Contradiction is a kind of logical judgment of error in things said. It applies when things said are either syntactically or semantically incompatible with one another. To be incompatible is to be incapable of “properly” coexisting in a single context or unity of apperception. Aristotle strongly emphasized this normative aspect of the principle of noncontradiction.
In the syntactic case, the concern is with purely formal rules for the well-formedness of expressions. A syntactic contradiction would be something like “A, and also not-A”, where either A and not-A have both been explicitly said, or both are implied by things that have been said. In this case, we need know nothing at all about the meaning of “A”. We are only concerned with generic rules for the application of logical operators like “and” and “not”.
In the semantic case, contradiction involves the specific meanings of concrete expressions, applied together to some one meant reality. Unlike the syntactic case, background knowledge is essential to judging whether or not meanings can compatibly coexist. We may also think we know the whole story when we don’t. New facts or understandings may change our generalizations and schemas of classification. (See also Interpretation; Error.)
Nothing follows from the principle of noncontradiction alone. Given some inputs, we can judge whether or not they are contradictory — by rigorous analysis in the syntactic case, and up to some level of practical confidence in the semantic case.
Hegel sometimes used the word “contradiction” in an idiosyncratic, highly metonymical or metaphorical way, straining language to the breaking point as part of a larger effort to draw out the complexities and subtleties involved in applying logic to concrete meanings and the real world, when no vocabulary existed for many of the subtleties involved. (See also Three Logical Moments.)
Some people, mainly Marxists, have talked about real-world conflict and social injustice as “contradictions” objectively existing in the world. Conflict and injustice are very real, but it is a misunderstanding of Hegelian dialectic and an inappropriate mixing of levels to associate them directly with contradiction. (See also Contradiction vs Polarity.)
Especially since the mid-20th century, many authors have pointed out common errors and issues associated with too-easy assumptions about identity. (See also Aristotelian Identity.) The Žižekian school has developed a sophisticated variant of the old talk about objective contradictions, by explaining it largely in terms of the issues with identity. If this were just a new metonymical or metaphorical usage in the style of Hegel, we could simply note that “contradiction” is being said in a nonstandard way, and move on. But unfortunately, the Žižekians have gone further, and also claimed that the logical principle of noncontradiction ultimately fails to hold, even though this logical (or illogical) claim is not necessary to address the social concerns that according to them need to be addressed, or to explain the things that according to them need to be explained. (See Split Subject, Contradiction.) We have to be very careful in moving back and forth between very different levels of analysis like this.
Just as on an interpersonal level we can reduce conflict by omitting those too-easy assumptions about identity, omitting those assumptions with respect to things said — and thus making more distinctions — also greatly reduces the potential for logical contradiction.
It is a category mistake to talk about contradiction driving events. Actual change does not result in contradiction either. Different things are true at different times, and the explanation for that is not “contradiction” but change.
Why is this important? The simple answer is that denial of the principle of noncontradiction allows someone to argue absolutely anything, including nonsensical and false things, and to sophistically respond to any refutation by simply introducing more inconsistency. This rejection of responsibility effectively ends the possibility of dialogue.
There ought to be no conflict between social criticism and the possibility of dialogue. Social criticism should be based on shareable, rational analysis. It may be unreasonable to suppose that all social issues can be resolved through dialogue (see Stubborn Refusal), but I do think all those concerned with doing something about those issues ought to be able to resolve their differences through dialogue.
I think Brandom has made an epic contribution in this area by finding a new way to simultaneously affirm — as Aristotle implicitly anticipated long ago — both the world’s recalcitrance to mastery and identity and its fundamentally rational, intelligible character. (See also Self-Evidence?)