Reply: authority is a type of power: the type that involves issuing orders which are obeyed, expressing preferences which others conform to, etc.
(iii) If someone in authority has to resort to using their power to get others to do something then that person has lost their authority.
Reply: One type of power/authority has been lost (the ability to get people to do things by command, persuasion, threat, etc), but another type has not been lost (the ability to get people to do things by force). This is just a distinction between different means by which someone can exercise authority.
(iv) There are two distinct notions of authority, de facto and de jure. The former may amount to no more than power, but the latter involves the idea of rightfulness. This is illustrated by the meaningful sentence, "The school authorities had lost authority over the pupils." This sentence means that de facto authority had been lost by the de jure authorities.
Reply: None of the so called "justifications" of authority really justify the use of power, therefore all "de jure" authority is really just de facto. The apparent "legitimacy" of an authority is just a matter of how accustomed people are to it, in other words whether they have acquired a habit of obedience. For example, the authority of a democratically elected government at best amounts to the brute force of the majority.
(v) A charismatic authority can sometimes be just a channel or mouthpiece for the will or interests of their followers.
Reply: If this is ever so, surely such a person is not really an authority? A mouthpiece is no more than a passive instrument. Usually the charismatic figure is exercising some influence over their followers, rather than just reflecting what is already there.
(vi) Winch argued that authority is simply freely accepting some external constraint upon one's behaviour, and that this is involved in any meaningful human action. To be meaningful, any action must be rule-governed. A rule is an external standard of correctness/incorrectness. Any free choice involves the assumption of a correct/incorrect decision, and there is no relevant difference between accepting the authority of a rule and accepting the authority of a person.
Reply: a rule in itself has no authority; it is purely abstract, and it is itself inert. It has no authority because in itself it has no power.
(vii) Power is something you take for yourself, authority is given to you.
Reply: To exercise your authority you are nevertheless exercising influence, i.e. affecting people's behaviour, which amounts to a form of power. Perhaps exercising the power is more justified if it has been given to you, rather than grabbed by you, but it is still power. Also, how have you managed to get people to give you this power? You must in some way have influenced them to do so, which itself is an exercising of power.
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Authority is not the same as power
In a very broad sense, exercising "power" or "influence" could be defined as "affecting what another person does". In this sense obviously authority is a form of power, since exercising authority must Involve affecting what someone else does. However, the notion of authority still seems not to fit more specific senses of power or influence, such as:
(a) COERCION, when a person doing x would not have chosen to do it but for a threat of something even worse;
(b) RATIONAL PERSUASION, when a person is induced to do x by coming to accept the rationality of some argument for doing it;
(c) NON-RATIONAL PERSUASION, e.g. nagging, propaganda, advertising, etc;
(d) SUBCONSCIOUS INFLUENCE, e.g. making a film which portrays an activity in a positive or negative light.
A suggestion might be that all forms of authority involve someone deferring to someone else's judgement; i.e. to accept authority is to cede one's own judgement in a certain area.
Often this involves an acceptance of "They know best." This applies most clearly to the authority of an expert, but also to some extent to papal and to parental authority.
In the case of a democratic representative, you perhaps defer to their judgement because they represent the delegated will of the electorate. You might think their decisions misguided, but, in so far as they exercise authority over you, you defer to their judgement.
There can be many reasons for deferring to someone else's judgement, including persuasion (whether it is rational or non-rational persuasion), subconscious influence, or just a feeling of trust, or perhaps even by brute coercion. In other words, there may be specific forms of power behind authority. There is still a distinction between exercising a specific form of power directly, and exercising power via exercising authority, i.e. via getting someone to accept, or defer to, your judgement.
Of course, many cases of authority involve power as well. For example, we might conform to a "No skating" sign partly because we respect the authorities' judgement that it is dangerous, and partly because we know there's a risk of a hefty fine.
On the other hand, one feature of the "power = authority" view remains, namely that there is no intrinsic legitimacy to authority. The notion involves no built-in rightness about it. Just like exercising specific forms of power, getting someone to accept your authority (getting them to defer to your judgement) may or may not be justified.