Hume’s (1739) *Treatise* famously argued that we cannot infer an “ought” from an “is”. This has presented an enduring problem for science: how should we produce a set of recommendations for what should be done following the results of a study? If a new cancer treatment dramatically improves remission rates, should study authors simply shrug, present the results, and leave the recommendations to politicians? What if a treatment causes significant harms – can we recommend that the treatment be banned? Or suppose we have ideas for future studies that should be carried out and want to summarise them in the conclusions…? Even doing this would be ruled out by Hume.

The solution, if it is one, is that any recommendations require a set of premises stating our values. These values necessarily assert something beyond the evidence, for instance that if a treatment is effective then it should be provided by the health service. In practice, such values are often left implicit and assumed to be shared with readers. But there are interesting examples where it is apparently possible to draw an is-ought inference without assuming values.

One example, due to Mavrodes (1964), begins with the premise

If we ought to do *A*, then it is possible to do *A*.

This seems reasonable enough. It would, for instance, be horribly dystopian to require that people behave a particular way if it were impossible for them to do so. Games like chess and tennis have rules that are possible – if they were impossible then it would make playing the games challenging. Let’s see what happens if we apply a little logic to this premise.

Sentences of the form

If *A*, then *B*

are equivalent to those of the contrapositive form

If not-*B*, then not-*A*

This can be seen in the truth table below, where 1 denotes *true* and 0 denotes *false*. The values of the last two columns are equivalent:

A | B | not-A | not-B | If A, then B | If not-B, then not-A |
---|---|---|---|---|---|

1 | 1 | 0 | 0 | 1 | 1 |

1 | 0 | 0 | 1 | 0 | 0 |

0 | 1 | 1 | 0 | 1 | 1 |

0 | 0 | 1 | 1 | 1 | 1 |

Together, this means that if we accept the premise

If we ought to do *A*, then it is possible to do *A,*

and the rules of classical logic, we must also accept

If it is not possible to do *A*, then it is not the case that we ought to do *A*.

But here we have an antecedent that is an “is” and a consequent that is an “ought”: logic has licenced an is-ought!

Worry not: there has been debate in the literature… See Gillian Russell (2021) for a recent analysis.

### References

Mavrodes, G. I. (1964). “Is” and “Ought.” *Analysis*, *25*(2), 42–44.

Russell, G. (2021). How to Prove Hume’s Law. Journal of Philosophical Logic. In press.