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Philosophy 302: Ethics
The Hedonistic Calculus 

Abstract:  A modified hedonistic calculus is sketched along the lines first proposed by Bentham and Mill.  The major problem encountered is the quantification of pleasure.

I. Bentham's method of estimating pleasures and pains can be applied to egoistic hedonism. With the addition of the utilitarian factor "extent" of pleasure, the hedonism can be extended to any number of persons.
  1. Utilitarianism is the moral theory that an action is morally right if and only if it is productive of the most utility (happiness, pleasure) for the greatest number of persons.  Bentham believed the right act is the act which of all those open to the agent, will actually or probably produce the greatest amount of pleasure in the world-at-large. Pleasure and pain form the basis of the standard of right and wrong.

  2. Bentham lists benefit, advantage, good, or happiness as suitable paraphrases of pleasure. The good of the community is simply the sum of the pleasures of the individuals who compose it.

  3. The main problem for the calculus is calculating the interpersonal utility comparison using cardinal utility measurement rather than ordinal measurement.


  4. John Stuart Mill's addition of the quality of pleasures later (in terms of higher and lower pleasures) is neglected for the moment since his distinction is patently qualitative rather than quantitative.


II. The Modified Hedonistic Calculus:
  1. The major factors of sensations of pleasure and pain resulting from an action as outlined by Bentham are summarized by these variables. 

    The first four variables (intensity, duration, certainty, and propinquity) show the value of the pleasure or the pain "considered by itself."  This phrase implies Bentham did not see pleasure and pain as polar concepts or contraries.

    The next two variables (fecundity and purity) are properties of the event or action produced by the pleasure or pain-—not properties of the pleasure or pain, itself.


    1. Intensity (I)--How intense is the pleasure or pain?

    2. Duration (D)--How long does the pleasure of pain last?

    3. Certainty (C)--What is the probability that the pleasure or pain will occur?

    4. Propinquity (nearness or remoteness) (N)--How far off in the future is the pleasure or pain?

    5. Fecundity (F)--What is the probability that the pleasure will lead to other pleasures?

    6. Purity (P)--What is the probability that the pain will lead to other pains?

    7. Extent (E)--How many persons are affected by the pleasure?


  2. How are the individual factors to be quantified or measured?


    1. Intensity (I)--Bentham apparently thought intensity would vary from zero to infinity, but psychological data indicates an upper threshold of pleasure; hence, we can use an ordinal relation from 0 to 10. Pain could be measured in the same manner, where for both pleasure and pain, 0 represents indifference.

      E.g.
      ,  a contemporary approach in the psychophysics of pain is reported by Price, et al.: "Someone who is having pain is asked to match the perceived intensity of the pain to a scale. This can be done in a variety of ways. For example, one can match words or numbers to pain intensity, or match an intensity of experimental pain to that of clinical pain, or use more than one of these procedures. … The clinician or investigator … provides the scaling procedures, records the reported values, and uses a measurement method that has been chosen to be reliable and valid." Donald C. Price, et al. "Psychophysical Approaches to Measurement of the Dimensions and Stages of Pain" in Dennis C. Turk and Ronald Melzack, Handbook of Pain Assessment (New York: Guilford Press, 2001) 54.


    2. Duration (D)--We can use increments of time: seconds, minutes, and so forth. The time interval from the perceived beginning of the pleasure until the end of the pleasure in question. (Bentham did not consider the notion of psychological or subjective time as opposed to clock or physical time. E.g., as Simone de Beauvoir wrote, "Our private inward experience does not tell us the number of our years, no fresh perception comes into being to show us the decline of years." Simone de Beauvoir, The Coming of Age (New York: Warner Books, 1973), 420.)


    3. Certainty (C)--The assigned probability can be drawn from records of out past experience and records from persons similar to us. What proportion of times has the pleasure followed actions of the kind under consideration?  


    4. Propinquity (nearness or remoteness) (N)--We can set up a future indifference curve based on a "store of satisfaction" such as money (q. v., below).  Propinquity of pleasure depends upon how long one must wait for the pleasure to occur. 

      Homework Assignment: How does the propinquity factor function in the modified hedonistic calculus? Specifically how does an individual's propinquity indifference curve yield an approximate weighting for a general equation of hedonism? (See below III, D).


    5. Fecundity (F)--The probability that the pleasure or pain will lead to other pleasure or pain of the same kind can be drawn from records of our own past experience and the past experience of others like us. Also, as the Epicureans noted, many extreme pleasures are not likely to be followed by other pleasures.


    6. Purity (P)--Bentham writes, "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."


    7. Extent (E)--The total amount of utility or pleasure can be had by summing a similar calculation for every other person who is affected by the action in question.

III.  How can the general equation be set up?
  1. In setting up the calculus, we need to make a number of assumptions about preferences and satisfaction in order to insure consistency. Initially, the following assumptions or principles seem reasonable to assume when faced with choices concerning pleasure and pain ...


    1. We can decide when we prefer one thing to another or whether we are indifferent.


    2. Our preferences are transitive. If I prefer activity A over activity B, and activity B over activity C, then I prefer activity A over activity C.   Our preferences as any given moment are transitive, but these preferences can differ at different times.  Consider, for example, the "law of diminishing utility" or "satiable wants": "The total utility of a thing to anyone (that is, the total pleasure or other benefit it yields him) increases with every increase in his stock of it, but not as fast as his stock increases." (Alfred Marshall, Principles of Economics (London: Macmillan, 1890), 168.)


    3. An egoistic hedonist prefers more pleasure to less pleasure and less pain to more pain.


  2. The magnitude of a pleasure (the product of the dimension of pleasure) is the duration multiplied by the intensity of the pleasure. (Try working out specific examples for (1) studying, (2) sleeping late, and (3) going to a small party.)
  3. D  ×   I

  4. The certainty is a factor of probability that the pleasure will occur. The expected pleasure value equals the probability multiplied by the magnitude of the pleasure.
  5. P  ×  M

  6. How do we get a weighting for the propinquity? I.e., the nearness or remoteness of the pleasure? How can we quantify our feelings about pleasures in the future? One possibility is to set up a future indifference curve based on a "store of satisfaction" such as money.


    1. Ask yourself the following questions: Would you rather receive $1 in one week or $10 in one month?


      1. If you prefer $10 in one month, then ask yourself about $9, $8, and so forth, until you are indifferent about when you receive the money.

      2. If you would prefer $1 in one week to $10 in one month, then ask yourself about $11 in one month, $12, $13, and so forth until you are indifferent about whether you receive the money in one week or one month.

      3. Ask the similar question about whether you prefer $1 in one week to x dollars in six months. Continue the questioning for one year, five years, ten years, twenty years.


    2. When you are finished, you should have a set of amounts of money corresponding to the various time periods. Each item numbered below represents one student's values:


    3. Question Time Amount in Dollars
      1. one week 1
      2. one month 10
      3.  six months 44
      4. one year 110
      5. five years 700
      6. ten years 5,000
      7. twenty years 12,000


    4. The student is indifferent about questions 1 through 7.  That is, this student is unable to decide which one of these is preferred. (Obviously, your list will differ in the amounts in dollars, and you should also be indifferent about the values you adopt.)


    5. After you have finished a chart like the one in 2 above, plot these values on a sheet of graph paper. (Note nonlinear scales are used in the pop-up button just below.)


    6. The graph shows your propinquity indifference. How can the information in this graph be used in the general equation on which we are working?  Let's use the slope of the curve at the same time period under consideration? (Note the similarity to indifference curves and marginal utility of interest rates in economics.)


  7. The fecundity of the action for that individual could be determined by the summation of the measure of the sensations of pleasure and pain for that individual which follow the initial sensation of pleasure and pain.

  8. The purity of the sensation of pleasure and pain resulting from the action would be the summation of the measures from the complex of individual pleasures and pains resulting from the action. This factor might be expresses as a ratio of pleasure to pain.  Since pleasure is scaled from 0 to 10 and pain is scaled from 0 to -10 a positive ratio would represent an aggregate purity value in the pleasure range, and a negative ratio would represent a purity value in the negative range.  It's quite possible that empirical measurements of pain and pleasure would result in a logarithmic relation rather than a scalar one.

  9. The factor of the extent is employed by repeating the above calculation for each person affected by the original action in question. The result would be the sum of the results of the each calculation with respect to the number of the individuals in the community under examination. 

    (If we were to assume deterministic natural laws, then there would have to be a calculation not only for each person who is alive at the time of the action but also for every person who will live in the future—a calculation that could only be vaguely estimated.)


IV. What would the final equation look like? Add to and develop the following first suggestion;

{ N [ C (I × D) ]  +   Nf  [ C( If  × Df ) ] }

where the subscript "f " indicates future assignments.  How is the purity factor to be assessed?  How do we assess extent?


Recommended Sources

On this site:

"Happiness the Greatest Good" : Edited reading from Jeremy Bentham's Principles of Morals and Legislation in Introduction to Ethical Studies.
"Happiness is the Greatest Good": Tutorial notes on the Bentham reading cited above. 
Quiz on Bentham: True-False quiz on the reading and notes on Bentham.

On the Web:

Consequentialism: A discussion in the Stanford Encyclopedia of Philosophy on the doctrine that the rightness of an action depends upon the consequences of the action. The classic utilitarianism of Bentham and Mill is discussed as a paradigm case by Walter Sinnott-Armstrong. 
Utilitarianism: A brief but useful discussion of the history of utilitarian and hedonistic theories from the Catholic Encyclopedia written by James R. Fox.
Utilitarianism: The entry from the classic 1911 Encyclopedia Britannica discussing the ethical doctrine and history of utilitarianism.

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