I'm reading something on the topic of logic and one of the exercises asked me to convince myself that a contradictory statement was true. I could not convince myself of this and now I am curious about whether this illustrates some innate characteristic of noncontradiction in the human mind.
You need to distinguish at least three cases here.
Case #1: To believe consciously, at the same time , and with full understanding A and not A. In other words, this is the case where simultaneously believe that the world is flat and the world is spherical at the same time. This seems difficult if not impossible and might represent a feature of the human mind that we call "rationality." This is what Kant believed.
Case #2: To believe A and B which happen to be contradictory. This is called "cognitive dissonance" and can happen all the time. This is a classic problem in the philosophy of language. Specifically, it's actually a joke about Sartre where they ask if you can simultaneously hold to the claims:
Where it turns out that they are the same. The question that arises and this relates to Case #1 is whether you can hold these beliefs even in light of knowing that they are the same person. There are several different ways of accounting for this but I think the easiest one is to qualify what you hate and love and realize the content of the novels and philosophy differ in a meaningful regard.
A second version of case 2 is Hesperus and Phosphorus. These are two names for Venus which refer to the evening star and morning star respectively (the Greeks believed they were different). The problem in the philosophy of language is again whether reference matters. Is everything true of one true of the other? Frege writes on it extensively, Kripke has his own idea of a solution too. The basic problem has to do with what is going in naming.
Case #3: To believe approximation A which we know to be false. This actually seems perfectly rational and is common behavior in science. I for instance would use classical orbital theory when describing most types of chemical bonds (at least ones that involve the common non-metals), but I know it is not in fact true.
So really it's going to be hard to believe Case #1 type contradictions and that might say something about our minds. But we can also be quite resilient to drawing logical inferences when in Case #2 circumstances. Witness the large number of people who stay in abusive relationships.
"Why can't humans believe contradictions?" is a question of psychology rather than a question of logic. The core reason is cognitive dissonance.
Contrary to the answer by @virmaior, cognitive dissonance is not "To believe A and B which happen to be contradictory." Cognitive dissonance describes "the feeling of discomfort that results from holding two conflicting beliefs." (Although "conflicting" is a broader category than "contradictory", any beliefs that are contradictory are also in conflict with each other.) The essence of cognitive dissonance is not the conflicting beliefs but the psychological discomfort.
A lot of the time we find it difficult to put up with the discomfort and so prefer to change our beliefs. But most of us live through times when we might be aware of the discomfort but not aware of the contradictory beliefs that cause it. At times we are conscious of believing something but the way we act would indicate that we believe it's negation. At other times we might try to deny the discomfort (which creates another contradictory belief because we actually know at some level that we are uncomfortable) or mask it so that the two contradictory beliefs never confront each other in our consciousness.
But putting it simply, humans can believe contradictions but we tend not to because doing so makes us uncomfortable.
I agree with the other users who believe that the context suggested by the term "contradictions" in this question is too broad. From a standpoint that is more narrow (and more formal), it would be impossible to believe that a proposition that one believes correctly describes a state of affairs in reality (i.e., a proposition that one believes to be true) is, at the exact same time, also not true according to precisely the same criterion for determining truth. It is only possible to believe one or the other of the two alternatives, but (in consideration of the Principle of Non-Contradiction) not both at the same time.
There's an ongoing discussing here between comments and answers about the definition of contradiction as asked by the OP. Since the question asked about the realm of logic, I'm going to provide an answer assuming that contradiction means the fact of statement A and not A to be held true at the same time.
OP asked why is it that humans cannot believe in contradictions, in the most literal sense of A and not A at the same time.
My take on it is that minds hold belief systems, complex networks of differently related facts with different certainty levels and relationships. This complexity of the networks is that gives rise to cognitive dissonance: facts that may have relationships to other facts which are contradictory indirectly.
The purpose of this network of beliefs is to be able to understand the world, predict it and interact with it.
Facts are as simple as logic proposes, little bits of knowledge of "X is true" and "Y is false". "X is true AND X is false" provides no information whatsoever, because nothing can be inferred from it, and endangers what related facts mean to it. A implies B, not A implies C, A and not A endangers the causality of A to B and not A to C.
This is why contradictions have this strange feeling into our mind: it endangers most of what we know. If statements are generic enough so that anything could be derived from them, contradictions on these statements are generic enough so that we may question everything we know. "This statement is false." is a great example.
(Notice I include paradoxes in contradictions, because the heart of a paradox is displaying one truth that contradicts itself.)
Formally Virmaior is entirely correct. Formally, a contradiction is the joint believe that the statement A is B is both true and false at the same time and in the same respect. Formally A is B means that B is predicated of A, is said of A in some sense of being (with respect to the Aristotelian categories of being; see here for a simple analysis of his categories.
From the point of view of Aristotelian logic, the essential fact is that of the relationship between metaphysics, which addresses the question: 'What is true of reality?' and epistemology, which addresses the question 'How do I know what is true of reality?'. The latter depends on the former, and not the other way round. The nature of reality must (if we wish to avoid error) determine our way of thinking about reality. The law of non-contradiction has therefore two aspects: 1. That reality (or any part of it) is what it is, has an identity determined by its nature. 2. Then to properly grasp or identify the facts of existence, one must grasp the law of identity as primary: A is A. A thing is what it is. Therefore ones epistemology, to avoid conceptual contradictions, must be based on the law of identity.
But one's concept formation is open to volitional choice, so there is no physical law forcing our minds to be logical or non-contradictory. However, when we discover that we are holding a contradiction, we have a choice: 'What shall I do about this contradictory set of beliefs?' Again, we are free to choose if, when, and how to resolve such contradictions or not to seek such resolutions. Why choose to resolve contradictions? That depends on how much we value being true to the facts of reality as against how much we value our existing beliefs. Again we must choose. Most important is the fact that reality contains no contradictions, so if we want to achieve true knowledge and the values that depend on such knowledge, we must care about resolving known contradictions insofar as we know them to have direct bearing on our consciously held values. So, in the end, yes. We may choose to hold or ignore contradictory beliefs.
I love you and I hate you. That's a contradiction that's often true. I want this piece of candy and I want to be skinny. That's another. Life is full of contradictions. The very definition of sanity is that you can get up every day and function despite all of life's contradictions.
If you were a complete sociopath you would be uninterested in and unmoved by the collective weight of human suffering.
If you were totally empathic with every living thing, and personally experienced the suffering of every creature in the universe, you'd die of grief the moment you woke up.
A sane person cares about human suffering; but not so much that it affects their own ability to care for themselves. A sane person must be part sociopath and part saint. That's another of life's contradictions.
It's only in abstract realms of thought that contradictions are false. In everyday life, they're often true. If you're told person X is shy, then if you know anything about human nature you know there's an exhibitionist hiding inside. And conversely if you see an exhibitionist, you know that inside they consider themselves shy.
Literally everything in life is the opposite of the way it appears. Everything is a contradiction.
This is why computers can not be conscious. Computers regard contradictions as false. How are you going to program a computer to understand that I'm tired and I don't want to go to bed? That I want to go to heaven and I don't want to die?
Logic is terribly limited in that it can only assign the value of FALSE to a contradiction. Life generally assigned contradictions a value of TRUE. Let's see anyone program that!
As I understand it, the OP question can be answered by the case represented by... I believe the earth is flat, and I believe the earth is not flat. These two statements form a contradiction (by definition). I am a human (by definition). Do I, or don't I, believe in this contradiction? The answers are: 1)I do believe that the two statements form a contradiction. 2)The contradiction, itself, can not be true or false. 3)Each contradictory statement can be true or false. 4) I can freely choose to believe which contradictory statement is true for me, but I can not believe that both (mutually exclusive) statements are true "at the same time" (by definition). This shows that the premise of the main question is false, humans can believe in contradictions.
In regard to convincing myself that a contradictory statement can be true, I can use the same example. For most people living in the 1200s, the statement "the earth is flat" was believed to be true. For most people living in 2014, the statement, "the earth is not flat," is believed to be true. Thus, I convinced myself that a contradictory statement can be true.
Is it really so hard to believe two contradictory things at the same time? It may be irrational to do so - so let's examine some of the irrational moments of life. For example, have you ever experienced:
There is nothing behind me in the dark watching me and wishing me harm.
Fears and phobias are irrational aspects of our personalities that can affect us even when we rationally know that there is nothing to be afraid of. Have you NEVER quickened your pace despite knowing full well that there's nothing that could possibly be there to cause you harm? Never laughed at yourself for being silly while still moving towards your destination just a tad quicker than one would call 'leisurely'