ABSTRACT
Classical logic in its current form was established by Frege in the late nineteenth century, but there are fundamental assumptions within this logic that may be traced back to classical Greece, in particular to the philosophical school of Plato. One of the key assumptions of the Platonistic approach is that we may ascribe truth to a statement quite independently of our ability to demonstrate that truth. This is nowhere more obvious than in the statement of the excluded middle described in Chapter 1: https://www.w3.org/1998/Math/MathML"> p ∨ ¬ p = t r u e https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203211991/94c3b92a-0a16-4a4e-9014-02b93290d004/content/math_569_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> This equivalence allows truth to be deduced from a statement p even when statement p is totally nonsensical. If, for example, p represents the statement “mermaids have magic powers” we have no method of demonstrating that this proposition is either true or false, yet the excluded middle proposition above still evaluates to true. There is an assumption in classical logic that somehow, somewhere one of the two familiar truth values can be bestowed on every possible statement. Intuitionistic logic on the other hand requires, in this particular case, that we either produce a mermaid with magic powers or we demonstrate that a mermaid cannot have magic powers. In either case we must make it clear which disjunct we have proven because the classical claim that it must be one or the other is no longer available.
