3203. Introduction to Artificial Intelligence
Issues in Reasoning
Though the AI research in reasoning has produced a lot of results,
there are still many remaining issues, and they show the limitation of the
traditional "mathematical logic" when applied outside mathematics.
1. UncertaintiesClassical logic (such as first-order predicate
calculus) are certain in several aspects, whereas the actual human
reasoning is uncertain.
Meaning of term:
Truth of statement:
- In classic logic, the meaning of an atomic (or primary) term is
determined by its denotation according to an interpretation,
therefore it does not change as the system is running. On the contrary, the
meaning of a term in the human mind often does not have a clear denotation, and its
meaning changes according to experience and
Example: What is "game"?
- In classic logic, when a compound (or defined) term is introduced, its
meaning is completely determined by its definition, which reduces its
meaning into the meaning of its components and the operator/structure that
joins the components. On the contrary, the meaning of a compound term in the 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
Process of inference:
- In classic 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 classic logic, the truth value of a statement does not change over
time. However, people often revise their beliefs after getting new
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 classic 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 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.
you ever waited for "inspiration" for your writing assignment?
- In traditional reasoning systems, how a conclusion is derived is
accurately explainable and repeatable. On the contrary, a human
mind often generates conclusions whose source cannot be backtracked.
Example: Have you ever said "I don't know why I believe that. It's just my
- In traditional reasoning systems, every inference process has a
pre-specified goal, and the process is 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
2. Non-deductive inferenceAll the inference rules of traditional logic
are deduction rules, which are truth-preserving, that is,
the truth of the premises guarantee the truth of the conclusion. In a sense, in
deduction the information in a conclusion is already in the premises, and the
inference rules just reveal what is previously implicit.
For example, from
"Robins are birds" and "Birds have feather", it is valid to derive "Robins have
The problem is, in human reasoning, there are other inference patterns (or
rules), that are not truth-preserving in the traditional sense.
The above non-deductive rules do not
guarantee the truth of the conclusion even when the truth of the premises can be
assumed. Therefore, they are not valid rules in traditional logics. 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 standards, then in
what sense they are better than arbitrary guesses?
- Induction produces generalizations from special cases.
Example: from "Robins are birds" and "Robins have feather" to derive "Birds
- Abduction produces explanations for given cases.
Example: from "Birds have feather" and "Robins have feather" to derive "Robins
- Analogy produces similarity-based judgments.
from "Swallows are similar to robins" and "Robins have feather" to derive
"Swallows have feather".
3. Various paradoxesTraditional logic, when used outside mathematics,
generate conclusions that are different from what people usually do, so the
"logically correct" conclusions are sometimes "intuitively wrong". Such a case
is often called a "paradox".
Attempts to resolve
the above issues will be introduced in the following lectures.
- 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 classical 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: Suppose there are four cards showing A, B,
4, and 7, respectively, and your task is to decide the truth of the following
rule: "If a card has a vowel on one side, then it has an even number on the
other side." Which cards should you turn over? On this task, few people do
what first-order predicate logic tells us to do.