Guilhem Lecouteux

In Search of Lost Nudges

Keywords: nudges, libertarian paternalism, Parfit, hyperbolic discounting, identity, psychological connectedness

this paper discusses the validity of nudges to tackle time-inconsistent behaviours. I show that libertarian paternalism is grounded on a peculiar model of personal identity, and that the argument according to which nudges may improve one's self-assessed well-being can be seriously questioned. I show that time inconsistencies do not necessarily reveal that the decision maker is irrational: they can also be the result of discounting over the degree of psychological connectedness between our successive selves rather than over time (Parfit 1984, Reasons and Persons, Oxford University Press). Time inconsistency can call for paternalism if and only if we accept that an individual is characterised by stable "true" preferences over time-dependent outcomes, and that she is rationally required to make time-consistent choices. This model is descriptively and normatively questionable. I then argue that behavioural findings may still justify paternalistic interventions, but on a non-welfarist basis.

Introduction

Standard economic theory is built on the assumption that people act as if seeking to satisfy stable and coherent preferences, and are instrumentally rational given their beliefs and those preferences. Behavioural economics however provides evidence that real individuals are subject to numerous decision biases, such as optimism and overconfidence [START_REF] Sunstein | Selective Fatalism[END_REF], loss aversion [START_REF] Kahneman | Prospect Theory: an Analysis of Decision under Risk[END_REF] or status quo biases [START_REF] Samuelson | Status Quo Bias in Decision Making[END_REF]. The most common interpretation of behavioural findings is that real individuals -unlike rational Homo economicus -are not good at decision making, and often make mistakes: they would in fact choose differently had they "complete information, unlimited cognitive abilities, and no lack of self-control" (Sunstein andThaler 2003, p.1162). This observation is the starting point of a general argument against Mill's Harm Principle (1859), according to which "the only purpose for which power can be rightfully exercised over any member of a civilized community, against his will, is to prevent harm to others. His own good, either physical or moral, is not sufficient warrant". [START_REF] Conly | Against Autonomy. Justifying Coercive Paternalism[END_REF] argues for instance that our poor instrumental reasoning prevents us from achieving our own goals: it is therefore in our interest to accept that the government may prevent us from acting in accordance with our decisions.

The proponents of libertarian paternalism (Sunstein andThaler 2003, Thaler andSuntein 2008, henceforth "TS") defend this paternalistic position by arguing that paternalism is actually inevitable. People are highly sensitive to framing effects (Kahneman and Tversky 2000), and their decisions are influenced by features of the choice environment that are not relevant from the perspective of the social planner (Bernheim and Rangel 2009, p.55). Since there is no "neutral" frame, any way of presenting a situation of choice will necessarily make an option more salient than another one (by defining for instance a default option, a first option in a list of alternatives, etc.). TS argue then that the individual in charge of the design of the choice environment -the choice architect -should choose the choice architecture so as to help people to improve their well-being, "as judged by themselves" (TS, p.5). The logic of libertarian paternalism (henceforth "LP") is therefore that people aim to choose the options that will make them better off, but that -due to human fallibility -they often make non-rational choices, and miss their objectives. Since the choice architect has the possibility of slightly influencing people's choices, she should nudge them so that they achieve in fine their goals. A "nudge" is then defined as "any aspect of the choice architecture that alters people's behavior in a predictable way without forbidding any option or significantly changing their economic incentives" (TS, p.6). Since the individuals are not forced to choose the option the choice architect wants them to choose, Sunstein and Thaler argue that nudges preserve individuals' freedom of choice.

As an illustration, imagine the savings choices of an optimistic individual, Oscar. When hired for his first job, Oscar had to decide how much he wanted to save for his retirement. He was then convinced that he would quickly get a higher salary, and therefore -so as to smooth his consumption over time -preferred to consume a relatively large proportion of his income at the beginning of his career.

He knew that this strategy could be risky, and therefore planned to decrease his level of consumption in the future if he could not manage to get a better-paid job.

Unfortunately for him, Oscar never earned a significantly higher salary, and -due to inertia in his consumption habits -was also unable to keep his initial commitment. He then ended up with a quite low old age pension. Oscar now regrets his past decision and thinks that, were he able to turn back the clock, he would change his choice, based on better information about his ability to keep commitments and not misled any more by his optimistic expectations.

Behavioural economists know that many individuals, like Oscar, are too optimistic (concerning their beliefs about getting a higher salary, but also about keeping their commitments) and are probably making a mistake when saving little for their retirement. We should therefore nudge them today such that they will benefit from higher savings when retired. This can be done by exploiting their bounded rationality through framing effects: we could for instance create a default option such that they will be inclined to save more without significantly limiting their freedom of choice [START_REF] Thaler | Save More Tomorrow: Using Behavioral Economics to Increase Employee Savings[END_REF]).

The two central claims of LP are therefore that (i) we can make the individuals better off, "as judged by themselves", since they can suffer from making mistakes, and (ii) we can do so while preserving their freedom of choice. The first claim characterises the paternalistic dimension of nudges, while the second claim ensures that nudges are compatible with libertarian principles. Several authors however question the claim that nudges are actually paternalistic, such as Hausman and Welch (2010, p.136), who argue that most of the nudges defended by TS are cases of rational persuasion. Since the choice architect is not supposed to impose her own normative views on the individuals, and since individuals are not constrained by the choice architect, they claim that TS's definition of "paternalism" is unsatisfactory -LP would be better understood as beneficence. Thaler and Sunstein (2003, p.175) indeed define a policy as paternalistic if "it is selected with the goal of influencing the choices of affected parties in a way that will make those parties better off": LP is therefore a means paternalism rather than an ends paternalism (Sunstein 2014, pp.19-20). Unlike ends paternalists who pursue objectives that are different from the ends of the individuals, means paternalists want to help people to achieve their own ends.

The object of this paper is to highlight that, within the context of intertemporal choices, LP implicitly relies on an implausible model of identity, and that referring to a more complex and plausible model may seriously undermine its paternalistic claim (according to which nudges can improve our own well-being). I firstly highlight that imposing a nudge on young-Oscar on the basis that retired-Oscar would have regretted his past choice requires accepting that (1) Oscar's true preferences remain stable over time, (2) retired-Oscar retrospectively knows what was in the interest of young-Oscar, and (3) Oscar is rationally required to use a constant discount factor when comparing future utilities (section 2). I then suggest analysing Oscar's choice in terms of Parfit's notion of psychological connectedness, and show that the regrets of retired-Oscar cannot justify unambiguously a paternalistic intervention on young-Oscar (section 3). I finally discuss the possibility of justifying paternalism on a non-welfarist basis (section 4).

Regrets and mistakes

Libertarian paternalists argue that the choice architect should nudge young-Oscar today, because young-Oscar is making a mistake when saving only a small proportion of his income. We can legitimately assume that retired-Oscar is likely to regret the choice of young-Oscar because, today, many individuals are in the situation of retired-Oscar, i.e. they did not save a lot for their retirement and now regret their past savings choices. "Oscar" should therefore be seen as a statistically representative individual 1 . A nudge in this situation can be seen as a form of means paternalism if and only if young-Oscar would agree with the policy if he knew that he is likely to regret his choice later. I show in this section that this condition, although quite intuitive, is true if and only if we accept a rather implausible model of individual identity.

The economic analysis of intertemporal choices assumes that the individual "Oscar" is a set of transient selves representing a decision maker at different dates. For sake of clarity, I will denote by t-Oscar the self of Oscar at date t. Each t-Oscar has preferences over time-dependent outcomes2 (x, n) -the promise to receive the outcome x at date (t+n). LP is grounded on the idea that t-Oscar can be characterised by two different types of preferences: his true preferences -the counterfactual preferences on which he would act if he had "complete information, unlimited cognitive abilities, and no lack of self-control", 1 We can notice that a few individuals may suffer from the nudge (those who truly prefer to consume a lot today): the legitimacy of nudges may therefore be questioned in this situation, since nothing justifies a priori that such exceptional individuals can be sacrificed to the benefit of the greatest number. See Bovens (2009, p.211) on this point.

whose associated utility function determines t-Oscar's welfare -and his revealed preferences -the preferences that are revealed by his choices (or equivalently, the preferences that determine his choices). LP's paternalistic claim relies on the assumption that there exists a discrepancy between young-Oscar's true and revealed preferences: young-Oscar should therefore be nudged such that he satisfies in fine his true preferences. Retired-Oscar's regrets are then taken as an indicator of the mistake of young-Oscar.

However, if retired-Oscar regrets the choice of young-Oscar, then it means that if young-Oscar had chosen differently, retired-Oscar would be better off. But under which conditions do we know that improving the well-being of retired-Oscar is actually in the interest of young-Oscar (and therefore that nudging young-Oscar on the basis that it will benefit retired-Oscar is actually in young-Oscar's interest)? We suggest that accepting the claim that retired-Oscar's regrets are a sufficient reason to nudge young-Oscar requires accepting the three following hypotheses:

(1) all the t-Oscars have the same true preferences,

(2) retired-Oscar's revealed preferences correspond to young-Oscar's true preferences,

(3) each self t-Oscar is rationally required to make time-consistent choices.

Those conditions (and the model of individual identity that supports them) are implicit in LP, and are not explicitly endorsed by its proponents: I nevertheless suggest that those conditions are necessary for LP's paternalistic claim, although they are both descriptively and normatively questionable.

If condition (1) is not verified, then time inconsistency does not matter from the perspective of young-Oscar: the reason why retired-Oscar regrets young-Oscar's choice is simply that his true preferences changed over time -and not that young-Oscar's choice was irrational. Indeed, allowing for preference changes with ageing may justify that retired-Oscar's regrets are consistent with a rational choice of young-Oscar. The paternalistic claim therefore requires the stability of t-Oscar's true preferences over time3 . Suppose therefore that condition (1) is verified. The reason why retired-Oscar disagrees with young-Oscar's choice is that they do not have the same revealed preferences, although they share the same true preferences. However, so as to be sure that young-Oscar would benefit from the nudge suggested by retired-Oscar, we need to assume that retired-Oscar's revealed preferences correspond to young-Oscar's true preferences (condition ( 2)): we should therefore assume that young-Oscar is mistaken, while retired-Oscar has a correct ex post assessment of young-Oscar's choice. We can for instance consider that young-Oscar's perception of his own interest is biased by a present bias (O'Donoghue and Rabin 1999), and has a tendency to put relatively higher weights on immediate outcomes -this would be the reason why he chose to postpone his savings effort.

Hyperbolic discounting -the tendency to care relatively less about the outcomes distant in time, as involved by a present bias -is indeed a well-documented phenomenon [START_REF] Frederick | Time Discounting and Time Preference: A Critical Review[END_REF], and would explain young-Oscar's time inconsistent choice.

The difficulty of this argument is that nothing justifies a priori the rationality of retired-Oscar -in particular if we assume that young-Oscar presents a present bias. It is therefore not certain that retired-Oscar's revealed preferences actually correspond to his true preferences (and therefore, by condition (1), to young-Oscar's true preferences). Retired-Oscar may for instance present a bias towards the future (Parfit 1984, p.165). Suppose for instance that the satisfaction of young-Oscar's true preferences actually implied saving little for his retirement.

When retired-Oscar is reminded of the pleasant life he had when he was young, he should accept that his low pension is the legitimate cost for his past consumption.

But if retired-Oscar is biased towards the future, then, when comparing the expectation of a future consumption of €100 with the memory of a past consumption of €100, he will prefer the future consumption of €100. What retired-Oscar really wants in this situation is to go back 40 years earlier to change his decisions, and then immediately enjoy the long term benefits of his choice 40 years later: the regrets he expresses today do not necessarily mean that his past consumption was a mistake, since the cost supported by young-Oscar is almost imperceptible from retired-Oscar's perspective. The effort that young-Oscar perceived as "intolerable" became retrospectively "unimportant" for retired-Oscar.

In this situation, saying that retired-Oscar would have agreed with being nudged when he was young does not mean that it would have been true for young-Oscar.

Nothing therefore justifies a priori the empirical validity of condition (2).

Suppose now that conditions (1) and ( 2) are descriptively accurate. The last condition implicitly stated by LP to ensure that retired-Oscar's regrets reveal the irrationality of young-Oscar's choice is that young-Oscar is rationally required to make time-consistent choices, i.e. to make choices with which all his future selves would agree. Unlike conditions (1) and (2), condition (3) is a normative rather than descriptive condition: it indeed states how t-Oscar is rationally required to discount his future utilities, under TS's conditions of unlimited cognitive abilities, perfect information and complete self-control. LP is indeed a "prescriptive approach", i.e. it is an "[attempt] to offer advice on how people can improve their decision making and get closer to the normative ideal" (Thaler and Benartzi 2004, p.S167, our emphasis). The exponential discounting model [START_REF] Samuelson | A Note on the Measurement of Utility[END_REF], which forbids time-inconsistent behaviours, is generally considered as a relevant normative model of intertemporal choices (and is implicitly considered as such by LP, since time-inconsistent choices are not allowed). This claim is for instance defended by O'Donoghue and Rabin (1999), according to whom time inconsistency leads to important welfare losses, and therefore that an exponential discounting may be preferable in terms of welfare (welfare is defined within their framework in a "long-run perspective" (p.113), with an equal weighing of the utilities of each period). Furthermore, we can notice that, if condition (1) is true, then, under TS's conditions of perfect rationality and complete information, (3) is necessarily true. If t0-Oscar is rational (in the sense that he ought to choose his action so as to satisfy his true preferences), then he should take the exact same decision when comparing two outcomes in t0 and t1, or in tn and tn+1. He indeed knows that, in n periods, he will face the situation of choice he faces today in the first case. If we assume that Oscar prefers €100 immediately to €100 in the future, then we can define a discount factor δ0,0 < 1 such that t0-Oscar is indifferent between €100* δ0,0 in t0 and €100 in t1 (a discount factor δt,n should be read as the discount factor used by t-Oscar when comparing two outcomes at dates (t+n) and (t+n+1)). Hyperbolic discounting means that t0-Oscar tends to give a relatively higher value to immediate rewards or costs than more distant ones: the discount factor δ0,n between two dates tn and tn+1 therefore increases when n increases, i.e. when the delay between the choice and its realisation increases. t0-Oscar therefore chooses his level of savings as if he believed that he would be more patient in the future than he is today: but since Oscar's true preferences remain stable over time (condition (1)), then retired-Oscar will regret the choice of young-Oscar, since he is exactly as patient as were young-Oscar. Hyperbolic discounting therefore implies that young-Oscar will take decisions that retired-Oscar will regret: this therefore justifies a paternalistic intervention on young-Oscar in his own interest.

I have shown that the argument according to which retired-Oscar's regrets are a sufficient reason to nudge young-Oscar means that (1) those regrets are meaningful for young-Oscar, and therefore retired-Oscar and young-Oscar have the same true preferences, (2) retired-Oscar retrospectively knows what was in young-Oscar's best interest, and (3) young-Oscar should have used a constant discount factor when choosing his level of savings. Accepting LP's argument therefore implies that we should also accept the idea that Oscar is simply a set of transient selves t-Oscar with constant true preferences over time. This picture of Oscar's identity seems quite implausible, since it imposes the stability of individual true preferences over time-dependent outcomes (from a descriptive perspective), as well as the superiority of the exponential discounting model (from a normative perspective). I will now show that an alternative account of Oscar's identity does not necessarily support the normativity of the exponential discounting model, and that it is possible to rationalise time inconsistent behaviours (nudging young-Oscar would therefore not be a case of means paternalism any more).

Does time inconsistency matter?

The object of this section is to investigate whether t-Oscar is rationally required to discount his future utilities with a constant discount factor or not. We should firstly notice that, although people actually discount future utilities, it is not clear whether they are rationally required to do so (and reciprocally, whether discounting one's future utilities is rational or not). Indeed, if we assume that Oscar's objective is that his life goes as well as possible, as a whole, then it is not certain that there is any decisive argument for discounting future utilities (see for instance [START_REF] Broome | Weighing Goods: Equality, Uncertainty, and Time[END_REF]). [START_REF] Frederick | Time Preference and Personal Identity[END_REF] notes that temporal neutrality (the claim that a person should give the same weight to all utilities, regardless of their temporal position) implicitly assumes "that all parts of one's future are equally parts of oneself; that there is a single, enduring, irreducible entity to whom all future utility can be ascribed" (p.90). If we accept the existence of such an irreducible entity, then t-Oscar should not discount his future utilities.

By opposition to this "simple" view of identity, [START_REF] Parfit | Reasons and Persons[END_REF], among others, offers a "complex" view, according to which such an irreducible entity does not exist: a person is a sequence of overlapping selves who are connected by different physical and psychological properties. Parfit suggests that there exists a relation of "psychological continuity" (p.206) between t0-Oscar and tn-Oscar: there exist strong psychological connections between t0-Oscar and t1-Oscar (such as shared memories, values, beliefs, desires...), as well as between t1-Oscar and t2-Oscar, until tn-1-Oscar and tn-Oscar. The relation of "strong psychological connectedness" is however not necessarily transitive: although Oscar can consider himself as the same individual than the individual he was yesterday, and that the one he will be tomorrow, he may not consider himself as the same individual than the one he was 10 years ago, nor than the one he will be in 10 years. Although there is some physical and psychological continuity between t-10-Oscar, t0-Oscar, and t10-Oscar, they are not necessarily the same person, because they do not necessarily share the same memories, values or preferences.

Parfit then argues that, from the standpoint of the decision maker, it is not personal identity but psychological connectedness that matters (Parfit 1984, p.245): knowing that your personality is likely to evolve over time, you are not rationally required to care as much about your further future than your closer one (p.158). If we accept the complex view, then the different selves of Oscar can be seen as different persons: it is therefore not irrational for t-Oscar to discount his future utilities, since his future selves are not entirely himself. The idea that my future selves are not "entirely" myself, although I may have a lot of common with them, can be captured by the notion of psychological distance between temporal selves. Psychological distance relates to the difficulty for people to experience the feelings and subjective states of others (either their future selves, [START_REF] Wilson | Affective Forecasting[END_REF], or other individuals, [START_REF] Andersen | Self-Knowledge and Social Inference: I. The Impact of Cognitive/Affective and Behavioral Data[END_REF]). [START_REF] Liberman | Psychological distance. Social psychology: Handbook of basic principles[END_REF] define for instance "psychologically distant things […] are those that are not present in the direct experience of reality" (p.353): at date t0, since t1-Oscar does not exist yet, t0-Oscar cannot directly experience t1-Oscar's utility. If t0-Oscar is unable to fully experience the utilities of his future selves, then he is legitimated to discount their utilities. The existence of a non-null psychological distance between me and my future selves may therefore rationally justify the discounting of my future utilities.

The question that follows is therefore the determination of a criterion to measure this psychological distance between temporal selves. Such a criterion may then give insights into how the individual is rationally required to discount future utilities. If we show that there exists a plausible measure of psychological distance such that hyperbolic discounting is not irrational, then the paternalistic claim of LP will be seriously undermined (the exponential discounting model would indeed not be the unique defensible normative model of intertemporal choice). Since the notion of psychological distance is related to the ability to experience the utilities of one's future selves, I suggest defining the psychological distance 0 ≤ d0,n ≤ 1 between t0-Oscar and tn-Oscar as the loss of welfare experienced by t0-Oscar when he gives €1 to tn-Oscar. The distance d0,n therefore captures the idea that t0-Oscar is less able to experience the utility of tn-Oscar than his own utility.

Consider now Parfit's description of Oscar's identity as a sequence of strongly psychologically connected selves. Suppose that the psychological connectedness between two successive selves can be measured by a parameter 0 ≤ β ≤ 1. We can for instance interpret β as follows: while t0-Oscar agrees with 100% of the choices he makes today because they are motivated by preferences, values, desires that he considers as being his own), he cannot be sure to agree with more than β % of the choices made by t1-Oscar, because a fraction (1-β) of the choices made by t1-Oscar are motivated by preferences, values or desires that t0-Oscar does not recognise as being his own5 . Since t0-Oscar only benefits from €β n when €1 is given to tn-Oscar, the psychological distance between t0-Oscar and tn-Oscar is d0,n = (1 -β n ). We can then deduce the discount factor used by t0-Oscar:

δ0,n = 1 -[(1 -β n+1 ) -(1 -β n )] δ0,n = 1 -β n (1-β)
The discount factor δ0,n increases with n: t-Oscar therefore discounts his future utilities as if he believed that his future selves would be more patient than him. The psychological distance (as perceived by t0-Oscar) between selves indeed tends to diminish as n increases. From the perspective of t0-Oscar, t40-Oscar and t41-Oscar are for instance almost the same person -a person who is quite different from t0-Oscar. It therefore does not cost anything for t0-Oscar to impose an important effort on t40-Oscar to the benefice of t41-Oscar.

Reconsider now the decision faced by young-Oscar a bit differently. Oscar is hired for his first job and must decide how much to save for his retirement. He also cares about poverty in the third world and intends to give a part of his salary to a charity. He also knows that he is likely to lose his charitable aspirations while getting older and wealthier. He therefore consciously decides to save a smaller proportion of his current income to be able to give more to the charity today, and imposes on his future selves greater savings efforts, since he cares less about his further selves with whom he does not identify. Within this context, the choice of young-Oscar is not irrational any more: the discount factor he applies to weight his future selves is indeed not a discount factor with respect to time, but a discount factor with respect to psychological connectedness. This implies in particular that the claim that retired-Oscar will regret young-Oscar's choices is not a sufficient reason to nudge young-Oscar today, since it does not mean that young-Oscar's choice was irrational.

LP claims that we should prevent people from being irrational (this is precisely the objective of means paternalism): since it is possible to rationalise time inconsistent behaviours by considering that what matters for t-Oscar is not the satisfaction of some stable true preferences but his degree of psychological connectedness with his future selves, we cannot defend nudges on the basis that they help young-Oscar to make better choices for himself. Nudging young-Oscar is indeed likely to cause some harm to young-Oscar and to benefit retired-Oscar: the well-being of young-Oscar will however be increased if and only if he has sufficiently strong psychological connections with retired-Oscar.

Justifying paternalism

LP and nudges are often justified by claiming that we should protect individuals from their own mistakes while respecting their subjectivity and freedom of choice.

The justification of nudges in intertemporal choices (on the basis of personal regret) however requires accepting that (1) people are defined by stable true preferences, (2) they have a correct assessment of their past choices, and (3) they should use a constant discount factor when comparing future utilities. I suggested that conditions (1) and (2) are descriptively inaccurate, while the normative claim of condition (3) -when we assume that a person is unified by the existence of her stable true preferences -implies that the only rationalisable time preferences would be temporal neutrality. I showed that an alternative account of individual identity such as Parfit's idea of psychological connectedness does not necessarily support the same paternalistic conclusions: it is indeed not necessarily irrational to care less about one's further future if we know that one's identity is likely to evolve over time.

Note that justifying imprudent behaviours (such as saving a small proportion of one's income) on the basis that they are not irrational does not imply that paternalism is not justifiable. Suppose for instance that a reason why young-Oscar prefers to consume more today is that he decides to start smoking. Although this behaviour can be rationalised because young-Oscar is not rationally required to care about retired-Oscar's we can argue that young-Oscar directly causes some harm to retired-Oscar (imprudent behaviours can therefore be morally wrong (Parfit 1984, p.318)). If intertemporal choices can be seen as choices involving several individuals, we can appeal to Mill's Harm Principle to justify a paternalistic intervention on young-Oscar. A new difficulty then arises, viz. which self of Oscar should be privileged. The issue here is that modelling Oscar's choice as a game between multiple selves with their own preferences (that are nonetheless similar for two relatively close selves) does not provide a concept of welfare that could be applied to the individual as an enduring agent. In a multiple selves model, the determination of a normative criterion on welfarist grounds is therefore not straightforward, since we do not have at our disposal such a notion of enduring agent, and we have no reason a priori to privilege a self over another.

It should however be noticed that, unlike within Parfit's analysis of identity, changes in preferences, values, or desires can also be initiated by the agent, rather than merely experienced. [START_REF] Korsgaard | Personal Identity and the Unity of Agency: A Kantian Response to Parfit[END_REF] for instance argues that persons are unified by the continuity of agency of their successive temporal selves, each of them being an active agent, contributing to the shaping of their own identity: what matters from this perspective is not psychological connectedness, but "the view of myself as an agent, as one who chooses and lives a particular life" (p. 23). This argument relies on the Kantian position that we may view ourselves not only as objects of theoretical understanding (the passive loci of our experiences) but also as agents, as "the thinkers of our thoughts, and the originators of our actions" (Korsgaard 1989, p.18). What matters for Oscar is therefore the life (in the sense of long-term commitments) he has chosen, as an autonomous agent, and not the experience of individuals with whom he is psychologically connected.

In this perspective, what behavioural economics tells us is not that real individuals are poor decision makers (who should be helped by the choice architect), but that they lack of autonomy, since they are likely to ruin their longterm plans due to irrelevant framing effects. I consider here a reasonsresponsiveness account of autonomy, according to which "an agent does not really govern herself unless her motives, or the mental processes that produce them, are responsive to a sufficiently wide range of reasons for and against behaving as she does" [START_REF] Buss | Personal Autonomy[END_REF]. The normative issue faced by boundedly rational individuals is not that they are not able to satisfy some hypothetical true preferences, but that their choices can be shaped by reasons they are not aware of (such as framing effects, a present bias… see [START_REF] Hausman | To Nudge or Not to Nudge?[END_REF] for a similar argument).

Acknowledging the right for the individual to govern herself therefore implies that it is not the satisfaction of one's true preferences that matters, but instead the possibility to choose one's own preferences and identity. Possible measures in this direction would typically consist in educating the individuals by warning them of the existence of framing effects, or more generally of the diverse sociopsychological biases that are likely to affect their choices. A complementary set of measures would then consist in providing to the individuals the means to make and enforce their commitments. So as to offer a philosophically and psychologically coherent solution to cases like Oscar's savings choice, normative economists should probably leave the welfarist background of libertarian paternalism, and focus instead on more deontological criteria.

Throughout the rest of the paper, I will simply say "t-Oscar preferences" instead of "t-Oscar preferences over time-dependent outcomes".

I will discuss in section

the possibility to justify paternalistic interventions if condition (1) is not verified, since the choice of t-Oscar can be seen as a choice involving several individuals (with different true preferences). This will however not be a case of means paternalism, and will not support the paternalistic claim of LP.

A solution would be to consider t0-Oscar's probability of dying before tn: since there is a probability that tn-Oscar does not exist, t0-Oscar would then be able to discount the utility of his future selves. This argument cannot however ensure that the difference (d0,n+1 -d0,n) remains constant over time: this would indeed require that Oscar has the same probability of dying at each period, which is highly implausible.

[START_REF] Frederick | Time Preference and Personal Identity[END_REF] stresses the difficulty of defining an objective measure of psychological connectedness. He for instance measures β by asking to subjects in an experiment to "rate how similar you expect to be in the future compared to how you are now, and how similar you were in the past compared to how you are now. By similar, I mean characteristics such as personality, temperament, likes and dislikes, beliefs, values, ambitions, goals, ideals, etc." on a scale from 0 (completely different) to 100 (exactly the same).

We can now redefine the discount factor δ0,n in terms of psychological distance. Recall that t0-Oscar is indifferent between €100 in tn+1 and €100*δ0,n in tn. The difference between those two outcomes can be interpreted as the cost supported by t0-Oscar when tn-Oscar gives €100 to tn+1-Oscar. This cost can therefore simply be measured as the difference between what t0-Oscar would experience if he gave €100 to tn-Oscar -i.e. €100* (1-d0,n) by definition of the psychological distance -and what he would experience if he gave €100 to tn+1-Oscar -i.e. €100* (1-d0,n+1). We have therefore:

From the perspective of t0-Oscar, if two temporal selves are relatively close, then the discount factor δ0,n tends to 1 (t0-Oscar therefore assigns similar weights to selves who are relatively close, from his perspective).

Consider firstly LP's description of Oscar's identity. So as to ensure that condition (3) is verified, we must have a measure of the psychological distance between two successive selves such that (d0,n+1 -d0,n) does not depend on n (otherwise Oscar is likely to make time-inconsistent choices). Furthermore, we can notice that, if t-Oscar is rational and does not hold false beliefs, then he knows that condition (1) is true. He could therefore perfectly anticipate the experience of his future selves: the psychological distance between the different selves t-Oscar is therefore necessarily null, implying that δt,n = 1, for all t and n. LP's conditions therefore imply that t-Oscar is rationally required to be temporally neutral 4 .


Alex D Johnson

Kyle A Paterson

Jake C Spiteri

Sergey A Denisov

Gediminas Jonusauskas

Arnaud A Tron

Nathan D Mcclenaghan
email: nathan.mcclenaghan@u-bordeaux.fr

David C Magri
email: david.magri@um.edu.mt

Water-soluble naphthalimide-based 'Pourbaix sensors': pH and redox-activated fluorescent AND logic gates based on photoinduced electron transfer

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Introduction

The development of photoinduced electron transfer (PET) sensors and switches, which began as curiosity-based science, [START_REF] Silva | Molecular Logic-based Computation[END_REF] is now state-of-the-art technology used in screening catalytic reactions, 2 detecting chemical warfare agents, 3 and measuring blood analyte 4 and glucose 5 levels. The basic concept is the engineering of molecules in a modular format according to a 'fluorophorespacer-receptor' configuration. 6 The modular construction is amenable to modifications with respect to both format and functionalities. 7 The predictable modulation of the luminescence has been further exploited to create switchable molecular systems with two or more states that emulate familiar electronic devices such as logic gate circuits 8 and triodes. 9 Recently we proposed a new class of photo-redox-ionic logic gates termed 'Pourbaix sensors'. 10 This class of molecular logic gate integrates the 'fluorophore-spacer-receptor' motif with a 'fluorophorespacer-electron-donor' design. 11 Our first examples consisted of a 'fluorophore-spacer 1 -receptor-spacer 2 -electron-donor' arrangement (Fig. 1) with a central anthracene fluorophore. 10 With the aim of improving water solubility and the fluorescence enhancement ratio, among other properties, we recently designed a molecular device with the structurally unsymmetrical naphthalimide fluorophore based on a 'receptor-spacer 1 -fluorophore-spacer 2electron-donor' format. 12 Fig. 1 A schematic diagram of a 'fluorophore-spacer 1 -receptorspacer 2 -electron-donor' design and the molecular structures of 1 and 2.

Naphthalimides are a remarkable class of fluorophore due to a strong internal charge transfer (ICT) character, which results in a large dipole moment in the excited state and an intense photoelectric field effect. 13 Depending on the magnitude and sign of the generated electric field, intramolecular photoinduced electron transfer can be either inhibited or accelerated. Due to the presence of this intrinsic phenomenon, we were interested in exploring the potential of naphthalimide-based 'Pourbaix sensors' with a 'fluorophore-spacer 1 -receptor-spacer 2 -electrondonor.' In addition, this design is amenable to the incorporation of an anchoring/linking unit at the imide position (R group) for covalent immobilisation to surfaces. 4,14 The relationship between pH and potential (pE) has long been known to be paramount to our fundamental understanding of metal ion solubility in geology, metallurgy and medicine. [START_REF] Pourbaix | Atlas of Electrochemical Equilibria in Aqueous Solutions[END_REF] We envision that molecular devices responsive to pH and redox chemistry could have a multitude of future state-of-the-art applications from corrosion detection, 16 histochemistry, [START_REF] Johnson | The Molecular Probes Handbook: A Guide to Fluorescent Probes and Labeling Technologies[END_REF] blood disorder diagnostics [START_REF] Magri | Supramolecular Systems in Biomedical Fields[END_REF] to anticancer agents. [START_REF] Gellerman | [END_REF] In fact, research is on-going to develop pH and redox-responsive micelles as intracellular drug delivery systems 20 and photodynamic photosensitizers. 21 We now report two novel fluorescent molecular AND logic gates modulated by redox potential and pH in aqueous methanol solution. 'Pourbaix sensors' 1 and 2 consist of an alkylated 1,8-naphthalimide (fluorophore) connected to a tertiary amine (proton receptor/electron donor) by a flexible ethylene spacer, the amine being linked by a methylene spacer to a ferrocene moiety (electron donor). The different alkyl spacers are deliberately incorporated into the design allowing for conformational mobility. An enhanced fluorescence output is observed upon simultaneous protonation of the tertiary amine with acid and oxidation of the ferrocene moiety with a suitable oxidant, in this study Fe 3+ .

Results and discussion

The naphthalimide-based logic gates 1 and 2 were prepared by a three-step procedure (Scheme 1). Condensation of methylamine with the anhydride precursor in tetrahydrofuran (THF) followed by amination with N-methylethylenediamine yielded 4-N-methylethylenediamine-1,8-N-methylnaphthalimide in 45% yield as the protonated salt. The final synthetic step involved reductive amination with ferrocenecarboxaldehyde and sodium triacetoxyborohydride in 1,2-dichloroethane (DCE) in the presence of triethylamine to give 1 in 20% yield. Purification was achieved by column chromatography and recrystallisation (see Experimental section). The 'Pourbaix sensors' 1 and 2 were characterised by 1 H and 13 C NMR, IR and HRMS (Fig. S1-S6, ESI †).

The UV-visible absorption spectra were measured in 1 : 1 (v/v) methanol/water and neat methanol solutions. Broad structureless absorption bands are observed with maxima at 439 nm and 440 nm in methanol, and at 446 nm and 450 nm in 1 : 1 (v/v) methanol/water for 1 and 2, respectively. Addition of acid (10 À3 M) to a solution of 1 results in hypsochromic shifts of 19 nm and 16 nm to maxima at 420 nm and 430 nm in 1 : 1 (v/v) methanol/water and methanol with no change in the absorption intensity. Similarly, 2 exhibits hypsochromic shifts of ca. 18 nm. Isosbestic points are observed at 438 nm and 437 nm in 1 : 1 (v/v) methanol/water for 1 and 2, respectively (Fig. S7 and S9, ESI †). The addition of acid results in the formation of a less stable Frank-Condon charge transfer excited state on protonation of the tertiary amine at the 4-position. 22 Intramolecular hydrogen bonding between the protonated tertiary amine and the NH amine and/ or an electrostatic field effect could contribute to the destabilization of the ICT state. The addition of more than 80 mM Fe 3+ significantly perturbs the UV-visible absorbance intensity (Fig. S8 and S10, ESI †). The photophysical data of 1 and 2 as well as 3 12 (Scheme 2) are provided in Table 1.

The Boolean logic characteristics of 1 and 2 were examined in 1 : 1 (v/v) methanol/water and methanol. The molecules display a fluorescence enhancement accompanied by a hypsochromic shift of 14 nm on addition of high threshold concentration levels of H + and Fe 3+ . Three 'off' states are ascribed to an ICT and/or PET from either the ferrocene moiety and/or the tertiary amine to the excited state naphthalimide fluorophore. Table 2 summarizes the AND logic truth table including the relative fluorescence quantum yields (F f ) of 1 and 2.

The Weller equation was used to predict the theoretical feasibility for the PET pathways. [START_REF] Weller | DG PET = E OX À E RED À E S À e 2 /er where E OX is the oxidation potential of ferrocene (0.45 eV) or the tertiary amine (1.15 eV) and E RED[END_REF] The singlet energy from the intersection of the excitation and emission spectra at 468 nm is 2.55 eV in 1 : 1 (v/v) methanol/water. 24 The calculated driving forces for PET from the tertiary amine and ferrocene to the excited state naphthalimide are 0.07 eV and À0.63 eV, respectively. These calculations suggest that PET from ferrocene to the excited state fluorophore should be highly favorably, while PET/ICT from the amine to the naphthalimide is sluggish. However, a repulsive negative field effect at the imide nitrogen atom is an additional kinetic barrier for PET from ferrocene to the naphthalimide as with 3, which is not accounted for in the driving force calculation. 13 On the opposite end, a positive electric field effect at the 4-amino position contributes to enhance the rate of PET from the tertiary amine to the naphthalimide.

The 'Pourbaix sensors' 1 and 2 display some definite advantages over our previous anthracene prototypes 10 including a longer emission maxima and solubility in aqueous methanol. The F f of 1 and 2 in the 'on' state is significantly higher than in the three 'off ' states in both methanol and 1 : 1 (v/v) methanol/water (Table 2, entry 4). The switching ratio between the maximum and second highest fluorescence output is on average a four-fold enhancement. Compared to AND logic gate 3 with an alternative arrangement consisting of an 'electron-donor-spacer 1fluorophore-spacer 2 -receptor', the maximum F f of 1 and 2 and fluorescence switching ratios are lower, but no less significant. 12 Our results are similar to some extent to a related molecule 4 with a rigid piperazine bridge connecting a carboxyferrocene by a methylene spacer (Scheme 2). [START_REF] Gan | Some photophysical data for 4 in acetonitrile: l Abs = 398 nm (log e = 4.05), l Flu = 521 nm (F em = 0.0061). In acid l Abs = 373 nm ([END_REF] The F f of 4 is 0.091 on oxidation of the carboxyferrocene moiety in acetonitrile. Upon protonation of the piperazine moiety the F f is 0.38. However, the simultaneous oxidation and protonation of 4 in accordance to the function of an AND logic gate was not reported. [START_REF] Gan | Some photophysical data for 4 in acetonitrile: l Abs = 398 nm (log e = 4.05), l Flu = 521 nm (F em = 0.0061). In acid l Abs = 373 nm ([END_REF] The similarity in the output response ratio of 4 suggests that 1 and 2 adopt a conformation such that the ferrocene moiety is distant from the naphthalimide fluorophore. Coincidentally, the same platform was incorporated into a three-input INHIBIT logic gate. [START_REF] Li | [END_REF] Nonetheless, clear digital switching between the 'off' states and the green fluorescent 'on' state of 1 and 2 is distinguishable with the naked eye (Fig. 2a and Fig. S11,ESI †).

Fluorimetric titration experiments were performed in 1 : 1 (v/v) methanol/water and methanol solution. The proton concentration in 1: 1 (v/v) methanol/water was monitored with a calibrated pH meter, while titrations in methanol were performed by adding aliquots of known acid concentration. An increase in the [H + ] resulted in sigmoidal-shaped profiles between pH 10 and 7 followed by a plateau region till pH 3 (Fig. S15 a 10 À5 M 1 excited at the isosbestic point of 438 nm and 432 nm. is obtained, short lifetime components of 21 ps (497%) and 25 ps (499%) were dominant for 1 and 3 with respect to the longer unquenched components, respectively. The longer component corresponds to the intrinsic lifetime of the naphthalimide fluorophore, while the shorter lifetime results when an ICT/PET from the ferrocene moiety and/or the tertiary amine to the excited state fluorophore is operational. On both oxidation and protonation, the deactivation of the PET emission quenching channels provides a single fluorescence lifetime of 8.5 ns and 7.3 ns for 1 and 3, respectively (Fig. 2b and Fig. S20, ESI †). [START_REF] Gan | A model compound of 3 with a methylated naphthalimide and methylpiperazine was reported to have lifetimes of 9.0 ns (80.6%) and 0[END_REF] Further information on the photophysical behaviour of the excited naphthalimides was sought via femtosecond transient absorption studies. Firstly, studies on ferrocene-free aminepiperazine 5 (Fig. 3) showed that immediately after excitation (o1 ps) a major charge density shift occurs within the chromophore (ground state bleaching at 450 nm, stimulated emission centred at 540 nm and positive absorption at 420-450 nm and 650-750 nm) testifying to the ICT character of the chromophore. 28 The transient signals rapidly evolve with time (B50 ps), which is ascribed to a rapid electron transfer from the nitrogen atom of the piperazine group bearing a methyl group.

The presence of the ferrocene moiety on the aminonaphthalimide chromophore in 3 leads to an increase in the transient signal recovery rate (B5 ps, Fig. 4) with respect to 5. The corresponding faster excited state process in 3 is attributed to a faster electron transfer from ferrocene to the neighbouring excited chromophore.

In the case of 1, relaxation of the stimulated emission occurs with a time constant of 20 ps, which characterises the photoinduced electron transfer rate, and ground state bleaching recovery with 40 ps. This time constant can be related to back electron transfer regenerating the initial ground state. Absence of the transient absorption band at 420-450 nm can be understood by comparison of the tertiary amine electron donor/ proton acceptor in 1 with the methylpiperazine of 3 and 5 (Fig. 5). Indeed, it was previously suggested that a difference in behaviour between piperazine and alkylamine substituents is anticipated based on greater steric encumbrance in the case of the former, leading to an inherent enhanced twisting in the ground state, which results in a more efficient twisted ICT state. 22 Both 1 and 5 exhibit a slower initial PET compared to 3, which has the ferrocene electron donor in closer proximity to the naphthalimide fluorophore.

Conclusions

In summary, we have designed and synthesised two novel 'Pourbaix sensors' based on a naphthalimide fluorophore according to a 'fluorophore-spacer 1 -receptor-spacer 2 -electrondonor' design. The molecules respond to redox potential and pH in accordance to two-input AND logic by a fluorescence signal. Protonation of the tertiary amine and oxidation of the ferrocene moieties prevents intramolecular charge transfer and photoinduced electron transfer resulting in a four-fold fluorescence enhancement in mixed aqueous methanol and methanol solutions. Time-resolved spectroscopic measurements confirm the efficient electron transfer processes towards the excited ICT chromophore on the tens of picosecond timescale involving ferrocene as well as amines, as anticipated from estimated thermodynamic driving forces. Although Fe 3+ is used as the oxidant and methanesulfonic acid as the proton source, in principle, other oxidants and acids with suitable properties should give a similar output. Our results contribute to the emerging number of intelligent molecular and supramolecular devices responsive to oxidants and pH. 11,29,30 We are currently exploring the use of naphthalimide-based 'Pourbaix sensors' for molecular biosensing and environmental diagnostic applications.

Experimental

Instrumentation 1 H and 13 C NMR spectra were collected at 298 K on a Bruker Avance III HD NMR spectrometer with a multinuclear 5 mm PABBO probe operating at 500.13 MHz and 125.76 MHz for 1 H and 13 C, respectively. Samples were measured in CDCl 3 (0.03% v/v TMS), acetone-d 6 (0.03% v/v TMS) or MeOD-d 4 (0.03% v/v TMS). Melting points were measured on a Stuart SMP11 melting point apparatus and are uncorrected. Infrared (IR) spectra were recorded as KBr discs, neat films on NaCl plates or in a solution cell with a Shimadzu IR Affinity-1 spectrophotometer calibrated at 1601 cm À1 using polystyrene. pH measurements were performed using a pH meter calibrated with standard buffer solutions at pH 4.00 and 7.00. UV-visible absorption spectra and steady-state fluorescence spectra were recorded on a Jasco V-650 spectrophotometer and Jasco FP-8300 spectrofluorimeter using 1 cm quartz cuvettes. The time-resolved luminescence set-up (Fig. S18, ESI †) used a frequency tripled Nd:YAG amplified laser system (30 ps, 30 mJ@1064 nm, 20 Hz, Ekspla model PL 2143) output with an optical parametric generator (Ekspla model PG 401) with tunable excitation pulses in the range 410-2300 nm. The subpicosecond time scale set-up was based on a femtosecond 1 kHz Ti:sapphire system (Fig. S19, ESI †). HRMS spectra were conducted by Medac Ltd.

Synthetic procedures

Synthesis of 1. 4-N-Methylethylenediamine-1,8-methylnaphthalimide (220 mg, 0.77 mmol) was dissolved in 10 mL of dry 1,2-dichloroethane with 170 mg (0.79 mmol) of ferrocenecarboxaldehyde, 220 mg (1.0 mmol) of sodium triacetoxyborohydride and 0.21 ml (1.5 mmol) of triethylamine in the dark at room temperature for 6 days. The reaction was monitored by TLC. The product R f (methanol) is 0.46. The reaction mixture was diluted with 1,2-dichloroethane and washed with 10% aqueous NaHCO 3 . The organic phase was dried over anhydrous magnesium sulphate, filtered and the solvent removed on a rotary evaporator. The crude brown oil was purified by column chromatography on silica gel using methanol as the eluent and recrystallised from ethyl acetate to yield 75 mg of pure product 1 (20% yield). M.p. 133-136 1C; 1 H NMR (500 MHz, CDCl 3 , ppm): d H 8.59 (d, J = 7.3 Hz, 1H), 8.45 (d, J = 8.4 Hz, 1H), 7.97 (d, J = 8.5 Hz, 1H), 7.60 (m, 1H), 6.62 (d, J = 8.4 Hz, 1H), 6.15 (s, 1H), 4.22 (t, J = 1.8 Hz, 2H), 4.17 (t, J = 1.8 Hz, 2H), 4.12 (s, 5H), 3.55 (s, 3H), 3.50 (s, 2H), 3.30 (m, 2H), 2.77 (t, J = 5.8 Hz, 2H), 2.29 (s, 3H); 13 C NMR (126 MHz, CDCl 3 , ppm): d C 165. 1, 164.5, 149.7, 134.6, 131.0, 129.7, 126.5, 124.5, 122.9, 120.4, 109.9, 104.4, 83.2, 70.0, 68.6, 68.3, 57.1, 53.3, 41.6, 40.0, 26.8; n max (NaCl/cm À1 ): 3381, 3092, 2941-2799, 1682, 1643, 1582, 1418, 1533, 1396, 1362, 1281, 1242, 1188, 1105, 1045 1026, 822; HRMS (ESI TOF) m/z calculated C 27 H 28 N 3 O 2 Fe [M + H] + 482.1531, found 482.1527.

Synthesis of 2. 4-N-Methylethylenediamine-1,8-butylnaphthalimide (97 mg, 0.30 mmol) and ferrocenecarboxaldehyde (70 mg, 0.33 mmol) were dissolved in 6 mL of dry 1,2-dichloroethane and stirred at room temperature with sodium triacetoxyborohydride (260 mg, 1.2 mmol) and 3 Å molecular sieves in the dark for three days. The reaction mixture was diluted with dichloromethane, washed twice with saturated NaHCO 3 (2 Â 25 mL) and once with distilled water (25 mL). The organic phase was dried over anhydrous sodium sulphate, filtered, and the solvent removed on a rotary evaporator. The crude product was precipitated from diethyl ether to yield 20 mg of orange solid 2 (10% yield). M.p. 118-120 1C; 1 H NMR (500 MHz, CDCl 3 , ppm): d H 8.57 (d, J = 7.2 Hz, 1H), 8.43 (d, J = 8.4 Hz, 1H), 7.99 (d, J = 8.3 Hz, 1H), 7.58 (m, 1H), 6.61 (d, J = 8.4 Hz, 1H), 6.14 (s, 1H, NH), 4.21-4.12 (m, 11H, FeCp 2 ), 3.50 (s, 2H), 3.30 (m, 2H), 2.75 (t, J = 5.7 Hz, 2H), 2.29 (s, 3H), 1.71 (quintet, J = 7.3 Hz, 2H), 1.45 (sextet, J = 7.3 Hz, 2H), 0.97 (t, J = 7.3 Hz, 3H); 13 C NMR (126 MHz, CDCl 3 , ppm): 8, 164.2, 149.6, 134.5, 131.0, 129.8, 126.4, 124.5, 123.1, 120.4, 110.0, 104.4, 83.1, 70.0, 68.6, 68.3, 57.1, 53.4, 41.6, 40.0, 39.9, 30.3, 20.5, 13.9; IR (solvent cell, cm À1 ): 3619, 3015, 2974, 2893, 1684, 1636, 1584, 1533, 1506, 1456, 1395, 1362, 1240, 1047

d C 164.

  , ESI †). Fitting of the data to the equation log[(I max À I)/(I À I min )] = Àlog[H + ] + log b H + yields a log b H + of 9.2 and 8.7 for 1 and 2, respectively, in 1 : 1 (v/v) methanol/water (Fig. S12-S15, ESI †).The log b H + of 2 in methanol was found to be an order of magnitude higher (Table1and Fig.S16, S17, ESI †). An apparent log b Fe 3+ of 4.2 was determined for 1 in 1 : 1 (v/v) methanol/water at pH 4.Time-resolved fluorescence spectroscopy was used to measure the fluorescence lifetimes of 1 and 3 under the four logic states in 1 : 1 (v/v) methanol/water. At pH 11, where complete deprotonation is anticipated, and hence fully quenched emission

  Fig. 2 (a) Vials with 1 under the four conditions for AND logic in 1 : 1 (v/v) MeOH/H 2 O under 365 nm UV illumination (see Table 2) (b) fluorescence decay curve and residual time delay (c) 3D plot at pH 4 with 60 mM Fe 3+ in 1 : 1 (v/v) MeOH/H 2 O, l ex = 438 nm.

Fig. 3

 3 Fig. 3 Transient absorption map of 5 in acetonitrile (l exc = 400 nm).

Fig. 4

 4 Fig. 4 Transient absorption map of 3 in acetonitrile (l exc = 400 nm).

Fig. 5

 5 Fig. 5 Transient absorption map of 1 in acetonitrile (l exc = 400 nm).

Table 1

 1 Summary of various photophysical parameters of 1-3 determined by UV-visible absorption and steady-state fluorescence spectroscopy a Molar absorptivity e in L mol À1 cm À1 . d Acidic pH adjusted with CH 3 SO 3 H; values in nm. e Dl taken as the Stokes shift between the absorption and fluorescence spectra at pH 3 in presence of 50 mM Fe 3+ . f Determined by log[(I max À I)/(I À I min )] = Àlog[H + ] + log b H + from fluorescence emission spectra. Fluorescence emission spectra obtained by excitation at l Isos . nd = not determined.

		1	1	2	2	3	3
		MeOH H 2 O/MeOH MeOH H 2 O/MeOH MeOH H 2 O/MeOH
	b 439 l Abs pH 12 c 4.25 log e pH 12 d 420 l Abs pH 3 log e pH 3 c 4.23	446 4.22 430 4.22	440 4.28 422 4.27	450 4.26 432 4.32	399 4.10 378 4.12	405 4.04 386 4.08
	l Isos /nm 433	438	432	437	390	399
	l Flu /nm 527 Dl/nm e 107	526 96	530 108	535 103	521 143	525 139
	pb H + * f	nd	9.2	9.9	8.7	nd	6.6
	a 4 AE 1 mM 1-3 in MeOH or 1 : 1 (v/v) MeOH/H 2 O solutions. Some data for 3 is originally reported in ref. 12. b Basic pH adjusted with 0.1 M
	NaOH; values in nm.

c

Table 2

 2 Truth table of AND logic gate 1 and 2 a

	Input 1 (H + ) d	Input 2 (Fe 3+ ) e	Output emission 1 b (F f ) f	Output emission 2 c (F f ) f
	0 (low)	0 (low)	0 (low, 0.001)	0 (low, 0.012)
	1 (high)	0 (low)	0 (low, 0.008)	0 (low, 0.017)
	0 (low)	1 (high)	0 (low, 0.001)	0 (low, 0.012)
	1 (high)	1 (high)	1 (high, 0.040)	1 (high, 0.050)

  ; HRMS (ESI TOF) m/z calculated C 30 H 33 N 3 O 2 Fe [M + H] + 542.2000, found 542.2005.

Acknowledgements

This work was supported by the University of Malta and ERDF grant 309CT 3097/2013 ''Strengthening of the Organic, Inorganic, Physical Chemistry Facilities''. Prof. Robert M. Borg is acknowledged for assistance with the acquisition of the NMR spectra. The Embassy of France to Malta, the CNRS, ANR FOSET and the MCST (Malta Council for Science and Technology) are acknowledged for financial aid. The Ministry for the Economy, Investment and Small Business is thanked for the 2014 Malta Science Innovation Award.