The one [the logician] studies the science of drawing conclusions, the other [the mathematician] the science which draws necessary conclusions.

It has often been argued that absolute scepticism is self-contradictory; but this is a mistake: and even if it were not so, it would be no argument against the absolute sceptic, inasmuch as he does not admit that no contradictory propositions are true. Indeed, it would be impossible to move such a man, for his scepticism consists in considering every argument and never deciding upon its validity; he would, therefore, act in this way in reference to the arguments brought against him.

All the progress we have made in philosophy ... is the result of that methodical skepticism which is the element of human freedom.

Every new concept first comes to the mind in a judgment.

It is a common observation that a science first begins to be exact when it is quantitatively treated. What are called the exact sciences are no others than the mathematical ones.

Third, consider the insistency of an idea. The insistency of a past idea with reference to the present is a quantity which is less, the further back that past idea is, and rises to infinity as the past idea is brought up into coincidence with the present.

Bad reasoning as well as good reasoning is possible; and this fact is the foundation of the practical side of logic.