[do] -- implication
In Chapter 1 of Principles of Mathematics, Bertrand Russell states — Pure mathematics is the class of all propositions of the form ‘p implies q’, where p and q are propositions”. But in Chapter 2, he cheerfully claims that “A definition of implication is quite impossible”. What is going on here?
Intuitively, the statement “p implies q” is a “promise” or “guarantee”. If someone gives p, then he must receive q in return. If p is true, then q is assured. But since these definitions employ an if / then (i.e. implication) structure, they are circular. We still do not know what implication actually is.
How do we break out of this vicious cycle? Other posts advocate approaching questions in cognitive science from a more physical perspective. Thought is best modeled by *geometric shapes* or *movement patterns*. Expressed schematically, implication states the following.
<x y> <y z>
<x z>
This picture shows <x y> combining with <y z> to form <x z>. In other words, we create a “chain” linking x and z together through the common element y. In this template, x and z represent the “if” and the “then”. The symbol y represents the binding of the two together. This journey from x to z is the essence of implication.
This is actually nothing more than the familiar transitive property. Implication may be a mere subspecies of transitivity.