Dialethic Dialectic

by Carl Pierer

HegelHistorically, formal logic and Hegel's philosophy's relation has been dominated by antipathy. Classical logic, developing from Aristotelian logic to the Frege/Russell logic of the 20th century, has largely rejected Hegel because of his overt embrace of contradictions. Hegel, vice-versa, has not been too charitable to the formal logic of his day. In the second half of the 20th century, however, formal logic has developed massively and in various directions. One of these, paraconsistent logics, have attempted to accommodate contradictions. Classical logic is anathema to contradictions, due to the explosion principle, a.k.a. ex falso quodlibet. A sketch of this principle is the following: since the classical or is non-exclusive, if we start with a true proposition A, the disjunct A or B is true for any proposition B. So, if we have A&~A, we get that A is true and hence A or B is true. But since ~A is true, too, from A or B we get that B must be true. Hence anything follows from a contradiction, or so the classical (and subsequently the Frege/Russell logic) claims. So contradictions seem to be a rather bad thing.

Now, paraconsistent logics deny this explosion principle. There are different ways of doing this, but we will stick with Priest's way in his (Priest, 1989). His is a dialethic interpretation, meaning that he thinks there are sentences that are both true and false. This has some interesting consequences. Note, first of all, that this does not mean that all sentences are true and false. Most importantly, most classical notions are indeed preserve. So, we have, for propositions A and B:

  • ~A is true implies A is false
  • A is true implies ~A is false
  • A&B is true only if A is true and B is true
  • A&B is false only if at least one of A or B is false

These are quite orthodox. Now, of course, on the dialethic point of view, A could be both true and false, and suppose B is true. Then A&B is both true and false. Next, we need the notion of logical consequence, which Priest defines also quite classically:

A is a logical truth just if A is (at least) true under all assignments of values.

A is a logical consequence of B just if every assignment of values that make B (at least) true makes A (at least) true.(Priest, 1989)

What does this mean exactly?

Like what you're reading? Don't keep it to yourself!
Share on Facebook
Tweet about this on Twitter
Share on Reddit
Share on LinkedIn
Email this to someone