Logical consequence is a notion that every person who reasons must possess, at least implicitly. To give a precise and accurate characterization of this notion is the fundamental task of logic. In a similar way, the notion of effectivity is a concept that anyone with a basic training in mathematics possesses, and the most fundamental task of a theory of computability is to give a precise characterization of this notion. The problem concerning effectivity was solved (at least to the satisfaction of most people) in the 1930s, almost as soon as it was raised, by the work of Turing, Church, and others. By contrast, the correct and precise characterization of logical consequence has been hotly contested through the two and a half thousand-year history of logic.

A tempting solution to problems of semantic vagueness and to the Liar Paradox is an appeal to truth-value gaps. It is tempting to say, for example, that, where Harry is a borderline case of bald, the sentence

1. (1) Harry is bald

is neither true nor false: it is in the ‘gap’ between these two values, and perhaps deserves a third truth-value. Similarly with the Liar Paradox. Consider the following Liar sentence:

1. (2) (2) is false.

That is, sentence (2) says of itself that it is false. If we accept the Tarskian schema

where ‘S’ is a name of a sentence ‘p,’ we are led into paradox. Both the assumption that (2) is true, and the assumption that (2) is false lead us, via (T), to

1. (3) (2) is true if and only if (2) is false.

Given this result, a natural reaction is to place (2) in a ‘gap’ between true and false.