# The Reasonable Effectiveness of
Diverse Evaluation Data

Lora Aroyo
Google Research
New York, US
<EMAIL_ADDRESS>
Mark Díaz
Google Research
New York, US
<EMAIL_ADDRESS>
Christopher Homan
Google Research
New York, US
<EMAIL_ADDRESS>
Vinodkumar Prabhakaran
Google Research
San Francisco, US
<EMAIL_ADDRESS>
Alex Taylor
Google Research
London, UK
<EMAIL_ADDRESS>
Ding Wang
Google Research
Singapore, Singapore
<EMAIL_ADDRESS>

###### Abstract

In this paper, we present findings from an semi-experimental exploration of
rater diversity and its influence on safety annotations of conversations
generated by humans talking to a generative AI-chat bot. We find significant
differences in judgments produced by raters from different geographic regions
and annotation platforms, and correlate these perspectives with demographic
sub-groups. Our work helps define best practices in model development–
specifically human evaluation of generative models– on the backdrop of growing
work on sociotechnical AI evaluations.

##  1 Introduction

In their 2009 paper “The Unreasonable Effectiveness of Data” [4], Alon Halevy,
Peter Norvig, and Fernando Perreira urge ML researchers to “follow the data
and see where it leads.” Since then, we have at an unprecedented scale amassed
data for the purpose of machine learning. Yet, data quality—including the
quality of human-collected data—has been left behind.

While there is a substantial body of work focusing on the reliability of human
raters when performing evaluations, there are few [2, 5, 3, 6] if any studies
investigating how the characteristics of rater pools impact ratings. That is,
we know little about annotators’ individual characteristics (such as
nationality, gender, education, race) and how they might influence the way
they label data. This matters because, in seeking to build fair and
responsible AI systems, we should anticipate potential biases that may emerge
as a result of differences across user populations, and evaluation data should
represent a variety of populations in order to better reflect viewpoints among
real-world stakeholders and build diversity aware ground truth datasets[1].

Responding to the _Data-Centric AI_ call to study impacts of data on AI
systems [7], we present findings from a semi-experimental exploration of rater
diversity and its influence on safety annotations of chat bot conversations.
We report results from a large-scale rater diversity study performed on a
sample of 990 conversations generated by humans conversing with a generative
AI-chatbot. We collected safety labels from a pool of 96 raters (recruited
from two rating platforms covering a range of socio-economic subgroups) using
24 safety questions. Rather than requesting the typical number of ratings (a
single rating or three-to-five ratings per conversation), we collected 40
ratings per safety question and repeated the experiment after 4 months. We
analyzed:

  * •

the variance, disagreement and reliability within and across different
demographic slices.

  * •

the majority and minority perspectives on the data items.

  * •

the consistency of raters across replications of the experiment.

The contribution and implications of our research are threefold. First, we
propose a methodological approach for assessing the influences of rater
diversity on data labels. Second, we demonstrate that raters’ backgrounds do
have an impact on the labels they provide. Third, we illustrate that there are
discrepancies in rater consistency across two replications. As such, we
contribute to two topics in this workshop: (1) repeatability of human
evaluations and (2) methods for assessing the quality and the reliability of
human evaluations.

##  2 Experimental Design for Diverse Human Evaluation

The overall research question guiding our study was: Do rater demographics
help us improve reliability of annotations and ultimately provide guidance for
collecting representative data for AI development and evaluation?

### Data:

The input data for this study was sampled from a 8K conversations dataset
(composing 48K turns in total) generated by humans interacting with a
generative AI-chatbot. In our study, the sample of 990 conversations was
annotated for 24 safety characteristics (plus one general question about
conversation understandability) by 40 unique raters per conversation, per
safety characteristic. This annotation experiment was repeated with a
different pool of 96 raters (including 19 overlapping raters in both
replications) 4 months later. The total number of annotations gathered is
close to 2 million (990K in each repetition).

### Task:

The safety evaluation task used a UI designed for the safety characteristics
as defined in Appendix. Each rater was asked to rate the final chatbot
response in each conversation as _Unsafe_ , _Safe_ or _Unsure_. These answers
were given for each of the 24 safety questions, which were organized in three
groups (Q2-Q4 below), plus one question (Q1) checking the overall
comprehensibility of the conversation.

  * •

Q1: Evaluate the whole conversation

  * •

Q2: Evaluate the last bot utterance for “Harmful Content”

  * •

Q3: Evaluate the last bot utterance for “Content with Unfair Bias”

  * •

Q4: Evaluate the last bot utterance for “Misinformation and Political
References”

### Data Collection:

We collected the ratings in two phases with an interval of four months. In
both phases, we recruited 96 unique raters from two rater pools. All raters
performed the task independently and used the same annotation template. All
raters were asked to complete an optional demographic survey (e.g. gender,
ethnicity, education level, age group, and native language). All questions in
the demographic survey gave raters the option to select "Prefer not to
answer". We also collected data about the average annotation time per
conversation and the total time each rater spent annotating.

### Raters:

In Phase 1, the breakdown of raters was: 71 in Pool 1 (42 India, 29 US), 25 in
Pool 2 (12 India, 13 US). For Phase 2, the break down was: 72 in Pool 1 (40
India, 32 US), 24 in Pool 2 (12 India, 12 US). 19 of the raters participated
in both phases (5 from Pool 1, 14 from Pool 2; the 5 raters from Pool 1 are
all in US, 6 out of the 14 from Pool 2 are in India and 8 in US; 9 identify as
female and 10 as male). In this paper, we report results from the Phase 2,
however we compare the two phases to measure consistency for the raters who
participated in both.

##  3 Results

We present four key high-level observations from this study that contribute to
our understanding of reliability of human evaluations, and its relation to
diversity among raters.

### Unreliability of gold labels:

The left side of Fig. 1 shows the difference between the number of raters
saying _Unsafe_ vs. _Safe_ for each of the 990 conversations in our data. For
around a quarter of the conversations, the number of _Unsafe_ and _Safe_
responses per conversation are quite similar, i.e., between 15-25 votes on
either side. If only 3 to 5 annotators were to rate each item, as is common
practice among researchers and practitioners building annotated datasets, this
level of observed disagreement in these conversations may easily be lost. This
suggests that majority-based (or even ‘unanimous’) gold labels may be
unreliable for a significant portion of the data, if the replication per item
is low. This is a critical issue, since many evaluation tasks, even related to
sensitive topics such as online safety, use such majority-based gold
annotations to measure rater and model performance.

Figure 1: Left side: Conversations are arranged horizontally, ranked by the
difference between _Unsafe_ and _Safe_ votes. The y-axis shows this
difference. The red square points to roughly a quarter of the conversations
with nearly equal numbers of _Unsafe_ and _Safe_ votes. Right side: Histogram
of the number of times each of the 19 raters who rated items in both phases
disagreed with themselves.

### Inadequate intra-rater consistency:

We now test whether the 19 raters who were present in both phases were
consistent in their ratings. For the subset of items each of them annotated in
both phases, we measured the number of times they disagreed with themselves
(i.e., between phases) at least once — considering all 25 questions
separately. The right side of Fig. 1 shows the histogram of disagreements for
these 19 raters. Eleven of these raters disagreed with themselves at least
ten, and as many as 3,199, times. This is another concerning finding that
suggests there are extraneous factors that may significantly influence the
consistency in raters’ responses across different sittings at different points
in time.

### Disparate within-group coherence across subgroups:

Despite the issues with consistency and reliability, we observed significant
patterns in rater behaviour within and across the various subgroups we
considered. For this analysis, we modeled each annotator’s response to a
conversation as a 72-dimensional response vector that captures the one-hot
encoding of the {UNSAFE, SAFE, UNSURE} answers for each of the 24 safety
questions (Q2-Q4). This allows us to calculate the pair-wise distance between
the response vectors of two raters as a metric for how strongly they disagreed
with one another on any particular conversation prompt.

Fig. 2 shows the average _hamming distance_ between all pairs of response
vectors for each conversation, averaged across raters within different
subgroups of raters. We observe that the average within-group rating distances
vary substantially across groups.

Lower hamming distance between a subgroup and _All raters_ means that the
subgroup is consistent within itself and different than all raters. The
results show disparities in agreement along three demographics - between US
and Indian raters, between Pool 1 and Pool 2 and between female and make
raters. In particular, US male raters in Pool 1 behaved more similarly among
themselves than any of the other groups studied.

Figure 2: The hamming distance metric on gender, locale and pool slices

### Cross-group differences between subgroups:

The above analysis provides only a partial picture, one that captures within-
group distances but says nothing about whether the rating behaviors of a
certain group of raters is more likely to be similar to others in the same
subgroup than to those outside the subgroup. For instance, low within-group
distance suggests that a particular subgroup has a coherent perspective on the
task. If two different subgroups along a diversity axis (say, gender) exhibit
such high within-group coherence, but also have low cross-group distance, it
suggests that this particular diversity axis may not have any substantial
influence in the context of this task. However, if two groups that have low
within-group distance, also have high cross-group distance, it suggests that
the diversity axis is a substantial differentiator for the task.

Fig. 3 shows the within-group and cross-group distance along the locale (IN
vs. US), pool (Pool 1 vs. Pool 2), and gender axes. The results show that
while US raters produced significantly more similar ratings with other US
raters, compared to IN raters, on average. In the case of gender, female
raters produced ratings that are very similar to each other, and significantly
dissimilar to the ratings produced by male raters. Moreover, there was not
much variance in the average distance across different female rater pairs,
whereas male raters exhibited high variance across pairs in how much they
disagreed with one another. While we observe some difference between the Pool
1 and Pool 2, those differences are not statistically significant.

Figure 3: Average pairwise hamming distance across different locale, platform,
gender slices. The number of pairs within each group is specified in
parenthesis.

##  4 Reflections

In this paper we are excited to share just a few of the high-level results
from the presented work. These results offer a clear indication that raters’
demographics and the pool from which raters have been recruited have an impact
on labelling tasks. Because the analysis we have done thus far is relatively
coarse-grained, we believe that slicing further into the ethnicity, native
languages and age groups of the raters is likely to reveal further insights
and provide additional evidence of systematic differences between different
groupings of raters. We will be conducting this detailed analysis with the
ethnicity, age group and native language data that accompanies our data corpus
and reporting results in upcoming publications.

As we propose a methodology for assessing the influences of rater diversity on
data labels, our future work will also focus on determining the optimal number
of raters per conversation and to what extent the impacts of rater diversity
can be captured in smaller numbers of raters. This will be done in order to
improve dataset generation methods that aim to address rater diversity.

Finally, we recognize more work is needed to help distinguish _good_ from
_bad_ disagreement. In our work, this could be done by correlating the
temporal data with other behavioral traits in raters across the two
replications. Ultimately, this would extend our methodology to include an
approach for studying outliers and different annotation perspectives.

## References

  * [1] Lora Aroyo and Chris Welty.  Truth is a lie: Crowd truth and the seven myths of human annotation.  AI Magazine, 36(1):15–24, 2015.
  * [2] Mark Díaz, Isaac Johnson, Amanda Lazar, Anne Marie Piper, and Darren Gergle.  Addressing age-related bias in sentiment analysis.  In Proceedings of the 2018 chi conference on human factors in computing systems, pages 1–14, 2018.
  * [3] Nitesh Goyal, Ian Kivlichan, Rachel Rosen, and Lucy Vasserman.  Is your toxicity my toxicity? exploring the impact of rater identity on toxicity annotation.  arXiv preprint arXiv:2205.00501, 2022.
  * [4] Alon Halevy, Peter Norvig, and Fernando Pereira.  The unreasonable effectiveness of data.  IEEE intelligent systems, 24(2):8–12, 2009.
  * [5] Yiwei Luo, Dallas Card, and Dan Jurafsky.  Detecting stance in media on global warming.  arXiv preprint arXiv:2010.15149, 2020.
  * [6] Vinodkumar Prabhakaran, Aida Mostafazadeh Davani, and Mark Diaz.  On releasing annotator-level labels and information in datasets.  In Proceedings of The Joint 15th Linguistic Annotation Workshop (LAW) and 3rd Designing Meaning Representations (DMR) Workshop, pages 133–138, 2021.
  * [7] Nithya Sambasivan, Shivani Kapania, Hannah Highfill, Diana Akrong, Praveen Kumar Paritosh, and Lora Aroyo.  "everyone wants to do the model work, not the data work": Data cascades in high-stakes ai.  2021\.


# The Complexity of Checking Partial Total Positivity111This work was
supported by the 2018 National Science Foundation Grant DMS #0751964.

Daniel Carter Department of Mathematics, Princeton University, Princeton, NJ
08544 Charles Johnson Department of Mathematics, College of William and Mary,
Williamsburg, VA 23187-8795<EMAIL_ADDRESS>
(2020-2021)

###### Abstract

We prove that checking if a partial matrix is partial totally positive is co-
NP-complete. This contrasts with checking a conventional matrix for total
positivity, for which we provide a cubic time algorithm. Checking partial sign
regularity with any signature, including partial total nonnegativity, is also
co-NP-complete. Finally, we prove that checking partial total positivity in a
partial matrix with logarithmically many unspecified entries may be done in
polynomial time.

##  1 Background

An $m$-by-$n$ matrix $A$ is totally positive (TP) or totally nonnegative (TN)
if all of its minors (determinants of square submatrices) are positive or
nonnegative, respectively. Naïvely, a matrix has exponentially many minors
that must be checked to verify it is TP or TN. However, due to Fekete’s
Theorem, only the contiguous minors, that is, minors whose row and column
index sets are consecutive, must be checked to verify total positivity. Even
better, only the initial minors (contiguous minors whose row or column set
includes 1) need to be checked [FJ11]. The number of initial minors in an
$m$-by-$n$ matrix is precisely $mn$, and calculating determinants may be done
in $O(n^{2.373})$ time, so this means there is a polynomial time algorithm for
checking total positivity [AW20]. In Section 2, we demonstrate a simple cubic-
time algorithm to check total positivity based on Dodgson condensation.

A partial matrix is one in which some entries are unspecified and free to be
chosen. A completion of a partial matrix is a choice of values for the
unspecified entries, resulting in a conventional matrix of the same size. The
TP completion problem asks for which partial matrices there exists a
completion that is TP [FJ11]. A partial matrix is partial TP if all of its
minors that consist only of specified entries are positive. Clearly a
necessary condition for there to be a TP completion of a partial matrix is
that the partial matrix is partial TP. This is not sufficient in general. In
Section 3, we prove that determining if a partial matrix is partial TP lies in
the complexity class co-NP-complete, in stark contrast to the case for
conventional matrices. We actually prove a more general result for strongly
and weakly sign regular matrices that includes checking partial TP and TN as
special cases.

Finally, in Section 4, we provide an algorithm for checking partial TP which
runs in exponential time in the number of unspecified entries and polynomial
time in the size of the partial matrix. This generalizes to any matrix
property which is inherited by all submatrices.

Our principal observation about the complexity of checking partial TP is
clearly relevant to the solution of TP completion problems.

##  2 TP in Cubic Time

Dodgson condensation is a method for computing the determinant of a matrix by
repeatedly applying the Desnanot-Jacobi-Sylvester identity

$\det(M)=\frac{\det(M_{1}^{1})\det(M_{n}^{n})-\det(M_{1}^{n})\det(M_{n}^{1})}{\det(M_{1,n}^{1,n})},$

in which $M$ is an $n$-by-$n$ matrix and
$M_{a_{1},a_{2},...}^{b_{1},b_{2},...}$ denotes the submatrix obtained by
deleting rows $a_{1},a_{2},...$ and columns $b_{1},b_{2},...$. Dodgson
condensation calculates the determinants of ever-larger contiguous minors
until the determinant of the whole matrix is found, as follows:

  1. 1.

Set $A\leftarrow M$.

  2. 2.

Let matrix $B$ be $(n-1)$-by-$(n-1)$ with entries
$b_{i,j}=a_{i,j}a_{i+1,j+1}-a_{i,j+1}a_{i+1,j}$. Evidently, the entries of $B$
are the contiguous $2$-by-$2$ minors of $M$.

  3. 3.

Let matrix $C$ be $(n-2)$-by-$(n-2)$ with entries
$c_{i,j}=(b_{i,j}b_{i+1,j+1}-b_{i,j+1}b_{i+1,j})/a_{i+1,j+1}$. In light of the
identity mentioned above, the entries of $C$ are the contiguous $3$-by-$3$
minors of $M$.

  4. 4.

Set $A\leftarrow B$ and $B\leftarrow C$, and repeat step 3 to obtain a new
matrix $C$ of dimension one less than before. This matrix consists of the
contiguous minors of $M$ one size greater than before. Do this until the
resulting matrix is $1$-by-$1$. Its entry is the determinant of the original
matrix $M$.

If ever an interior entry of the previous matrix is zero, step 2 is ill-
defined. However, note that we compute every contiguous minor of the matrix at
some point in the algorithm, so if any entry of any of the matrices is
nonpositive, we can immediately conclude the original matrix was not TP. If
instead the above algorithm terminates with all entries encountered positive,
then all contiguous minors of the original matrix were positive, and it is TP.
It is clear that we may apply the same algorithm on non-square matrices,
stopping when we obtain a matrix with either only one row or only one column.

Analysis of the complexity of this algorithm is straightforward. To get the
contiguous $(k+1)$-by-$(k+1)$ minors from the contiguous $k$-by-$k$ minors,
this algorithm performs two multiplications, one subtraction, and one division
per $(k+1)$-by-$(k+1)$ minor. If the matrix is $m$-by-$n$ with $m\leq n$,
there are $(m-k)(n-k)$ such minors, and the largest minors are $m$-by-$m$ (so
the largest value of $k$ is $m-1$). Therefore the asymptotic runtime is given
by

$\sum_{k=1}^{m-1}(m-k)(n-k)c,$

with $c$ the cost of performing the arithmetic operations. This sum is
$(3m^{2}n-m^{3}-3mn+m)\cdot c/6$, which is $O(m^{2}n)$. For square matrices
this is cubic time.

Note there is no general analogue to Fekete’s Theorem for total nonnegativity,
so the cubic time algorithm presented here cannot be straightforwardly adapted
to get a cubic time algorithm for determining TN. There are non-TN matrices
where all contiguous minors have nonnegative determinant, such as the one
below:

$\begin{bmatrix}1&0&1\\\ 1&0&0\end{bmatrix}.$

##  3 Partial TP is Co-NP-complete

A matrix is called weakly sign regular (WSR) with signature $(e_{i})$,
$e_{i}\in\\{-1,1\\}$ for all $i$, if all of its $k$-by-$k$ minors $A$ satisfy
$e_{k}A\geq 0$ for all $k$. If the minors satisfy $e_{k}A>0$ for all $k$, the
matrix is strictly sign regular (SSR). Partial matrices are partial WSR if
their $k$-by-$k$ specified minors satisfy $e_{k}A\geq 0$ for all $k$ and
similarly for partial SSR. Note that TP (TN) is the special case of SSR (WSR)
with $e_{i}=1$ for all $i$.

A biclique is an induced subgraph of a bipartite graph that is a complete
bipartite graph; a biclique is balanced if it has the same number of vertices
in both parts. The balanced biclique problem asks if there is a balanced
biclique in a given bipartite graph with at least $2k$ for given integer $k$.
The balanced biclique problem is shown to be NP-complete in [Joh87] by a
transformation from the clique problem.

We reduce the balanced biclique problem to instances of the complement of
partial $(e_{i})$-SSR and partial $(e_{i})$-WSR, the problems of checking if a
given partial matrix is partial SSR/WSR with a fixed signature $(e_{i})$. This
shows that partial $(e_{i})$-SSR and partial $(e_{i})$-WSR are co-NP-hard. It
is easy to show they are also co-NP, so they are co-NP-complete. As special
cases, both partial TN and partial TP are also co-NP-complete.

First, we recall a known fact that may be found in [And87] and elsewhere. It
may be proven inductively by “bordering” [FJ11].

######  Fact.

For any positive integers $m$ and $n$ and signature $(e_{i})$, there exists an
$m$-by-$n$ SSR (WSR) matrix with signature $(e_{i})$.

We then have our main theorem:

######  Theorem.

Both partial $(e_{i})$-SSR and partial $(e_{i})$-WSR are co-NP-complete for
any signature $(e_{i})$.

###### Proof.

Co-NP-hard: Given a bipartite graph $G$ with parts
$U=\\{u_{1},u_{2},\dots,u_{m}\\}$ and $V=\\{v_{1},v_{2},\dots,v_{n}\\}$ and an
integer $k$, we construct a particular $m$-by-$n$ partial matrix $M$. First,
let $X$ be an $m$-by-$n$ SSR matrix with signature
$(f_{i})=(e_{1},e_{2},\dots,e_{k-1},-e_{k},-e_{k+1},\dots)$. If
$\\{v_{i},u_{j}\\}$ is an edge of $G$, let $m_{ij}$ be specified and equal to
$x_{ij}$; otherwise let $m_{ij}$ be unspecified.

Now $G$ contains a balanced biclique of size $k$ or greater if and only if $M$
has a $k$-by-$k$ specified minor $A$. This minor, if it exists, satisfies
$f_{k}A>0$. This means $M$ is not partial SSR (or WSR) with signature
$(e_{i})$, since it has a $k$-by-$k$ minor with $-e_{k}A\not>0$
($-e_{k}A\not\geq 0$). Thus, partial $(e_{i})$-SSR (WSR) is co-NP-hard.

Co-NP: Given a partial matrix $M$ which is not partial SSR (WSR) with a given
signature $(e_{i})$, one can provide a witness $k$-by-$k$ minor $A$ that can
be computed in polynomial time which does not satisfy $e_{k}A>0$ ($e_{k}A\geq
0$), so partial $(e_{i})$-SSR (WSR) is co-NP.

We conclude partial $(e_{i})$-SSR (WSR) is co-NP-complete. ∎

Of interest is a special case of this theorem:

######  Corollary.

Both partial TP and partial TN are co-NP-complete.

###### Proof.

The partial TP problem is precisely the partial $(1,1,\dots)$-SSR problem, and
the partial TN problem is precisely the partial $(1,1,\dots)$-WSR problem. ∎

##  4 Algorithms for Checking Partial TP

Consider a partial matrix with one unspecified entry at the $(i,j)$ position.
Any submatrix with only specified entries that includes row $i$ cannot include
column $j$ due to this unspecified entry, and similarly if it includes column
$j$ then it cannot include row $i$. This means to check if this partial matrix
is partial TP, it is sufficient to check if the two conventional matrices
formed by removing row $i$ or column $j$ are TP. These matrices are denoted
$M_{i}$ and $M^{j}$, following the notation in Section 2.

This generalizes to the case of general partial matrices:

######  Theorem.

If a partial matrix $M$ has an unspecified entry at the $(i,j)$ position, then
it is partial TP if and only if the two partial matrices $M_{i}$ and $M^{j}$
are partial TP.

###### Proof.

Any fully specified square submatrix of $M$ must lie entirely within $M_{i}$
or $M^{j}$ (or both), since it cannot include the $(i,j)$ entry. Hence if both
of these partial matrices are partial TP, any specified square submatrix of
$M$ has positive determinant, so $M$ is partial TP. The converse is trivial. ∎

This immediately gives a recursive algorithm to check partial TP:

  1. 1.

If the partial matrix $M$ has no unspecified entries, then check if it is TP
using the cubic time algorithm presented previously.

  2. 2.

Otherwise, say the $(i,j)$ entry is unspecified. Recursively check $M_{i}$ or
$M^{j}$ for partial TP; the original matrix is partial TP if and only if both
submatrices are partial TP.

It is clear this algorithm runs in time $O(2^{x}n^{3})$, where $x$ is the
number of unspecified entries in the partial matrix, since each unspecified
entry generates at most two subproblems with at most $x-1$ unspecified
entries, for a total of at most $2^{x}$ subproblems. Consequently, if $x\leq
c\log n$ for a fixed $c$, this algorithm runs in polynomial time, with the
degree depending on $c$. It is also worth noting that the number of specified
minors of an $m$-by-$n$ partial matrix is at most
$\sum_{k=1}^{\min(m,n)}\binom{m}{k}\binom{n}{k}=\binom{m+n}{n}-1\leq 2^{m+n}$,
so this algorithm is only faster than brute-force checking every specified
minor in the case that there are relatively few unspecified entries.

It is also clear that this generalizes to the case of checking partial SSR or
partial WSR for any sign signature, including partial TN, or indeed to any
matrix property that is inherited by all submatrices. If one has an algorithm
to check a matrix for such a property $P$ that runs in time $O(f(n))$, then
the appropriate modification of the above algorithm will check partial
matrices for “partial $P$” in time $O(2^{x}f(n))$.

Furthermore, one can connect this result to the bipartite graph construction
of the previous section. A maximal biclique is a bliclique that is not a
subgraph of any larger bliclique. Then we have the following more general
theorem:

######  Theorem.

Let $G$ be the bipartite graph corresponding to a partial matrix $M$. Then $M$
is partial TP iff each of the submatrices corresponding to the maximal
blicliques of $G$ are TP.

###### Proof.

Each fully specified square submatrix of $M$ corresponds to a balanced
biclique in $G$. Each balanced biclique is either maximal or a subgraph of a
larger biclique, so if all of the maximal bicliques correspond to TP
submatrices of $M$, then every specified minor is positive, by virtue of being
a minor of a TP submatrix. The converse is trivial. ∎

Therefore any algorithm to enumerate maximal bicliques of a bipartite graph
translates into an algorithm for efficiently checking partial TP, simply by
checking for TP in each maximal biclique (and it will also inherit a cubic
factor from this check).

This is a generalization of the algorithm presented earlier in this section
because that algorithm may be obtained from the observation that, if vertices
$a_{i}$ and $b_{j}$ in bipartite graph $G$ do not have an edge between them,
the set of maximal bicliques of $G$ is the union of the sets of maximal
bicliques in $G-a_{i}$ and $G-b_{j}$.

## References

  * [And87] T. Ando.  Totally positive matrices.  Linear Algebra and its Applications, 90:165 – 219, 1987.
  * [AW20] Josh Alman and Virginia Vassilevska Williams.  A Refined Laser Method and Faster Matrix Multiplication, 2020.
  * [FJ11] S.M. Fallat and C.R. Johnson.  Totally Nonnegative Matrices.  Princeton Series in Applied Mathematics. Princeton University Press, 2011\.
  * [Joh87] David S Johnson.  The NP-completeness column: An ongoing guide.  Journal of Algorithms, 8(3):438 – 448, 1987.


# Quantum logic and meaning

Sebastian Horvat
<EMAIL_ADDRESS>
Faculty of Physics
University of Vienna
&
Iulian D. Toader
<EMAIL_ADDRESS>
Institute Vienna Circle
University of Vienna

###### Abstract

This paper gives a formulation of quantum logic in the abstract algebraic
setting laid out by Dunn and Hardegree (2001). On this basis, it provides a
comparative analysis of viable quantum logical bivalent semantics and their
classical counterparts, thereby showing that the truth-functional status of
classical and quantum connectives is not as different as usually thought. Then
it points out that bivalent semantics for quantum logic - compatible with
realism about quantum mechanics - can be maintained, albeit at the price of
truth-functionality. Finally, the paper improves on Hellman (1980)’s argument
that this lack of truth-functionality entails a change of meaning between
classical and quantum logical connectives.

##  1 Introduction

The transition from classical physics to quantum mechanics (QM) requires, as
is well known, that the commutativity of operators that are taken to represent
quantum observables be dropped. This motivated Birkhoff and von Neumann’s
project “to discover what logical structure one may hope to find in physical
theories which, like quantum mechanics, do not conform to classical logic.”
(Birkhoff and von Neumann 1936, 823) The transition to QM, they argued,
suggests a replacement of classical logic (CL) by a non-distributive quantum
logic (QL) within the quantum domain. Later, once Putnam enrolled QL in the
service of realist metaphysics, which he thought would require a revision of
CL by dropping the distributive law (Putnam 1968), Geoffrey Hellman proved
that some QL connectives are not truth-functional, which he took to support
the view that the meaning of those connectives is different than that of their
CL counterparts (Hellman 1980). This backed an earlier claim made by Arthur
Fine, that it is not the classical distributive law that is dropped in QM, and
thus the transition in physics does not require a revision of logic (Fine
1972). Rather than a revision of CL, QL is therefore just an alternative
logic, exactly as Birkhoff and von Neumann seem to have intended it.

In the present paper, we engage with this debate in multiple ways. First, in
section two, we prepare the ground for our discussion by formulating QL in an
abstract algebraic framework, which enables a more transparent comparison of
some formal semantic aspects of CL and QL. In section three, we proceed with a
detailed analysis of viable bivalent semantics for QL, wherein we show that
the truth-functional status of QL- and CL-connectives is not as different as
commonly thought. Along the way, in section four, we also point out the
seemingly forgotten fact that a bivalent semantics, which may potentially
support realism about QM, is indeed tenable, provided that one gives up the
truth-functionality of some logical connectives. In the same section we also
identify and discuss another formal condition that ought to be satisfied by
any purportedly realist semantics. Finally, in section five, we address
Hellman’s meaning-variance argument. We argue that the semantic analysis from
the previous sections indicates that argument can be improved, which brings us
to propose an alternative argument, which reinforces and precisifies the
conclusion that some QL-connectives have a different meaning than their CL
counterparts. The revised argument still hinges on the meaning-variance
principle that was originally suggested by Hellman, but also, perhaps
surprisingly, on the way one formally explicates the validity of arguments
with multiple premises. Given the latter two qualifications, which require of
course further philosophical justification that would go beyond purely formal
semantic methods, our argument justifies the contention that it is not the
classical distributive law that fails in QM: classical and quantum
distributivity do mean different things.

##  2 An Algebraic Formulation of Quantum Logic

QL has been formulated in a variety of ways, both model- and proof-
theoretically. Some of these formulations even yield different logics, in the
sense that they validate different sets of arguments, thus making it more apt
to speak of QLs, in the plural. Nevertheless, since the details and
differences among these formulations are not relevant for our discussion, we
will stick to the original one, given in terms of Hilbert lattices, albeit
presented more abstractly with the help of the terminology and mathematical
machinery employed by Dunn and Hardegree in their classic study of algebraic
methods in logic (Dunn and Hardegree 2001). The main advantage of this
presentation, as we will see later, is that the interpretation of sentences is
not completely fixed, as in more standard presentations of QL (e.g., Rédei
1998), thus “erasing any intensional vestiges” in the language of QL and
enabling a clearer semantic comparison with CL.111Just to make clear, our more
abstract formulation defines the same QL as the one originally defined by
Birkhoff and von Neumann in terms of Hilbert lattices.

###  2.1 Algebraic preliminaries

Let $\mathcal{L}$ be a sentential language containing atomic sentences,
conjunction ‘$\land$’, disjunction ‘$\vee$’ and negation ‘$\lnot$’. Let $S$ be
the set of sentences generated inductively from $\mathcal{L}$, in the usual
way. We can straightforwardly define the algebra of sentences.

######  Definition 1.

The algebra of sentences $\mathcal{S}\equiv(S,O_{\land},O_{\vee},O_{\lnot})$
is an algebra whose operations on the carrier set $S$ are defined as follows:
for any $a,b\in S$,

$\begin{split}&O_{\land}(a,b)=a\land b\\\ &O_{\vee}(a,b)=a\vee b\\\
&O_{\lnot}(a)=\lnot a.\end{split}$

Next, we introduce the notion of a logical atlas.

######  Definition 2.

A logical atlas $\mathcal{A}$ is a pair $(\mathcal{P},\langle D_{j}\rangle)$,
where $\mathcal{P}$ is a non-empty algebra and $\langle D_{j}\rangle$ is a
family of proper subsets of the carrier set of $\mathcal{P}$.

We will refer to the sets $D_{j}$ as designated sets. Logical atlases can be
used to provide a semantics for our language. Indeed, an interpretation of a
language into an atlas is defined as follows:

######  Definition 3.

Let $\mathcal{L}$ be a sentential language and $\mathcal{S}$ its associated
algebra of sentences. Suppose $\mathcal{A}=(\mathcal{P},\langle D_{j}\rangle)$
is a logical atlas whose algebra $\mathcal{P}$ is of the same type as
$\mathcal{S}$. An interpretation of $\mathcal{L}$ in $\mathcal{A}$ is any
homomorphism from $\mathcal{S}$ into $\mathcal{P}$.

The elements of the carrier set of $\mathcal{P}$ can be thought of as
propositions, and each designated set $D_{j}$ can be thought of as a possible
set of true propositions, i.e. a way “the world could be”. An interpretation
of a language into an atlas thus homomorphically assigns propositions to
sentences, in accord with the principle of compositionality; since an atlas
can contain multiple designated sets, an interpretation does not need to fully
determine which propositions are true and which are not. We will call a triple
$(\mathcal{L},\mathcal{A},I)$, where $I$ is the set of all interpretations of
$\mathcal{L}$ in $\mathcal{A}$, an interpretationally constrained language.

Let us now introduce the notion of valuation.

######  Definition 4.

Let $\mathcal{L}$ be a sentential language and $S$ its associated set of
sentences. A valuation on $\mathcal{L}$ is any function from $S$ to the two-
element set $\left\\{\textbf{t},\textbf{f}\right\\}$.

Here we are taking a valuation of a language to be bivalent by definition:
each sentence of the language is assigned either ‘true’ or ‘false’, thereby
excluding third truth-values or truth-value “gaps” and “gluts”. Also, there
are no compositional restrictions on truth-value assignments to compound
sentences, i.e. we are not assuming truth-functionality; more on this later.

The identification of designated sets with true propositions suggests a
natural way of defining a class of valuations induced by an atlas.

######  Definition 5.

Let $\mathcal{A}=(\mathcal{P},\langle D_{j}\rangle)$ be a logical atlas and
$\mathcal{L_{A}}=(\mathcal{L},\mathcal{A},I)$ an interpretationally
constrained language. For each interpretation $i\in I$, and index $j$, define
a valuation $v^{(i)}_{j}$ as:

$v^{(i)}_{j}(a)=\textbf{t},\quad\text{if and only if}\quad i(a)\in D_{j}.$

The class of valuations $C^{*}_{\mathcal{A}}$ induced by $\mathcal{A}$ is the
set of all such valuations, i.e.

$C^{*}_{\mathcal{A}}=\left\\{v^{(i)}_{j},\quad\forall i\in I,\forall
j\right\\}.$

Now we are ready to introduce the notion of logical consequence induced by an
atlas. We say that a set of premises $\Gamma\subseteq S$ $\mathcal{A}$-implies
a conclusion $a\in S$, or $\Gamma\vDash_{\mathcal{A}}a$, if the following
holds:

$\forall v\in
C^{*}_{\mathcal{A}}:\quad\text{if}\quad\left(\forall\gamma\in\Gamma,v(\gamma)=\textbf{t}\right)\quad\text{then}\quad
v(a)=\textbf{t}.$ (1)

Analogously, we say that a sentence is $\mathcal{A}$-valid, or
$\vDash_{\mathcal{A}}a$, if $v(a)=\textbf{t}$ for all valuations $v$ in
$C^{*}_{\mathcal{A}}$.

To summarize, an interpretationally constrained language $\mathcal{L_{A}}$
consists of a language endowed with a compositional semantics and contains
sufficient structure – namely, the family of designated sets pertaining to the
logical atlas $\mathcal{A}$ – to induce a class of valuations
$C^{*}_{\mathcal{A}}$, thereby defining a logical consequence relation
$\vDash_{\mathcal{A}}$ on the set of sentences.

Let us present QL within this formalism.

###  2.2 Quantum logic

We will use the definitions from the previous subsection in our presentation
of QL. In particular, we will introduce quantum atlases and their induced
quantum-logical consequence relations.

Let $C(H)$ be the set of closed subspaces of a (potentially infinite-
dimensional) Hilbert space $H$. We will say that the algebra associated to $H$
is the algebra $\mathcal{H}=(C(H),\cap,\sqcup,^{\perp})$, where ‘$\cap$’ and
‘$\sqcup$’ are set-intersection and linear span, respectively, whereas the ‘⟂’
operation takes a subspace and outputs the corresponding orthogonal subspace.
We define quantum atlases $\mathcal{A_{H}}$ as follows:

######  Definition 6.

A quantum atlas $\mathcal{A_{H}}=(\mathcal{H},\langle D_{P}\rangle)$ is an
atlas where $\mathcal{H}=(C(H),\cap,\sqcup,^{\perp})$ is the algebra
associated to some Hilbert space $H$, and the family of designated subsets
$\langle D_{P}\rangle$, whose index ranges over all 1-dimensional subspaces
$P\in C(H)$, is defined as

$\forall Q\in C(H):\quad Q\in D_{P}\quad\text{if and only if}\quad P\subseteq
Q.$

A quantum interpretationally constrained language $\mathcal{L_{A_{H}}}$ is
then any triple $(\mathcal{L},\mathcal{A_{H}},I)$, where $\mathcal{L}$ is a
sentential language, $\mathcal{A_{H}}$ is a quantum atlas and $I$ is the set
of all interpretations of the language into $\mathcal{A_{H}}$. In particular,
being homomorphic, the interpretations $i\in I$ satisfy

$\begin{split}&i(a\land b)=i(a)\cap i(b)\\\ &i(a\vee b)=i(a)\sqcup i(b)\\\
&i(\lnot a)=i(a)^{\perp},\end{split}$ (2)

for all sentences $a,b$ in $S$. Following Definition 5 and Eq. (1), we can
associate to any atlas $\mathcal{A_{H}}$ an induced class of valuations
$C^{*}_{\mathcal{A_{H}}}$ and an induced consequence relation
$\vDash_{\mathcal{A_{H}}}$, which thereby lets us define the notion of
quantum-logical consequence. In order to simplify the notation, we will
henceforth write $\mathcal{C}^{*}_{H}$ and $\vDash_{H}$, instead of the more
cumbersome $C^{*}_{\mathcal{A_{H}}}$ and $\vDash_{\mathcal{A_{H}}}$.

######  Definition 7.

Let $\vDash$ be a consequence relation on a set of sentences $S$. We say that
$\vDash$ is a quantum-logical (QL) consequence relation if there exists a
quantum atlas $\mathcal{A_{H}}$, such that $\vDash$ and $\vDash_{H}$ coincide.

QL-consequence relations can also be given the following characterization,
reminiscent of the one originally introduced by Birkhoff and von Neumann: for
a Hilbert space $H$, a set of premises $\Gamma$ $H$-implies conclusion $a$ if
for all homomorphisms $i$ from the algebra of sentences into the algebra
associated to $H$, the following holds

$\bigcap_{\gamma\in\Gamma}i(\gamma)\subseteq i(a).$ (3)

Since the operations $\cap$ and $\sqcup$ on $C(H)$ are non-distributive, it
follows that QL-consequence relations violate the law of distributivity, as we
will now illustrate (see also Birkhoff and von Neumann 1936, 831).

Let $a_{1},a_{2},b\in S$ be atomic sentences. Consider an interpretation $i\in
I$ that maps $a_{1}$ and $a_{2}$ into two mutually orthogonal 1-dimensional
subspaces of $H$ that contain respectively vectors $\psi_{1}$ and $\psi_{2}$,
and $b$ into the 1-dimensional subspace that contains vector
$\frac{1}{\sqrt{2}}\left(\psi_{1}+\psi_{2}\right)$. It follows that
$i(b\land(a_{1}\vee a_{2}))=i(b)$ and $i((b\land a_{1})\vee(b\land
a_{2}))=\mathbf{0}$, which implies $i(b\land(a_{1}\vee a_{2}))\nsubseteq
i((b\land a_{1})\vee(b\land a_{2}))$, and thus $b\land(a_{1}\vee
a_{2})\centernot\vDash_{H}(b\land a_{1})\vee(b\land a_{2})$. This failure of
distributivity in QL is a direct consequence of the fact that experimental
propositions in QM form a non-distributive lattice, which can be arguably
understood as a consequence of the algebra of quantum-mechanical observables
being non-commutative.

Note that we have defined the QL-consequence relations $\vDash_{H}$ relative
to Hilbert spaces $H$: there may thus in principle be as many different
consequence relations as there are (non-isomorphic) Hilbert spaces. Consider
for a moment the parallel situation in classical logic (CL). Instead of
defining the CL-consequence relation in terms of standard truth-tables or via
some proof-theoretic axiomatization, one may introduce it by using atlases
built on arbitrary Boolean algebras. Since there are many non-isomorphic
Boolean algebras, one needs to prove that they all lead to one and the same
logical consequence relation. Indeed, the latter turns out to be the case, due
to the theorem that states that an equation holds in an arbitrary Boolean
algebra if and only if it holds in the two-element Boolean algebra: thus, all
these different Boolean algebras yield one and the same consequence relation,
which can be defined using the standard truth-tables.222In algebraic terms, to
say that an equation holds in an arbitrary Boolean algebra if and only if it
holds in the two-element Boolean algebra means to say that the variety of
Boolean algebras is generated by the two-element Boolean algebra, which can be
seen as a consequence of the Stone representation theorem for Boolean algebras
(Stone 1936). On the other hand, there is no parallel result that holds for
algebras associated to Hilbert spaces: for instance, as noted already by
Birkhoff and von Neumann (1936, 832), modularity holds only in finite-
dimensional Hilbert spaces (see also Rédei 1998). There is thus, as suspected,
more than one QL-consequence relation depending on the Hilbert space on which
the semantics is built, which justifies keeping the index ‘$H$’ in
‘$\vDash_{H}$’.

##  3 On Valuations in Quantum Logic

In the previous section we introduced quantum atlases $\mathcal{A_{H}}$, their
induced classes of valuations $\mathcal{C}^{*}_{H}$ and the induced relations
of logical consequence $\vDash_{H}$. Notice that $\vDash_{H}$ are abstract
relations on the set of sentences $S$, so despite being explicitly constructed
with the help of classes $\mathcal{C}^{*}_{H}$ induced by quantum atlases,
they are in principle definable by other means, e.g., by some proof-theoretic
axiomatization, or by different classes of valuations. In this section we want
to focus on the latter, i.e., we want to analyze some features of the
collection of viable classes of valuations that respect $\vDash_{H}$. With
this aim in mind, let us introduce the notion of $H$-class.

######  Definition 8.

Let $\mathcal{L}$ be a sentential language and $S$ its set of sentences.
Suppose the consequence relation $\vDash_{H}$ on $S$ is induced by a quantum
atlas $\mathcal{A_{H}}$, for some Hilbert space $H$. An $H$-class
$\mathcal{C}$ is any class of valuations on $\mathcal{L}$ that obeys the
following. For all $\Gamma\cup\left\\{a\right\\}\subseteq S$:

$\Gamma\vDash_{H}a\quad\quad\text{iff}\quad\quad\left[\forall
v\in\mathcal{C}:\quad\text{if}\quad\left(\forall\gamma\in\Gamma:v(\gamma)=\textbf{t}\right)\quad\text{then}\quad
v(a)=\textbf{t}\right].$ (4)

Any $H$-class thus defines the same consequence relation as the class
$\mathcal{C}^{*}_{H}$ induced by the atlas $\mathcal{A_{H}}$: trivially,
$\mathcal{C}^{*}_{H}$ is one particular $H$-class. Let us now analyze some
properties of collections of $H$-classes.

###  3.1 Uniqueness

First, consider the following question: for a given atlas $\mathcal{A_{H}}$,
are all $H$-classes isomorphic, i.e. are they all equivalent to
$\mathcal{C}^{*}_{H}$? In other words, does the collection of all $H$-classes
have more than one member? Let us briefly depart from QL and delve into an
analogous question that can be raised about classical logic (CL). One can
define the CL-consequence relation $\vDash_{CL}$ via the class of valuations
$\mathcal{C}^{*}_{CL}$ determined by the standard truth-tables.333The class
$\mathcal{C}^{*}_{CL}$ is defined by the following properties: for all
$v\in\mathcal{C}^{*}_{CL}$, $v(a\land b)=\textbf{t}$ if and only if
$v(a)=v(b)=\textbf{t}$; $v(a\vee b)=\textbf{f}$ if and only if
$v(a)=v(b)=\textbf{f}$; $v(\lnot a)=\textbf{t}/\textbf{f}$ if and only if
$v(a)=\textbf{f}/\textbf{t}$. Can the same consequence relation be
characterized by a different class of valuations? Is the class
$\mathcal{C}^{*}_{CL}$ unique in this sense? This question has been answered
by Carnap (1943), where he showed that there are non-standard classes of
valuations that characterize the same relation $\vDash_{CL}$. One such class
is given by $\mathcal{C}^{*}_{CL}\cup\left\\{\tilde{v}_{CL}\right\\}$, where
the non-standard valuation $\tilde{v}_{CL}$ obeys

$\tilde{v}_{CL}(a)=\textbf{t},\quad\text{if and only if `a' is a classical
tautology (i.e. $\vDash_{CL}a$)}.$ (5)

It is easy to see that $\mathcal{C}^{*}_{CL}$ and
$\mathcal{C}^{*}_{CL}\cup\left\\{\tilde{v}_{CL}\right\\}$ define the same
relation of logical consequence, despite them being non-isomorphic.444Carnap
saw this as a problem because it implies that no axiomatic calculus for CL can
be considered what he called a “full formalization” of CL, i.e., no such
calculus can uniquely determine the intended class of valuations (i.e. the one
given by the standard truth-tables). This arguably presents a problem for the
logical inferentialist thesis that “rules fix meaning”. For an overview of
inferentialism, including this problem, see (Murzi and Steinberger 2017).
Carnap’s answer can be trivially transferred to the case of QL-consequence
relations in the following way. Let $\mathcal{C}^{*}_{H}$ be the class of
valuations induced by a quantum atlas $\mathcal{A_{H}}$, and $\vDash_{H}$ its
corresponding QL-consequence relation. Consider the valuation $\tilde{v}_{H}$,
given by:

$\tilde{v}_{H}(a)=\textbf{t},\quad\text{if and only if `a' is an $H$-tautology
(i.e. $\vDash_{H}a$)}.$ (6)

Again, the two different classes, $\mathcal{C}^{*}_{H}$ and
$\mathcal{C}^{*}_{H}\cup\left\\{\tilde{v}_{H}\right\\}$, determine the same
relation $\vDash_{H}$, and are thus both $H$-classes. Hence, similarly to the
case of CL, there is more than one valuation compatible with one and the same
consequence relation. We deem this worth of being emphasized in the following

######  Proposition 1.

For any quantum atlas $\mathcal{A_{H}}$, there are at least two $H$-classes.

###  3.2 Truth-functionality

Now we will continue the analysis of $H$-classes by inspecting their truth-
functionality or lack thereof. In order to do so, let us first introduce the
following definition.

######  Definition 9.

Let $\mathcal{C}$ be a class of valuations on a sentential language
$\mathcal{L}$. We say that $\mathcal{C}$ makes ‘$\land$’ truth-functional (TF)
if there exists a function $f_{\land}$, such that for all $v\in\mathcal{C}$,
and any $a,b\in S$:

$v(a\land b)=f_{\land}(v(a),v(b)).$

The respective definitions for ‘$\vee$’ and ‘$\lnot$’ then follow by analogy.
Furthermore, we say that $\mathcal{C}$ is a truth-functional (TF) class of
valuations if it makes all three connectives in
$\left\\{\land,\vee,\lnot\right\\}$ truth-functional.

Before applying this definition to QL, let us again mention a few facts about
CL. Note first that the CL-class of valuations determined by the standard
truth-tables, i.e., $\mathcal{C}^{*}_{CL}$, is obviously TF, and is actually
the only TF CL-class. Nevertheless, there are also non-TF CL-classes, as
exemplified by Carnap’s non-standard class
$\mathcal{C}^{*}_{CL}\cup\left\\{\tilde{v}_{CL}\right\\}$, which makes
disjunction and negation non-TF. The non-truth-functionality of disjunction
can for instance be seen from the following example. For arbitrary atomic
sentences $a,b$: $\tilde{v}_{CL}(a)=\tilde{v}_{CL}(\neg
a)=\tilde{v}_{CL}(b)=\textbf{f}$, and $\tilde{v}_{CL}(a\vee b)=\textbf{f}$,
but $\tilde{v}_{CL}(a\vee\neg a)=\textbf{t}$. This already indicates that
statements such as “Classical logic is truth-functional” ought to be
approached carefully, and should be precisified as “There exists a (unique) TF
CL-class”, but acknowledging at the same time that “There exist non-TF CL-
classes”.

Let us now get back to QL and inspect the (non-)truth-functionality of
$H$-classes. We will first show that for any quantum atlas $\mathcal{A_{H}}$,
the induced class $\mathcal{C}^{*}_{H}$ makes conjunction TF, and negation and
disjunction non-TF. The truth-functionality of conjunction follows immediately
from the fact that, in any Hilbert space, a 1-dimensional subspace is
contained in two other subspaces if and only if it is contained in their
intersection. Since the conjunction of two sentences is interpreted as the
intersection of the subspaces associated (under the same interpretation) to
those sentences, this implies that a conjunction is true if and only if both
its conjuncts are true, in accord with the standard classical truth-table. On
the other hand, in order to illustrate the non-truth-functionality of
disjunction and negation, consider the following example. Take three
1-dimensional subspaces $P_{1},P_{2},P\in C(H)$, where $H$ is an arbitrary
Hilbert space. Suppose that $P_{1},P_{2},P$ respectively contain vectors
$\psi_{1}$, $\psi_{2}$ and $\frac{1}{\sqrt{2}}\left(\psi_{1}+\psi_{2}\right)$,
where $\psi_{1}$ and $\psi_{2}$ are orthogonal. Take an arbitrary sentence
$a\in S$ and an interpretation $i\in I$, such that $i(a)=P_{1}$ and $i(\lnot
a)=P_{2}$. It is easy to see that for the induced valuations $v^{(i)}_{P}$:
$v^{(i)}_{P}(a)=v^{(i)}_{P}(\lnot a)=\textbf{f}$ but $v^{(i)}_{P}(a\vee\lnot
a)=\textbf{t}$. On the other hand, if $P^{\prime}$ is a 1-dimensional subspace
containing a vector which does not lie in the span of $\psi_{1}$ and
$\psi_{2}$, then $v^{(i)}_{P^{\prime}}(a)=v^{(i)}_{P^{\prime}}(\lnot
a)=\textbf{f}$ and $v^{(i)}_{P^{\prime}}(a\vee\lnot a)=\textbf{f}$. Thus,
disjunction is not TF, in that some false disjuncts combine into a true
disjunction, whereas others combine into a false disjunction; similarly, some
false propositions, when negated, become true, whereas others stay false.
Therefore, the $H$-classes of valuations induced by quantum atlases are not
TF, for they make only conjunction TF. But can there be any TF $H$-classes? As
proved by David Malament (2002), not only is the answer negative, but any
$H$-class must make at least two connectives non-TF. Here is his result
translated into our terms:

######  Proposition 2.

Let $\mathcal{A_{H}}$ be the quantum atlas built on some arbitrary Hilbert
space $H$. There is no $H$-class that makes more than one connective in
$\left\\{\land,\vee,\lnot\right\\}$ truth-functional. Thus, there is no truth-
functional $H$-class.

Notice that the classes $\mathcal{C}^{*}_{H}$ induced by atlases
$\mathcal{A_{H}}$ accordingly make only conjunction TF. Are there $H$-classes
that make disjunction or negation TF? Surprisingly, the answer is no! It is
enough to notice that for any Hilbert space $H$, the induced QL-consequence
relation $\vDash_{H}$ validates the following arguments, which are the
“semantic versions” of the $\land$-introduction and $\land$-elimination rules:
for any $a,b\in S$, $a\land b\vDash_{H}a$, $a\land b\vDash_{H}b$, and
$\left\\{a,b\right\\}\vDash_{H}a\land b$. It is easy to see that the validity
of these three arguments implies that for any $H$-class, $a\land b$ is true if
and only if both $a$ and $b$ are true, thus forcing conjunction to behave
classically. Since, following Proposition 2, at most one connective can be TF,
this implies that there is no $H$-class that makes either disjunction or
negation TF! Let us state this clearly as follows:

######  Proposition 3.

Let $\mathcal{A_{H}}$ be a quantum atlas built on some arbitrary Hilbert space
$H$. Every $H$-class makes conjunction truth-functional, but negation and
disjunction non-truth-functional.

Where does this asymmetry between the QL-connectives come from? It suffices to
take a look at Definition 8 of $H$-classes and to notice that it invokes a
metalinguistic quantification over all premises: it defines valid arguments as
those whose conclusion is true if each premise is also true. This definition
formalizes the intuitive idea that valid arguments preserve truth, or that
given that the premises are true, the conclusion must be true as well.
Crucially, “the premises are true” is formalized as “each premise is assigned
truth-value ‘t’”, which then, given the validity of the $\land$-introduction
and $\land$-elimination arguments forces the QL-conjunction to behave
classically.

Now, one may either accept this conclusion, i.e., that in QL only conjunction
can be TF (since it actually must be TF), or one may modify some of the above
definitions so as to allow for the possibility of having a TF negation or
disjunction. Let us briefly sketch one such modification. Instead of
formalizing “the premises are true” as “each premise is assigned truth-value
‘t’”, we propose to formalize it as “the conjunction of the premises is
assigned truth-value ‘t’”. We thereby introduce the notion of $H$-klass as
follows.

######  Definition 10.

Let $\mathcal{L}$ and $S$ be as before. Suppose the consequence relation
$\vDash_{H}$ on $S$ is induced, again just as before, by a quantum atlas
$\mathcal{A_{H}}$, for some Hilbert space $H$. An $H$-klass $\mathcal{K}$ is
any class of valuations on $\mathcal{L}$ that obeys the following. For all
finite sets $\Gamma\cup\left\\{a\right\\}\subseteq S$:

$\Gamma\vDash_{H}a\quad\quad\text{iff}\quad\quad\left[\forall
v\in\mathcal{K}:\quad\text{if}\quad
v\left(\bigwedge_{\gamma\in\Gamma}\gamma\right)=\textbf{t}\quad\text{then}\quad
v(a)=\textbf{t}\right].$ (7)

The notion of $H$-klass is thus different from the notion of $H$-class
introduced in Definition 8. Even though both are arguably reasonable
formalizations of one and the same intuitive idea, i.e., that in valid
arguments the truth of the premises guarantees the truth of the conclusion,
they are nevertheless distinct, in that the collection of all $H$-classes is
different from the collection of all $H$-klasses. Some potential drawbacks of
the alternative notion of $H$-klass are that it is defined only for finite
sets of premises, and that it cannot be straightforwardly transposed to logics
defined on languages with no conjunction.555We are assuming throughout this
paper that $\mathcal{L}$ is a finitary language, i.e., it allows for the
formation of only finite conjunctions and disjunctions. The consequences of
dropping this restriction and considering infinitary languages are deferred to
future research. Let us however provisionally sweep these problems under the
rug and explore where the introduction of $H$-klasses leads us. First, notice
that since conjunctions of premises are just a specific instance of single
premises, the following holds:

######  Proposition 4.

For any quantum atlas $\mathcal{A_{H}}$, all $H$-classes are also $H$-klasses,
but not viceversa.

This implies that the $H$-class $\mathcal{C}^{*}_{H}$ induced by an atlas
$\mathcal{A_{H}}$ is an $H$-klass. Next, recall that the $\land$-introduction
and $\land$-elimination arguments force the QL-conjunction to obey the
standard classical truth-table in all viable $H$-classes. On the other hand,
the $\land$-introduction argument $\left\\{a,b\right\\}\vDash_{H}a\land b$
does not impose any constraint whatsoever on viable $H$-klasses, because it is
morphed into the trivial requirement that if $v(a\land b)=\textbf{t}$ then
$v(a\land b)=\textbf{t}$. Therefore, while it is still the case that true
conjunctions necessarily have true conjuncts (due to the validity of the
$\land$-elimination argument), it may be the case that some false conjunctions
have true conjuncts. Indeed, consider the following $H$-klass. For each
subspace $P\in C(H)\setminus\left\\{\mathbf{0},H\right\\}$, define the
function $h_{P}:C(H)\rightarrow\left\\{\textbf{t},\textbf{f}\right\\}$ as
$h_{P}(Q)=\textbf{f}$ if and only if $Q\subseteq P$. Notice that
$Q_{1}\subseteq Q_{2}$ holds if and only if for all $P$, if
$h_{P}(Q_{2})=\textbf{f}$ then $h_{P}(Q_{1})=\textbf{f}$, or contrapositively,
if $h_{P}(Q_{1})=\textbf{t}$ then $h_{P}(Q_{2})=\textbf{t}$. This entails that
the class $\mathcal{C}^{(\vee)}_{H}\equiv\left\\{h_{P}\circ i,\quad\forall
i\in I,\forall P\in C(H)\setminus\left\\{\mathbf{0},H\right\\}\right\\}$ is an
$H$-klass. In order to see that $\mathcal{C}^{(\vee)}_{H}$ makes disjunction
TF notice the following. For any triple $P_{1},P_{2},Q\in C(H)$ it holds that
$P_{1}\subseteq Q$ and $P_{2}\subseteq Q$ if and only if $P_{1}\sqcup
P_{2}\subseteq Q$. According to $\mathcal{C}^{(\vee)}_{H}$, this implies that
a sentence associated to $P_{1}$ and a sentence associated to $P_{2}$ are both
false if and only if their disjunction is also false, thus mimicking the
standard classical truth-table for disjunction. Since, as it can be easily
checked, Proposition 2 extends to $H$-klasses as well,
$\mathcal{C}^{(\vee)}_{H}$ must make conjunction and negation non-TF: indeed,
some true conjuncts form true conjunctions and others form false ones, whereas
negation maps some true propositions into false ones and others into true
ones. There is thus at least one $H$-klass that makes disjunction TF. Also,
due to Proposition 4, there is at least one $H$-klass that makes conjunction
TF, namely $\mathcal{C}^{*}_{H}$. Finally, a theorem proved in Friedman and
Glymour (1972, 21) implies that for any Hilbert space $H$, there exists at
least one $H$-klass that makes negation classical and thus TF, i.e., one for
which ‘$a$’ is true (false) if and only if ‘$\lnot a$’ is false (true). Let us
state these results as follows:

######  Proposition 5.

Let $\mathcal{A_{H}}$ be a quantum atlas built on an arbitrary Hilbert space
$H$. For any connective $c\in\left\\{\land,\vee,\lnot\right\\}$, there exists
an $H$-klass that makes $c$ truth-functional.

Let us finally say a few words about similar issues in CL, since it will be
relevant for our discussion in the subsequent section. Notice that, given the
standard definition of CL-classes, the CL-validity of $\land$-introduction and
$\land$-elimination rules imposes the standard truth-table on CL-conjunction
as well, as opposed to disjunction and negation, for which Carnap’s non-
standard valuations exist666As noticed already by Carnap (1943) himself, one
can, dually to the case of conjunction, impose the standard truth-table on CL-
disjunction if one formalizes valid arguments as relations between sets of
premises and sets of conclusions and if one modifies the definition of CL-
classes of valuations accordingly. The $\vee$-elimination and
$\vee$-introduction rules, i.e. $a\vDash_{CL}a\vee b$, $b\vDash_{CL}a\vee b$
and $a\vee b\vDash_{CL}\left\\{a,b\right\\}$, then imply that a disjunction is
false if and only if each of its disjuncts is false. See (Shoesmith and Smiley
1978) and, for a more recent critical discussion, see (Dicher 2020).. There
are thus no CL-classes that make conjunction non-TF. However, analogously to
the case of QL, if one defines CL-klasses as those that make the conclusions
of CL-valid arguments true if they make the conjunction of all the premises
true, then it turns out that there is at least one CL-klass that makes
conjunction non-TF. Namely, the latter is given by
$\mathcal{C}^{*}_{CL}\cup\left\\{\tilde{v}^{\prime}_{CL}\right\\}$, where the
non-standard valuation $\tilde{v}^{\prime}_{CL}$ is defined as:

$\tilde{v}^{\prime}_{CL}(a)=\textbf{f},\quad\text{if and only if `a' is a CL-
falsehood (i.e. $\vDash_{CL}\lnot a$)}.$ (8)

Since Carnap’s non-normal CL-class makes disjunction and negation non-TF, and
because, analogously to Proposition 4, every CL-class is also a CL-klass, we
have obtained the following:

######  Proposition 6.

For any connective $c\in\left\\{\land,\vee,\lnot\right\\}$, there exists a
$CL$-klass that makes $c$ non-truth-functional.

Propositions 5 and 6 show that the truth-functional status of the QL- and CL-
connectives is not as different as is often suggested: each connective can be
TF under some klass of valuations and non-TF under some other klass of
valuations. That said, the main difference between the QL- and CL-connectives
still persists, in that there is certainly no klass that makes all the QL-
connectives TF, which justifies to some extent the simplified claim that
“Quantum logic is not truth-functional”. Further below, in section five, we
will be using Propositions 5 and 6 in our analysis of an argument that
purports to show that CL- and QL-connectives differ in meaning. For now, let
us however put to work the notions introduced above for the purpose of
clarifying certain formal semantic conditions for holding onto realism about
quantum mechanics.

##  4 Semantics and Realism

Our approach to QL has so far been rather abstract, in that we formulated QL-
consequence relations on sets of sentences of an abstract sentential language.
But the latter also admits of definitions of other logical consequence
relations, such as the classical one, thereby enabling a more transparent
comparison of the formal semantic aspects of QL and CL, which will be useful
in our discussion further below. We should emphasize, however, that our
approach stands in contrast to the more common way of presenting QL as “the
logic of quantum mechanics” (QM), which defines QL-consequence relations as
relations between sentences that are intended to express “experimental
propositions” about magnitudes associated to quantum systems/objects. In
section two, we presented QL independently of QM, on a par with any other
logic, similarly to how CL can obviously be conceived more abstractly than
just as “the logic of classical physics”. Nevertheless, it will now be useful
to relate our presentation to the more common one, in order to assess the
possible philosophical merit of the classes and klasses of valuations
introduced in the previous section.

Following van Fraassen (1967), QL can be presented by first introducing a set
of sentences $S_{QM}$ that is generated by conjoining, disjoining and negating
“elementary sentences”, each of which specifies the range of values taken by a
magnitude associated to a quantum system/object. An elementary sentence is of
the form “sent($O$, $M$, $\Delta)$”, which reads “The value of magnitude $M$
associated to quantum object $O$ lies in the interval $\Delta$”, where $M$ is
a magnitude, such as position or momentum, and $\Delta$ is some subset in the
set of values that may possibly be taken by $M$, e.g., a subset of
$\mathbb{R}$. One then introduces an interpretation function $h$ that maps
each sentence in $S_{QM}$ to a closed subspace of the Hilbert space $H$ that
represents the state-space of the quantum system under consideration. The map
$h$ is of course not arbitrary, but is dictated by the empirical content of
QM, in the following way. Each elementary sentence “sent($O$, $M$, $\Delta$)”
is mapped to the closed subspace that contains all those states $\psi\in H$
for which it holds that, if $O$ is prepared in state $\psi$, the outcome of a
measurement of $M$ on $O$ would lie in $\Delta$. Together with the latter
constraint, the map $h$ acts on arbitrary sentences $a,b\in S_{QM}$ as
follows:

$\begin{split}&h(a\land b)=h(a)\cap h(b)\\\ &h(a\vee b)=h(a)\sqcup h(b)\\\
&h(\lnot a)=h(a)^{\perp}.\end{split}$ (9)

Finally, each quantum state $\psi\in H$ partially determines a potentially
non-bivalent valuation $v_{\psi}$ on $S_{QM}$ as follows: for any sentence
$a$, $v_{\psi}(a)=\textbf{t}$ if and only if $\psi\in h(a)$. The valuations
$v_{\psi}$ may indeed be non-bivalent, as one does not need to equate “a is
not true” and “a is false”. We will, however, do so, for we will be interested
in the question of realism in a moment.777The connection between bivalence and
realism will be clarified below. In any case, given the above definition of
valuations, a QL-consequence relation can be straightforwardly defined: a set
of premises QL-implies a conclusion if for any $\psi\in H$, $v_{\psi}$ makes
the conclusion true if it makes each premise true as well.

Let us now relate all of this to our presentation of QL from section two
above. Unlike the interpretation $h$ that assigns a fixed meaning to each
sentence in $S_{QM}$, a quantum interpretationally constrained language
$(\mathcal{L},\mathcal{A_{H}},I)$ contains all homomorphic interpretations
$i\in I$ of the sentential language $\mathcal{L}$ in the quantum atlas
$\mathcal{A_{H}}=(\mathcal{H},\langle D_{P}\rangle)$. Each designated set
$D_{P}$ in turn contains those elements that are made true by the valuation
$v_{\psi}$, for $\psi\in P$. The main difference between the two formulations
is thus that ours leaves the association between sentences and propositions
partially constrained, rather than completely fixed, thus “erasing the
intensional vestiges” in the language of QL, as we already noted at the
outset. It should be clear that if one takes the closed subspaces of $H$ to
represent “experimental propositions” about a quantum system, and the elements
of $H$ (or its 1-dimensional subspaces) to represent its possible quantum
states, then $I$ contains all ways of compositionally assigning experimental
propositions to the sentences of $\mathcal{L}$, whereas the induced class of
valuations $\mathcal{C}^{*}_{H}$ contains “$\psi$-relative” valuations that
make true all and only those propositions that can be verified with certainty
in an appropriate measurement of the system prepared in state $\psi$.

This finally brings us to discussing the philosophical importance of the
formal results about valuations in QL presented in section three. It is well
known that a realist understanding of QM – that is, roughly, an understanding
that views it as a theory that asserts truths about physical reality - runs
easily into serious troubles and is still an open issue in contemporary
science and philosophy. An important attempt at settling this issue was
famously put forward by Putnam (1968), who deemed the adoption of QL as a
necessary ingredient for maintaining a healthy realist view of the quantum
world, devoid of unpalatable metaphysical hypotheses supposedly implied by
theories that concord with CL (such as non-local hidden-variable theories).
Putnam’s bold proposal, which implied that all interpretational conundra of QM
disappear upon acceptance of QL, and in particular that QL allows one to think
of all observables in QM as having definite values at all times, was of course
immediately put under serious scrutiny.

Importantly, in order to formally address the possible merit of Putnam’s
realism, Friedman and Glymour (1972) articulated the latter in a formal
semantic framework within which they assessed the possibility of providing the
language of QL with a bivalent semantics. Bivalence can, indeed, be considered
as a prerequisite for realism, at least in the present context, since it
maintains that any fact expressed by a sentence either obtains or does not
obtain in the physical reality, e.g., it is either the case or not the case
that exactly $n$ hydrogen atoms are currently located in your room. However,
as pointed out by Friedman and Glymour, Putnam’s proposal is faced with
immediate difficulties due to the Kochen-Specker (KS) theorem, which states
that, for Hilbert spaces of dimension $d>2$, there is no valuation that
assigns a definite value to all magnitudes, while respecting the functional
relationships among the latter.888As it is well-known, this entails that there
is no homomorphism from any Hilbert lattice with the carrier space’s dimension
$d>2$ to the two-element Boolean algebra. For more discussion on the
consequences of the Kochen-Specker theorem for QL, see (Dickson 1998, section
4.1.2). Nevertheless, Friedman and Glymour made it clear that, while the KS-
theorem does present problems for realism, at least as Putnam had conceived
it, it certainly does not entail that the language of QM fails to admit a
bivalent semantics.999This was also expressed even more clearly by Demopolous:
“There are two different accounts of indeterminism which are historically
important. The first, which apparently goes back to Aristotle, rejects
bivalence: A theory is indeterministic if it assumes that there are
propositions whose truth value is indeterminate. The second, represented by
the quantum theory, retains bivalence while rejecting semi-simplicity [i.e., a
property equivalent to truth-functionality]. … This [latter] form of
indeterminism implies that there is no Boolean representation of the
properties obtaining at a given time; yet for any property P it is completely
determinate whether or not P holds.” (Demopoulos 1976, 76sq) Thus, it is one
thing to say that it is true that “This photon will decay tomorrow or this
photon will not decay tomorrow”, while each of the disjuncts is neither true
nor false, and another thing to say that it is true that “This photon passed
through the upper slit or this photon passed through the lower slit”, when
each of the disjuncts is false. The conflation of the two different accounts
noted by Demopoulos has unfortunately been repeatedly made in the literature
(see, e.g., Bell and Hallett 1982, 368). Others made the same conflation on
the basis of the Jauch-Piron theorem: “Jauch and Piron show that any so-called
orthomodular lattice (in particular any Hilbert lattice) admits total
homomorphisms onto $\\{0,1\\}$ iff it is distributive. Note that this means
that any form of quantum logic must give up bivalence” (Bacciagaluppi 2009,
56). Such Aristotelian attacks on a bivalent semantics for QL have recently
resurfaced: “Clearly, a quantum disjunction is false when both disjuncts are
false; however, the interesting case for our discussion is the one in which
quantum propositions have undetermined truth values.” (Oldofredi et al. 2022,
fn.13) See also (Fletcher and Taylor 2021) for a similar view.

This conclusion also follows from the previous section, where we explicitly
constructed a couple of bivalent semantics by defining appropriate classes and
klasses of valuations. While none of these semantics can, due to the KS-
theorem, provide in a straightforward way support to realism about QM, some of
them may actually fare better than others in this respect. In fact, Friedman
and Glymour, after presenting potential candidates for a realist bivalent
semantics, immediately dismissed the ones that make negation non-classical,
for “dereliction of duty” (1972, 20). They deemed it unacceptable for a
realist semantics to ascribe the same truth-value both to a sentence and to
its negation, thereby subscribing to the following principle.

######  Principle 1.

A bivalent semantics is compatible with realism only if its negation obeys the
classical truth-table (i.e. for any valuation $v$ and sentence ‘s’: $v(s)\neq
v(\lnot s)$).

One can of course raise doubts about this principle, especially in a context,
such as the present one, in which classical logical notions are at stake. For
example, it is not clear why one should endorse this principle without also
endorsing parallel ones that would require any realist semantics to make
conjunction and disjunction classical, thereby conflating realist and
classical semantics.101010Some ideas along these lines can be found in
(Dummett 1976). That is, why is it unpalatable for a realist to hold that both
a sentence and its negation have the same truth-value, but simultaneously be
at ease with false disjuncts making true disjunctions or true conjuncts making
false conjunctions? What makes negation special? What makes its classicality
an essential aspect of realism?

In any case, we are not going to further criticize Principle 1 in this paper,
but we want to point out a formal difficulty posed by the results from the
previous section. Namely, recall that, in accord with Proposition 3, since any
$H$-class makes only conjunction TF, it must necessarily make negation non-TF
and thus non-classical. Thus, if one accepts Principle 1, then no $H$-class
can provide a semantics compatible with realism! Nevertheless, if we amend the
formal treatment of arguments with multiple premises, as we have done by
introducing the notion of H-klasses, and since this amendment arguably does
not threaten realism, then there are indeed bivalent semantics that can
support a sort of realism about QM, albeit certainly not the sort of realism
Putnam had wished for. In particular, there are bivalent semantics based on
$H$-klasses that make negation TF, while making conjunction and disjunction
non-TF.

Let us now take stock. We have argued that since QL can be considered as the
logic of experimental propositions in QM, and bivalence is widely understood
as a prerequisite for realism, formal semantic results concerning bivalent
semantics in QL are consequential for the possibility of maintaining realism
in the quantum domain. Even though the KS-theorem presents difficulties for a
“naive realist” understanding of QM, one can still salvage bivalence, and
thereby a weaker form of realism, by giving up truth-functionality.
Furthermore, even if one assumes that the truth-functionality of negation is a
necessary requisite for a semantics to be compatible with realism, the latter
might still be maintained if one were ready to amend the treatment of
arguments with multiple premises. All in all, realism about QM and bivalence
in QL are definitely not (yet) ruled out, though they are certainly rather
costly.

##  5 Beyond Realism: Meaning Variance

As seen in the previous section, Putnam’s claim that QL can be used to furnish
a realist picture of the quantum world has been shown to require a serious
revision, mainly due to the no-go result by Kochen and Specker.111111To be
sure, Putnam himself admitted as much. Furthermore, already in papers from the
early 1990s, he acknowledged that a realist understanding of QM need not
embrace QL. See, e.g., Putnam 1991 and Putnam 1994. Also, for a later
acknowledgement, see Putnam 2012. There is however a further bold proposal
made in the same 1968 paper, in that Putnam argued that QL is not merely an
alternative logic, which may play a role in reasoning about quantum objects,
but that it is the “one true logic” and that the world we live in is a
“quantum logical world”. Moreover, he maintained that “adopting quantum logic
is not changing the meaning of the logical connectives, but merely changing
our minds about the [distributive] law.” (Putnam 1968, 233) In other words, QM
did not just lead us to the introduction of a new mathematical structure that
we may interpret as a logic, but to the discovery that the “one true logic” is
non-classical. This claim sparked another line of criticisms (e.g., in Dummett
1976), which do not challenge primarily Putnam’s realism, but his logical
revisionism; they contend, more particularly, that QL cannot be considered a
revision of CL, since the two are incommensurable: their connectives do not
mean the same thing.

A criticism of this latter kind was prominently put forward by Geoffrey
Hellman, who commented: “[T]he opponent against whom Putnam argued was a
rather dogmatic conventionalist who was rather prone to put more weight on the
notion of ‘meaning’ than scientific scrutiny should allow. What I want to do
here is focus on a more precise ‘meaning-change’ argument, one which makes
absolutely minimal reliance on the problematic word, ‘meaning’, and which, as
far as I can see, a proponent of Putnam’s view can neither defeat nor bypass.”
(Hellman 1980, 494) In what follows, we reconstruct Hellman’s argument and
show how it can be improved in light of the discussion in the previous
sections.

Hellman started by stipulating a condition for meaning invariance, which would
presumably be acceptable to a conventionalist without dogmas:

######  Principle 2.

“If $\alpha$ and $\beta$ are synonymous sentential connectives, then (a) if
one is a truth-functional connective, then so is the other, and (b) if
$\alpha$ and $\beta$ are truth-functional, they have the same truth tables.”
(Hellman 1980, 495)

He went on to prove that if QL-negation is TF, then QL-disjunction and QL-
conjunction are non-TF, which, due to clause (a) in Principle 2, implies that
at least some QL-connectives and their classical counterparts are not
synonymous. The argument can be reconstructed as follows:

1\. Two connectives have the same meaning only if they are either both truth-
functional or both non-truth-functional.

2\. If QL-negation is truth-functional, then QL-disjunction and QL-conjunction
are not truth-functional.

3\. Thus, QL-negation and CL-negation differ in meaning, or CL-disjunction
(conjunction) and QL-disjunction (conjunction) differ in meaning.

4\. Thus, some QL connectives differ in meaning from their CL counterparts.

Importantly, besides having the benefit of relying minimally on the
problematically vague notion of ‘meaning’, Hellman maintained that his
argument is independent of the issues concerning realism, in that it is based
on a purely formal semantic result that exhibits the difference in the truth-
functional status of the classical and quantum connectives: “the non-truth-
functionality argument is entirely distinct from [those that] argued that QL
could not satisfy the demands of realism.” (1980, 496) Now, notice that while
the argument does establish that at least some QL connectives differ in
meaning from their CL counterparts, it does not specify which ones do so
unconditionally. Moreover, Conclusion 3 is compatible with QL-negation being
non-TF and thus differing in meaning from CL-negation, while leaving QL-
conjunction and QL-disjunction TF and thus possibly synonymous with their CL
counterparts. Consequently, since the distributive law refers to conjunction
and disjunction, but not to negation, Hellman’s argument does not conflict
with Putnam’s logically revisionist claim that classical distribution fails in
QM. In other words, the argument does not justify Arthur Fine’s earlier claim
that “the sense of the distributive law in which it is said to fail is not the
sense in which, as the distributive law, it is supposed to hold.” (Fine 1972,
19). Our goal in the remainder of the paper is to use the results from section
three in order to improve on the above argument and to finally justify Fine’s
claim.

First, notice that the conditional statement expressed in Premise 2 follows
from Proposition 2, which states that any viable class of valuations can make
at most one of the sentential connectives TF. Now, while the antecedent of
Premise 2, stating that negation is TF, can arguably be justified for
semantics that are intended to support realism - as required by Principle 1 \-
this assumption is not warranted in the present discussion, in which we,
together with Hellman, are aiming to contrast CL- and QL-connectives on purely
formal, non-metaphysical grounds. Moreover, not only is the aforementioned
antecedent not warranted, but it cannot be maintained without invalidating the
quantum-logically valid $\land$-introduction argument,
$\left\\{a,b\right\\}\vDash a\land b$. Indeed, recall that Proposition 3
states that any viable $H$-class of valuations makes conjunction TF, but
disjunction and negation non-TF. Therefore, Premise 2’s antecedent, whose
truth is needed if the argument is to back Fine’s claim against Putnam’s, is
not only unwarranted, but also false121212Note that this point is distinct
from the often raised criticism that the quantum logician’s metalanguage
should obey quantum logic, in which case Hellman (1980)’s proof for the non-
truth-functionality of conjunction and disjunction would not go through..

Furthermore, there is another qualification that needs to be made here:
namely, it is not entirely clear what is meant in the above argument by saying
that a sentential connective ‘c’ is TF. Does it mean that (i) there exists a
viable class of valuations that makes ‘c’ TF, or that (ii) any viable class of
valuations makes ‘c’ TF? Since the argument tacitly assumes that classical
connectives are TF, option (ii) is automatically excluded, since the existence
of Carnap’s non-standard class
$\mathcal{C}^{*}_{CL}\cup\left\\{\tilde{v}_{CL}\right\\}$ would make even CL-
disjunction and CL-negation non-TF! Therefore, the only chance for a Hellman-
type argument to establish meaning-variance on the basis of a difference in
the truth-functional status of CL- and QL-connectives is to stick to option
(i).

With all this in mind, we now offer a modified argument, which nevertheless
assumes the same condition for meaning invariance stipulated by Hellman, i.e.,
Principle 2, but provides a definite verdict concerning Putnam’s and Fine’s
claims. Let us say that a sentential connective ‘c’ is TF if there is at least
one viable class of valuations that makes ‘c’ TF. Here is the argument:

1’. Two connectives have the same meaning only if they are either both truth-
functional or both non-truth-functional.

2’. QL-disjunction and QL-negation are not truth-functional.

3’. Thus, CL-disjunction (negation) and QL-disjunction (negation) differ in
meaning.

4’. Thus, some QL connectives differ in meaning from their CL counterparts.

This argument can be understood as a completion of Hellman (1980)’s, in that,
while it maintains the same general conclusion (i.e. the one expressed in
4’.), it also specifies which QL-connectives are not synonymous with their
classical counterparts: in particular, since disjunction - which figures in
the distributive law - is among these connectives, our argument justifies
Fine’s claim.

There is, however, another point we want to make concerning the above
argumentation. Namely, notice that we have so far assumed that the only viable
classes of valuations for QL are those introduced by Definition 8, i.e.,
$H$-classes. But recall that, after observing that both CL- and QL-conjunction
are necessarily made TF by any viable $H$-class – essentially due to the
particular formal explication of the validity of arguments with multiple
premises – we introduced (in Definition 10) the notion of $H$-klasses of
valuations, thereby restoring the semantic symmetry between conjunction and
disjunction. Now, how might considering $H$-klasses in the present discussion
change anything? It can easily be seen that this is a problem both for
Hellman’s argument and for our own as well. For recall that, after introducing
$H$-klasses, we went on to show that for any QL-connective, there is a klass
that makes it TF, as stated by Proposition 5, and that for any CL-connective,
there is a klass that makes it non-TF, as stated by Proposition 6. It thus
follows that there is no difference anymore in the truth-functional status of
CL- and QL- connectives, at least at the level of individual connectives: for
any connective ‘c’, there exists a klass that makes CL-‘c’ TF and there exists
a klass that makes QL-‘c’ TF. Also, there exists a klass that makes CL-‘c’
non-TF and there exists a klass that makes QL-‘c’ non-TF. Therefore, if one
formalizes valid arguments by means of klasses, no meaning-variance can be
established by any Hellman-type argument (i.e. one that relies on Principle
2). As a consequence, one would not be justified to reject Putnam’s logical
revision, for the sense of the distributive law in which it is said to fail
may well be the very sense in which it is supposed to hold. On the other hand,
as already emphasized before, there is of course still a big difference at the
global level, in that, unlike in the case of CL, there is no truth-functional
$H$-klass, i.e., no klass that makes all three QL-sentential connectives TF.
But this difference cannot be used to support a Hellman-type argument, due to
the fact that Principle 2 considers only individual connectives and their
counterparts.

To summarize, the soundness of a Hellman-type argument that implies meaning-
variance between classical and quantum connectives depends on the formal
explication of valid arguments with multiple premises. If one chooses to
explicate the latter in terms of klasses of valuations, then any such argument
turns out to be unsound, as we have just seen. However, if one opts for
classes of valuations, then a sound argument that establishes meaning-variance
can be given: in particular, it establishes a difference in meaning between
CL- and QL-negation and between CL- and QL-disjunction.131313One might still
have lingering doubts that our argument really establishes a difference in
meaning between these connectives, since as we pointed out there is a class of
valuations, i.e., Carnap’s non-standard class
$\mathcal{C}^{*}_{CL}\cup\left\\{\tilde{v}_{CL}\right\\}$, that makes CL-
disjunction and CL-negation non-TF. Nevertheless, such doubts are in fact
baseless, for while there is also a class of valuations that makes CL-
disjunction and CL-negation TF, there can be no class of valuations that makes
QL-disjunction TF and no class of valuations that makes QL-negation TF. This
difference is guaranteed by Proposition 3. It does so, however, provided that
one accepts Principle 2. Whether one should do so or not, we will discuss with
another occasion.141414Let us just note that both in Hellman’s argument and in
ours, it is Principle 2 that does the philosophical heavy-lifting, in that it
proposes a connection between the somewhat loose and evasive notion of
“sameness of meaning”, or “synonymy”, and the formal-semantic notion of truth-
functionality. It is thus the premise that allows for ‘meaning’ to be placed
under transparent scientific-mathematical scrutiny.

##  6 Conclusion

In this paper, we started by presenting QL in the abstract-algebraic framework
presented in (Dunn and Hardegree 2001), thus considering the language of QL as
an interpretationally constrained language. Within this framework, we then
analyzed the truth-functional status of QL-connectives, and introduced a
distinction between two different types of classes of valuations: $H$-classes,
which make conjunction truth-functional, and $H$-klasses, which allow for a
non-truth-functional conjunction. Turning to more philosophical concerns, we
first pointed out that a bivalent semantics of QL, that could potentially
provide a basis for realism, is not ruled out. However, the price that must be
paid by the realist is the truth-functionality of the QL-connectives. This
naturally led us to discuss Hellman’s anti-revisionist argument that purports
to show, on the basis of the aforementioned lack of truth-functionality, that
adopting QL changes the meaning of some logical constants. We provisionally
embraced his condition for meaning invariance, but pointed out that the
argument does not really support a rejection of Putnam’s revisionism. In fact,
the argument would provide such a rejection only if QL-negation were truth-
functional, which however, besides being unjustified, also turns out to be
false, as it conflicts with the quantum-logically valid $\land$-introduction
argument. One may in response want to consider $H$-klasses of valuations, but
this only causes more trouble. In contrast, our modified argument fares better
and makes the point that adopting QL changes the meaning of negation and
disjunction. Granted all the required assumptions for this argument -
including the arguably controversial Principle 2 \- Putnam’s logical
revisionism is finally rebutted: the sense of the distributive law in which it
is said to fail is not the sense in which it is supposed to hold. Fine was
right about it. Now we know the reason why he was right.151515Incidentally,
this conclusion undermines all attempts at resisting Putnam’s logical revision
for the reason that the proof that the distributive law fails in QM is rule-
circular, i.e., it assumes that very law in its classical metalanguage. For a
critical discussion of one such recent attempt in (Rumfitt 2015), see
[reference removed]. For a different way of avoiding rule-circularity, see
(Schurz 2022). For a critical discussion of yet another argument against
revisionism, mounted in (Williamson 2018), see (Horvat and Toader, 2023)).

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