# Modal logic, ethics, and obligation

Recently, I posted about necessary truth, the logic of belief, and epistemic logic. I would like to follow up on that one more time by discussing deontic logic, the logic of obligation and moral action. We can capture this concept using the 4 rules of D4 modal logic. The first 3 of these are the same as those I used for belief. I am replacing the previous modal operators with  Ⓞ  which is intended to be read as “it is obligatory that” (hence the O in the circle):

• if P is any tautology, then  Ⓞ P
• if  Ⓞ P  and  Ⓞ (PQ)  then  Ⓞ Q
• if  Ⓞ P  then  Ⓞ Ⓞ P
• if  Ⓞ P  then  ~ Ⓞ ~P

where  ~ Ⓞ ~P  is read as “~ P is not obligatory,” i.e. “P is permissible.” For those who prefer words rather than symbols:

• if P is any tautology, then P is obligatory
• if P and (P implies Q) are both obligatory, then Q is obligatory
• if P is obligatory, then it is obligatory that P is obligatory
• if P is obligatory, then P is permissible

For these rules as they stand, the only things that are obligatory are necessary truths like 2 + 2 = 4. This is because you can’t get an “ought” from an “is.” Apart from the first rule, there is no way of introducing a  Ⓞ  symbol out of nowhere. Consequently, if we are to reason about ethics and morality, we must begin with some deontic axioms that already contain the  Ⓞ   symbol. For people of faith, these deontic axioms may be given by God, as in the 10 Comandments, which include:

Ⓞ  you do not murder.
Ⓞ  you do not commit adultery.
Ⓞ  you do not steal.
Ⓞ  you do not bear false witness against your neighbor.

Immanuel Kant famously introduced the categorical imperative, a deontic axiom which Kant thought implied all the other moral rules, and thus provided the smallest possible set of deontic axioms:

Ⓞ  [you] act only according to that maxim whereby you can, at the same time, will that it should become a universal law.

Others have suggested the greatest happiness of the greatest number as a principle. Fyodor Dostoevsky, William James, and Ursula Le Guin are among those who have explained the problem with this:

Tell me yourself, I challenge your answer. Imagine that you are creating a fabric of human destiny with the object of making men happy in the end, giving them peace and rest at last, but that it was essential and inevitable to torture to death only one tiny creature – that baby beating its breast with its fist, for instance – and to found that edifice on its unavenged tears, would you consent to be the architect on those conditions?” (Fyodor Dostoevsky, “The Grand Inquisitor,” in The Brothers Karamazov, 1880; 4.35 on Goodreads)

Or if the hypothesis were offered us of a world in which Messrs. Fourier’s and Bellamy’s and Morris’s Utopias should all be outdone and millions kept permanently happy on the one simple condition that a certain lost soul on the far-off edge of things should lead a life of lonely torture, what except a specifical and independent sort of emotion can it be which would make us immediately feel, even though an impulse arose within us to clutch at the happiness so offered, how hideous a thing would be its enjoyment when deliberately accepted as the fruit of such a bargain?” (William James, “The Moral Philosopher and the Moral Life,” 1891)

Some of them understand why, and some do not, but they all understand that their happiness, the beauty of their city, the tenderness of their friendships, the health of their children, the wisdom of their scholars, the skill of their makers, even the abundance of their harvest and the kindly weathers of their skies, depend wholly on this child’s abominable misery.” (Ursula K. Le Guin, “The Ones Who Walk Away from Omelas,” 1973; reprinted in The Wind’s Twelve Quarters, 1975; 4.05 on Goodreads)

The meaning of deontic statements can be described using Kripke semantics, which exploits the idea of possible worlds (i.e. alternate universes). To say that some statement is obligatory is to say that the statement would be true in better possible worlds (we write w1 → w2 to mean that w2 is a better possible world than w1).

In any given world v, the statement  Ⓞ P  is equivalent to :

• P  is true in all better worlds wi (i.e. all those with v → wi)

Likewise, in any given world v, the statement  ~ Ⓞ ~P  (P is permissible) is equivalent to:

• P  is true in at least one better world wi (i.e. one with v → wi)

The rules of deontic logic imply two conditions on these arrows between possible worlds:

• if  w1 → w2 → w3  then  w1 → w3  (i.e. chains of arrows are treated like arrows too)
• in every world v there is at least one arrow  v → w  (i.e. chains of arrows don’t stop; this includes the case of  v → v)

A number of philosophers have suggested that deontic logic leads to paradoxes. In all cases that I have seen, these “paradoxes” have involved simple errors in the use of deontic logic – errors that become obvious when the deontic statements are translated into statements about possible worlds.

There are limitations to deontic logic, however. For example, if we say that it is obligatory not to steal, this means that, in all better possible worlds, nobody steals. If we also say that it is obligatory to punish thieves, this means that, in all better possible worlds, thieves are punished. However, if it is obligatory not to steal, better possible worlds have no thieves, so the two statements do not combine well.

Some people would, no doubt, suggest that fiction like that of Dostoevsky is a better tool than logic for exploring such issues. In cases where the writer is a genius, they are probably right.

In this post series: logic of necessary truth, logic of belief, logic of knowledge, logic of obligation

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