Adding premisses to a valid inference in traditional and modern logic

Sophia asked:

I have this question in this logic course that I am taking and have been stuck on it for quite a while, the answer is true but have yet to figure out exactly why.

True or false?

Answer by Helier Robinson

Modern symbolic logic allows things that traditional logic does not. Your (iii) is true in modern logic and false in traditional logic. Writing ‘If p then q’ as p->q then p->q is valid traditionally only if the truth of p necessitates the truth of q, while in modern logic it is valid if it is a tautology; that is, it is always true. The difference here is that being a tautology is a sufficient condition for validity in modern logic, but only a necessary condition in traditional logic. So in modern logic if p is true and q is true then p->q is valid; while in traditional logic p->q is valid if, given that p is true then q must be true. So in modern logic if we have p->q is valid then, by the truth-table definition of p->q, q is true and p is either true or false; if we now add an extra premise, r, to get (p&r)->q then p&r is true if p and r are both true, and otherwise false. But since q remains true, it does not matter whether p&q is true or false, since (p&r)->q is true. hence valid.

This peculiarity of modern logic arises from the fact that it is based on Boolean algebra, which is an algebra of only two numbers, 0 and 1. Ordinary arithmetical operations work in this algebra, with two exceptions: 1+1=1 and 0-1=0. The algebra is perfectly consistent, and the application of it to computer switching works perfectly, but the application of it to logic does not. In modern logic 0 stands for false and 1 stands for true. Boolean addition then becomes logical disjunction and Boolean multiplication becomes logical conjunction; this works perfectly, and is the main justification for basing logic on this algebra; but two other Boolean operations, interpreted as implication and equivalence, do not. As a result you get such strange things as being able to deduce anything you please from a false proposition (and, worse, anything you please from a contradiction) and that any two propositions are equivalent if they have the same truth values. And these peculiarities extend in derivative logics: in quantificational logic anything you say of a non-existent thing is true, as in ‘All mermaids are butterflies’ and in modal logic you can deduce anything you please from an impossible proposition. It is a quite extraordinary fact that in the twentieth century Boolean logic was regarded as legitimate by almost all philosophers. It still is, for that matter. But feel free to reject it if you want to.

Helier Robinson is Emeritus Professor in Philosophy, University of Guelph. His books are available from http://SharebooksPublishing.com.