On April 10, 2014, in response to a question, Stephen Maitzen wrote: "I can't see how there could be any law more fundamental than the law of non-contradiction (LNC)." I thought that there were entire logical systems developed in which the law of non-contradiction was assumed not to be valid, and it also seems like "real life" suggests that the law of non-contradiction does not necessarily apply to physical systems. Perhaps I am not understanding the law correctly? Is it that at most one of these statements is true? Either "P is true" or "P is not true"? or is it that at most one of theses statements is true? Either "P is true" or "~P is true"? In physics, if you take filters that polarize light, and place two at right angles to each other, no light gets through. Yet if you take a third filter at a 45 degree angle to the first two, and insert it between the two existing filters, then some light gets through. Based on this experiment, it seems like the law of non-contradiction cannot be true in this situation. Thanks for any clarity you might bring!

I won't address the issue about physics, but yes: There are plenty of logical systems that allow for the possibility of true contradictions. For the most part, these are motivated by various sorts of paradoxes, such as the liar paradox (which has to do with truth) or the Sorities paradox (vagueness) or Russell's paradox (set theory). But there can be, and have been, deeper motivations, connected with questions about the limitations of human thought, and even Buddhist notions about the nature of ultimate reality.

If you're interested in that sort of issue, have a look at Graham Priest's book In Contradiction or his more recent book Towards Non-Being, which is on a slightly different but related topic.

I'll add that my own view is that contradictions cannot be true and that, even if they could, that would not help us solve the sorts of paradoxes I mentioned. But that doesn't mean such views aren't worth taking seriously. I could be wrong!!!

There are real-life situations in which contradictions can appear.

Consider a deductive AI knowledge base that can use classical logic to infer new information from stored information (think of IBM's Watson, for example).

Suppose that a user tells the system that, say, the Yankees won the World Series in 1998. (It doesn't matter whether this is true or false.) Suppose that another user tells it that the Yankees lost that year.

Now the system "believes" a contradiction. So, by classical logic, it could infer anything whatsoever. This is not a good situation for it to be in.

One way out is to replace its classical-logic inference engine with a relevance-logic inference engine that can handle contradictions.

For example, the SNePS Belief Revision system will detect contradictions and ask the user to remove one of the contradictory propositions. (It can also try to do this itself, if it has supplementary information about the relative trustworthiness of the sources of the contradictory propositions.)

For more information, see:

Martins, João P. & Shapiro, Stuart C. (1988), "A Model for Belief Revision", Artificial Intelligence 35(1): 25-79.

Ari I. Fogel & Stuart C. Shapiro, "On the Use of Epistemic Ordering Functions as Decision Criteria for Automated and Assisted Belief Revision in SNePS: (Preliminary Report)". In Sebastian Sardina & Stavros Vassos, Eds., Proc. 9th Int'l Workshop on Non-Monotonic Reasoning, Action, and Change (NRAC'11), Technical Report RMIT-TR-11-02, School of Computer Science and Information Technology, RMIT University, Melbourne, Australia, July, 2011, 31-38.

Essential SNePS Readings (online)

Just for clarity, and not that Prof Rapaport needs me to tell him this, but it is important to distinguish the question whether contradictions can be true from the question whether one can get oneself into a situation in which one was believed. I rather suspect that we most or even all of us have contradictory beliefs of one sort or another, and that might motivate the view that classical logic is not a good theory of how we ought always to reason. But as Gilbert Harman famously pointed out, it isn't obvious that logic should be in the business of formulating norms of reasoning. Maybe what it does is simply study the notion of truth-preservation. So classical logic might be a good theory of validity, but not a good theory of how to reason, and maybe paraconsistent or relevance logics (or probabilistic analogues thereof) are better theories of the latter.

For what it's worth, my own view is that Harman's point, though fundamentally correct, needs very careful handling and that, even in the presence of contradictory information, classical logic has more resources than is typically supposed. But that's a much larger issue.

Because the present questioner refers to my reply to Question 5536, I'll chime in here to clarify what I said there.

My point was about the fundamentality of LNC. I wrote, "I can't see how there could be any law more fundamental than the law of non-contradiction (LNC)." I gave the following reason: "Let F be any such law. If the claim 'F is more fundamental than LNC' is meaningful (and it may not be), then it conflicts with the claim 'F isn't more fundamental than LNC' -- but that reasoning, of course, depends on LNC." So that's why no law could be more fundamental than LNC, because LNC would need to be true before (in the sense of logical priority) the claim that some other law is more fundamental would even make sense.

If someone can make sense of the claim that some law is more fundamental than LNC, I'm all ears.

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