§1.  Three Kinds Of Signs (cont.)

I have taken pains to make my distinction of icons, indices, and tokens clear, in order to enunciate this proposition:  in a perfect system of logical notation signs of these several kinds must all be employed.  Without tokens there would be no generality in the statements, for they are the only general signs;  and generality is essential to reasoning.  Take, for example, the circles by which Euler represents the relations of terms.  They well fulfill the function of icons, but their want of generality and their incompetence to express propositions must have been felt by everybody who has used them.  Mr. Venn has, therefore, been led to add shading to them;  and this shading is a conventional sign of the nature of a token.  In algebra, the letters, both quantitative and functional, are of this nature.

But tokens alone do not state what is the subject of discourse;  and this can, in fact, not be described in general terms;  it can only be indicated.  The actual world cannot be distinguished from a world of imagination by any description.  Hence the need of pronouns and indices, and the more complicated the subject the greater the need of them.  The introduction of indices into the algebra of logic is the greatest merit of Mr. Mitchell’s system.  He writes to mean that the proposition is true of every object in the universe, and to mean that the same is true of some object.  This distinction can only be made in some such way as this.  Indices are also required to show in what manner other signs are connected together.

With these two kinds of signs alone any proposition can be expressed;  but it cannot be reasoned upon, for reasoning consists in the observation that where certain relations subsist certain others are found, and it accordingly requires the exhibition of the relations reasoned with in an icon.  It has long been a puzzle how it could be that, on the one hand, mathematics is purely deductive in its nature, and draws its conclusions apodictically, while on the other hand, it presents as rich and apparently unending a series of surprising discoveries as any observational science.  Various have been the attempts to solve the paradox by breaking down one or other of these assertions, but without success.  The truth, however, appears to be that all deductive reasoning, even simple syllogism, involves an element of observation;  namely, deduction consists in constructing an icon or diagram the relations of whose parts shall present a complete analogy with those of the parts of the object of reasoning, of experimenting upon this image in the imagination, and of observing the result so as to discover unnoticed and hidden relations among the parts.  (3.363).