In my Aristotle class this morning, we studied Aristotle’s most basic arche (or principle) for rational discourse. Aristotle argues that the principle of non-contradiction (PNC) is the most fundamental and necessary principle for intelligible discourse. The principle says that ~(A&~A)–that it is impossible for A and ~A to be true simultaneously. The PNC, he argues, is indispensable for rational discourse.
I asked the following question in class:
We now have sufficiently developed logical systems in which PNC is neither an axiom nor a theorem. These systems are called paraconsistent logics. In these systems PNC is not valid. Would the developments of these systems constitute undermining evidence against the purported special status of PNC?
My question was not well received. I was told that we are doing history of philosophy in my class. Because my question is ahistorical, it is not particularly conducive to the basic objective of the class, which is to gain a good understanding in Aristotle’s works. Moreover, I was told that I was being uncharitable to Aristotle. After all, at the time of Aristotle, paraconsistent logics were not developed.
I certainly do not think I was being uncharitable to Aristotle for I do think that Aristotle was justified in his commitment to the PNC. All I wanted to know was whether if Aristotle possessed knowledge about paraconsistent logics, he would remain justified in his robust commitment to PNC or not. Perhaps I did not articulate my question well enough or maybe I was simply misunderstood in this respect. As for my question being ahistorical and therefore against the spirit of the class, I partly agree. Usually, ahistorical questions are not conducive to the goals of a history of philosophy class. However, I think in trying to get an answer to my question, the nature of Aristotle’s commitment to the PNC would be elucidated. This is why I think my question while being ahistoric was still relevant and useful. I could not discuss the status of PNC given the development of paraconsistent logics in my class. However, I hope to get some feedback over here.
The principle of non-contradiction (PNC) is a valid principle in classical function-argument logics such as the Fregean logic and intuitionisitic logic. From a contradiction (or the violation of PNC) anything follows in these logics. In other words, within these systems, if we get a contradiction of the form (A&~A), we can derive any belief and it’s negation. This is bad! Let me show how from a contradiction and valid rules of classical logic we can derive any belief.
A-The sky is blue
- A&~A Assumption
- A Conjunction Elimination 1
- AvE Disjunction Introduction 2
- ~A Conjunction Elimination 1
- E Disjunction Syllogism 3, 4
From the sentence ‘The sky is blue and the sky is not blue’, we are able to derive ‘Superman exists’. This sentence is derived employing valid rules–conjunction elimination and disjunction syllogism–in classical logic. Similarly, we could have derived the negation of ‘Superman exists’. All we would have to do is replace E with ~E in the above derivation. Moreover, this can be done for any sentence. This is why, in classical logic at least, from a contradiction anything follows.
There are, however, sufficiently developed families of logics called paraconsistent logics in which the PNC is neither an axiom nor a theorem (a sentence traceable back to the axioms of a system). When we get a contradiction in such logical systems, an explosion of sentences does not follow. This is because one of the above inferences (such as disjunctive syllogism) is invalid in these systems. We can no longer, through valid forms of reasoning in this system, derive ‘Superman exists’ from ‘The sky is blue and it is not blue’. In other words, from the denial of PNC (or acceptance of a contradiction) one is no longer able to derive any proposition and it’s negation. The denial of PNC in such logical system is not devastating–rational discourse is still maintained.
Given that PNC is not an indispensable principle of reasoning in the sense that it is possible to construct systems of logics (or languages) without PNC being valid, it seems that had Aristotle known about these logics, he might have made his commitment to PNC less robust than he did in his Metaphysics. Alternatively, he might have argued against the development of all paraconsistent logics and say something like, “These logics are deviant systems. The correct logic is one in which PNC is analytic.” The difficulty for Aristotle to argue for one correct logic would be to somehow avoid circular justification. He would have to presume his logic in which PNC is valid to argue for the very logic whose epistemic justification is under question.