LOGIC signs for logical connectives are still in common use, though we use in this text instead of the horseshoe the sign ‘!’, which is nowadays preferred For conjunction Russell and Whitehead used a simple point, as in ‘p.q’; nowadays the ampersand, as in ‘p & q’, is commonly used instead Russell and Whitehead expressed universal quantification thus: ‘(x)F(x)’; and existential quantification thus: ‘(Ex)F(x)’ These symbols, too, are now in common use; the ‘E’ in existential quantification is sometimes printed in reverse The system of Principia is, like Frege’s, an axiomatic system in which logical truths are derived by rule from a handful of axioms The initial set of axioms, however, differs from Frege’s set, and whereas Frege had taken ‘if’ and ‘not’ as primitive connectives from which the others could be defined, Russell and Whitehead took ‘or’ and ‘not’ (which they called ‘logical constants’) as basic In fact many other sets of axioms are possible, with different primitive constants, and they were studied by logicians in the next decades But it soon came to be realized that axiomatic systems were not the only way, or even necessarily the best way, to give logic a rigorous form This was shown by Wittgenstein, who invented a formal device which, like many of those of Frege, passed into the logic textbooks, namely the truth-table It is possible to define the propositional connectives by setting out in a table the truth-conditions of propositions containing them Thus the table p T F T F q T T F F p&q T F F F represents that ‘p & q’ is true in the case in which ‘p’ and ‘q’ are both true, and false in the three other possible cases, namely (a) when ‘p’ is false and ‘q’ is true, (b) when ‘p’ is true and ‘q’ is false, (c) when ‘p’ and ‘q’ are both false The truth-value of ‘p & q’, as the table brings out, is determined by the truth-values of the component propositions ‘p’ and ‘q’; the compound proposition, we may say, is a truth-function of its constituents, and the possible combinations of the truth-values of the constituents set out the truthconditions for the compound proposition 111