These issues show the limitation of the traditional "mathematical logic" when applied outside mathematics.

1. Uncertainties

meaning of term:

• The meaning of an atomic term in traditional logic is determined according to an interpretation, therefore it does not change as the system is running. On the contrary, the meaning of a term in human mind often changes according to experience and context.
  Example: What is "game"?
• When a compound term is formed in traditional logic, its meaning is completely determined by its definition, which reduces its meaning into the meaning of its components and the operator/connector that joins the components. On the contrary, the meaning of a term in human mind often cannot be fully reduced to that of its components, though is still related to them.
  Example: Is a "blackboard" exactly a black board?

• In traditional logic, a statement is either true or false, but people often take truth value as a matter of degree.
  Example: Is "A bird can fly" true or false?
• In traditional logic, the truth value of a statement does not change over time. However, people often revise their beliefs after getting new information.
  Example: After learning that Tweety is a penguin, you may change some of your beliefs formed when you only know that it is a bird.
• In traditional logic, all useful inference must start with a consistent premise set, because a contradiction can lead to the "proof" of any arbitrary conclusion. On the contrary, the existence of a contradiction in human mind will not make the person to believe an arbitrary statement.
  Example: Have you ever had a (explicit or implicit) contradiction in your mind? Do you believe 1 + 1 = 3 at that time?

• In traditional reasoning systems, inference processes follow (deterministic) algorithms, therefore are predictable, that is, after each step, what will happen next is predetermined. On the other hand, human reasoning processes are often unpredictable, in the sense that sometimes a inference process "jumps" in an unanticipated direction.
  Example: Have you ever waited for "inspiration" for your writing assignment?
• In traditional reasoning systems, how a conclusion is derived is accurately explainable. On the contrary, human mind often generate conclusions whose source cannot be backtracked.
  Example: Have you ever said "I don't know why I believe that. It's just my intuition."?
• In traditional reasoning systems, every inference process has a pre-specified goal, and the process terminatable whenever its goal is achieved. However, though human reasoning processes are also guided by various goals, they often cannot be completely achieved.
  Example: Have you ever tried to find the goal of your life? When can you stop thinking about it?

2. Non-deductive inference

The problem is, in human reasoning, there are other inference patterns (or rules), where the conclusions contain information not available in the premises.

Induction produces generalizations from special cases.
Example: from "Robins are birds" and "Robins have feather" to derive "Birds have feather".

Abduction produces explanations for given cases.
Example: from "Birds have feather" and "Robins have feather" to derive "Robins are birds".

Analogy produces similarity-based judgments.
Example: from "Swallows are similar to robins" and "Robins have feather" to derive "Swallows have feather".

The above non-deductive rules do not guarantee the truth of the conclusion even when the truth of the premises can be supposed. Therefore, they are not valid rules in traditional logic. On the other hand, it is easy to see that these kinds of inference often happen in everyday thinking, and, especially, they play important roles in learning and creative thinking.  If they are not valid according to traditional theories, then in what sense they are better than arbitrary guesses?

3. Various paradoxes

Sorites paradox: No one grain of wheat can be identified as making the difference between being a heap and not being a heap. Given then that one grain of wheat does not make a heap, it would seem to follow that two do not, thus three do not, and so on. In the end it would appear that no amount of wheat can make a heap.

Implication paradox: Traditional logic uses "P → Q" to represent "If P, then Q". By definition, the implication proposition is true if P is false or if Q is true, but "If 1+1 = 3, then the Moon is made of cheese" and "If life exists on Mars, then robins have feather" don't sound right.

Confirmation paradox: In traditional logic, "Ravens are black" and "Non-black things are not Ravens" are equivalent, that is, they have the same truth value. If we want to extend truth value beyond merely true and false, and allow the system to learn the truth value of a statement gradually according to available evidence, then it is natural to take "black ravens" as positive evidence for "Ravens are black". For similar reasons, "non-black non-ravens" should be used as "Non-black things are not Ravens". But since the two statements are equivalent, "non-black non-ravens" (such as white sacks and red flowers) are also positive evidence for "Ravens are black".

Wason's selection task: A psychological experiment shows that human judgment is systematically different from the conclusion produced by traditional logic. Suppose that I show you four cards, showing A, B, 4, and 7, respectively. Suppose in addition that I give you the following rule to test: "If a card has a vowel on one side, then it has an even number on the other side." Which card or cards should you turn over in order to decide the truth value of the rule? On this task, few people do what first-order predicate logic tells us to do.