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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