Fifteen Eighty Four

Studying self-referring language is fun. This is the reason why so many philosophers talk about the logic of truth. When we talk about the truth or falsity of sentences, we use language to talk about itself. My book, The Logic of Entailment and its History (Cambridge University Press, 2024), is not about truth, but about another way in which language talks about itself and involves people thinking about their own reasoning.

Suppose that you want to prove some statement using logic. You might be doing a logical derivation for a course in logic, trying to prove a mathematical statement, trying to determine the content of a scientific theory, or whatever. What you might think is something like: If I can prove B from A, and then prove C from B, then I can prove C from A. This is a simple reasoning chain, and we all use inferences of this form from time to time. The conditional (the ‘if … then’) in this reasoning chain should also be thought of as telling us about what follows logically. Thus, we have an inference of the pattern: (proving B from A) and (proving C from B) logically entails that we can prove C from A.

This notion of logical entailment itself, from the logical point of view, is the same notion that is expressed by ‘proving’ and ‘we can prove’. But what is this notion?

The conceptions of entailment that I look at are captured by connectives – logical operators that take formulas in the logic and create new formulas. Other examples of connectives are conjunction, disjunction, and negation. Where ® is the entailment connective, ‘A®B’ says that we can prove B from A and the reasoning chain that I give above can be represented by ‘((A®B)&(B®C))®(A®C)’.

One traditional way of representing entailment in a formal logic is to use standard modal logic and treat entailment as strict implication. Strict implication is necessitated material implication (and material implication is the implication of classical propositional calculus, the logic that we learn in an introductory logic course). Strict implication, however, has a serious flaw. Suppose that what you want to prove (C in the example above) is really a theorem of the logic (like a tautology of propositional calculus). Then any proposition strictly implies C.

Suppose that C is some famous theorem but that all of its proofs are really difficult (like Fermat’s Last Theorem in number theory). If you believe that entailment is strict implication you can reason as follows: If I can give a proof that the Earth is round entails the Earth has some shape and that the Earth has some shape entails that C, then the Earth is round entails C. You can prove each of the steps in this chain, if entailment is strict implication. But the reasoning here seems pretty lousy. Just because C is a logical truth should mean that ‘the Earth has some shape’ really entails it.

Thus we need a much more subtle notion of entailment. I suggest in my book that we adopt a relevant notion of entailment. The notion of relevance that I support requires the real use of a statement in a proof if it is to be thought of as active in the entailment. This conception of real use prevents us from holding that just any statement can be used to prove any logical truth.

In the book, I draw a strong connection between the general concept of entailment and an understanding of the way in which scientific theories are structured. If a statement A is in a theory and A entails B then B is also in that theory, at least implicitly. I construct a formal semantical theory to make precise this concept of entailment. In the formal semantics, instead of using possible worlds as the points at which statements are deemed true or false, I have theories. A theory here does not have to be anything as sophisticated as a theory of physics, but it can be an everyday theory that captures some of one’s beliefs about the world.

Some of the theories used in the semantics are theories about entailment. Some of them are correct (in the sense that they match what the semantics says about reasoning overall) and some of them get this wrong. But we can reason about incorrect theories as easily as we can reason about correct ones. In this way, I hope to have constructed a theory of entailment that explains how we can reason about our own reasoning (in the sense of proving things) but that also has this sort of self-reference at the heart of the semantical theory itself.