Section 3

We must pronounce whether it pertains to the same science to study both the so-called axioms in mathematics and substance, or to different sciences. It is obvious that the investigation of these axioms too pertains to one science, namely the science of the philosopher; for they apply to all existing things, and not to a particular class separate and distinct from the rest. Moreover all thinkers employ them—because they are axioms of Being qua Being, and every genus possesses Being—

but employ them only in so far as their purposes require; i.e., so far as the genus extends about which they are carrying out their proofs. Hence since these axioms apply to all things qua Being (for this is what is common to them), it is the function of him who studies Being qua Being to investigate them as well.

For this reason no one who is pursuing a particular inquiry—neither a geometrician nor an arithmetician—attempts to state whether they are true or false; but some of the physicists did so, quite naturally; for they alone professed to investigate nature as a whole, and Being.

But inasmuch as there is a more ultimate type of thinker than the natural philosopher (for nature is only a genus of Being), the investigation of these axioms too will belong to the universal thinker who studies the primary reality. Natural philosophy is a kind of Wisdom, but not the primary kind.

As for the attempts of some of those who discuss how the truth should be received, they are due to lack of training in logic; for they should understand these things before they approach their task, and not investigate while they are still learning.

Clearly then it is the function of the philosopher, i.e. the student of the whole of reality in its essential nature, to investigate also the principles of syllogistic reasoning. And it is proper for him who best understands each class of subject to be able to state the most certain principles of that subject; so that he who understands the modes of Being qua Being should be able to state the most certain principles of all things.

Now this person is the philosopher, and the most certain principle of all is that about which one cannot be mistaken; for such a principle must be both the most familiar (for it is about the unfamiliar that errors are always made), and not based on hypothesis.

For the principle which the student of any form of Being must grasp is no hypothesis; and that which a man must know if he knows anything he must bring with him to his task.

Clearly, then, it is a principle of this kind that is the most certain of all principles. Let us next state what this principle is.

It is impossible for the same attribute at once to belong and not to belong to the same thing and in the same relation; and we must add any further qualifications that may be necessary to meet logical objections. This is the most certain of all principles, since it possesses the required definition;

for it is impossible for anyone to suppose that the same thing is and is not, as some imagine that Heraclitus says 1 —for what a man says does not necessarily represent what he believes.

And if it is impossible for contrary attributes to belong at the same time to the same subject (the usual qualifications must be added to this premiss also), and an opinion which contradicts another is contrary to it, then clearly it is impossible for the same man to suppose at the same time that the same thing is and is not; for the man who made this error would entertain two contrary opinions at the same time.

Hence all men who are demonstrating anything refer back to this as an ultimate belief; for it is by nature the starting-point of all the other axioms as well.