Notation and First-Order Logic

Abstract

An elementary statement (or atomic statement) is a sentence with a subject and a verb (and sometimes an object) but no connectives (and, or, not, if-then, if-and-only-if). See Subsection 2.1.1 for a more rigorous definition. Elementary statements are joined together using connectives.

...Even though his most important research contributions were in topology, Brouwer never gave courses on topology, but always on—and only on—the foundations of intuitionism. It seemed that he was no longer convinced of his results in topology because they were not correct from the point of view of intuitionism, and he judged everything that he had done before, his greatest output, false according to his philosophy.—B.L. Van der Waerden

Most mathematicians would find it hard to believe that there could be any serious controversy about the foundations of mathematics, any controversy whose outcome could significantly affect their own mathematical activity. —Erret Bishop

The world is all that that is the case.—Ludwig Wittgenstein

When Aristotle was asked how much educated men were superior to the uneducated, he replied, “As much as the living are to the dead.” —Dionysus of Halicarnassus

The world is a totality of facts, not things. —Ludwig Wittgenstein

Mathematics takes us into the region of absolute necessity, to which not only the actual world, but every possible world, must conform. —Bertrand Russell

If the only tool you have is a hammer, then every problem you see looks like a nail. —Confucian saying