Proof and Falsity, by Nils Kürbis

“Not”, or its equivalent, is possibly the most important word of all. Its addition to a sentence turns the meaning on its head: changing a truth into a falsehood and a falsehood into a truth. If it’s true that today is Thursday, then it’s false that today is not Thursday. But what exactly does “not” mean? This is a complex and intricate matter, as Kürbis proves in this detailed and at times technical study of negation.

One answer is that meaning is found in use, so we should look at how the word “not” is used. Within formal logic, there is an approach named proof-theoretic semantics that does exactly this, looking at the use of negation in deductive arguments. This seems especially apt for a word such as “not”, since there is nothing that it names. You cannot point at “not” in a way that you could point at the Eiffel Tower or a table. There is only use.

A problem occupying much of Kürbis’ time is how “not” can be introduced into a logical system. The other connectives, or logical constants, “and”, “or” and “if, then”, have relatively simple introductions. For example, assume A is true and B is true, then it is true that A and B. The introduction of “not” is more troublesome, however. Suppose we try to introduce it by saying that if A entails something absurd, then not-A must be true. The problem is that absurdity is usually understood along the lines of both X and not-X being true at the same time, which means that the notion of negation must already be understood in order for negation to be introduced. The proposal is circular.

There are a number of strategies for overcoming this problem and salvaging the proof-theoretic thesis that meaning is use. The way forward is to accept a further undefined primitive specifically to account for the introduction of negation. But which primitive should that be? We could add incompatibility to our account, whereby if A and B are incompatible, then they cannot both be true. But surely the notion of negation is better understood and more basic than that of incompatibility. Instead, we could just accept negation as a primitive of the system, but then it has no introduction rules at all. Or we might consider a primitive notion of denial, where to say not-A is really to deny A. Finally, and this is Kürbis’ preferred choice, we can have a logic that has a primitive notion of falsity in addition to that of truth. To say not-A is to say that A is false. This is not a new idea, but the demonstration is ingenious.

While Proof and Falsity is a deep, difficult and challenging book, it is nevertheless clear and accessible to those with only a rudimentary background in logic. It certainly succeeds in showing that the little word “not” is one of the hardest to understand. Surprisingly, there is no engagement with Laurence Horn’s A Natural History of Negation (1989), with which it can be most closely compared. Philosophy isn’t always easy but, as Kürbis shows, we can at least avoid making it more complicated than necessary, and for this he deserves great credit.

Stephen Mumford is professor of metaphysics at Durham University.

Proof and Falsity
By Nils Kürbis
Cambridge University Press
320pp, £75.00
ISBN 9781108481304
Published 9 May 2019