LOGIC natural numbers, in such a way that every relationship between two formulae of the logical system corresponds to a relation between the numbers thus associated In particular, if a set of formulae A, B, C is a proof of a formula D, then there will be a specific numerical relationship between the Goădel numbers of the four formulae He then went on to construct a formula that could only have a proof in the system if the relevant Goădel numbers violated the laws of arithmetic The formula must therefore be unprovable; yet Goădel could show, from outside the system, that it was a true formula We might think to remedy this problem by adding the unprovable formula as an axiom to the system; but this will enable another, different, unprovable formula to be constructed, and so on ad infinitum We have to conclude that arithmetic is incomplete and incompletable Even if a system is complete, it does not follow that there will always be a way of deciding whether or not a particular formula is valid Production of a proof will of course prove that it is; but failure to produce a proof does not prove that it is invalid For propositional calculus, there is such a decision procedure: the truth-table method will show whether something is or is not a tautology Arithmetic, being incompletable, a fortiori is undecidable But between propositional logic and arithmetic, what of first-order predicate logic, which Goădel had shown to be complete: is there a decision procedure there? The painstaking work of logicians showed that parts of the system were decidable, but that there can be no decision procedure for the entire calculus, nor can we give a satisfactory rubric to determine which parts are decidable and which are not Modern Modal Logic Meanwhile, other logicians were studying a branch of logic that had been neglected since the Middle Ages, modal logic Modal logic is the logic of the notions of necessity and possibility Its study in modern times dates from the work of C I Lewis in 1918, who approached it via the theory of implication What is it for a proposition p to imply a proposition q? Russell and Whitehead treated their horseshoe sign (the truth-functional ‘if ’) as a sign of implication, on the grounds that ‘If p and p ! q then q’ was a valid inference But they realized that it was an odd form of implication—it entails, for instance, that any false proposition implies every proposition—and so they gave it the name 116