and sum to 1.  In other words, W is a convex combination of a set of projections along mutually orthogonal vectors.  We call the equation W = Σb(i)P[u(i)] an orthogonal decomposition of W.  

What about truth?  We need to extend that notion by the same criterion that was used for pure states, namely that the probability of a certain measurement outcome equals 1.  

What is certain in state W = (1/2)P[u] + (1/2)P[t] must be what is certain regardless of whether the actual pure state is u or t. So that should identify the subspaces which are true in W.

But now the geometric complexities return.  If u and t both lie in subspace S then so do all linear combinations of u and t.  So we should look rather to all the vectors v such that, if the relevant measurement probability is 1 in W then it is 1 in pure state v.  Happily those vectors form a subspace, the support of W.  If W = Σb(i)P[u(i)], then that is the subspace [{u(i)}]. This, as it happens, is also the image space of W, the least subspace that contains the range of W. (Note 4.)

It is clear then how the notion of truth generalizes:

            Subspace S is true in W exactly if the support of W is part of S

And we do have some redundancy again, because of the disappearance of any probabilities short of certainty, since truth is construed following von Neumann.  For every subspace is the support of some pure or mixed state, and for any mixed state that is not pure there are infinitely many mixed states with the same support.

While a pure state P[u] has no refinements but itself, if v is any vector in the support of W then P[v] is a refinement of W.  And in general, if W’ is a statistical operator whose support is part of W’s support, then W’ is a refinement of W.  

So we have here a non-trivial refinement relation.

Note: the geometric complexities.  I introduced mixed states in a way seen in text books, that for example W = (1/2)P[u] + (1/2)P[t] represents a situation in which the state is either u or t, with equal probabilities .  That is certainly one use (note 5).

But an ‘ignorance interpretation’ of mixtures in general is not tenable. The first reason is that orthogonal decomposition of a statistical operator is not unique.  If W = (1/2)P[u] + (1/2)P[t] and W = (1/2)P[v] + (1/2)P[w] then it would in general be self-contradictory to say that the state is really either u or t, and that it is also really v or w.  For nothing can be in two pure states at once.  Secondly, W has non-orthogonal decompositions as well.  And there is a third reason, having to do with interaction.  

All of this has to do with the non-classical aspects of quantum mechanics.  Well, good!  For if everything became classical at this point, we’d lose the solution to (Puzzle *).

7.   An open question

So, if we identify what Holliday and Mandelkern call possibilities as quantum states, we have ways to represent such situations as depicted in (Puzzle *), and we have a non-trivial refinement relation.

But there is much more to their theory.  It’s a real question, whether continuing with quantum-mechanical states we could find a model of their theory.  Hmmm …. 

NOTES

  1. In textbooks and in practice this redundancy is eliminated by the statement that pure states are represented by unit vectors (vectors of length 1).  In foundations it is more convenient to say that all vectors represent pure states, but multiples of a vector represent the same state.
  2. See e.g. page 49 in Birkhoff, Garrett (1948) Lattice Theory.  Second edition.  New York: American Mathematical Society.  For a more extensive discussion see the third edition of 1967, Chapter V section 7. 
  3. Holliday, W. and M. Mandelkern (2022)  “The orthologic of epistemic modals”.  https://arxiv.org/abs/2203.02872v3
  4. For the details about statistical operators used in this discussion see my Quantum Mechanics pages 160-162.
  5. See P. J. E. Peebles’ brief discussion of the Stern-Gerlach experiment, on page 240 of his textbook Quantum Mechanics, Princeton 1992.  Peebles is very careful, when he introduces mixed states starting on page 237 (well beyond what a first year course would get to, I imagine!) not to imply that an ignorance interpretation would be generally tenable.  But the section begins by pointing to cases of ignorance in order to motivate the introduction of mixtures:  “it is generally the case …[that] the state vector is not known: one can only say that the state vector is one of some statistical ensemble of possibilities.”

Hilbert logics (2): Quine, quantifiers, and modalities

Bas van FraassenLeave a comment

Thesis: all our most familiar logics, both sentential and quantificational, are Hilbert logics.

In Henkin’s paper (cited in the preceding post) his main target was not sentential but predicate logic. Given that the proof of the Deduction Theorem hinges on Modus Ponens being the sole rule of inference, there is an obvious question. Isn’t

Universal Generalization. from ⊢Fa to infer ⊢(x)Fx

an indispensable rule as well?

Moreover, Henkin indicates that his topic here is not just classical and intuitionistic quantification theory, but applies to quantified modal logic. At that time the concept of a normal modal logic had not yet been introduced. The main modal logics were Lewis and Langford’s S1 through S5, and in retrospect we can say that of these only S4 and S5 were normal. But as typically formulated now, the normal modal logics have a rule that is similar to Universal Generalization, namely

Necessitation. from ⊢A to infer ⊢□A

It is important, and easy to see, that these rules are quite different from Modus Ponens, which says

from A and A ⊃ B to infer B

or, using the deducibility symbol,

A, A ⊃ B ⊢ B

Universal Generalization and Necessitation do not provide inferences from premises to conclusions. Instead, what they do, in effect, is to add lots to the stock of axioms, that is, to the lines in a deduction that do not need to be inferred from previous lines.

It is in fact possible in both cases to have a formulation that is a Hilbert logic, and here I would like to explore one way in which that can be shown.

Quantifier logic a la Quine

For quantificational logic this was made this beautifully clear by Quine in his Mathematical Logic (revised edition 1951). His system of quantifier logic with identity, called ML, does not have the rule of Universal Generalization, and Modus Ponens is its only rule of inference. But he introduced a more nuanced notion of axiom schemata.

First he defines the notion of closure: if A is a well-formed formula which has in it free variables x, y, z, … then the closure of A is the statement which results when universal quantifiers are prefixed to A to bind those variables. So the closure of A looks like this: (x)(y)(z) …A.

Each axiom schema then defines a large class of statements, which are the closures of instances of the displayed form. For example for the class he calls C he displays the form

A ⊃ (x)A, where A is a formula in which x is not a free variable,

and C is the class of all closures of well-formed formulas that have this form. Examples would be (y)(Fy ⊃ (x)Fy), (z)(y)(Rzy ⊃ (x)Rzy), and so forth.

It is clear then that no axiom, and hence no theorem, has free variables in it. Quine’s language did not have individual constants. So it is not even possible to formulate a non-trivial version of Universal Generalization for ML itself. But more to the point, if individual constants are added, we have a choice, and for each choice we arrive at a complete quantificational logic without adding the Universal Generalization rule. If we add the axiom schema

(Ey)(y = a)

the result is the familiar classical predicate logic. Intuitively, each individual constant is the name of a real thing. If we do not add anything then the result is free logic. In that logic, again intuitively, the only existence assumption built it (as it was for Quine) is that there is something rather than nothing. But the individual constants can be non-referring terms, like our “Pegasus”.

In both cases it is provable as a meta-theorem that if ⊢Fa then ⊢(x)Fx. The need for the rule of Universal Generalization has been neatly by-passed by Quine’s tactic.

Modal logic

So how does it stand with the modal logics? Ruth Barcan Marcus wrote the seminal papers on quantified modal logic in the years 1946-1953. She proved the usual Deduction Theorem for material implication for S4 and S5, as formulated by Lewis and Langford, and showed that it could not be proved for S1 or S2.

In our present context, where the family of normal modal logics has a standard formulation, the rule Necessitation plays a salient role. Can Quine’s tactic be implemented here, mutatis mutandis?

The weakest normal logic is system K, formulated as follows with one axiom schema and three rules:

R0. If p is a theorem of classical propositional logic then ⊢ p

R1. p, p ⊃ q ⊢ q

R2. If ⊢ p then ⊢ □p

A1. ⊢ □(p ⊃q) ⊃ (□p ⊃ □q)

Only R1, modus ponens, is an ordinary rule of inference. Rule R0 can be replaced by the axiom schemata of a Hilbert logic formulation of sentential logic. But R2 is Necessitation.

Let’s adapt Quine’s tactic as follows. If A is a formula then call the necessitations of A all formulas that consist of A preceded by zero or more symbols □. And introduce the more nuanced notion of the use of schemata as follows: each axiom schema introduces a large class of axioms, namely all the necessitations of instances of the displayed form. So for example the schemata that can replace R0 include the form (p ⊃ p), and this means that the axioms include all sentences of form (p ⊃ p), □(p ⊃ p), □□(p⊃p), …, and do forth.

Lets say that K* is the modal logic which is like K except for lacking rule R2, but with the axiom schemata understood in this new way. Then it is easy to prove that K* and K have the same theorems (see Appendix). And K* is a Hilbert logic, that is, a Hilbert logic formulation of K.

The same can be done, in the same way, for the other normal modal logics.So, it follows from this that Henkin was quite right, if we read him anachronistically as referring to the normal modal logics: the Deduction Theorem is provable for all of them.

Conclusion Both the normal modal logics and quantificational logics (classical, normal modal, and intuitionistic) are Hilbert logics.

Historical note. Ruth Barcan Marcus who pioneered quantified modal logic, proved the Deduction Theorem for S4 and S5, and showed that it could not be proved for S1 or S2 (Journal of Symbolic Logic 18 (1953): 234-236). For the completeness of ML with names added as free logic, van Fraassen (Zeitschr. Math. Logic und Grundl. der Math. 12 (1966): 219-234).

APPENDIX

K* and K have precisely the same theorems. The proof is by induction.

Suppose A is a theorem of K; then there is a proof S = S(1), …., S(n) in K which has A as its last line. Change this proof into the sequence T = T(1), …, T(k), … , T(n) by moving the axioms that appear in S to the beginning, though kept in the same order they had in S: these are the lines T(1) …, T(k). The lines T(k+1), … , T(n) are the remaining lines of S in the same order in which they appeared in S.

T is also a proof of A in K, for the axioms need no justification, and if another line of S followed from preceding lines by modus ponens, that same line appearing in T follows from lines that precede it by modus ponens as well.

Claim: There is a proof in K* of □A.

Proof.

Each of □T(i), for in = 1, …, k is an axiom, for T(i) is a necessitation of axiom schema A1, and therefore so is □T(i).

Hypothesis of induction: for k < m ≤ n, and for all j < m, there is a proof of □T(j) in K*.

In S the line S(c) which appears as T(j) in T came from preceding lines S(a) and S(b) by modus ponens, so S(b) was [S(a) ⊃ S(c)]. Thus by hypothesis there are also proofs of □S(a) and □[S(a) ⊃ S(c)]. From these, by an instance of axiom A1. and two applications of modus ponens, □S(c) follows. That conclusion is the same as □T(j).

Therefore, by induction, there is a proof in K* of □T(i), for 1 ≤ i ≤ n; hence of □T(n), which is □A.