LOGIC mortal’ is a true sentence, we can say that the function ‘ is mortal’ holds true for the argument ‘Socrates’ To express generality we need a symbol to indicate that a certain function holds true no matter what its argument is Adapting the notation that Frege introduced, logicians write (x)(x is mortal) to signify that no matter what name is attached as an argument to the function ‘ is mortal’, the function holds true The notation can be read as ‘For all x, x is mortal’ and it is equivalent to the statement that everything whatever is mortal This notation for generality can be applied in all the different ways in which sentences can be analysed into function and argument Thus ‘(x)(God is greater than x)’ is equivalent to ‘God is greater than everything’ It can be combined with a sign for negation (‘$’) to produce notations equivalent to sentences containing ‘no’ and ‘none’ Thus ‘(x) $ (x is immortal)’ ¼ ‘For all x, it is not the case that x is immortal’ ¼ ‘Nothing is immortal’ To render a sentence containing expressions like ‘some’ Frege exploited the equivalence, long accepted by logicians, between (for example) ‘Some Romans were cowards’ and ‘Not all Romans were not cowards’ Thus ‘Some things are mortal’ ¼ ‘It is not the case that nothing is mortal’ ¼ ‘$ (x) $(x is mortal)’ For convenience his followers used, for ‘some’, a sign ‘(Ex)’ as equivalent to ‘$ (x) $’ Frege’s notation, and its abbreviation, can be used to make statements about the existence of things of different kinds ‘(Ex)(x is a horse)’, for instance, is tantamount to ‘There are horses’ (provided, as Frege notes, that this sentence is understood as covering also the case where there is only one horse) Frege believed that objects of all kinds were nameable—numbers, for instance, were named by numerals—and the argument places in his logical notation can be filled with the name of anything whatever Consequently ‘(x)(x is mortal)’ means not just that everyone is mortal, but that everything whatever is mortal So understood, it is a false proposition, because, for instance, the number ten is not mortal It is rare, in fact, for us to want to make statements of such unrestricted generality It is much more common for us to want to say that everything of a certain kind has a certain property, or that everything that has a certain given property also has a certain other property ‘All men are mortal’ or ‘What goes up must come down’ are examples of typical universal 104