Utilitarianism

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Utilitarianism is the ethical paradigm that judges the morality of an action based on whether it maximizes the good over the bad, which is typically determined via a utility function.

[Ethics is] the rules and precepts for human conduct, by the observance of which [a happy existence] might be, to the greatest extent possible, secured.

Utilitarianism by John Stuart Mill

Discussion

The progenitors of classic utilitarianism, Mill and Bentham, sought to free up moral discourse and its role in political policy from the strictures of adherence to tradition, which the paradigms of deontology and virtue ethics seemed to promote. They wished to provide scientific grounds for determining which courses of action and policies would do the most moral good. This shifts the focus from determining what a good person is or what our duties are to that of determining what is good in general terms. The utility function represents the beneficial quality of a situation or object.

The debate is then what the utility function should actually measure and how to aggregate personal and societal utility functions. For example, are ‘higher’ or ‘lower’ pleasures more significant for a happy existence? Does utility fundamentally boil down to pleasures and pains as proponents of hedonistic utilitarianism hold? As an optimization problem, does minimizing pain unilaterally take higher precedence than maximizing pleasure as negative utilitarianism holds? Preference utilitarianism holds that preference fulfillment is to be maximized rather than happiness itself. A practical challenge is how to deal with comparative and contradictory utility functions. So far, each choice seems to have its own pros and cons.

Utilitarian philosophies are usually consequentialist, which is the claim that the moral rightness of an act depends only on its consequences. This can appear to be an explicit rejection of any virtue and duty-based grounding of moral good. Consequentialism limits the scope of possible utility functions.

A definitional difficulty is whether to consider the paradigm of utilitarianism as being defined by a central teleological goal of “the greatest happiness for all” or to consider “utility function-based optimization” as the defining feature of the paradigm. Given the aforementioned debates on what “happiness” means, I believe the more general, paradigmatic approach is to focus on utility function-based optimization, which could be called optimizationism1David Weinberger presents an argument that we should prioritize optimized ethical performance over being able to explain why our choices are good in the case of applied AI systems..

In SUMO, I define a generic class of utility functions that map formulas into real numbers. I define two kinds of utilitarian theories: those that assign utility to formulas and those that compare the utility of two formulas. For a finite preference ordering, one can represent the preferences via a utility function if the ordering is complete and transitive2See Wikipedia’s Utility entry. In the case of infinite options, the preference ordering must also be continuous.. Some (trivial) mappings between utilitarian sentences and value judgment sentences are presented. Hedonic and Consequentialist versions of utilitarianism are defined by additional meaning postulates about the utility functions involved.

SUMO

(documentation Utilitarianism EnglishLanguage "Utilitarianism is the ethical paradigm that judges the morality of an action 
based on whether it maximizes the good over the bad, which is typically determined via a utility function.")
(subclass Utilitarianism Ethics)

(documentation UtilitarianTheory EnglishLanguage "A set of sentences dealing with the utility of behaviors.")
(subclass UtilitarianTheory MoralTheory)

(theoryPhilosophyPair Utilitarianism UtilitarianTheory)

(documentation UtilitarianSentence EnglishLanguage "A sentence of the variety of a utilitarian theory.")
(subclass UtilitarianSentence MoralSentence)

(<=> 
  (instance ?U UtilitarianTheory)
  (forall (?S)
    (=>
      (element ?S ?U)
      (instance ?S UtilitarianSentence))))

Utilitarianism is defined as a proposition, a subclass of Ethics. Utilitarian theory is defined as a set of utilitarian sentences. 

(documentation SimpleUtilitarianSentence EnglishLanguage "A sentence that assigns or compares the value of situations described by formulas.")      
(subclass SimpleUtilitarianSentence UtilitarianSentence) 

(documentation UtilityAssignmentSentence EnglishLanguage "A Sentence that assigns a (real) number value to a situation described by a formula.")
(subclass UtilityAssignmentSentence SimpleUtilitarianSentence)

(documentation UtilityComparisonSentence EnglishLanguage "A sentence that compares the value of two situations described by formulas.")
(subclass UtilityComparisonSentence SimpleUtilitarianSentence)

(<=>
  (instance ?SENTENCE UtilitarianSentence)
  (exists (?SUS)
    (and 
      (instance ?SUS SimpleUtilitarianSentence)
      (part ?SUS ?SENTENCE))))

(<=>
  (instance ?SENTENCE SimpleUtilitarianSentence)
  (or
    (instance ?SENTENCE UtilityComparisonSentence)
    (instance ?SENTENCE UtilityAssignmentSentence)))

Utilitarian sentences are those that contain a simple utilitarian sentence as a part. Every simple utilitarian sentence is either a utility comparison sentence or a utility assignment sentence3Maybe they should also be ‘simple’!.

(documentation UtilityFormulaFn EnglishLanguage "A UnaryFunction that maps Formulas to the net utility of that which is described.  Typically, the formula should refer to an action.")
(subclass UtilityFormulaFn PartialValuedRelation)
(subclass UtilityFormulaFn UnaryFunction)

(=>
    (instance ?UF UtilityFormulaFn)
    (and
        (domain ?UF 1 Formula)
        (range ?UF RealNumber)))

Every instance of a utility formula function is a total-valued relation and unary function that maps a formula to a real number. The idea is that constraints as to how to formulaically describe that which is evaluated can vary by use-case and theory, e.g., are actions or situations evaluated?

(<=>
  (instance ?SENTENCE UtilityAssignmentSentence)
  (exists (?FORMULA ?VALUE ?UF)
    (and 
      (equal ?SENTENCE (equal (AssignmentFn ?UF ?FORMULA) ?VALUE))
      (instance ?UF UtilityFormulaFn)
      (instance ?FORMULA Formula)
      (instance ?VALUE RealNumber))))

Utility assignment sentences are of the form, “utility(FORMULA) is VALUE”, that is, “the utility of that which is described by this formula is this value”4Strictly speaking, the utility function is applied to the syntactic formula. The intended interpretation is probably that the utility function evaluates formulas in light of a semantic interpretation., where VALUE is a number.

(<=> 
  (instance ?SENTENCE UtilityComparisonSentence)
  (exists (?FORMULA1 ?FORMULA2 ?COMPARATOR ?UF)
    (and
      (instance ?FORMULA1 Formula)
      (instance ?FORMULA2 Formula)
      (instance ?UF UtilityFormulaFn)
      (or
            (equal ?COMPARATOR greaterThan)
            (equal ?COMPARATOR lessThan)
            (equal ?COMPARATOR greaterThanOrEqualTo)
            (equal ?COMPARATOR lessThanOrEqualTo)
            (equal ?COMPARATOR equal))
      (equal ?SENTENCE (AssignmentFn ?COMPARATOR (AssignmentFn ?UF ?FORMULA1) (AssignmentFn ?UF ?FORMULA2))))))

Utility comparison sentences express the comparison of the utility values of two formulas: the comparator can be “greater than”, “less than”, “greater than or equal to”, “less than or equal to”, or “equals”5It seems that one would like to also include “incomparable” to the list..

(documentation UtilityAssignmentToValueJudgmentSentenceFn EnglishLanguage "A UnaryFunction that maps utility assignment sentences into simple value judgment sentences.")
(domain UtilityAssignmentToValueJudgmentSentenceFn 1 UtilityAssignmentSentence)
(range UtilityAssignmentToValueJudgmentSentenceFn SimpleValueJudgmentSentence)
(instance UtilityAssignmentToValueJudgmentSentenceFn TotalValuedRelation)
(instance UtilityAssignmentToValueJudgmentSentenceFn UnaryFunction)

(=> 
  (and 
    (equal (UtilityAssignmentToValueJudgmentSentenceFn ?UAS) ?VJS)
    (equal ?UAS (equal (AssignmentFn ?UF ?FORMULA) ?VALUE))
    (instance ?UF UtilityFormulaFn)
    (instance ?FORMULA Formula)
    (instance ?VALUE Number))
  (and
    (=>
      (greaterThan ?VALUE 0)
      (equal ?VJS
        (modalAttribute ?FORMULA MorallyGood)))
    (=>
      (lessThan ?VALUE 0)
      (equal ?VJS
        (modalAttribute ?FORMULA MorallyBad)))
    (=>
      (equal ?VALUE 0)
      (equal ?VJS 
        (modalAttribute ?FORMULA MorallyNeutral)))))

One way to ‘interpret’ utility assignment sentences in the language of value judgment sentences is to consider any value greater than zero as denoting moral goodness, any value less than zero as denoting moral badness, and a value of zero as denoting moral neutrality. One could similarly use any other partition of the real numbers into 2-3 contiguous subsets6Two subsets in the case that one may not wish to involve the notion of moral neutrality.. 

(documentation SimpleValueJudgmentToUtilityAssignmentSentenceFn EnglishLanguage "A UnaryFunction that maps value judgment sentences to utility assignment sentences.")
(domain SimpleValueJudgmentToUtilityAssignmentSentenceFn 1 SimpleValueJudgmentSentence)
(range SimpleValueJudgmentToUtilityAssignmentSentenceFn UtilityAssignmentSentence)
(instance SimpleValueJudgmentToUtilityAssignmentSentenceFn TotalValuedRelation)
(instance SimpleValueJudgmentToUtilityAssignmentSentenceFn UnaryFunction)

(=> 
  (and 
    (equal (SimpleValueJudgmentToUtilityAssignmentSentenceFn ?VJS) ?UAS)
    (equal ?VJS (modalAttribute ?FORMULA ?MORALATTRIBUTE))
    (instance ?FORMULA Formula)
    (instance ?MORALATTRIBUTE MoralAttribute)
    (instance ?UF UtilityFormulaFn))
  (and
    (=>
      (equal ?MORALATTRIBUTE MorallyGood)
      (equal ?UAS 
        (equal (AssignmentFn ?UF ?FORMULA) 1)))
    (=>
      (equal ?MORALATTRIBUTE MorallyBad)
      (equal ?UAS
        (equal (AssignmentFn ?UF ?FORMULA) -1)))
    (=>
      (equal ?MORALATTRIBUTE MorallyNeutral)
      (equal ?UAS
        (equal (AssignmentFn ?UF ?FORMULA) 0)))))

A simple translation from moral value judgments into utility assignment sentences is to assign the value of “1” to every morally good formula, “-1” to every morally bad formula, and “0” to every morally neutral formula.

(documentation SimpleValueJudgmentToUtilityAssignmentLikelihoodSentence EnglishLanguage "A UnaryFunction that maps value judgment sentences to utility assignment likelihood sentences.")
(domain SimpleValueJudgmentToUtilityAssignmentLikelihoodSentence 1 SimpleValueJudgmentSentence)
(range SimpleValueJudgmentToUtilityAssignmentLikelihoodSentence UtilitarianSentence)
(instance SimpleValueJudgmentToUtilityAssignmentLikelihoodSentence TotalValuedRelation)
(instance SimpleValueJudgmentToUtilityAssignmentLikelihoodSentence UnaryFunction)

(=> 
  (and 
    (equal (SimpleValueJudgmentToUtilityAssignmentLikelihoodSentence ?VJS) ?UAS)
    (equal ?VJS (modalAttribute ?FORMULA ?MORALATTRIBUTE))
    (instance ?FORMULA Formula)
    (instance ?MORALATTRIBUTE MoralAttribute)
    (instance ?UF UtilityFormulaFn))
  (and
    (=>
      (equal ?MORALATTRIBUTE MorallyGood)
      (equal ?UAS
        (modalAttribute 
          (greaterThan (AssignmentFn ?UF ?FORMULA) 0) Likely)))
    (=>
      (equal ?MORALATTRIBUTE MorallyBad)
      (equal ?UAS
        (modalAttribute 
          (lessThan (AssignmentFn ?UF ?FORMULA) 0) Likely)))
    (=>
      (equal ?MORALATTRIBUTE MorallyNeutral)
      (equal ?UAS
        (modalAttribute 
          (equal (AssignmentFn ?UF ?FORMULA) 0) Likely)))))

An alternative approach is to consider moral value judgments to be probabilistic of nature: that is, if a formula is morally good, then it is likely to have a utility greater than zero. This could be because a value judgment needs to be made of a whole class of actions even though there are edge cases where an instance of the class produces negative consequences.

(documentation SimpleValueJudgmentToUtilityComparisonSentence EnglishLanguage "A UnaryFunction that maps value judgment sentences to utility comparison sentences.")
(domain SimpleValueJudgmentToUtilityComparisonSentence 1 SimpleValueJudgmentSentence)
(range SimpleValueJudgmentToUtilityComparisonSentence UtilitarianSentence)
(instance SimpleValueJudgmentToUtilityComparisonSentence TotalValuedRelation)
(instance SimpleValueJudgmentToUtilityComparisonSentence UnaryFunction)

(=> 
  (and 
    (equal (SimpleValueJudgmentToUtilityComparisonSentence ?VJS) ?UCS)
    (equal ?VJS (modalAttribute ?FORMULA ?MORALATTRIBUTE))
    (instance ?FORMULA Formula)
    (instance ?MORALATTRIBUTE MoralAttribute)
    (instance ?UF UtilityFormulaFn)
    (equal ?SITUATION (SituationFormulaFn ?FORMULA)))
  (and
    (=>
      (equal ?MORALATTRIBUTE MorallyGood)
      (equal ?UCS
        (modalAttribute
          (forall (?F)
            (=> 
              (exists (?AGENT ?CP)
                (and
                  (capableInSituation ?CP agent ?AGENT ?SITUATION)
                  (realizesFormulaSubclass ?CP ?F)))
              (greaterThanOrEqualTo (AssignmentFn ?UF ?FORMULA) (AssignmentFn ?UF ?F)))) Likely)))
    (=>
      (equal ?MORALATTRIBUTE MorallyBad)
      (equal ?UCS
        (modalAttribute 
          (exists (?F ?AGENT ?CP)
            (and 
              (capableInSituation ?CP agent ?AGENT ?SITUATION)
              (realizesFormulaSubclass ?CP ?F)
              (lessThan (AssignmentFn ?UF ?FORMULA) (AssignmentFn ?UF ?F)))) Likely)))
    (=>
      (equal ?MORALATTRIBUTE MorallyNeutral)
      (equal ?UCS
        (modalAttribute 
          (equal (AssignmentFn ?UF ?FORMULA) 0) Likely)))))

I found it tricky to translate moral value judgments into utility comparison sentences: can one claim that a moral value judgment about one formula implies anything about other formulas? 

One idea is to consider the situation referred to by the formula and to take every other formula realizing an action in this situation. To state that a formula is morally good in this situation implies that it is likely of greater than or equal utility to the utility of the formula realizing any (other) action. In simple terms, “an action is good if it is the best that one can do.” And, “an action is bad if there exists a (significantly) better option”.

As can be seen in the case of moral neutrality, I lazily didn’t think of a translation. One approach would be to define “approximate equality” and to claim that all other options are approximately equal in value. Another option would be to define a neutral situation of zero value to compare with. 

It’s noteworthy that the translation in essence defines the scope of a particular moral theory.

(documentation UtilityComparisonToValueJudgmentSentence EnglishLanguage "A UnaryFunction that maps utility comparison sentences to value judgment sentences.")
(domain UtilityComparisonToValueJudgmentSentence 1 UtilityComparisonSentence)
(range UtilityComparisonToValueJudgmentSentence ValueJudgmentSentence)
(instance UtilityComparisonToValueJudgmentSentence TotalValuedRelation)
(instance UtilityComparisonToValueJudgmentSentence UnaryFunction)

(=> 
  (and 
    (equal (UtilityComparisonToValueJudgmentSentence ?UCS) ?VJS)
    (equal ?UCS (?COMPARATOR (AssignmentFn ?UF ?FORMULA1) (AssignmentFn ?UF ?FORMULA2)))
    (instance ?FORMULA1 Formula)
    (instance ?FORMULA2 Formula)
    (instance ?UF UtilityFormulaFn))
  (equal ?VJS 
        (?COMPARATOR 
          (probabilityFn (modalAttribute ?FORMULA1 MorallyGood)) 
          (probabilityFn (modalAttribute ?FORMULA2 MorallyGood)))))

A weak translation of utility comparison sentences into value judgment sentences is to translate the comparator into a comparison of the likelihoods that the formulas are judged as morally good7This begs the question of how the probabilistic judgments are grounded..

A simpler translation is the following in the case of the comparators, “greater than”, “greater than or equal”, and “equal”: if the utility of F1 is greater than the utility of F2, then if F2 is morally good, F1 will be morally good, too.

(documentation UtilityComparisonToValueJudgmentSentence2 EnglishLanguage "A UnaryFunction that maps utility comparison sentences to value judgment sentences.")
(domain UtilityComparisonToValueJudgmentSentence2 1 UtilityComparisonSentence)
(range UtilityComparisonToValueJudgmentSentence2 ValueJudgmentSentence)
(instance UtilityComparisonToValueJudgmentSentence2 PartialValuedRelation)
(instance UtilityComparisonToValueJudgmentSentence2 UnaryFunction)

(=> 
  (and 
    (equal (UtilityComparisonToValueJudgmentSentence2 ?UCS) ?VJS)
    (equal ?UCS (AssignmentFn ?COMPARATOR (AssignmentFn ?UF ?FORMULA1) (AssignmentFn ?UF ?FORMULA2)))
    (instance ?FORMULA1 Formula)
    (instance ?FORMULA2 Formula)
    (instance ?UF UtilityFormulaFn)
    (or
        (equal ?COMPARATOR greaterThan)
        (equal ?COMPARATOR greaterThanOrEqualTo)
        (equal ?COMPARATOR equal)))
  (equal ?VJS 
          (=>
            (modalAttribute ?FORMULA2 MorallyGood)) 
            (modalAttribute ?FORMULA1 MorallyGood)))