You will have heard that David Hume claimed ethical oughts cannot be derived from any factual is (empirical observation). This is called Hume’s Guillotine, presumably for the great number of people who lost their heads thinking about it. Many other authors, such as Kant and Kierkegaard, follow on Hume.
There are two important consequences of Hume’s Guillotine: (1) it is true, and (2) it does not mean much. It certainly does not mean that there are not any objectively true ethical (moral) statements. It does not, therefore, mean that “anything goes” or that all morality is relative.
Here is a statement equivalent to Hume’s Guillotine, but for mathematics: “You cannot derive mathematical theorems from any factual is.” Call this Euclid’s Catapult, if you like; named for a popular engine of killing in Euclid’s day.
Euclid’s Catapult is also true, and it also doesn’t mean much. It also certainly does not mean there are no objectively true mathematical statements.
The difference, or differences between the two statements, for there are two, is that no one loses his head over mathematics, and are happy to believe mathematical statements are, or at any rate can be, objective. All also know mistakes can be made in math, like ethics, because people make mistakes, but this does not diminish their trust in the objectivity of math per se.
Unlike with the Guillotine, it’s a fair bet most have never heard of the Catapult. Yet after accepting Hume, many rush to embrace the non-empirical “ought” that “there are no and can be no objective ethical statements”. Which is absurd. That would be like, on hearing about Euclid’s Catapult, people lose their faith that 1 + 1 = 2.
Let’s handle the important consequences of the Guillotine in reverse order. We’ll look first at David Stove’s proof (as in proof) that because we cannot derive ethical oughts from any factual is, conditional on any factual is alone, it does not follow that there are no objective ethical truths. Then we’ll look at authors in the vein of Alasdair MacIntyre (following Aquinas, etc.) who tried to show how we can know objective ethical truths. Not all are satisfied by these attempts. So we’ll highlight a tacit premise of these arguments which sets them right, the same premise that allows mathematical objective truths to be produced.
Stove’s Argument
Let’s review David Stove’s paper “On Hume’s Is-Ought Thesis“. It would be best to read this paper, for it is tight and short and complete. But perhaps not easy. I only here cover the highlights.
To understand it, we need one small bit of logical notation, which will be well familiar to regular readers. And this is “Pr(h|e)”. The h and e are statements or propositions. So this reads “the probability h is true given or accepting or conditional on believing evidence e is true.” This works for any h and e. (These may be quite difficult and long compound propositions,)
We could work without this notation: it adds nothing to the validity of any argument. Yet once you get in the habit of reading it, you discover it greatly shortens exposition.
Hume’s Guillotine (says Stove, and we agree) is
(1) “For any factual statement e and any ethical statement h, h is not deducible from e,”
which can be written in our and Stove’s notation (the numbering everywhere is his)
(3) “For any factual statement e and any ethical h, Pr(h|e) < 1.”
This expresses at least two things: (a) we cannot say, given e alone, Pr(h|e) = 1, which would prove h is true given e (the basis of Hume’s Guillotine), and (b) it is possible h is false.
We also need to understand that (what Stove called) deductivism is false. This is the idea that only deductive arguments are valid, or “count”. If deductivism were true, then it would be false, because there is no way to demonstrate deductively the rules of deduction are themselves true. All deductions are based on the tacit premise that deduction produces valid conclusions. Believing in deductivism can lead to a total skepticism, in which the idea that nothing can be known is entertained. Another Hume speciality. And also self-defeating, because to argue, and believe, nothing can be known is to express at least one thing that is known.
We next move to the fallibility of certain kinds of inductive arguments; namely, induction-probability arguments. These are of the form: “for any e and h such that the argument from e to h is inductive, Pr(h|e)<1.” Here Stove did not err, but he was not careful in separating kinds of induction, which we do below.
Stove devoted a great part of his career to showing induction, largely induction-probability, is rational. Here again, Hume was an inductive skeptic, as most know. We haven’t the space here to review Stove’s arguments disproving Hume’s deductivism, which led to Hume’s skepticism, and will take them as read: deductivism is false and induction is rational, for at least the reasons given above.
Now, even if it is true that ethical statements cannot be deduced from factual statements, the reverse is not true: we can deduce factual from ethical statements. We can write (these are more-or-less direct quotes from Stove):
(10) Pr(Socrates is a man|t)<1
where t is any tautology. We need this t for technical reasons, and most importantly to show we cannot deduce factual statements from logical truths (all tautologies are logical truths). Statement (10) is true (not just the probability number, but the entire statement (10) itself is true). This one (11) is also true:
(11) Pr(Socrates is a man|Socrates is a good man & t)=1.
Stove: “Whence the ethical ‘Socrates is a good man’ is not irrelevant, but on the contrary favourably relevant to the factual ‘Socrates is a man’, in relation to a tautology.” So we can deduce factual statements from ethical statements.
The statements (1) and (3) above are judgments of non-deducibility. If they are true, as many believe, they are true “not in virtue of the relation of any statement to the actual universe; but just in virtue of the relation between the two statements which the judgment of non-deducibility mentions. Hence a judgment of non-deducibility, if true, is a logical truth.”
That’s a long way (and much we skipped) to say that if (1) and (3) are true, then they are logical truths (and not factual truths). This is the key premise. Be sure you have it. Which, I would imagine, most do, perhaps not because of my summary, but because it seems to them obvious (1) and (3) are true à la Hume. I repeat, (1) and (3), if they are true, then they are logical truths.
From this follows two more statements (again using Stove’s numbering, and recalling (10)):
(15) “For any ethical h, h is not deducible from (1) :
“and
15′) For any ethical h, h is not deducible from (3).”
In other words, if we accept (1) or (3), or we don’t but we assume for the sake of argument they are true, then it follows we cannot deduce any ethical statement from the assumption the Guillotine is true. To repeat: no ethical statement follows from the Guillotine.
These conclusions are important, because beliefs inconsistent with them are widespread. A few years ago I read in an undergraduate essay words to this effect, (unfortunately I did not make a copy of the exact words): “Since no ethical statement can be deduced from a factual one, it follows that we can do what we like.” Now clearly, the second “can” here was an ethical one: “we can do what we like” was a version of the ‘universal permission’:
(16) Anything is morally permissible.
And (16) is clearly an ethical statement; indeed, it is only an extreme expression of an ethical attitude which has recently been quite common in the west, viz. liberalism or permissiveness.
In other words, (16), and many versions and permutations of it, are ethical statements people claim to have deduced from Hume’s Guillotine. Which deductions are thus invalid. Indeed, Stove amplifies this (using logical truths about irrelevancy) to prove “Hume’s non-deducibility theses (1) and (3) not only have no ethical statements among their consequences, but are even irrelevant to every ethical statement.” My emphasis.
Which means that those who take Hume’s Guillotine as proving the following are wrong:
“(20) Ethical statements cannot be true or false.”
In plain words, (20) is false: ethical statements can be true or they can be false. It is only that they cannot be true or false conditionally only on factual statements.
MacIntyre’s Argument
Since (20) is false, we can look for true ethical statements (and false ones!) and for what it is that makes knowledge of them true or false. Alasdair MacIntyre (in After Virtue; third edition), AN Prior, and many others, usually following St Thomas, believe they have provided the basis.
MacIntyre (p 57), challenging Hume’s Guillotine directly, expands on an example from Prior in which, they claim, an ethical ought is derived from a factual statement. Mixing Stove’s and MacIntyre’s notation, this is:
(a) e = “He is a sea captain”
where e is obviously a factual statement; which brings us to
(b) h = “‘He ought to do whatever a sea-captain ought to do.”
Says MacIntyre:
(c) Pr(h|e) = 1.
His reason is this. In the definition of sea captain are such abilities as navigating, piloting a boat to safety and the like. His (a) includes the idea that if one is a sea captain, one likely (but not certainly) has the abilities of a sea captain. It is not certainly because, say, a person given the title of sea captain might be a person hired only for her demographic persuasion or is a political appointee. But regardless of actual ability, the definition includes or contains the ends—the final cause, the teleological nature—of a sea captain.
The argument then moves from “has the abilities” to “ought to have the abilities”, because if one did not have the abilities, one could not fulfill the role of a sea captain; in short, one could not be a good sea captain.
MacIntyre then applies the same reasoning to watches and crops.
From such factual premises as ‘This watch is grossly inaccurate and irregular in time-keeping’ and ‘This watch is too heavy to carry about comfortably’, the evaluative conclusion validly follows that ‘This is a bad watch’. From such factual premises as ‘He gets a better yield for this crop per acre than any farmer in the district’, ‘He has the most effective programme of soil renewal yet known’ and ‘His dairy herd wins all the first prizes at the agricultural shows’, the evaluative conclusion validly follows that ‘He is a good farmer’.
Both of these arguments are valid because of the special character of the concepts of a watch and of a farmer. Such concepts are functional concepts; that is to say, we define both ‘watch’ and ‘farmer’ in terms of the purpose or function which a watch or a farmer are characteristically expected to serve. It follows that the concept of a watch cannot be defined independently of the concept of a good watch nor the concept of a farmer independently of that of a good farmer; and that the criterion of something’s being a watch and the criterion of something’s being a good watch-and so also for ‘farmer’ and for all other functional concepts-are not independent of each other. Now clearly both sets of criteria-as is evidenced by the examples given in the last paragraph-are factual.
So oughts can be had from some factual is. And yet, and yet: something gives the impression of being there but unacknowledged.
Mathematical Interlude
Before we come to it, here are three elementary (sort of) mathematical equations that are true:
- 1+1 = 2,
- 169/13 = 13,
- 10^(1,000,000) – 10.1^(1,000,000) < -2 x 10^(1,000,000).
Many (!) more can be given. These are all true and objective (in the sense that they are not true solely because of desire or wish). How do we know? How can we prove rather than just assert these things?
We start by noticing that if we have one object, such as a bottle of Winking Owl cabernet (now a shocking $4.39 per), and someone gives us another, we now have more objects, indeed, two. This is factual statement. And there are many other similar factual statements, such as “There are in times of abundance six ways to arrange three bottles”. But mathematical theorems are said to hold for all possibilities (such as “x + x = 2x”), even those possibilities that were never checked by enumerating or arranging objects, and to hold even when it is impossible to check, such as in the last equation (#3). If you think you can check, good luck arranging, or rearranging, 10^million objects.
In other words, all the equations above are like this: h_1 = “1 + 1 = 2” or h_3 = “10^(1,000,000) – 10.1^(1,000,000) < -2 x 10^(1,000,000)”. For the first we have e = “this object and this object”. For the second, we do not have the particular observation (factual) e, but we have something even better, to be introduced in a moment. But suppose we did have an e to accompany h_3. In neither case is it true
(d) “Pr(h|e) = 1.”
In other words, (d) is not true; this equation does not hold. Though this statement is true: “Pr(e|e) = 1”. Understand the difference: “Pr(h|e) = 1” says all such h are true for any real or even imaginary or even no objects, whereas “Pr(e|e) = 1” is a mere confirmation of what one has seen.
This statement, for any of these h above, is also true:
(e) “Pr(h|e & A) = 1.”
Where A is this (the whole block):
- Let there be natural numbers N: 0 is a natural number. There are other natural numbers which we’ll call n.
- Let S(n) be the successor function of n, i.e. S(n) is the next number after n. For all the n which are natural numbers, their successors are also natural numbers.
- But 0 is not a successor; there is no n in the natural numbers such that S(n) = 0.
- The successor function is injective: that is, if both m and n are natural numbers, and if S(m) = S(n), then m = n.
- Induction axiom: If K are numbers inside the natural numbers N, and that 0 is in K, and for all the n inside K, S(n) is inside K, then K = N.
These are arithmetic’s axioms. They are all statements which are believed true by every mathematician that considers them. They are written in a somewhat confusing manner—who needs a “successor” function!—because where would professionals be without their jargon. But this jargon will come to most of you if you spend a moment with it. The key step is the last, that induction. It says for the rules we discovered by factual statements, like “this object and that object make two objects” and so on, extensions of them (theorems) are true for all statements about natural numbers, even the ones that are or that can never be verified with objects (such as: “x + x = 2x for all x which are natural numbers”).
There is empirical support for Step 5, and indeed for all the Steps, but the proof of the last one arises by induction-intellection, the highest and surest form of knowledge we have. We move to from finite instances to Infinity itself, if you like. It is impossible, for any of those who considered these things, to even think of counter-examples that fit the definitions in the first Steps. (The book to read is Jim Franklin’s An Aristotelian Realist Philosophy of Mathematics: Mathematics as the Science of Quantity and Structure.)
I have emphasized that Step 5 is true, and known to be so for those who have given it thought. Most people have not given it thought, and never will. Some will given it brief consideration and say “You’re crazy, none of this stuff is necessary, I know how many beers I’ve had, so don’t give me any of this ‘you need proof’ nonsense.” Others will be happy to let mathematicians have their axioms, trusting they will do a good job with them, but they’ll never really follow any theorem. Some will try to force-fit true square equations into round Realities. Others, like Euler himself, will grasp the truth of what follows from A, but when it comes down it, will make mistakes when proffering theorems, such as when he said an obvious (to him) extension of Fermat’s last theorem was true (it is not).
Then there are endless examples of people saying things like “8 x 6 = 54”. And many more, such as when a scientist says, “Behold my proved-true equation! It shows how life is impossible.” These only show that mistakes can be made in math, both in derivation and application. Again, nobody takes these mistakes as evidence, or proof, that therefore math cannot at least sometimes make objectively true statements, both in derivation and application.
It may now be obvious how the rules for deduction itself arose, and why we place such trust in logic: from the same ways as mathematics, and using the same kinds of induction. That is, even logic and reason itself are known to be true by the power of induction-intellection. Induction cannot be escaped.
Back To The Sea
Let’s return to the briny deep and the claim
(f) “Pr(He ought to do what sea captains do|He is a sea captain) = 1.”
We are in the same situation as with mathematic’s beginning. Something is off; there is a whiff of circularity. Facts about a sea captain’s duty are used to say one who is (or claims to be) a sea captain ought to be a good one. But that’s a restatement of “sea captains ought to do what sea captains do.”
What has happened was this: Men went to sea. It was found, by trial and error, what worked, in reaching desired goals, and what did not work. From this finite sample, the end, the telos, the nature, the essence of the sea captain was inferred, and inferred in exactly the same way the axioms in mathematics were: by using induction. Once sea captain was defined, it forever contained inside its definition that induction about the proper end of sea captaining.
It remains only to prove that the goal was itself objective. Because all we have right now something like
(g) “Pr(He ought to do what sea captains do|He is a sea captain, so defined & It’s good to get to one’s destination) = 1”,
which is valid, and where I indicate the induction about the ends of a sea captain (or watch or farmer or etc.) as “so defined.” We still have to explain why the premise “It’s good to get to one’s destination” is there. It’s needed to save the argument. The argument (g) would only be local truth and not objectively true unless that premise itself were true. Is it? And how can we know? (A local truth is arises from a deduction where the premises themselves may not be true, e.g. Pr(Elsie is a cow and has three heads | All cows have three heads) = 1.)
People only became sea captains because of the goal of crossing seas, people only created watches for the goal of dividing time, and they only farmed for the goal of producing food. All these activities, and those goals, were had for the sake of the same thing: they fulfilled the ends of man.
We know it is man’s nature to eat, to want to live, to find ways to survive. We infer those ends, these natures, from finite observations inductively, the same way we infer that man by nature has two legs and is born to die (“All men are mortal”). There is thus no difference in the operation of the intellect in providing sure and certain foundations in ethics or mathematics and even logic itself. Things and acts that are in accord with man’s nature or essence are good. Things and acts that are discordant, we infer inductively, are bad.
Ethics and morality would thus seem to be easy: find the ends of sea captains, watches, farmers, and man himself, and then fill in the blanks, start producing theorems like in math. What are these ends? Alas, beyond a handful of indisputable fundamentals, these are not always so easy to see. All know we need air, gravity, food, sun, and temperature in a certain window, and that we need other people to make more people. But that doesn’t mean we always know the proper ethical way to get these things. Ethics can rely on circumstance to a great degree. Acts can have consequences which have consequences and so forth. Acts which are right in one circumstance can be wrong in another, as all know.
Mistakes are often made in ethics, like in math. We might easily liken the practice of ethics to a middle school algebra test in forlorn and remote benighted locale, like Detroit. It’s not only frequent mistakes that are analogous. Even the essence in mathematics, it may surprise you, can be less than clear: not all axioms are plain. There is one in particular, called the “axiom of choice“, that still finds widespread disagreement among mathematicians. If this consternation is found in what is, after all, a very small thing, consider the ethical difficulties of mixing peoples from varying cultures and (let us say) biology. There is man, then there are men: Not all men are the same; indeed, some of them are women. Capabilities, qualities, abilities are not uniform.
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