LOGIC Therefore, All Germans are Europeans Some Germans are blonde Some Europeans are blonde and invalid inferences such as Therefore, All cows are mammals Some mammals are quadrupeds All cows are quadrupeds Though both these inferences have true conclusions, only the first is valid, that is to say, only the first is an inference of a form that will never lead from true premisses to a false conclusion Syllogistic, in fact, covers only a small proportion of the forms of valid reasoning In Anthony Trollope’s The Prime Minister the Duchess of Omnium is anxious to place a favourite of hers as Member of Parliament for the borough of Silverbridge, which has traditionally been in the gift of the Dukes of Omnium He tells us that she ‘had a little syllogism in her head as to the Duke ruling the borough, the Duke’s wife ruling the Duke, and therefore the Duke’s wife ruling the borough’ The Duchess’s reasoning is perfectly valid, but it is not a syllogism, and cannot be formulated as one This is because her reasoning depends on the fact that ‘rules’ is a transitive relation (if A rules B and B rules C, then A does indeed rule C), while syllogistic is a system designed to deal only with subject–predicate sentences, and not rich enough to cope with relational statements A further weakness of syllogistic was that it could not cope with inferences in which words like ‘all’ or ‘some’ occurred not in the subject place but somewhere in the grammatical predicate The rules would not determine the validity of inferences that contained premisses such as ‘All politicians tell some lies’ or ‘Nobody can speak every language’ in cases where the inference turned on the word ‘some’ in the first sentence or the word ‘every’ in the second Frege devised a system to overcome these difficulties, which he expounded first in his Begriffsschrift The first step was to replace the grammatical notions of subject and predicate with new logical notions, which Frege called ‘argument’ and ‘function’ In the sentence ‘Wellington defeated Napoleon’ grammarians would say (or used to say) that ‘Wellington’ was the subject and ‘defeated Napoleon’ the predicate Frege’s introduction of 101