A class means all things which have a certain list of attributes stated in the definition.May we not then infer other properties from the definition?May not mortality,for example,be deducible from the other attributes of man?The assumption that we can do so is connected with the fallacy most characteristic of the misuse of the syllogism.It is plain that we may create as many classes as we please,and make names for combinations of attributes which have no actual,or even no possible,existence.
Any inferences which we make on the strength of such classification must be nugatory or simply tautologous.I show that a certain proposition follows from my definition;but that gives no guarantee for its conformity to the realities behind the definition.Your 'proof'that a man is mortal means simply that if he is not mortal you don't call him a man.The syllogism treated on that system becomes simply an elaborate series of devices for begging the question.From such methods arise all the futilities of scholasticism,and the doctrine of essences which,though Locke confuted it,(31)has 'never ceased to poison philosophy.'(32)It may,I suppose,be taken for granted that the syllogism was constantly applied to cover such fallacies,and so far Mill is on safe ground.The theory,however,leads him to a characteristic point.Already in the early review (January 1828),he had criticised Whately's account of definition.A 'real definition,'as Whately had said,'explains and unfolds "the nature"of the thing defined,whereas a "nominal definition"only explains the name.'Whately goes on to point out that the only real definitions in this sense are the mathematical definitions.
It is impossible to discover the properties of a thing,a man,or a plant from the definition.If it were possible,we might proceed to 'evolve a camel from the depths of our consciousness,'and nobody now professes to be equal to that feat.When,however,we 'define'a circle or a line and so forth,we make assertions from which we can deduce the whole theory of geometry.Ageometrical figure represents a vast complex of truths,mutually implying each other,and all deducible from a few simple definitions.The middle term is not the name of a simple thing,or of a thing which has a certain set of coexisting attributes,but a word expressive of a whole system of reciprocal relations.
If one property entitles me to say that a certain figure is a circle,I am virtually declaring that it has innumerable other properties,and I am thus able to make inferences which,although implicitly given,are not perceived till explicitly stated.By assigning a thing to a class,I say in this case that I may make any one of an indefinite number of propositions about it,all mutually implying each other,and requiring the highest faculties for combining and evolving.Pure mathematics give the one great example of a vast body of truths reached by purely deductive processes.They appear to be evolved from certain simple and self-evident truths.Can they,then,be explained as simply empirical?Do we know the properties of a circle as we know the properties of gold,simply by combining records of previous experience?Or can we admit that this great system of truth is all evolved out of 'definitions'?
Mill scents in Whately's doctrine a taint of a priori assumption,and accordingly meets it by a direct contradiction.Ageometrical definition,he says,is no more a 'real'definition than the definition of a camel.No definition whatever can 'unfold the nature'of a thing.He states this in his review,though it was at a later period,(33)when meditating upon a passage of Dugald Stewart,that he perceived the full consequences of his own position.In answering Whately,he had said that all definitions were 'nominal.'A 'real definition'means that to the definition proper we add the statement that there is a thing corresponding to the name.(34)The definition itself is a 'mere identical proposition,'from which we can learn nothing as to facts.But it may be accompanied by a postulate which 'covertly asserts a fact,'and from the fact may follow consequences of any degree of importance.This distinction between the definition and the postulate may be exhibited,as he remarks,by substituting 'means'for 'is.'If we say:a centaur 'means'a being half man and half horse,we give a pure definition.If we say:a man 'is'a featherless biped,our statement includes the definition --man 'means'featherless biped;but if we said no more,no inference could be made as to facts.If we are really to increase our knowledge by using this definition,we must add the 'covert'assertion that such featherless bipeds exist.The mathematical case is identical,Stewart had argued that geometrical propositions followed,not from the axioms but,from the definitions.From the bare axiom that if equals be added to equals the wholes are equal,you can infer nothing.You must also perceive the particular figures which are compared.Of course the truth of the axioms must be admitted;but they do not specify the first principles from which geometry is evolved.In other words,geometry implies 'intuition,'not the a priori 'intuitions'to which Mill objected,but the direct perception of the spatial relations.We must see the figure as well as admit the self-evident axiom.
Mill,on considering this argument,thought that Stewart had stopped at a half truth.(35)He ought to have got rid of the definitions as well as the axioms.Every demonstration in Euclid,says Mill,might be carried on without them.When we argue from a diagram in which there is a circle,we do not really refer to circles in general,but only to the particular circle before us.
