A Logico-Epistemic Investigation of Frauchiger and Renner’s Paradox

The scientific literature on Wigner’s Friend extended paradox rapidly grew in the last years. A sign that Frauchiger and Renner (2018)’s argument caught an important point. Indeed, they conclude that either we must abandon the universal validity of quantum mechanics, or a certain kind of traditional objective knowledge is impossible. We investigate this contradiction through a logico-epistemic toolbox. We show that abandoning the transmissibility of knowledge, as proposed by many kinds of relational approaches to quantum mechanics, is a heavy epistemological renouncement. Perhaps, it is better to bite the bullet and accept Frauchiger and Renner’s contradiction, until a new revolutionary solution will appear.


Introduction
In a groundbreaking paper, Frauchiger and Renner [11] -FR hereafter -originally propose a more sophisticated version of Wigner's friend paradox [29]. In the original version of the paradox, a Friend of Wigner measures a dichotomic observable in a superposed state in an isolated laboratory, finding a certain result. In contrast, if Wigner, which is out of the laboratory, would apply quantum mechanics to the whole system -composed of the laboratory, his Friend and the measured observable -Wigner would find that the system his Friend is measuring inside the laboratory is still superposed. Thus, the physical situation is different for Wigner and his Friend. The conclusion is that, to avoid the contradiction if quantum mechanics is a universal 1 theory, the only reason why the Friend observes the collapse of the superposition could be the intervention of his consciousness on the physical system. 2 is not the same as negative introspection. Again, with an example like the preceding one, is it reasonable to suppose that Darwin believed that he did not believe that DNA's structure is helicoidal? Clearly not.
Even if Boge's theorems are very interesting, it seems to us that he assumes too much. In justifying the use of KT45 and BD45, Boge quotes many computer scientists. It is well-known [27:14] that epistemic logical systems for computer scientists comprehend 5 -since, generally speaking, a machine has no problem implementing negative introspection -whereas this is not a reasonable assumption from the point of view of the philosophy of science.
Let us consider 5 from the semantic point of view as well. 5 corresponds to the symmetry of the accessibility relations between possible worlds. Symmetry would mean that if Alice looks at a possible world where she acquires either new knowledge or new beliefs, the actual world must be accessible to her in the new cognitive situation. This conclusion is highly implausible since, with the new pieces of knowledge or beliefs, Alice could have precluded her access to the actual world. Again, an example can clarify the point. Before Hubble's discovered that some nebulas are galaxies, we lived in a world from which two worlds were accessible: one in which Milky Way was the only galaxy and one in which there were many galaxies. In the new cognitive world, where we know that there are millions of galaxies, the old one becomes epistemically inaccessible.
After this brief introduction, we now turn to the core of the paper. We start by presenting Deutsch' [6] version of Wigner's Friend in "Deutsch's experiment". Then, we discuss, in "The assumptions of the new experiment", FR's assumptions from a logico-epistemic perspective. Next, we present, in "The new experiment" and "The inference", a simplified form of FR's argument. A brief discussion on how the different interpretations of quantum mechanics faces FR's paradox ends the paper ("FR impact on the interpretation of quantum mechanics"). Conclusions follow suit.

Deutsch's experiment
NR emphasizes that FR's argument is very useful to evaluate the different interpretations of quantum mechanics. We completely agree on this issue. And indeed, they repeatedly highlighted in the paper that FR's argument shows that physics, rather than philosophy, can decide among the different interpretations of quantum mechanics. This could be a good point, but, as we will show in what follows, our unsatisfaction with quantum mechanics is not only physical but philosophical as well. 5 The gist of our argument indeed is that FR sheds light once more on the fact that quantum mechanics presents important unsolved problems. This does not mean that we must abandon the theory, only that we must not forget that no theory is the end of the road of science.
Moreover, NR note that many thought experiments are based on a sort of double role of an entity, both as an agent and as a physical system. Indeed, this double role is present both in Maxwell's Demon and in Wigner's Friend. The Demon and the Friend are both agents and physical systems. Furthermore, the solution of Maxwell's Demon is a correct physical representation of the Demon's alleged agency. NR propose a definition of an agent as something that could be completely physical, as a machine, which measures and register the results of measurements. We stick with this definition. For it seems not a good idea to do physics with muddled concepts such as that of mind, consciousness, agency, etc. It is better to avoid such notions, not because they are irrelevant, but because there is not yet a good scientific definition of what we are speaking about. Now, it is useful to dwell a little on NR reformulation of Deutsch's form of Wigner's Friend paradox. Suppose that Alice measures, in the so-called computational basis | 1⟩ and | 0⟩, the state: Leaving apart the environment for the sake of simplicity, let us call "Z" the considered observable, and then Alice will find that either Z = 0 or Z = 1. On the contrary, Wigner, applying quantum mechanics to Alice's isolated lab, will find, with an obvious notation: Wigner's Friend paradox is simply the diversity between (2) and the determinate result observed by Alice. Deutsch adds another registration instrument shared by Wigner and Alice: let us call it the "book". Alice, after having measured, writes in the book that she gets an outcome -without specifying which. Therefore, the book could be either empty or filled. After Wigner received the information that Alice has measured, he switches on a control, which undoes the measurement in the lab. In principle, even if not in practice, this is possible. After making this inversion, the possible results are two: either in the lab -after Alice's measurement -there was still an actual superposition, then the undoing brings back the lab in the state: Or, if the state is no longer superposed, Wigner will find either This in principle possible experiment would discriminate physically between different interpretations of quantum mechanics. Indeed, if Wigner will find one of the states (4), we must find a good physical explanation of the collapse. On the contrary, if Wigner finds (3'), the many-worlds interpretation, that is quantum mechanics without collapse, is favoured.
From this last statement, it is possible to catch a glimpse of how the contradiction is built in FR's argument. Indeed, if one finds a method to bring outside the information that Alice achieved an outcome, maintaining that of the superposition of the whole system, the conceptual tension between Alice's and Wigner's point of view -typical of Wigner's Friend experiment -would be wholly ascribed to Wigner. In this way, the tension becomes a contradiction in Wigner's knowledge. The experimental set proposed by FR reaches exactly this aim.

The assumptions of the new experiment
To reach the result of having both the cake -superposition -and eating it -Alice's outcome -we must double Wigner's experiment. Therefore, we have two isolated labs -Alice's and Bob's -and two external experimenters -Wigner and Bell. To reach the situation in which Bell eat the cake and has it -the contradiction -there must be partial communication among the four experimenters. Not only, but they must also make some inferences. These inferences, as we will see, could be performed also by a machine. Therefore, no kind of human direct intervention is involved. Moreover, we will also see that these inferences are logically indisputable.
The whole argument is based on three (main) assumptions. The first is: Q. The agents implement quantum mechanical rules to describe each possible system and they predict and retrodict outcomes using Born's rule.
Note that in the original FR's paper, Q was not assumed in this universal form. This point is highlighted by Nurgalieva and Del Rio [18], that criticize FR, emphasizing that the argument works only on this universal basis. Q means that if the outcome of a given observable Z assumes with certainty the value z -based on quantum mechanics -then Z = z is certain.
At this point, it is useful to establish which interpretations of quantum theory violate Q. Copenhagen interpretation clearly does, 6 since it distinguishes between what is quantum and what is classical. On the contrary Q -as already said -must be universal. Moreover, Bohmian mechanics is, for different reasons, also against Q, since at the micro level many unknown facts happen, which are not registered by the Born rule. 7 Finally, all objective collapse theories, such as GRW, do not accept Q since they maintain that at the macroscopic level quantum mechanics does not hold. 8 On the contrary, Q is satisfied by the Many-worlds interpretation.
The second assumption of FR's paradox is: C. If an experimenter 1 is certain that another experimenter 2 has found "Z = z", then experimenter 1 is certain that "Z = z".
Pictorial of principle C 6 NR correctly distinguishes the different forms of "Copenhagenism": the subjective Copenhagenist puts Heisenberg's cut where it is most convenient, so perhaps, one can maintain that s/he respects Q. On the contrary, the objective Copenhagenist puts the cut in a determinate position, therefore violating Q. 7 In fact, there are possible Bohmian answers to FR's argument. See Sudbery [23], Tausk [24], Lazarovici and Hubert, [15]. 8 Also at the microscopic level, but it is not statistically relevant.
Let us consider a justification of C based on epistemic logic (see Fig. 1). Suppose that W 2 is the set of worlds epistemically accessible to experimenter 2 . If experimenter 2 knows that "Z = z", then in each w belonging to W 2 "Z = z" holds. Let us consider the set of worlds epistemically accessible to experimenter 1 , that is, W 1 . If experimenter 1 knows that experimenter 2 knows that "Z = z", then in each w belonging to W 2 holds that "experimenter 2 knows 'Z = z'". Now, take any w belonging to W 1 ; in it "experimenter 2 knows 'Z = z'" holds, therefore if w belongs to W 2 , then in w "Z = z" holds, therefore experimenter 1 knows that "Z = z".
This argument shows that C could be violated only if: 1. There is no world accessible to experimenter 1 . 2. 2 there is no world accessible to experimenter 2 .
1. and 2. trivialize the notion of knowledge, since they hold if and only if the truth of a sentence is equivalent to the knowledge of it; that is, iff "p ↔ Kp". For this reason, 1. and 2. must be discarded, as possible cognitive situation. On the other hand, 3. is less radical, but it seems like Gorgia's strongly sceptical third dictum attributed to him: "He says that nothing is; and if [scil. something] is, it is unknowable; and if [scil. something] both is and is knowable, it cannot be indicated to other people." (our italics, LM, D26). Indeed, even 3. seems untenable for a non-sceptical theory of knowledge. Therefore, we conclude that C -sometimes called "transmissibility of knowledge" -is a very solid principle. 9 Despite this, C is explicitly denied by QBist [5]. Furthermore, according to NR, all relational approaches in a certain sense reject C. Indeed, Rovelli's relational quantum mechanics (RQM) and Copenhagen's subjective interpretations seem of this kind. 10 Considering the above argument we proposed in support of C, it seems that abandoning C boils down to renouncing to do science in a certain intersubjective way. For this reason, we conclude that this kind of interpretation after FR must be accepted only after a long investigation. We will come back to this point in what follows.
The last central assumption of FR's paradox is: S. An experimenter registers only one value of a measured observable.
In FR, it appeared that S could be eliminated in the Many-worlds interpretation. Moreover, since Q and C seem not negotiable, the conclusion could be that FR favoured Many-worlds. On the contrary, NR maintain that S is accepted by all standard interpretations of quantum mechanics. For this reason, we will no longer be concerned with this premise.
We conclude this part with a general consideration. All three premises of FR's argument could be physically implemented. That is, one can imagine machines able to realize the tasks they represent. Therefore, assuming Q, C and S does not mean exiting from the realm of physics. It is relevant to note that all three assumptions have also a normative character, in the sense that they could be thought of as valid principles. And their normative character is an important part of the thought experiment since from them it is possible to deduce a contradiction. This means that -as emphasized by NR -physics proposes to us that there is a problem. Physics, not only philosophy, because the three principles could be physically implemented. Therefore, it seems that our theory makes possible something physically impossible.

The new experiment
In Fig. 2, the reader can see a pictorial of the experimental setting. Alice and Bob stay in their respective isolated labs. They interact only when Alice sends Bob the state A. Out of Alice's lab there is Wigner with a hypothetic instrument to measure Alice's lab. Out of Bob's lab, there is Bell with a hypothetic instrument to measure Bob's lab in the basis | ok⟩ B , | fail⟩ B . Wigner can communicate with Bell.

Alice measures the state A:
in the computational basis | 0⟩, | 1⟩. Alice registers the result (application of S). Then, Alice prepares the new state A according to this rule: If she found | 0⟩ she prepares | 0⟩. If she found | 1⟩ she prepares the state 1 Finally, Alice sends A to Bob in the other lab. Bob measures A in the computational basis. Bob will find | 0⟩ with probability 2/3 and | 1⟩ with probability 1/3. Bob registers the result obtained.

Wigner measures Alice's lab in the basis:
where | lab A0 ⟩ and | lab A1 ⟩ are the states of Alice's lab in the respective cases in which she finds | 0⟩ and | 1⟩. Wigner registers the result (application of S).

Bell measures Bob's lab in the basis:
where | lab B0 ⟩ and | lab B1 ⟩ are the states of Bob's lab respectively in the cases Bob finds either | 0⟩ or | 1⟩. Bell registers the result of his measurement (application of S).

The inference
Let us consider the case in which Alice finds | 1⟩. Therefore, she prepares L in the state: This means that Alice can deduce that Bob's state must be (application of Q): Hence, Alice concludes that Bell will find | fail⟩ B (application of S). Therefore, Alice can register that she found | 1⟩ and Bell certainly will find| fail⟩ B . Bob received (7), therefore she can conclude from the setting of the experiment that Alice measured | 1⟩. Bob, applying Q and S can establish that Alice knows that Bell will find| fail⟩ B . Hence, by applying C, Bob can conclude that Bell will find | fail⟩ B .
Let us now consider the case in which Wigner observes| ok⟩ A . It is possible -although not trivial -to show that in those cases, Wigner, on the basis of Q, can show that Bob measured | 1⟩. Indeed, by applying Q, S and C, Wigner can be certain that Bell will find| fail⟩ B . Wigner communicates this result to Bell, which is now sure of finding | fail⟩ B (by applying C).
Let us consider the cases in which Wigner finds | ok⟩ A and Bell finds| ok⟩ B . It is not difficult to show that these couple of results happen with probability 1/12, that is, 1 time every 12 runs of the experiment on average. This means that sometimes Bell must accept a contradiction, that is he knows that the result of his measurement must be| fail⟩ B , but he finds| ok⟩ B . On the other hand, if he finds | ok⟩ B he knows that the outcome is | ok⟩ B (application of S). Therefore, Bell knows that on average 1 time out of 12 the result of his experiment is both | fail⟩ B and| ok⟩ B , against S. Contradiction!

FR impact on the interpretation of quantum mechanics
As already emphasized in "The assumptions of the new experiment", many approaches deny Q: viz. Objective Copenhagen interpretation, Bohmian mechanics and Collapse theories. The problem with these perspectives is that in more than half a century of research we did not find a trace of evidence favouring the limit of applicability of quantum mechanics. Recently, also the case raised by Hawking against quantum mechanics in black hole physics seems deflated [4].
As emphasized also by NR (p. 193) "However, with the development of quantum technologies, more and more complex systems are investigated, and so far no indications have been found that quantum theory could be inaccurate on larger scales." This fact compels us to take into serious consideration that our capacity of knowing cannot respect C.
On the contrary, many approaches do not pose limits to the applicability of quantum mechanics: Relational quantum mechanics, Many-worlds interpretation, Subjective Copenhagen interpretation and QBism. These interpretations of quantum mechanics reject C, albeit they do so in different ways. As shown in "The assumptions of the new experiment", these approaches endorse a sceptical point of view. Scepticism, like several interesting philosophical theses, could be tested empirically [26]. Indeed, it could be that certain features we believe knowledge must have -as C, for instance -are not achievable. In other words, it could be that the large empirical confirmation of quantum mechanics is saying to us that it is not possible to reach the form of shared knowledge we aimed at. Nevertheless, before accepting this sceptical conclusion one must investigate every alternative. As shown by Don 14, pp. 245 ff.), the most important reason why Einstein did not trust quantum mechanics was exactly the relational character of their states, which makes objective knowledge problematic. That Einstein was not persuaded of the possibility of making physics in this relational context is not an argument. Indeed, Howard himself in the following section (ibidem), and Muller [17], advocate a structural metaphysics for quantum mechanics. Nevertheless, it must be emphasized that if quantum entities would be relations, this seems to pose serious limitations to our objective knowledge, as shown by the influence of this fact on the validity of C. Perhaps an example can clarify all the scepticism implicit in the physical negation of C.
Let us consider three different physical systems S 1 , S 2 and S 3 . Between S 1 and S 2 the external and asymmetrical relation "knows" holds: "S 2 knows S 1 ." "External" means that knows is not reducible in any sense to the properties of S 1 and S 2 . Between S 1 and S 3 knows holds as well; that is "S 3 knows S 1 ." This set makes it possible that S 2 and S 3 know different things about S 2 . Here it is the scepticism favoured by the universality of quantum mechanics.
QBism and RQM deny explicitly C, but other relational approaches are in the same boat. For instance, as emphasized by NR, also Many-worlds must deny C, in the sense that what is known in one branch cannot be known in another. 11 In other terms, if it is not possible to ascribe a state to something independently of the measurement apparatus, at the end of the day, objective knowledge as we conceived it before the advent of quantum mechanics is not an achievable goal.
There is a peculiar many-worlds interpretation, which deserves our attention.
Zurek [31] attempts ante litteram to give body to the idea that C must be violated. More precisely, in his perspective, the "worlds" are only the different aspects of a physical system an experimenter with her apparatus can register. In other words, an apparatus applied to a certain system establishes a division between what is apparent and what must be dispersed in the other "worlds", that is in the environment. In this perspective, the same physical system would differently appear to different observers for precise physical reasons. Nevertheless, as emphasized also by Schlosshauer [22], decoherence is not a general solution to the measurement problem in quantum mechanics, but an ongoing program of opening the black box orthodox perspective had built on the collapse of the wave function.

Conclusion
Then, what we must do? Are we sure that we must accept the scepticism implicit in the negation of C? The history of physics presents plenty of cases in which our knowledge of physical phenomena radically changed. For instance, Aristotle thought the violent movement must be necessarily connected with a mover -with the theological exception of the unmoved mover. This position probably depended on the fact that he did not have the notion of inertial mass. Therefore, the discovery of inertial mass by Galilei, Descartes and Newton considered movement a non-relational fact. Why something similar cannot happen in the case of the relational character of quantum entities?
Perhaps we can learn a lesson from EPR story. Indeed, the problem about nonlocality raised by EPR has been resolved in favour of quantum non-separability, else the ascription of the Nobel Prize to Aspect, Clauser and Zeilinger for having confirmed the violation of Bell's inequality would have not occurred. Nevertheless, Maldacena [16] presents the ER-EPR conjecture as a sort of explanation of Bell's correlations. Indeed, it could be that where there are correlations, there is a spacetime deformation, which connects (apparently) distant spacetime points. This perspective is highly speculative, but it shows that the progress of physics often follows unexpected roads. The same happened with the action at distance introduced by Newton in 1686 and explained by Einstein in 1915. If Maldacena is right, exactly as in the case of Newton's gravity, the weirdness of Bell's correlation will be explained in an altogether surprising framework. Why the same could not happen in the case of the so-called measurement problem?
FR show once more that there is a conflict between the universality of quantum mechanics and the determinateness of our knowledge of measurement outcomes. We have no serious experimental reason to abandon Q, and neither to abandon C seems a mild epistemological renounce. There is a third possibility on the table: we must learn to live with the contradiction. FR show us that there is a contradiction between the universality of quantum mechanics and a certain form of objective knowledge. It is not necessary to hurry up in abandoning one of the two horns of the dilemma. Science is always in fieri. And this becoming is characterized by long periods during which contradictions are in the agenda. Another example is the one emphasized by [21], cap. 5): today's student of physics in the first lecture of quantum field theory learns that spacetime is flat, whereas in the second lecture of general relativity s/he realizes that spacetime is curved. 12 This does not mean that we must accept contradiction; we must accept only that for a certain period of research we live with a contradiction. We are not endorsing dialetheism (Priest et al., 2022), that is the thesis that reality is contradictory, but only that the representation of our knowledge at a certain time could be [2].
Author contributions Vincenzo Fano wrote the first draft of the paper, and prepared the figures. Preliminary work on Wigner's paradox has been carried out by Gino Tarozzi; the formal aspect of epistemic logic and its application to the quantum domain has been done by Vincenzo Fano. The implications for the different interpretations of quantum mechanics is the result of the work of Alberto Corti. The main thesis of the paper and the analysis of the paradox is the result of the shared work of the three authors, as the paper has been revised several times by each co-author.
Funding Open access funding provided by Università degli Studi di Urbino Carlo Bo within the CRUI-CARE Agreement.

Competing interests The authors declare no competing interests.
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