applying $-\text{log }$ to the right hand side of the denominator identity
(2.1) gives

$\sum_{\begin{subarray}{c}\ell\in\mathbb{N}\\\
\ell|\mathbb{k}\end{subarray}}\frac{1}{\ell}\operatorname{mult}\big{(}\eta\left(\mathbb{k}/\ell\right)\big{)}=|\pi^{G}_{\mathbb{k}}(q)[q]|$
(5.6)

if $\beta(\mathbb{k})\in\Delta_{0}^{+}$ and

$\sum_{\begin{subarray}{c}\ell\in\mathbb{N}\\\
\ell|\mathbb{k}\end{subarray}}\frac{(-1)^{l+1}}{\ell}\operatorname{mult}\big{(}\eta\left(\mathbb{k}/\ell\right)\big{)}=|\pi^{G}_{\mathbb{k}}(q)[q]|$
(5.7)

if $\beta(\mathbb{k})\in\Delta_{1}^{+}$.

The statement of the corollary is now an easy consequence of the following
Möbius inversion formula: $g(d)=\sum_{d|n}f(d)\iff
f(n)=\sum_{d|n}\mu(\frac{n}{d})g(d)$ where $\mu$ is the möbius function. ∎

######  Example 10.

Consider the BKM superalgebra $\mathfrak{g}$ and the root space
$\eta(\mathbb{k})=3\alpha_{3}+3\alpha_{6}\in\Delta_{+}^{1}$ from Example 2.
The $\mathbb{k}$-chromatic polynomial of the quasi Dynkin diagram $G$ of
$\mathfrak{g}$ is equal to

$\pi^{G}_{\mathbb{k}}(q)=\binom{q}{3}\binom{q-3}{3}=\frac{1}{3!3!}q(q-1)(q-2)(q-3)(q-4)(q-5).$

By Corollary 1, since $\eta(\mathbb{k})$ is odd,

$\displaystyle\operatorname{mult}(\eta(\mathbb{k}))$
$\displaystyle=\sum\limits_{\ell|\mathbb{k}}\frac{(-1)^{l+1}\mu(\ell)}{\ell}\
|\pi^{G}_{\mathbb{k}/\ell}(q)[q]|$
$\displaystyle=|\pi_{\mathbb{k}}^{G}(q)[q]|+\frac{\mu(3)}{3}|\pi_{\mathbb{k}^{\prime}}^{G}(q)[q]|\text{
where }\mathbb{k}^{\prime}=(0,0,1,0,0,1)$
$\displaystyle=\frac{10}{3}-\frac{1}{3}$ $\displaystyle=3$

######  Example 11.

Consider the BKM superalgebra $\mathfrak{g}$ from the previous example. Let
$\mathbb{k}=(2,1,0,1,2,0)\in\mathbb{Z}_{+}[I]$. Then
$\eta(\mathbb{k})=2\alpha_{1}+\alpha_{2}+\alpha_{4}+2\alpha_{5}\in\Delta_{+}^{0}$.
We have

$\operatorname{mult}(\eta(\mathbb{k}))=\sum\limits_{\ell|\mathbb{k}}\frac{\mu(\ell)}{\ell}\
|\pi^{G}_{\mathbb{k}/\ell}(q)[q]|$

This implies that
$\operatorname{mult}(\eta(\mathbb{k}))=|\pi^{G}_{\mathbb{k}}(q)[q]|$. We have
$\pi^{G}_{\mathbb{k}}(q)=\frac{1}{4}q(q-1)^{3}(q-2)^{2}$. Therefore
$\operatorname{mult}(\eta(\mathbb{k}))=1$.

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# Quantum theory is logically inevitable

Lars M. Johansen  Department of Science and Industry Systems, University of
South-Eastern Norway,Hasbergs vei 36, Kongsberg, N-3616, Norway  E-mail:
<EMAIL_ADDRESS>
###### Abstract

General relativity required the abandonment of Euclidean geometry. Here we
show that quantum theory requires the abandonment of classical logic. We show
that the Hilbert space representation of quantum theory is logically
inevitable. There are no fundamental principles of quantum theory. We find an
inevitable generalization of classical logic. The conjunction can take
negative values, and reduces to the classical conjunction for order invariant
measurements. The expectation of the conjunction is a generalized joint
probability that can take negative values. A commutative conjunction leads to
the Hilbert space formalism of quantum theory. Quantum theory applies both to
microscopic and macroscopic systems. Classicality is represented by non-
negativity of the generalized joint probability. We illustrate this by
applying the logic to explain puzzling results in quantum cognition.

According to his research assistant Ernst Strauss [1], Albert Einstein once
stated that “what really interests me is whether God could have created the
world any differently; in other words, whether the demand for logical
simplicity leaves any freedom at all”.

Quantum theory is probably the greatest intellectual achievement of the past
century. It seems to be a general law of nature with universal validity. There
are no experiments that has shown any deviation from what it predicts.
Obviously, there is something very general in the foundations of the theory.
However, we don’t know what that is. The theory is founded on abstract Hilbert
space axioms with no direct connection with reality [2]. This has led to a
number of conceptual problems, in particular the so-called measurement
problem. Many interpretations of quantum theory have been suggested. But the
opinion on interpretation is more divided than ever [3, 4].

The axiomatic Hilbert space formulation of quantum theory was published by von
Neumann in his legendary textbook on the mathematical foundations of quantum
mechanics [2]. But soon after, he wrote in a letter to Birkhoff that “I do not
believe absolutely in Hilbert space any more” [5]. In a joint paper, they set
out to “discover what logical structure one may hope to find in physical
theories which, like quantum mechanics, do not conform to classical logic”
[6]. The paper initiated a long lasting research program in quantum logic.

The starting point for Birkhoff and von Neumann was to build the logic of
quantum theory on a specific mathematical structure. An extensive discussion
over mathematical structures and axiomatic foundations of the logic itself
ensued. It led, in the end, to a “labyrinth” of possible quantum logics [7].
Since the axiomatic foundations of the logic itself were perhaps even more
abstract than the Hilbert space foundations of quantum theory, the program
made little or no impact on mainstream physics [8]. The problem, originally
posed by Birkhoff and von Neumann, of the _operational_ significance of
logical connectives, never found an adequate solution [9]. The program did not
lead to the desired conceptual clarification of quantum theory [10], and
interest in the program has waned in recent years.

The search for the conceptual foundations of quantum theory has now turned
towards reconstructing quantum theory from intuitively comprehensible
principles. This has been inspired by the way Einstein constructed his theory
of relativity. The approach was initiated by Hardy, who derived quantum theory
from “five reasonable axioms” [11]. He considered the operational procedures
of preparations, transformations and measurements. This work gave rise to a
number of reconstruction efforts in a tradition now called generalised
probabilistic theories. Many reconstruction schemes have been found, in
particular on basis of informational principles (see e.g. [12]). However, a
unique identification of fundamental principles seems far away [13].

George Boole, the founder of the mathematical formulation of classical logic,
in his classic treatise “The Mathematical Analysis of Logic” [14] assumed that
“the result of two successive acts is unaffected by the order in which they
are performed”. He represented this by a commutativity relation. Otherwise, he
stated, “the entire mechanism of reasoning, nay the very laws and constitution
of the human intellect, would be vitally changed. A Logic might indeed exist,
but it would no longer be the Logic we possess” [14].

Boole was thinking about mental acts. We know today that mental acts are order
dependent. For example, people may respond differently to surveys depending on
the order of questions asked. This is a standard textbook material in research
methods (see e.g. [15]). In recent years, a new research area related to order
dependence in human cognition has developed (see e.g. [16]). And order
dependence is of course well known in quantum theory. For example, if the spin
of an electron is measured sequentially along two different directions, the
result will depend on the order of the operations [17].

## The logic of ordered questions

In this paper, when using the term “quantum theory” we will essentially be be
referring to the axiomatic Hilbert-space formulation of the theory. This is in
analogy with how it’s done in the field of quantum information (for a more
careful distinction, see [18]). We will not address unitary time development,
only the time development associated with measurements. This is where the
major conceptual problems of quantum theory appear.

Basic to our analysis is the notion of _yes-no questions_. Any experiment may
be formulated as a yes-no question. We may consider any event, as a subset of
sample space, and ask if the event happened or not. If the answer to the
question $A$ is “yes”, we assign the value $A=1$. If the answer is “no” we
assign the value $A=0$. Therefore, per definition, a yes-no question is
idempotent, $A^{2}=A$. Associated with a question $A$ is also the
complementary question $\bar{A}$. The answer to $\bar{A}$ is always the
opposite to the answer to $A$. We write this symbolically as

$A+\bar{A}=I_{A}$ (1)

where the value of $I_{A}$ is 1.

Consider now a sequence of two yes-no questions, $A$ and then $B$. We call
this a _sequential yes-no question_ , and represent it by the _sequential
conjunction_ $A\sqcap B$. The answer to a sequential question is also always
“yes” or “no”. It is “yes” if the answer to both questions is “yes”.
Otherwise, if the answer to one or both of the questions is “no”, the answer
to the sequential question is also “no”. Therefore, the value of $A\sqcap B$
is the value of $A$ multiplied with the value of $B$ obtained afterwards. In
contrast to Boole’s approach, we allow the product to be non-commutative,
allowing the question order to be of significance.

We now form the _exclusive disjunction_

$\displaystyle A\oplus B$ $\displaystyle=A\sqcap\bar{B}+\bar{A}\sqcap B.$ (2)

The combined sequential question $A\oplus B$ consists of two disjoint
sequential questions $A\sqcap\bar{B}$ and $\bar{A}\sqcap B$. The answer to
these two sequential questions can never be both “yes” at the same time.
Therefore, we can write the disjunction as a sum.

We now consider exclusive disjunctions of both question orders, $A\oplus B$
and $B\oplus A$. We substitute for the complementary questions. Since the
answer to a question and it’s complementary questions contains the same
information, the sequential conjunction can be treated as distributive in this
case. We therefore have

$\displaystyle A\oplus B-B\oplus A=2\left(B\wedge A-A\wedge B\right)$ (3)

where we have introduced the _logical conjunction_

$\displaystyle A\wedge B=A\sqcap B+\frac{1}{2}\left(B-B_{A}\right).$ (4)

We write down connectives only for one question order. For simplicity, we have
introduced the notation $B\sqcap I_{A}=B$ and $I_{A}\sqcap B=B_{A}$ for single
questions. Here $B_{A}$ refers to the question $B$ after a nonselective
question $A$, i.e. after disregarding the answer to the question $A$. It is
easily shown that the logical conjunction satisfies the marginality relations

$\displaystyle\begin{split}A\wedge B+A\wedge\bar{B}&=A,\\\ A\wedge
B+\bar{A}\wedge B&=B.\end{split}$ (5)

The last term on the r.h.s of Eq. (4) is the non-classical correction to
classical logic. This term is what differentiates the logic from a Boolean
algebra. The possible values for the logical conjunction ${A\wedge B}$ for all
possible answers to the questions $A$, $B$ and $B_{A}$ are given in table 1.
Notably, it may take negative values. If $B=B_{A}$ the logical conjunction is
Boolean.

Table 1: Value-table for logical connectives.

Conjunction $A\wedge B$
---
| $B=0$ | $B=1$
| $B_{A}=0$ | $B_{A}=1$ | $B_{A}=0$ | $B_{A}=1$
$A=0$ | 0 | $-\frac{1}{2}$ | $\frac{1}{2}$ | 0
$A=1$ | 0 | $\frac{1}{2}$ | $\frac{1}{2}$ | 1
Inclusive disjunction $A\vee B$
| $B=0$ | $B=1$
| $B_{A}=0$ | $B_{A}=1$ | $B_{A}=0$ | $B_{A}=1$
$A=0$ | 0 | $\frac{1}{2}$ | $\frac{1}{2}$ | 1
$A=1$ | 1 | $\frac{1}{2}$ | $\frac{3}{2}$ | 1

The possible values of the logical connectives as a result of answers to the
questions $A$, $B$ and $B_{A}$. Both connectives are four-valued. Boolean
truth-values are reproduced when $B_{A}=B$ (leftmost and rightmost columns).
Non-classical half-integer values appear when $B_{A}\neq B$. The conjunction
can take a minimum value of $-1/2$, and the inclusive disjunction can take a
maximum value of $3/2$.

By rearrangement and simplification of Eq. (3) we obtain equations of the form

$\displaystyle A\oplus B+2A\wedge B=A+B.$ (6)

In analogy with Boolean algebra, we can also _define_ the inclusive
disjunction as

$\displaystyle A\vee B=A+B-A\wedge B.$ (7)

From (6) we can then show that

$\displaystyle\begin{split}A\vee B&=A\oplus B+A\wedge B,\end{split}$ (8)

which also is in correspondence with classical logic.

The possible values for the inclusive disjunction $A{\vee}B$ for all possible
answers to the questions $A$, $B$ and $B_{A}$ are given in table 1. We note
that it can exceed unity. If $B=B_{A}$ the inclusive disjunction is also
Boolean.

## Logical joint probabilities

The expected value of an question $A$ is the _probability_ that the question
$A$ gives the answer $A=1$, $\langle A\rangle=P(A=1)$. ${\langle A\sqcap
B\rangle}$ is the sequential probability of first obtaining the answer $A=1$
and then $B=1$. The _logical joint probability_ $\langle A\wedge B\rangle$ is
the expected values of the conjunction (4).

The logical joint probability is a generalization of the real part of the
Kirkwood-Dirac quasi-probability distribution [19, 20]. The logical
conjunction (4) has been encountered before in the analysis of successive
projective measurements in quantum mechanics [21]. In this case, the logical
joint probabilities are order invariant. However, in the papers [21, 22], the
evolution due to observation was represented by the state. Here, the evolution
is represented by the questions. It has also been shown that this structure is
related to weak values [23, 22]. The logical joint probability can take
negative values [24, 25].

Wigner demonstrated that quantum states can be represented as quasi-
probabilities over phase space [26]. These quasi-probabilities can take
negative values for some quantum states. Soon after, Kirkwood derived a quasi-
probability over phase space that could take complex values [19]. This was
further generalized by Dirac to arbitrary pairs of observables [20]. The
Kirkwood-Dirac distribution is closely related to the concept of weak
measurements and weak values [23]. The weak value of an observable may exceed
the eigenvalue spectrum. This requires that the real part of the Kirkwood-
Dirac distribution is negative [24].

Figure 1: Order-invariant logical joint probabilities

In a Gallup poll conducted during September 6–7, 1997, people were asked ”Do
you generally think [Bill Clinton/Al Gore] is honest and trustworthy?” Half of
the 1002 respondents were asked about Clinton first and Gore afterwards, the
other half where asked the same questions in the opposite order. The results
exhibited a striking order effect. The data were examined in [27] and found to
satisfy a so-called quantum question equality [28]. We have reconstructed the
logical joint probabilities. Observed sequential probabilities are shown in
blue hues, and reconstructed logical joint probabilities are shown in red
hues. $C$ is the question if Clinton is honest and trustworthy, $G$ is the
question if Gore is honest and trustworthy. The order effect on sequential
probabilities is apparent. Nevertheless, we see that ${\langle C\wedge
G\rangle}$ and ${\langle G\wedge C\rangle}$ are virtually identical. The
observation is approximately ideal. The case $C=1,G=0$ is particularly
striking. Here both sequential probabilities are positive whereas the logical
joint probabilities are essentially zero. In fact, they are both slightly
negative, but the effect is not statistically significant.

In Ref. [28] a socalled “quantum question equality” was derived from quantum
mechanical principles. In our notation, it reads $\langle A\oplus
B\rangle=\langle B\oplus A\rangle$. This implies by (6) that $\langle A\wedge
B\rangle=\langle B\wedge A\rangle$. In [27] the quantum question equality was
tested on a number of opinion polls. It was found that a requirement for the
opinion polls to satisfy the quantum question equality was that questions
should be asked successively with no additional information inserted in or
between questions [27]. An example of a poll satisfying the quantum question
equality is shown in Fig. 1. We see that although the sequential probabilities
are strongly order dependent, the calculated logical joint probabilities are
nearly perfectly order invariant. We also note that all logical joint
probabilities are non-negative, which is typical of the classical regime.

## Commutative logic, Jordan algebras and Hilbert space

Eq. (3) and the logical conjunction (4) apply to questions of any form, and
regardless of anything that happens during or between the two questions. In
general, none of the logical connectives are commutative. But of particular
interest are commutative logical connectives. This would correspond to a
particular subclass of sequential questions. Thus, we _define_ sequential
questions as _ideal_ if the logical conjunction is commutative,

$A\wedge B=B\wedge A.$ (9)

Because of Eqs. (3) and (7), this also implies that the exclusive and
inclusive disjunction is commutative.

We conjecture that a commutative logic can be represented by an associative,
non-commutative algebra. By making the Lüders _ansatz_ [29]

$\displaystyle A\sqcap B\rightarrow A{\cdot}B{\cdot}A$ (10)

it follows that

$\displaystyle B_{A}=A\sqcap B+\bar{A}\sqcap B\rightarrow
A{\cdot}B{\cdot}A+\bar{A}{\cdot}B{\cdot}\bar{A}$ (11)

and hence

$\displaystyle A\wedge B\rightarrow A\circ B,$ (12)

where

$\displaystyle A\circ B=\frac{1}{2}\left(A{\cdot}B+B{\cdot}A\right)$ (13)

is the commutative Jordan product [30]. The ansatz also implies that

$\displaystyle A\oplus B\rightarrow
A{\cdot}\bar{B}{\cdot}A+\bar{A}{\cdot}B{\cdot}\bar{A}.$ (14)

It can be shown that also this expression is commutative. In sum, due to the
Lüders ansatz all connectives are commutative. Furthermore, the Jordan product
also satisfies the marginality relations (5).

Jordan algebras where the Jordan product takes the form (13) are called
_special_. Jordan algebras are a separate research field in mathematics [31].
Although Jordan algebras were originally proposed in physics, their physical
meaning has remained obscure until now. It has not been obvious why the
observables of a physical system should carry any physically meaningful
bilinear product [32]. However, a connection between the requirement of
symmetry of the exclusive disjunction and Jordan algebras was noted in Ref.
[33].

In a logical system, a question being asked twice should give the same answer.
Therefore, we require that ${A\sqcap A=A}$. This translates, by (10) to the
requirement $A{\cdot}A{\cdot}A=A$, which is satisfied for $A\cdot A=A$. For
the reverse order we have the requirement $B{\cdot}B=B$. It then follows that

$A\circ A+B\circ B=A+B=0\quad\Rightarrow\quad A=B=0.$ (15)

This means that the Jordan algebra is _formally real_. Jordan, von Neumann and
Wigner [34] showed that all special, finite-dimensional, formally real Jordan
algebras are direct sums of simple Jordan algebras of three different types:
self-adjoint parts of real, complex or quaternionic matrix algebras. These
results have also been generalized to Jordan algebras with an infinite number
of degrees of freedom [35, 36]. Thus, the axiomatic Hilbert space foundations
of quantum theory follow from a commutative logic.

A number of recent publications have also shown that formally real Jordan
algebras follow from various principles of probabilistic or information
theoretical character (see e.g. [37, 38, 39]).

Quantum theory is usually formulated in terms of a complex Hilbert space. We
are not discussing here whether some specific principles single out complex
Hilbert space. Discussions on this can be found elsewhere (see e.g. [40]).

## Discussion

We build the logic on the tautology (3). It is true for all questions
regardless of the answers. Interestingly, some philosophers have taken
tautologies as defining the laws of logic [41]. However, the possibility that
tautologies might redefine classical logical connectives does not seem to have
been contemplated.

Since the tautology (3) and the logical conjunction (4) apply to questions of
any sort, independently of what happens in or between the questions, the logic
also applies to generalized measurements as they are currently defined in
quantum mechanics. However, in this case logical connectives will not be
commutative.

## Acknowledgements

The author acknowledges valuable comments from Kjetil Børkje and Leon
Loveridge.

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