LOGIC of ‘material implication’ Lewis insisted that the only genuine implication was strict implication: p implies q only if it is impossible that p should be true and q false ‘p strictly implies q’, he maintained, was equivalent to ‘q follows logically from p’ He drew up axiomatic systems in which the sign for material implication was replaced by a new sign to represent strict implication, and these systems were the first formal systems of modal logic Strict implication struck many critics as being hardly less paradoxical than material implication, since an impossible proposition strictly implies every proposition, so that ‘If cats are dogs then pigs can fly’ comes out true Lewis’s modal researches, however, were interesting in their own right He offered five different axiom systems, which he numbered S1 to S5, and showed that each of the axiom sets was consistent and independent They vary in strength S1, for instance, does not allow a proof of ‘If p&q is possible, then p is possible and q is possible’ (which seems very plausible), while S5 contains ‘If p is possible, then p is necessarily possible’ (which seems rather dubious) In some ways the most interesting system is S4, which Goădel showed was equivalent to the logic of Principia Mathematica with the following additional axioms (reading ‘if ’ as material, not strict, implication): (1) If necessarily p, then p (2) If necessarily p, then (if necessarily [if p then q] then necessarily q) (3) If necessarily p, then necessarily necessarily p He added also a rule, that if ‘p’ was any thesis of the system, we can add also ‘necessarily p’ The system exploits the interdefinability of necessity (which he represented by the symbol ) and possibility (represented by ) As was well known in antiquity and the Middle Ages, ‘necessarily’ can be defined as ‘not possibly not’ and ‘possibly’ as ‘not necessarily not’ There are many statements that can be formulated within modal logic about whose truth-value there is no consensus among logicians The most contentious ones are those in which modal operators are iterated The system that Goădel axiomatized, S4, contains as derivable theses the two following formulae: ^ & If possibly possibly p, then possibly p If necessarily p, then necessarily necessarily p It does not, however, contain these two: 117