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The felicific calculus was an algorithm formulated by Jeremy Bentham for calculating the degree or amount of happiness that a specific action is likely to cause, and hence its degree of moral rightness. It is also known as the "Utility calculus", the "Hedonistic calculus" and the "Hedonic calculus".

The calculus was proposed by Bentham as part of his project of making morals amenable to scientific treatment. Since classical utilitarians considered that the rightness of an action was a function of the goodness of its consequences, and that the goodness of a state of affairs was itself a function of the happiness it contained, the felicific calculus could, in principle at least, establish the moral status of any considered act.

Variables, or vectors of the pleasures and pains included in this calculation—which Bentham called "elements" or "dimensions"—were:

  1. intensity
  2. duration
  3. certainty or uncertainty
  4. propinquity or remoteness
  5. fecundity: the probability it has of being followed by sensations of the same kind
  6. purity: the probability it has of not being followed by sensations of the opposite kind

To these six, which consider the pleasures and pains within the life of a person, Bentham added a seventh element, in order to account for possible variations among the number of people involved:

7. extent: the number of persons to whom it extends

Bentham's felicific calculus contained the following sequence of instructions on analysing an action:

  • Begin with any one person of those whose interests seem most immediately to be affected by it: and take an account,
    • Of the value of each distinguishable pleasure which appears to be produced by it in the first instance.
    • Of the value of each pain which appears to be produced by it in the first instance.
    • Of the value of each pleasure which appears to be produced by it after the first. This constitutes the fecundity of the first pleasure and the impurity of the first pain.
    • Of the value of each pain which appears to be produced by it after the first. This constitutes the fecundity of the first pain, and the impurity of the first pleasure.
  • Sum up all the values of all the pleasures on the one side, and those of all the pains on the other. The balance, if it be on the side of pleasure, will give the good tendency of the act upon the whole, with respect to the interests of that individual person; if on the side of pain, the bad tendency of it upon the whole.
  • Take an account of the number of persons whose interests appear to be concerned; and repeat the above process with respect to each. Sum up the numbers expressive of the degrees of good tendency, which the act has, with respect to each individual, in regard to whom the tendency of it is good upon the whole: do this again with respect to each individual, in regard to whom the tendency of it is good upon the whole: do this again with respect to each individual, in regard to whom the tendency of it is bad upon the whole. Take the balance which if on the side of pleasure, will give the general good tendency of the act, with respect to the total number or community of individuals concerned; if on the side of pain, the general evil tendency, with respect to the same community.

To make his proposal easier to remember, Bentham devised what he called a "mnemonic doggerel" (also referred to as "memoriter verses"), which synthesized "the whole fabric of morals and legislation":

Intense, long, certain, speedy, fruitful, pure—
Such marks in pleasures and in pains endure.
Such pleasures seek if private be thy end:
If it be public, wide let them extend
Such pains avoid, whichever be thy view:
If pains must come, let them extend to few.

An example of the felicific calculus in action is as follows.

Let's imagine you are a doctor driving to a patient, a young mother who is about to give birth. It looks like she will need a Caesarian section. It is late at night and you come across a car accident on the country road you are travelling on. Two cars are involved in the accident and both drivers are unconscious and have visible injuries. One of the men is the father of the child you are going to deliver, and the other man is very old. You do not know the extent of their injuries but in your opinion, without immediate medical help, one or both may die. You as a Utilitarian are now faced with one of three possible solutions:

  1. You help the young mother who's about to give birth.
  2. You help the young woman's husband.
  3. You help the old man.

The outcome of felicific calculus would suggest:

  1. Attending to the mother first is your primary concern as the doctor. The death of both mother and child is almost a certainty if you do not act now, whereas the death of the men is uncertain. Furthermore, the pain of the mother is clearly greater than that of the men at this time. There is a greater richness and purity in saving the life of a young child who has, in all probability, a long happy life ahead. Therefore the extent and duration of the utility created by these two people is a clear likelihood.
  2. Attending to the young husband is the next priority. The pleasures of a new family—its intensity, duration, extent, richness, and purity—are all clear probabilities. If, as the doctor, you attend him first his wife and child would in all probability die. The man would then experience pain. The pain experienced by the widowed husband is likely to outstrip any pleasure to be gained from continued life without his loved ones.
  3. Attending to the old man is the last priority. The duration and certainty of his future pleasure are questionable owing to his age—he has all but lived his life. This is sometimes known as the 'good innings' argument, according to which the older you are the less claim you have to life.

Certainly, the doctor should not be limited to the three choices. To maximize the felicific calculus, he should try to secure external help by calling another doctor to help the mother, and by asking people nearby and the emergency services to deal with the accident.

Some critics argue that the happiness of different people is incommensurable, and thus a felicific calculus is impossible in practice.

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