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Suppose that a person believes that all humans are guaranteed a set of rights that cannot be taken from them in any situation or circumstance (for example, the right to marry a person of your choice, the right to practice any religion...). Is this moral belief compatible with Harsanyi's axioms? (This is part of an essay prompt that I wanted some guidance on. Thanks!)

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Your question is a bit confused, because it mixes together several different things. For example, in the title, you mention Sen's Minimal Liberalism, but in the actual question, you don't mention Sen at all --- instead you talk in more general terms about "rights that cannot be taken ... in any situation". You mention "Harsanyi's axioms", but it isn't clear what you mean by this ---or what you mean by "compatible". So I will try to answer a couple of different interpretations of your question.

Presumably, by "Harsanyi's axioms", you mean the axioms in Harsanyi's Social Aggregation Theorem. (These are: (1) All individuals and the social planner have preferences over lotteries that satisfy the von Neumann-Morgenstern axioms, and hence admit expected utility representations; and (2) The social preferences over lotteries satisfy the Pareto axiom with respect to individual preferences over lotteries.) Harsanyi's Social Aggregation Theorem is an axiomatic characterization of utilitarianism. (Or at least, that is how Harsanyi interpreted it.) So I will take your question to really be asking: "Is utilitarianism compatible with an absolute, non-negotiable respect for individual rights?"

The answer is "No". And this has nothing to do with Harsanyi, or even modern axiomatic social welfare theory in general. The incompatibility of utilitarianism with absolute individual rights has been understood for two hundred years --indeed, this is one of the major philosophical objections to utilitarianism. Of course, utilitarianism generally promotes human rights, because utilitarianism wants to maximize sum-total utility, and violating people's rights tends to cause a massive loss of utility, so it is something you generally want to avoid, if possible. However, because it aggregates over the preferences of many people, utilitarianism has the feature that the desires of a very large group of people can override the desires ---or even the "rights" ---of a small group of people, in some situations. The moral philosophy literature is full of thought experiments where utilitarianism leads to some egregious violation of individual rights. (Of course, the same is true for almost every other social welfare function.)

But it is important to nuance this point. We want to maximize social welfare over a "feasible set" of possible social outcomes. Whether or not this will involve violating individual rights depends upon what is contained in this feasible set. One way to prevent utilitarianism from violating rights is to just exclude rights-violating outcomes from the feasible set. In the terminology of Robert Nozick, we can impose "side constraints" on the social optimization problem, so that certain policies or social outcomes are just off-limits.

Most moral philosophers who endorse utilitarianism are not fans of using "side constraints" to protect individual rights, because it ruins the beautiful conceptual simplicity and unity of pure utilitarianism. Most utilitarians don't actually believe that "rights" are part of the fundamental structure of morality ---except in the limited, instrumentalist sense that I described two paragraphs back. Jeremy Bentham (often seen as the originator of utilitarianism) once described the idea of "natural rights" as "nonsense upon stilts".

However, I am raising this issue because it has a bearing on whether or not Harsanyi's Theorem is "compatible" with some notion of rights. The reason is that Harsanyi's Theorem does not make any assumptions about the feasible set. It simply supposes that there is some set $X$ of "social alternatives", and every agent (i.e. the social planner and every individual) has preferences over the space $\Delta(X)$ of lotteries over $X$. So the question of whether Harsanyi's theorem is compatible with rights is simply the question: what is contained in the feasible set $X$? If $X$ contains outcomes that violate rights, then under some conditions, a maximization of utilitarian social welfare will choose these outcomes, so Harsanyi's Theorem is incompatible with rights. But if $X$ does not contain outcomes that violate rights, then Harsanyi's Theorem is incompatible with rights.

Turning to Sen: Sen's Minimal Liberalism axiom can in fact be seen as an assertion about what is in $X$. It stipulates that $X$ contains a pair of outcomes $x$ and $y$ such that (on the grounds of "liberty" or "rights") the social preference between $x$ and $y$ should be entirely determined be the preferences of some individual $i$. For example, it might be that $x$ and $y$ are two outcomes which are identical in every way except for what colour shirt $i$ is wearing. If we believe (plausibly) that people should have the "right" to chose their own clothing, then the social preference between $x$ and $y$ should be determined by $i$'s preferences between $x$ and $y$. This is what Sen's Minimal Liberalism Axiom is saying. Sen's Impossibility Theorem says that even this minimal liberalism condition is incompatible with the Pareto axiom.

However, notice that this is only an issue because somehow the set $X$ of social outcomes included information about what colour of clothing people can wear. So Sen's Minimal Liberalism is really saying, "$X$ contains outcomes which bear upon individual liberties, so that it is possible to violate individual rights by choosing the wrong element of $X$". In other words, Minimal Liberalism is saying something about what is contained in $X$. If we simply restrict $X$ to social outcomes where rights violations are not possible (i.e. impose "side constraints"), then the problem goes away.

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It depends on what you mean with Harsanyi's axioms. If you are referring to Harsanyi's social aggregation theorem, then you should note that it relies on some Pareto Condition. In Harsanyi's original version, it is Pareto Indifference, but there are versions of the social aggregation theorem that use Weak Pareto or Strong Pareto.

Sen's theorem on the impossibility of a Paretian liberal demonstrates that Weak Pareto is inconsistent with an axiom called "minimal liberalism".

These two clues should be enough to get started!

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