Category: Metaphysics

NIETZSCHE’S METAPHYSICS

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There have been many books and articles written about Nietzsche with the common thread among them being that they rarely agree on how to understand him. This could be said with many philosophers, but it is particularly true with Nietzsche. Not only are the interpretations not in agreement with one another, but they can differ very widely indeed, so that one can hardly see that they are speaking of the same philosopher. The cause for this lies partly in Nietzsche’s writing itself. Nietzsche is not a philosopher who can have a single position attributed to him on the basis of the sum of his texts, because in each text he is rethinking the issues which he has discussed before as well as coming up with new ideas. He is a creative philosopher — so his books differ very widely, in their content if not in their style. Interpretations of Nietzsche are particularly fraught when an author combines the material in the notebooks with theses that are articulated scattered in the published texts. Nietzsche is constantly trying out new ideas, and the notebooks are full of ideas that never made it into texts. Thus if one tries to combine the two, as most all interpreters do, what one will get is whatever the interpreter wants. One can manufacture in this way any interpretation of Nietzsche one likes, by selective quotation, and by incorporating the discarded material that is in the notebooks — from naturalist to projectionist to sceptic. There will always be contrary passages not cited or noted.

Thus most of the characteristic views that are attributed to Nietzsche are not true in the incautious way that they are usually stated. Take for example his view that ‘God is dead’, as asserted in The Gay Science, and then reasserted in Thus Spake Zarathustra. This is the view with which Nietzsche is most commonly associated in the popular mind, but it is simply not his view, not stated this sweepingly. His view, stated clearly in The Anti-Christ, is that he is not against the existence of all or any God, he is only opposed to the existence of the Christian God. It is the Christian God that he feels doesn’t exist, or perhaps one should say, as he does say, that the Christian God is not even worthy to exist. The Christian conception is an enfeebling concept of God. Thus:

47. — What sets us apart in not that we recognise no God, either in history or in nature or behind nature — but that we find that which has been reverenced as God not ‘godlike’ but pitiable, absurd, harmful, not merely an error but a crime against life . . . We deny God as God . . . If this God of the Christians were ever proved to us to exist, we should know even less how to believe in him. — In a formula: Deus, qualem Paulus creavit, dei negatio.

The Anti-Christ

In fact, in his own life, Nietzsche does seem to have had a belief, however tentative and experimental, in the god Dionysus. He was known to dance as Dionysus’ worshippers were meant to dance, but did so behind closed doors; he wrote poems to Dionysus as a God, he spoke sometimes of being Dionysus. Of course one could argue that this does not constitute evidence of a full belief in Dionysus as an existent being, that it is play-acting a belief — and perhaps that is how it should be interpreted. But one has to say his behaviour is surprisingly close to that of a devout acolyte. This is rarely ever said, although those who know Nietzsche’s works well can’t fail to know it — but the popular atheist view (promoted in the main by Walter Kaufmann) has been allowed to linger in the zeitgeist without qualification.

The same could be said of Nietzsche’s ideas about the Will to Power, present through Nietzsche’s early works and up to Zarathustra, but not emphasised in the later works. This was given a spurious centrality by being the title of a volume edited together from the Notebooks by Nietzsche’s sister Elizabeth as though it represented Nietzsche’s final thoughts, when it was no such thing. Kaufmann himself complained in the following terms.

The claim that these notes rather than the books Nietzsche finished represent his legacy is as untenable as the boast that this — or any — arrangement can claim the sanction of Nietzsche’s own intentions. WP p. xviii

WP p. xviii

Kaufmann’s judicious dismantling of the claims that can be made on the book’s behalf have not however led to the proper attitude of critical distance. The concept of the Will to Power has been pushed forward, along with the Superman, as though it represents Nietzsche’s most mature statement of his position — whereas it had been abandoned in 1888,

And yet for all this we find very clearly stated in the Twilight of the Idols the denial of the idea that Will exists in any form: Will is a psychological phantom, a chimera. What are we to make of this? The best they can be said is that the Will to Power is an idea that Nietzsche flirted with, on and off, but that his final (and published) view was that it, along with the will to anything, rested on a mistake. As he himself said in Twilight of the Idols, section 7 of The Four Great Errors:

The whole of the old-style psychology, the psychology of will, has as its precondition the desire of its authors, the priests at the head of the ancient communities, to create for themselves a right to ordain punishments — or their desire to create for God a right to do so. . . . Men were thought of as ‘free’ so that they could become guilty: consequently, every action had to be thought of as willed, the origin of every action as lying in the consciousness (— whereby the most  fundamental  falsification  in psychologicis was made into the very principle of psychology). . . . 

Twilight of the Idols, section 7 of The Four Great Errors:

Nietzsche goes on to deny other aspects of what has been called Folk Psychology. The idea that we can inspect our own thoughts and understand them, see them, take them as they appear. His view is that all of this is an invented nonsense. Beliefs, desires, thoughts, are a superficial description of phænomenon that we project onto our own consciousness as a way of explaining our own behaviour; they are a vulgar mental physics on a par with the Greek idea that everything is water and fire. The real origins of our behaviour are a mystery to us. Given this attitude as expressed in Twilight of the Idols it is inconceivable that he would title his final book ‘The Will to Power’.

Similar reservations can to be made about the concept of the Superman, or the Übermensch. At various points, particularly in The Improvers of Mankind, Nietzsche announces that mankind can’t be improved. Therefore, mankind can’t attain or be advised to attain to the level of the superman. What then is Nietzsche’s doctrine of the Superman? It seems to be, at best, a description of himself as Zarathustra, or within the Zarathustra mythos, rather than a view which is being recommended for others to adopt. He gives no instructions on how someone who is not Zarathustra might become a superman in terms of recommendations in which he believes, and his idea that these are the beliefs of  the superman do not then constitute a recommendation that others could follow. Let them become Zarathustra, if they can! Saying: ‘I teach the Superman’ is not helpful advice, if it is intended to be advice at all. It is an inspired rush of ecstatic poetry, in the Dionysian mode, but it is certainly not an instruction manual. If anything Nietzsche seems to be hoping that his words might bring the Superman forth, by a form of incantatory magic!

That which gives wings to asses and milks lionesses, all praise to that unruly spirit that comes to all present and all the mob like a storm-wind,

– that is enemy to all thistle-heads and prying noses and to all withered leaves and weeds: all praise to that wild, good, free storm-spirit that dances upon swamps and afflictions as upon meadows!

Zarathustra, Of the Higher Man, §20.

In fact, by the Twilight of the Idols, the concept of the Superman seems to have dropped out to a very large extent (it is mentioned only once), just as the Will to Power has dropped out. What we have arrived at in the final works of 1888 is a psychology where the mind is unknowable, and not within our scope to change. In a very clear sense we no longer know who we are, our internal states are completely mysterious, and only superficial manifestations, which may be wholly unlike their real causes, are evident to us. We could not bring anything forth in ourselves by any deliberate means. We lack a correct theory — or even a theory of a theory — of what we are.

In the same way: though Nietzsche obviously thinks that Christian morality is a very bad thing for mankind, it’s very unclear that he thinks that there is no morality at all. Sometimes he says things which very clearly suggests this. On the other hand, he often seems to be making moral recommendations, for example that courage and nobility are virtues, one might say Roman martial virtues — and it does seem as though the Superman, when that was being recommended, would have a different morality. (It may be rejoined that Nietzsche takes the Superman to have no morality at all, but is that really so?) It may be seen that this confusion runs through much of his work. It is present almost everywhere. If he is saying that there is no objective right and wrong and that what we believe here is mere projection then it makes no sense to say that Christian morality is an especially wrong morality. For on this view they should all be wrong equally including his own beliefs that Christian beliefs are morally wrong. Thus he swings between impossible extremes. In this way he sometimes comes across as a nihilist, and yet when he discusses nihilism, he finds it intolerable. He is a moralising anti-moralist — just as he is a religiously ecstatic anti-theist. But though he is opposed to Christian morality and in theory, celebrates a wild, impassioned rejection of Christian mercy or kindness, he himself seems not to have been given to barbarism or cruelty, or to condone it, even in the Übermensch. Out of this, a skilful interpreter can construct any view he or she wishes. And yet one can picture Nietzsche turning in his grave at some — perhaps most — of the interpretations that have been foisted on his words. (His reaction to Nazi Germany can easily be predicted from the damning things he has to say about Bizmarck’s Germany.)

Given the problems of understanding Nietzsche through the scattering of headline-making topics my intention here is to focus on what he says in his last work with a substantial amount of metaphysics: Twilight of the Idols. The metaphysics elaborated in this work is far deeper and more interesting than what has often been attributed to him. The aim is to show that his thinking on these matters was almost clairvoyant.

On the question of what does and does not exist he has this to say.

One knows my demand of philosophers that they place themselves beyond good and evil — that they have the illusion of moral judgement beneath them. This demand follows from an insight first formulated by me: that there are no moral facts whatever. Moral judgement has this in common with religious judgement that it believes in realities which do not exist. Morality is only an interpretation of certain phænomena, more precisely a misinterpretation.

Of course the idea that there are no moral facts was formulated long before Nietzsche, by David Hume, as Nietzsche was aware. (He knew Hume’s writings and referred to him directly in The Gay Science.) But the main point that I would make here is that this passage strongly suggests that there is a world, it is just that it is not a world that has spiritual or moral aspects to it. There are phænomena, and those phænomena can be interpreted wrongly by us to seem to have moral or spiritual aspects. Thus the question is: what was Nietzsche’s attitude to that world, to its metaphysics — even if he may have balked sometimes at this way of putting it.

Just as Nietzsche mirrored Hume in the latter’s denial of moral facts, so he also mirrored him in the claim that there are no necessary connections we can discern, that causation is something whose inner workings are mysterious to us. We see events regularly occurring, but not the why of it? And, just as in Hume, this is used as a way of showing the problem with the concept of will.

It is this which sees everywhere deed and doer; this which believes in Will as cause in general; this which believes in the ‘I,’ in the I  as Being, in the I  as substance, and which projects its belief in the I -substance on to all things — only thus does it create the concept ‘thing’. . . . Being is everywhere thought in,  foisted on, as cause; it is only from the concept ‘I’ that there follows, derivatively, the concept ‘Being’. . . . At the beginning stands the great fateful error that the will is something which produces an effect — that will is a faculty. . . . Today we know it is merely a word. . . . .

TWI, ‘Reason’ in Philosophy §5.

The ‘inner world’ is full of phantoms and false lights: the will is one of them. The will no longer moves anything, consequently no longer explains anything — it merely accompanies events, it can also be absent. The so-called ‘motive’: another error. Merely a surface phenomenon of consciousness, an accompaniment to an act, which conceals rather than exposes the antecedentia of the act. And as for the ego! It has become a fable, a fiction, a play on words: it has totally ceased to think, to feel and to will! . . . .

TWI, The Four Great Errors §3.

Nietzsche’s idea is that we have invented a phantom (Will) which is a non-existent intermediary between physical events. This illusory faculty is then used to create the secondary illusions of self (ego) and moral responsibility. This is not evading the idea of moral responsibility through a classical form of hard determinism, it is something much more radical with far deeper consequences. This is a metaphysical view which undercuts to a very large extent the earlier work that Nietzsche had done. Both the Superman and the Will to Power have been cut adrift. Even the perspectivism of Beyond Good and Evil is no longer maintainable. What is projection if not some kind of causal process in the mind’s interaction with the world?

It is no longer clear that we can say anything definitive about how events are caused, nor can we say anything definitive as to what cause is. The Holism, and the Monism suggest there is a serious difficulty in understanding whether there could be actions at all or indeed, change at all, just as there was for Parmenides himself. We seem to have in the Monism a return to the Eleatic problem. The view that we reach, then in Twilight of the Idols, could be said to be a repudiation of much of the earlier philosophy, either in whole, or in part. In this respect Nietzsche’s conclusions are more radical than were those of Hume, who did not see that he was undermining his own Empiricism — though I would argue that he was. Nietzsche sees clearly that empiricism — pursued to its logical conclusion — has undermined itself.

The whole of the alleged empiricism which affirmed them has gone to the devil! Thatis what follows! — And we had made a nice misuse of that ‘empiricism’, we hadcreated the world on the basis of it as a world of causes, as a world of will, as a world of spirit.

TWI, The Four Great Errors §3.

Obviously the target here can’t be Hume, but who is intended? Mach perhaps? Whatever the answer to that question, Nietzsche failed to draw the conclusion that his notion of projection had also been undermined. Thus he continued:

The oldest and longest-lived psychology was at work here — indeed it has done nothing else: every event was to it an action, every action the effect of a will, the world became for it a multiplicity of agents, an agent (‘subject’) foisted itself upon every event. Man projected his three ‘inner facts’, that in which he believed more firmly than in anything else, will, spirit, ego, outside himself — he derived the concept ‘being’ only from the concept ‘ego’, he posited ‘things’ as possessing being according to his own image, according to his concept of the ego as cause.

But how can it be known that we definitively have something in the mind that is being projected, rather than that we are seeing what is there, or we have no idea that either of these apply. Psychological explanations have no foundation, and thus its categories vanish. Projection must vanish also. The only thing that could sustain the idea of projection is the conviction that there is a world of merely ‘physical’ events that are devoid of these qualities. Is Nietzsche entitled to that conviction? If he is not then Nietzsche’s evolutionary story is just that — a statement of how things could be with nothing that compels belief.

This invites a logical question: is it possible to have a projection that is not a projection onto something? I would suggest that it is not. Projection not only requires a causal process to be activated it requires a screen on which to project. Projection is nothing other than a metaphor for a causal process in which a seeming reality appears on a surface to give the illusion of something existent. Without the screen, projection is not possible. (What about a hologram? it may be protested. A hologram projects one light field onto another light field where this second is scattered off objects, for example air. It would not be possible in empty space.) Thus the idea that there may be no screen at all is impossible. But then it may be asked if dreams are not projections? I would suggest that they are not: they are events in the sleeping brain, but no projection is involved. (If one has them while awake then they are confused with, or intermingle with, ordinary perceptions.) Thus a sweeping view that declares that everything is projection is not possible. Sometimes Nietzsche seems to be reaching for such a view but I think ultimately he realises that it is unintelligible.

Can we find other passages that broaden Nietzsche’s point?

We can start with the surprising statement Nietzsche makes about Heraclitus.

2.  I set apart with high honour the name of  Heraclitus. When the rest of the philosopher folk rejected the evidence of the senses because these showed plurality and change, he rejected their evidence because they showed things as if they possessed duration and unity. Heraclitus too was unjust to the senses, which lie neither in the way the Eleatics believe nor as he believed — they do not lie at all. It is what we make of their evidence that first introduces a lie into it, for example the lie of unity, the lie of materiality, of substance, of duration. . . . ‘Reason’ is the cause of our falsification of the evidence of the senses. In so far as the senses show becoming, passing away, change, they do not lie. . . . But Heraclitus will always be eternally right in this, that Being is an empty fiction. The ‘apparent’ world is the only one: the ‘real world’ is simply a lying addition. . . . *

TWI ‘Reason’ in Philosophy

We can see from this that he is very clearly against a dualistic metaphysics, what one could call a Two World-ism — and this is consistent from his earliest writings to the last.

(To be continued)

Problem of Truth in Quantum Mechanics

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This is the introduction of a paper of this name which is in print in the journal Global Philosophy (the article was submitted and accepted in the journal called Axiomathes, but just before the paper was sent back for proof-reading the journal changed its name to Global Philosophy — thus this is where it can be found at http://t https://doi.org/10.1007/s10516-023-09656-4 . A pre-print copy can be found at Academia.com https://www.academia.edu/95833087/The_Problem_of_Truth_in_Quantum_Mechanics

Please go to either of these sources if you wish to see how the argument of the paper progresses.

Abstract

 There is a large literature on the issue of the lack of properties (accidents) in quantum mechanics (the problem of “hidden variables”) and also on the indistinguishability of particles. Both issues were discussed as far back as the late 1920’s. However, the implications of these challenges to classical ontology were taken up rather late, in part in the ‘quantum set theory’ of Takeuti (1981), Finkelstein (1981) and the work of Decio Krause (1992) — and subsequent publications). But the problems created by quantum mechanics go far beyond set theory or the identity of indiscernibles (another subject that has been often discussed) — it extends, I argue, to our accounts of truth. To solve this problem, to have an approach to truth that facilitates a transition from a classical to a quantum ontology one must have a unified framework for them both. This is done within the context of a pluralist view of truthmaking, where the truthmakers are unified in having a monoidal structure.

The structure of the paper is as follows. After a brief introduction, the idea of a monoid is outlined (in §1) followed by a standard set of axioms that govern the truthmaker relation from elements of the monoid to the set of propositions. This is followed, in §2, by a discussion of how to have truthmakers for two kinds of necessities: tautologies and analytic truths. The next section, §3, then applies these ideas to quantum mechanics. It gives an account of quantum states and shows how these form a monoid. The final section then argues that quantum logic does not, despite what one might initially suspect, stand in the way of an account of quantum truth.  (end of Abstract)

Pluralism, or at least dualism, about truth has been the unacknowledged orthodoxy for the greater part of the history of philosophy. Most philosophers have recognised ‘truths of fact, or existence’ and those due to the ‘relation of ideas’, to put it in Hume’s way of speaking, synthetic and analytic relations to put it in Leibniz-Kant’s. Why unacknowledged? Perhaps because in thinking about truth it is the relation of the proposition to fact or existence that has dominated the mind, and so most traditional accounts have been cast in terms of agreement with how things are. As the medievals said: Qualitercumque significat esse, ita est (A proposition is true if, howsoever it signifies to be, so it is).(1) This dominance was at the forefront when Russell and Wittgenstein contributed to the unofficial birth of truthmaker theory the idea that truths require states of affairs (or facts, situations, circumstances — der Sachverhalten, in Wittgenstein’s original German) to make them true. What about the relations of ideas? These were consigned to emptiness — or at least mere definition. When Armstrong took up this truthmaker project it was states of affairs that were assumed to be the truthmakers and analytic sentences were for the most part left out of consideration (Armstrong (2004)).(2) But, for all that, truthmaker theorists were often unofficial dualists about truth.

That such a dualism can be carried into a more extensive pluralism has been argued by several authors in the last thirty years, notably Crispin Wright, Michael Lynch and Gila Sher. Here I focus on the version given by Sher (in Sher (2004), (2013)), since Sher sets her pluralism in the general context of the correspondence account of truth. Whereas truthmaker theory has often been accused of not giving an account of certain classes of truths, pluralists seek to leave no class of statement out of consideration. Armstrong was motivated by an ontological parsimony — to see how much one could do with just a physicalist ontology of states of affairs; Lynch is motivated by giving folk attributions of truth a weight equal to the scientific and his alethic pluralism unifies the different folk attributions by their sameness of functional role, this being given by the relation of the purported truths to beliefs and actions, rather than being constrained by ontological considerations. 

Sher, by contrast, though motivated by the correspondence intuition, is not wedded to a strict physicalism, but allows higher level abstract structures to be the ita est of the medieval formulation — in short, to be that to which true statements may correspond. This is consonant with the view of truthmaking which is taken in the present paper. The truthmaker idea should not be constrained by an ontological prejudice for what the truthmakers should be, at least not before we know what is. (One finds a different liberalisation in Griffiths (2015).) On the other hand, there should be a clear essence to truth, something that unifies the various plural instances into one clear kind. This is to protect us against including things that merely look like Truth: for example fervent belief.

Giving an account of analytic truths, however, requires that one make the truthmakers something other than states of affairs. I have described this elsewhere but I review it here, keeping the relation between truthmakers and propositions the same as in the standard case. What changes is the truthmaker basis. 

What is more, the structure of these two kinds of truthmakers can be seen to exhibit a very general mathematical form: they are both monoids — something resembling a group but with no requirement that there exist inverse elements. Monoids are quite basic in terms of their structure and it is hard to imagine a collection of truthmakers not exhibiting such a structure. If correct this means that we can abstract the idea of truthmaking to a relation between certain monoids and boolean algebras of propositions, where the nature of the mapping is given by the truthmaker relation. We could also say that truth is a representation of a monoid in a boolean algebra. This is its essence: it is a representation of a structure that lacks negative elements in a structure that possesses them. This representation can be understood as a “picture” in the sense of Wittgenstein’s Tractatus.(3) Pictures are Wittgenstein’s (loose) way of speaking of representations.(4)

But ultimately the present reason for wanting to construct a less restrictive truthmaker theory is due to a problem that looms over many theories of truth: the problem of truth in quantum mechanics. Specifically it looms for any theory of truth that depends upon a classical ontology of individualised objects and properties that are then mapped to (correspond with, form sequences that satisfy, etc) assertions, propositions, statements, or sentences. This will include Armstrong’s, Sher’s and Tarski’s theories — but also many others. This is because quantum mechanics has a different mathematical character to that which we find in a classical world view: there are states in a topological vector space and non-commutative sets of operators that act upon those states, with particles that fail to be individualised. How are we to make this into a basis for a truthmaker account?

It is interesting to note that one of the pioneers in the mathematical formulations of quantum mechanics, Hermann Weyl, was also one of the first to properly formulate the notion of truthmaking, some years before Wittgenstein and Russell. Here are his words, written in 1917. 

A judgment affirms a state of affairs. If this state of affairs obtains, then the judgment is true; otherwise. it is untrue. States of affairs involving properties are particularly important. (Indeed, logicians often made the mistake of ignoring every other state of affairs.) A judgment involving properties asserts that a certain object possesses a certain property, as in the example: “This leaf (given to me in a present act of perception) has this definite green colour (given to me in this very perception).” A property is always affiliated with a definite category of object in such a way that the proposition “a has that property” is meaningful, , expresses a judgment and thereby affirms a state of affairs, only if a is an object of that category.A judgment corresponds only to a meaningful proposition, a state of affairs only to a true judgement; a state of affairs, however, obtains — purely and simply. (Weyl 1917; 5) (5)

Unfortunately, however, Weyl never returned to this issue following the advent of quantum mechanics in 1925. 

The structure of the paper is as follows. In §1 the idea of a monoid is outlined followed by the standard axioms that govern the truthmaker relation from elements of the monoid to propositions. This is followed by a discussion of how to have truthmakers for two kinds of necessities: tautologies and analytic truths. The next section then applies these ideas to quantum mechanics. It gives an account of quantum states and shows how these form a monoid. The final section then argues that quantum logic does not, despite what one might initially suspect, stand in the way of a theory of quantum truth.

Notes: (1) Moody (1953) p. 102. It has to be said that, strictly, this formulation does not say anything as to what the proposition signifies to be and it may just as well be how ideas relate to one another as what the states of affairs are, even if the latter are what one tends to think of as intended. 

(2) This is not to say that Armstrong did not give a lot of space to necessary truths, including the mathematical, indeed he devoted a third of his book to the subject, chapters 7–10. 

(3) In the Tractatus, all the 2 and 3 series of propositions, e.g. 2.173: ‘A picture represents its subject from a position outside it. (Its standpoint is its representational form.) That is why a picture represents its subject correctly or incorrectly.’ Also Tractatus 4.1: ‘Propositions represent the existence and non- existence of states of affairs’. This last could almost be a motto for this paper. 

(4) For algebraic objects like groups and monoids, in fact associative algebras in general, there is a mathematical notion of representation where the algebra is represented by symmetries in a vector space (usually) (see Steinberg 2016, or for a general introduction Gowers (2009; IV.9) or Derksen and Weyman (2005)). This is not quite the sense of ‘representation’ used above, but the underlying principle is the same: one structure is represented in a different structure. It cannot be ruled out that the fuller notion may enter into the picture in the future. Into which structure we make the representation is something that can be adjusted to purpose. 

(5) Weyl acknowledges the prior work from 1900-1 of Edmund Husserl in the Logical Investigations,  and Husserl there relied heavily on the idea of states of affairs. The publication of Weyl’s work was delayed by the war. 

2  Monoids and Truthmaker Axioms

Our truthmakers will be a collection, denoted 

s,s‘,s”,…t,t‘,t”,…,u,vuantum

but also sometimes capitalised. On this collection there is an associative operation  ⊕ , i.e. 

s⊕(t⊕u) = (s⊕t)⊕u.

This operation will, in the classical case, also be commutative, i.e. 

s⊕t = t⊕s

(later this condition will be dropped.) Among this collection there is an element that is an identity for this operation. It is unique. I will denote it as  V. Thus, for all s, 

s⊕V = s.

With this operation the truthmakers form a monoid. A significant property of monoids is that elements need have no inverses under the given binary operation. A set on which an associative binary operation is defined is called a semigroup. A monoid is a semigroup with an identity element. Thus the theory of monoids is often folded into the theory of semigroups. The first significant results on semigroups did not come until 1928 (see Clifton and Preston (1964)) but since 1940 it has become a rapidly developing field.

Some familiar examples of monoids are: the natural numbers  N  with addition as the binary operation and 0 as the identity; also the natural numbers with multiplication as the operation and 1 as the identity element; the powerset  P(S)  of a non-empty set S with  ∪  as the binary operation and the empty set ∅ as the identity element. A simple example of a non-commutative monoid would be strings, i.e. a set of symbols/letters under the operation of concatenation. (Jacobson (2009) ch. 2.) 

Significantly, the operators in quantum mechanics form a monoid, as noted in von Neumann (1929) which can then also be made into a regular ring (Murray and von Neumann (1936) and von Neumann (1936)). This leads to the operator algebra point of view, to give it its more modern name. 

Additional relevant aspects of monoids will be gathered together in an appendix at the end of the paper.

As noted above, the usual idea in the literature is that truthmakers are, in Wittgenstein’s original term, der Sachverhalten — in translation: facts, or circumstances, or states of affairs.(6) This is the idea we start with. We begin by describing the relationship between the set of truthmakers and propositions.

Truthmaker Defn: 

• If s is a truthmaker for proposition A, designated s ⇝ A, then it is necessarily the case that if s obtains then A is true. 

The embedded conditional in the Truthmaker Definition is intended to be a natural conditional. Note that it cannot be strengthened to an ‘if and only if’ conditional, because A may also be true due to the obtaining of some other truthmaker. The existence of the truthmaker s may be said to be a sufficient condition for A to be true. This is often summarised in a slogan: truth depends upon being. This is slightly misleading, in that it suggests that we have here the dependence of one thing, Truth, upon another thing, Being. In fact it is the dependence of the obtaining of a property of a thing (a proposition’s truth) on another complex thing (usually a state of affairs).

We now have the main claim of truthmaker theory — a claim that may, as Russell said, seem so obvious that it is hard to imagine anyone denying it. (Russell 1956).

Primary Truthmaker Axiom: 

• For any true proposition A, and no untrue proposition, there is an s such that s ⇝ A. Conversely, for any s there is a proposition A that it makes true. 

In other words, no true proposition exists that is true independently of the truthmakers that obtain, nothing is true ‘all by itself’, no truth is an island. David Armstrong calls this principle truthmaker maximalism and assumes it — the present paper attempts to make it more plausible. On the other hand false propositions are false due to the absence of anything that might have made them true — they are islands. There are no “falsemakers”. And just as true propositions require truthmakers so also truthmakers require propositions that they make true: there are no truthmakers that are idle. (It is also worth noting that a proposition’s being true is also a state of affairs and thus a higher level truthmaker.)

We now relate the operation  ⊕  to the Boolean structure of the propositions.

Co-obtaining: 

• If s ⇝ A and t ⇝ B then s⊕t⇝A & B where s⊕tis the agglomeration (or mereological sum) of the truthmakers s and t, i.e. the co-obtaining of both truthmakers. 

Closure: 

• If s and t are both truthmakers then s⊕tis also a truthmaker. 

One-Sided Disjunction: 

• If s ⇝ A then s⇝A ∨ B, for any B. (The converse does not hold — though of course this does not mean that true disjunctions have no truthmakers, they do.) 

In older formulations of truthmaker theory it was supposed that truthmaking should be transmitted by classical logical entailment from one proposition to another (see Heathcote (2003) (2006) and Read (2000)). However this led to undesirable consequences, since any proposition entails a necessarily true proposition and thus any true proposition does also; this would mean that all of the truthmakers will become truthmakers for all and any necessary propositions. (This has been called truthmaker monism collapse.) This is not a consequence of the present view, for we explicitly reject the idea that truthmaking is transmitted by classical entailment. (Armstrong (2004) pp. 10–12 wanted to keep in some limited form of this entailment principle, but no specific proposal was made)

The monoid with the operation  ⊕  has an identity element V (for Void) which has the property that for any s we have 

s⊕V= s. 

This is analogous to the empty set, which is the identity element in the set-theoretic example of a monoid given on p. ? Co-obtaining has some interesting consequences. Since our monoid is commutative there exists a pre-ordering  ≤  on its elements that gives us the intuitive idea of smaller and larger truthmakers, given by 

s≤s‘ if there exists some t such that s⊕t=s‘.

This allows us to speak of truthmakers that are parts of other truthmakers. 

Parts of truthmakers: 

• A truthmaker s is a part of a truthmaker t if and only if s⊕tis just t. 

Enlargement: 

• If s is a truthmaker for A and s is a part of t then t is also a truthmaker for A. 

The final consequence is the existence of a Universe.

Universe: 

• The Universe (or totality) U of truthmakers is that for which, given any truthmaker s, s⊕U= U. It can be regarded as the agglomeration of all of the truthmakers and so by closure a truthmaker itself — thus all truthmakers are parts of U. (See Jacobson (2004).) (7)

As a consequence of these axioms we can also say what it is for a complement to a state of affairs s to exist. Thus  s‘ is a complement of s if and only if (1) no part of  s‘ (resp. s) is a part of s (resp.  s‘); and (2)  s‘ ⊕ s =U. It does not follow from this that every state of affairs has a complement.

From these axioms it follows trivially that negative existentials have truthmakers. Let p be ‹ there exists a Unicorn ›: p is false and so has no truthmaker. ∼p is true and so has a truthmaker. Since U has all truthmakers as parts, U is a truthmaker for ∼p. This was one of Wittgenstein’s insights in the Tractatus. At 2.05 he says, ‘The totality of existing states of affairs also determines which states of affairs do not exist.’ (8) Very often when we speak of the truthmaker for a proposition A we mean a minimal truthmaker — this can be defined as a truthmaker for which there are no proper parts that are also truthmakers for A. With this in mind we can note that even though U will always be a truthmaker for negative existentials it may not be the minimal truthmaker. Thus consider ‹ There are no Arctic penguins ›. If Arctic penguins existed then they would have to exist in the Arctic, or at least on Earth. So now repeat the above argument replacing U by the set of existing states of affairs constituting the Arctic (or Earth). The Arctic is then the truthmaker for the truth that there are no Arctic penguins. Other things are such that they will inevitably require all of U. There are no 10 GeV Higgs Bosons anywhere and so we can’t consider a state of affairs smaller than U itself. 

The example of the Arctic penguins is reminiscent of an example that Russell tried out on Wittgenstein. Before Wittgenstein formulated what we can now see is the correct view he held that negative existentials are one and all meaningless. On the 1st of November 1911 Russell in a lecture tried to get Wittgenstein to agree with the statement ‘There is no rhinoceros in this room’. Wittgenstein refused to agree. Russell responded by looking all around the room including under the desk while Wittgenstein said nothing. But then Russell in his lectures on logical atomism advocated negative facts (see Russell 1956) as a solution. (We don’t know if he had this view in 1911). This idea has been restated with more sophistication in Priest (2000) and Beall (2001). When he revived the notion of truthmaking David Armstrong advocated for a higher-order totalising fact (Armstrong (1997) and (2004)). Others had advocated absences (Martin (1997)). In this way truthmaker theory came to seem encumbered by an indenumerable number of phantom, causally inert, states of affairs. Metaphysics run rampant! The problem for these views was put sharply by George Molnar in his (2001).

The above Wittgensteinian solution to the negative existentials problem was restated in 2003. It was proposed in Heathcote (2003) and then in nearly the same terms by Lewis and Rosen (2003) with respect to Lewis’ own qua strategy; it was again put forward in 2004, in Niiniluoto (2004); and was defended in Cheyne and Pigden (2006) and again more recently in Griffith (2015). However, it seems to have escaped the notice of many critics, for it has not to my knowledge been either accepted, and the problem laid to rest, or refuted. (That said, a rather similar proposal was made by Cameron in his (2008), but adding the idea that U — the World, in Cameron’s terminology — necessarily lacks unicorns. He says: ‘Given truthmaker necessitarianism this commits me to the claim that the world is essentially lacking in unicorns, talking donkeys etc.’ (p. 413) I think this is a mistake, and it is certainly not part of the traditional view. Given the world as it is ‹ there are no unicorns ›is necessitated to be true; so if ‹ there are no unicorns ›had been true the world would have had to be different, in that there would then have been unicorns. That is all that follows — U is not essentially unicornless, it is contingently unicornless. Thus it should be noted that the states of affairs that make up U are for the most part contingent — thus forcing U to be contingent, for if one contingent state of affairs were different then U would be different.)(9)

This gives us the basis to speak of truthmaking for propositions including negative existentials. To extend it to sentences S in a given language L we have:

Truthmaker Axiom for Sentences: 

• i) If there exists an s such that s ⇝ S then S is true; ii) For any true sentence S that expresses a true proposition A, if s is such that s ⇝ A then s ⇝ S when S is so understood. 

We do not assume that for every true proposition A there will be some sentence S that expresses it, simply because sentences are finitary objects. Ambiguous sentences may have a truthmaker on one resolution of their ambiguity but not on another.(10)

One might ask: what is it about monoids that make them particularly suitable for characterising states of affairs? The answer is that in a monoid there is no need for there to be any negative elements, no inverses relative to the binary operation  ⊕ — there is no requirement that they exist. So there need be no members of the monoid which “negate” or “undo” other truthmakers under the application of the operation, as one would have to have if the structure were a group — just as in the monoid of the natural numbers under the operation of addition there are no additive inverses, no negative numbers (which is why  N  does not form a group under either addition or multiplication). So we also don’t have the following situation with truthmakers, with t being the negative of s: 

s⊕t = V.

If this situation were to occur the mere aggregation of the two truthmakers s and t would cancel one another to yield the Void. (V be it noted is a positive state of affairs.) Being is only positive, it is. It is propositions that bring in negation.(11) Thus we agree with the Tractatus at 2.04: ‘The totality of existing states of affairs is the world.’ Also, as already noted, agreed with is Tractatus 4.1: ‘Propositions represent the existence and non-existence of states of affairs’. Unknowingly, but presciently, Wittgenstein was characterising a commutative monoid.

The introduction of U, the universe of states of affairs, may tempt someone to the view that one could make do with U as a single truthmaker for all truths, particularly in the light of the Enlargement principle above (see for example, Shaffer (2010)). But simplicity is not everything and there are very good epistemological reasons for our needing to consider more fine-grained truthmakers — such as in addressing the Gettier problem (see Heathcote (2006) (2012) (2014)).

In the next section we take up the derivation of necessary truths from a different monoidal representation, before moving on to the problem of extending truthmaking to quantum claims.

Notes: (6) Armstrong characterises the relationship between truthmakers in terms of mereology (and others have followed suit). But mereological fusion generates a monoid structure; thus we lose nothing in speaking of the latter notion, which is accepted algebra. 

(7) Such an element is also called an absorbing element or an annihilating element — though this is only suggestive when the operation is analogous to multiplication. Wittgenstein also spoke of a ‘totality of facts’, at Tractatus 1.1. 

(8) Wittgenstein (1921). However from the part relation one can also see that there is no sense in which all states of affairs can be regarded as independent or separate from one another — thus the Tractatus’ proposition 2.061 cannot be accepted: ‘States of affairs are independent of one another.’ But the complement to a state of affairs s could be said to be independent of s . 

(9) It has to be noted that Cameron is trying to square his view with Lewisian metaphysical claims, such as Humean Supervenience and this may have caused him to adopt an account of truthmaking which is different to that found in the authors mentioned in footnote 6 below. 

(10) See Heathcote (2003). Many of these axioms go back in some form or other to Mulligan et al (1984). See also Armstrong (2004) and Read (2000) — with useful clarification on half-disjunction in the latter paper. See §4 below. The same denial of a full disjunction thesis goes back to Aristotle in On Interpretation. With Rodriguez-Pereyra (2006) and (2009), I reject a generalised entailment principle — for discussion see Jago (2009) and López de Sa (2009). 

(11) As long as we have a monoid with no negative elements we should not get paradox. It is reasonable to conjecture that contradictions such as the Liar paradox arise because there are negative elements in the truthmaker ground (in this case the monoid of sentences) as well as the objects (also the sentences) that we are mapping to. We leave this suggestive idea hanging. 

(Continued in the full pdf mentioned at the outset.)

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Review of Demopoulos 2022

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(Printed version is on Metascience doi: 10.1007/s11016-022-00807-8; an unedited pre-print is on Academia.com: https://www.academia.edu/86788063/Traces_in_Probability_Space )

Traces in Probability Space

William Demopoulos’ On Theories: Logical Empiricism and the Methodology of Modern Physics, Cambridge: Harvard University Press. xxiv + 247 pp. 

William Demopoulos died in May 2017, altogether too young, and with a manuscript nearing completion that needed some tidying editorial attention. This is the volume under review, and Demopoulos’ long-time friend and collaborator, Michael Friedman, has admirably completed the necessary editorial tasks. In addition he has contributed an Editor’s Foreword and Afterword to help give the chapters more continuity — and, more importantly in the case of the Afterword, to trace Demopoulos’ complex path in interpreting quantum theory. 

Since the quest to understand quantum mechanics spanned Demopoulos’ career and since nearly half of the present book is given over to it, we focus on this. 

In the late 1970s Demopoulos was one of the leading figures in the ‘Western Ontario’ school of quantum logic. It was an exciting period, even if one was unconvinced by the overall plan. (It was during this period that this reviewer, then a graduate student, became familiar with his work; one never had cause to regret reading his essays, which were always sharp and to the point.) By 1982 Demopoulos had become disenchanted with quantum logic, as he had realised that there were classical contradictory propositions that were nevertheless true in quantum logic! Unable to reconcile himself to this, for the next 22 years he turned away from quantum logic to focus on other, unrelated, issues.

He returned to quantum theory only around 2004, around the same time as Itamar Pitowsky published his paper interpreting quantum probabilities through Dutch book arguments (Pitowsky 2003). They were clearly in close communication and the association informs the discussion in the present book. 

The basic idea behind the proposal is as follows. Consider Minkowskian space-time and its group of rotations, the Lorentz group. The introduction of this structure takes a phenomenon that looks to be dynamical, namely Lorentz contraction, or time dilation, and reinterprets it as a kinematical phenomenon. It is nothing more than the result of a rotational change of frame. This might be called a structural explanation or, less happily, a principle explanation. In it, something that looks dynamical receives a purely geometrical explanation. We have a similar case, not mentioned by Demopoulos but worth having in front of us, in the Pauli exclusion principle. This originally also looked dynamical — a strange “force” that kept electrons apart from one another — and was then, almost immediately, given a geometrical interpretation: it was due to the reduction of the tensor product space to just the antisymmetric quotient (for fermions), which is now identified as the exterior power, a Grassmann algebra. Or we could consider the way that gravitation became nothing more than a kinematical phenomenon on a curved space-time. Such structural explanations have had considerable success in 20th Century physics! 

But how can we understand the baffling probabilities that occur in quantum theory, both for measurement outcomes and the distant correlations? These also have a faux dynamical character and thus look fit for a structural explanation. The crucial theorems for Demopoulos are those of Gleason and Kochen-Specker (KS) — the latter following from the former plus a (logical) compactness claim — and the relevant structure the projection lattice of subspaces of a Hilbert space. Gleason’s theorem tells us that a generalised measure defined on such a lattice of a Hilbert space (with dim ≥ 3) extends uniquely to a linear functional on all bounded operators. In quantum mechanical terms this means that it extends from pure states to apply also to mixed states. This probability measure is the trace of a density matrix. The pure states, the extremal points of the convex set of all states, are in 1 : 1 correspondence with the subspaces of a Hilbert space. This justifies in a striking way the Born Rule of probabilities for pure and mixed states: it ‘drops out’ of the formalism in a natural way. (In the exceptional case of two dimensions it is possible to have a 0 : 1 measure on the basis vectors that can be consistently continued to the whole space, called the ‘Bloch sphere’ — this is sometimes described as ‘colouring the sphere’ with just two colours.) 

Whereas Gleason’s theorem shows the global failure of a 0-1 measure, the KS theorem constructs quite small sub-algebras in which the failure is made manifest (Thm. 1 of Kochen and Specker 1967)). Over the years both Gleason’s and the KS theorem have been made progressively more tractable. For KS and a Hilbert space of dim = 4, the number of basis sets needed is only 9, with 20 vectors (q.v. Kernaghan (1994)). Even in dim = 3 it is possible to use only 31 vectors — down from the KS original 117. A consequence of Gleason’s theorem that is stronger than the KS theorem is this: if two pure states a and b are a part of a set Γ of pure states and if they are not orthogonal, then there is no probability measure on Γ that assigns both a and b a probability of zero or one (a truth value) unless they are both assigned 0, i.e. are both false. This is Pitowsky’s Indeterminacy thesis (Pitowsky 1998) as used by Demopoulos. 

In the final section of the KS paper they proved that there were classical tautologies that were quantum-false. Even the seemingly harmless associativity law will be satisfiable only under restricted circumstances. proved the dual of this in  (1977; 86). He argued that there were classical contradictions that were true. In particular he showed that a finite partial Boolean subalgebra of subspaces can be represented by a Boolean polynomial in 86 variables — less now of course in the light of Kernaghan (1994). For every assignment of either 0 or 1 this Boolean polynomial takes the value 0 – thus it is a classical contradiction. And yet it is satisfiable by some assignment of co-measurable quantum variables and so can be ‘true’. The conclusion Demopoulos reached was to believe that the semantic determinacy of quantum logic cannot be maintained — there have to be disjunctions that are true but where none of the disjuncts are true. And this has led him to the conclusion that the interpretation of the elements of the projection lattice as propositions about some particle must be false, for if it were not, quantum truth would have as a subset some classical contradictions. As he said: ‘Hence, if we think of the laws of logic as “laws of truth” — as a guide to our concept of truth — the quantum logical interpretation’s concept of truth cannot be our classical concept.’ (2010; 380). It would be a step into paraconsistency that Demopoulos is unwilling to take. (This is all, of course, in contrast to intuitionist truth that is a subset of the classical truths.)

This is apparently why Demopoulos abandoned quantum logic for more than 20 years.

But since 2004 he has returned, obviously having thought deeply on these matter in the intervening years. Quantum logic, as originally conceived, cannot work; quantum probability can work, provided that the probabilities are not understood as a measure on a lattice of subspaces conceived as propositions. The problem then is: how should the states be conceived? His answer is that they should be conceived as effects and such effects have probabilities of occurring. An effect is an event, the result of an interaction with another system. Demopoulos says of effects that they ‘are induced by an interaction with an experimental set-up and registered in the experimental apparatus.’ (Demopoulos 2010; 382) If there is no interaction Demopoulos avails himself of the memorable saying of Asher Peres, ‘Unperformed experiments have no results’. The probabilities are interpreted by Demopoulos in a Bayesian fashion using the notion of a Dutch book, but the underlying space is not the standard Boolean event space but the non-Boolean projection lattice and presumably its continuation to all density operators. This idea (Quantum Bayesianism, or “QBism”) was also set out in Pitowsky (2003) but it goes back to the beginning of quantum mechanics and one can find many comments in von Neumann’s work, including his letters, which would support it (albeit inconsistently). The idea is that we have here a structural explanation: the structural change is to the underlying event space.

An obvious question must arise at this point in the mind of any reader: what does this say about the EPR-type distant correlations? How does a Bayesian probability work in this case, where the correlated probabilities seem perfectly objective? This question is taken up in the book under review, based on the ideas we’ve already seen. Drawing on the analogy with the structural role of Minkowski space-time, Demopoulos puts it as follows: ‘… the probabilities exhibited by the EPR correlations are understood as a consequence of the Hilbert space structure of the properties of physical systems, rather than the effect of unknown local causes’ (p. 183). Underneath this there is a reliance on an unpublished formulation of Bell’s theorem and KS due to Pitowsky. On this formulation, if we have a finite set F of Hermitian operators, a state s and a probability distribution Ps on all of the eigenvalues for the Hermitian operators in F, and agreeing with QM probabilities for commuting members in F, then Bell’s theorem says that there is an s (the EPR state) and an F  (which in this case are local operators) for which there is no Ps ; and KS says that there is an F  such that for any s the Ps does not exist. Thus KS is not dependent on the choice of state and Bell’s theorem follows from KS and so also from Gleason’s theorem. then deploys this against Einstein’s ‘criterion of reality’ and on behalf of Bell’s analysis of the prospects for a completion of QM along the lines Einstein envisaged. All of this one can accept. But I still don’t quite see how the Bayesian framework plus the non-Boolean character of the quantum probability space explains the distant correlations. In the analogous case of Minkowski space-time there is a clear, provable, sense in which it explains a Lorentz contraction and time dilation. In the current case, Minkowski space-time both forbids causality and allows correlations. Does this suggest that another group is needed? Another space for the probabilities?

The cost of making probabilities Bayesian and personal is that one needs a rich theory of the mutual ‘effect’ of a quantum system on a measurement device — the objective causal event we are betting on. The problem here is that there is no such causal theory on offer. Moreover there may be good reasons why nothing is forthcoming. To describe such an interaction you need, at the very least, two Hilbert spaces under a tensor product. But even this simple requirement has posed a great obstacle for quantum lattices and their logics. This was the problem uncovered by Aerts, Foulis and Randall, Pulmannová, in the early 80s and discussed by Boris Ischi more recently. Under the product, one of the lattices is forced to be commutative. The underlying problem is that a tensor product is not a Cartesian product but is vastly larger. And then there is an additional problem that this reviewer has not seen addressed anywhere: we are really interested in the quotient spaces of the tensor product: the symmetric power (for bosons) and the exterior power (for fermions); where is the lattice product corresponding to these? Where the logics and the probabilities? And what about these quantum probabilities? It has been argued (by Gromov and Hrushovski) that probability, measure theory, has a monadic presupposition, and that this naturally creates a ‘space’ that is imposed from above on the algebra of sets. Relations between things are not fundamental, rather they are functions on this space. In a sense our spatial metaphysics makes everything ‘loose and separate’ from the outset, by assumption. The problem has been put this way: 

Thus pure probability logic sees, very precisely, the monadic space of the theory; and conditional probabilities of basic relations as functions on this space. Beyond this, or relative to it, the universe is boring; everything is locally statistically independent, no special features can be made out, and no new relations can be described. Hrushovski (2020; 393) 

What is missing is a probability theory that puts the right relations in from the get-go. Unrepresented structures, we might say, imitating Peres, have no interpretation. 

Unfortunately we do not have either Demopoulos or Pitowsky — who died within 7 years of one another — to address these issues. 

But a further interesting aspect of Demopoulos’ discussion in the book has to do with understanding what Einstein believed — and then, by contrast, with what Bohr believed. Commenting in 1949, in the Schilpp volume, Einstein noted that the best explanation of the EPR situation given by the ‘orthodox’ interpretation [the view that the ψ-function is complete] comes from Bohr: 

If the partial systems A and B form a total system which is described by its ψ-function ψ/(AB), there is no reason why any mutually independent existence (state of reality) should be ascribed to the partial systems A and B viewed separately, not even if the partial systems are spatially separated from each other at the particular time under consideration. Schilpp (1949) 

Demopoulos doubts that this is Bohr’s view. I think he is right about this. But even though Bohr in his published remarks never said anything resembling this (though von Neumann and did) we have good reasons to believe it to be true: the reduced density matrices of A and B are maximally mixed states, so there is a good sense in which no ‘independent existence (state of reality) should be ascribed to the partial systems’, no ‘being-thus’.

I have one further caveat: I think Demopoulos goes too far in trying to enlist Bohr as an anticipator of this kind of structural view he favours. Bohr showed no interest in the algebraic structure of quantum mechanics, preferring instead his own talk of ‘complementarity’, which had no algebraic interface. Bohr’s form of words that was meant to prove that the EPR argument is unsuccessful is, to put it politely, less than transparent: ‘But even at this stage [the stage at which the final measurement is made on a subsystem] there is essentially the question of an influence on the very conditions which define the possible types of predictions regarding the future behaviour of the system.’ Bohr (1935; 700) What is influencing what here, and when? Just as with the above QBism, we are missing a theorem — and Bohr showed no interest in proving such.

This issue aside, Demopoulos has crafted a thoughtful and interesting interpretation of quantum mechanics that completes his earlier work of the mid-70s; it must be hoped that it gets the attention it deserves. (Potential readers are recommended to have Demopoulos (2010) and (2012) on hand.) I turn now to a brief account of the remaining chapters.

The theme of scientific theories and how theoretical terms should be understood runs through the book and its basic tenets are outlined in the introduction. The stress here is on the importance of theory-mediated measurement, something first noted by Newton in the 3rd edition of his Principia, as against the partial interpretation of theories under the logical empiricists, Carnap in particular. The second chapter lays out this argument in detail. It threads through a number of familiar needles: the Putnam model-theoretic argument; the significance of Ramsey sentences; van Fraassen’s constructive empiricist view, as outlined in The Scientific Image; the Newman argument against Russell’s ‘causal theory of perception’ in Analysis of Matter — as resurrected in the 80s by Demopoulos and Friedman. The criticism of constructive empiricism has its roots in a trenchant 1982 review from Demopoulos of The Scientific Image and was continued in a 2003 article. The argument about unobservable entities is continued here with Demopoulos arguing for the inadequacy of a response to him by van Fraassen. This argument with van Fraassen is continued in chapter 2 on Jean Baptiste Perrin. The M.H.A. Newman 12-page argument against Russell is fascinating: Demopoulos has developed it over a number of publications since 1985 in such a way that it has come to seem like one of the central works of 20th century philosophy. It is a new point here that Frank Ramsey developed his idea of Ramsey sentences as a result of being confronted by early chapters of Russell’s Analysis of Matter. It is one of Demopoulos’ major contributions to our understanding.

All in all, this posthumous publication is a wonderful tribute to a very significant philosopher.

References

Bohr, N. 1935. Can Quantum-Mechanical Description of Physical Reality be considered Complete?, Physical Review, 48, 696–702.

Demopoulos, W. (1977). Completeness and realism in quantum mechanics. In: Butts, R., Hintikka, J. (eds.) Foundational Problems in the Special Sciences, 81–88. Reidel, Dordrecht.

Demopoulos, W. (2010). Effects and Propositions. Foundations of Physics, 40(4), 368–389. doi: 10.1007/s10701-009-9321-x 

Demopoulos, W. (2012). Generalized Probability Measures and the Framework of Effects. Ch. 13 of  Probability in Physics ed. Yemima Ben-Menahem and Meir Hemmo, The Frontiers Collection, Berlin: Springer-Verlag.

Hrushovski, E. (2020). On the Descriptive Power of Probability Logic, in M. Hemmo, O. Shenker (eds.), Quantum, Probability, Logic, Jerusalem Studies in Philosophy and History of Science. doi: 10.1007/978-3-030-34316-3_17

Kernaghan, M. (1994). Bell-Kochen-Specker theorem for 20 vectors. Journal of Physics A: Mathematical and General, 27(21), 829–830. doi: 10.1088/0305-4470/27/21/007 

Kochen, S. and Specker, E. P. (1967). The problem of hidden variables in quantum mechanics. J. Math. Mech. 17, 59–87. doi: 10.1512/iumj.1968.17.17004

Pitowsky, I. (1998). Infinite and finite Gleason’s theorems and the logic of indeterminacy. Journal of Mathematical Physics, 39, 218–228. doi: 10.1063/1.532334 

Pitowsky, I. (2003). Betting on the outcomes of measurements: a Bayesian theory of quantum probability. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 34(3), 395–414. doi: 10.1016/s1355-2198(03)00035-2 

Schilpp, P. A. (ed.) 1949. Albert Einstein, Philosopher-Scientist, Library of Living Philosophers Vol. VII, N.Y.: MJF Books.

Introduction

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The purpose of this blog is to say some things that don’t fit into the format of a traditional academic paper. There are many such things, and they would, in the past, have gone unsaid, or have been said only in the context of a discussion among academics over coffee. To the outside world such things are invisible — but they constitute the living world in which academic debates and disagreements exist and they constitute the invisible 0.9 iceberg as compared to the published tip (as it were). It is the existence of this larger context that makes it virtually impossible for anyone who is not within a University to understand what is going on in a particular discipline — and I would say this is especially true for philosophy. For example, just browsing a philosophy bookshelf in your local bookstore would give you no idea what it is philosophers do or what they are interested in — and this would be true even if the bookstore is well-stocked with recent publications. But this has the consequence that no one who is not in philosophy can really understand the subject, though many people will be deceived into thinking that they can and do.

This is a great pity. But it is more than that — it has divorced philosophy from the culture that surrounds it and consequently left that culture philosophy-less, clutching on to fragments of misunderstood eastern mysticism and wisdom literature, blended together with cat-memes and the saloon-bar sawdust of political outrage. I can’t possibly hope to reverse the damage that has already been done. But it may do something (an epsilon of something) to say a few things that are fragments of that wider philosophical discussion, not behind closed doors but out in the open. However I can only do that from within my own areas of interest, which are Philosophy of Physics, Philosophy of Mathematics, Epistemology and Metaphysics. Perhaps the best that could be hoped for is that people will understand their debt to Plato or Aristotle — or have a little more knowledge of what they were saying, rather than at present, where it is vanishing to zero.

As you can see, I’m a pessimist even in my optimism.