Predicate logic (述語論理) is a logic that expresses the properties of objects and the relations between them with symbols of the form "predicate(object)."
Plain sentence
Written in predicate logic
What it expresses
Kitty is a cat
Cat(Kitty)
A property of an object
Kitty is bigger than Tom
Bigger(Kitty, Tom)
A relation between objects
Every cat is an animal
∀x (Cat(x) → Animal(x))
For every x, if x is a cat then x is an animal (∀ = for all, → = if ... then)
There is a white cat
∃x (Cat(x) ∧ White(x))
At least one x is both a cat and white (∃ = there exists, ∧ = and)
For example, "Kitty is a cat" is written Cat(Kitty), where "Cat" is the predicate and "Kitty" is the object. Adding quantifiers (∀ and ∃) for "all" and "some" lets you write general rules that hold for many objects, which makes it the foundation of symbolic AI, where people write knowledge as symbols and rules and let the machine reason with it.
AnalogyIf an address book has fixed fields such as "name" and "relationship," a computer can pick out who is whose parent. Predicate logic is a way of putting knowledge into that kind of fixed format.
How to rememberPicture a function call: the predicate goes first, like a function name, and the things it talks about go inside the parentheses.
Exam points
Mix-upPropositional logic (命題論理) treats a whole sentence as one unit that is true or false, while predicate logic (述語論理) goes further and separates the objects in the sentence from their properties and relations.
Deductive reasoning (演繹推論) applies a general rule to an individual fact and draws a conclusion about that individual.
General rule: every mammal breathes with lungs
Individual fact: a whale is a mammal
Apply the general rule to the whale
Conclusion: a whale breathes with lungs
If both premises are true, the conclusion drawn is true as well.
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Its typical form is the syllogism, which reaches a conclusion from a general rule (the major premise) and an individual fact (the minor premise). If the premises are true and each step follows the rules of logic, the conclusion is guaranteed to be true. In symbolic AI, deductive reasoning applied to knowledge written in predicate logic derives facts that were never written down.
AnalogyIf you know a building is entirely non-smoking, you can tell that smoking is not allowed in any room you walk into without checking each one.
How to rememberDeduction comes from Latin for "lead down": you lead down from the general rule to the single case.
Exam points
ExamDeductive reasoning (演繹推論) takes a general rule (一般則) that covers the whole group and an individual fact as premises, and draws a conclusion that holds for that individual.
Mix-upA valid chain of reasoning and a true conclusion are separate things. If a premise is wrong, the conclusion is not necessarily true even when every step is valid.
Inductive reasoning (帰納推論) draws a general rule from what several individual cases have in common.
Observation: swan no. 1 is white
Observation: swan no. 2 is white too
Observation: hundreds of swans, all white
General rule: swans are white
Finding a black swan overturns the rule
If a counterexample turns up among cases not yet observed, the general rule no longer holds.
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The more cases you observe, the more likely the conclusion becomes, but it never guarantees the cases you have not yet seen. Machine learning, which finds patterns in data, is built on this same inductive idea.
AnalogyIt is like noticing that a shop you have visited a few times is always closed on Wednesdays and deciding Wednesday is its day off. If it is open one Wednesday, you have to rethink.
How to rememberInduction comes from Latin for "lead into": the individual cases lead you into one general rule.
Exam points
ExamThe conclusion of inductive reasoning (帰納推論) generalizes what the observed cases share, so a single case outside those observations that does not fit is enough to overturn it.
A rule-based (ルールベース) system makes decisions by applying facts to "if ... then ..." rules that people have written in advance.
Input facts: room 30 °C, humidity 75%
Match against rules: IF room above 28 °C AND humidity 70% or more THEN turn on cooling
Apply the rule whose conditions match
Output: turn on cooling (the rule used can be shown as grounds)
Output the conclusion part of the rule and show which rule was used.
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Each rule is a pair of IF (a condition) and THEN (a conclusion or action), and a system that turns an expert's knowledge into a set of rules so it can make judgments in the expert's place is called an expert system. It can easily show which rule served as the grounds for a decision, but writing and maintaining the rules takes effort, and it cannot handle situations its rules do not cover.
AnalogyIt is like using the troubleshooting table in an appliance manual, such as "if the lamp is blinking, clean the filter," to pick a fix from the symptom.
Exam points
Mix-upIn a rule-based (ルールベース) approach, people write the decision criteria and hand them to the machine, while machine learning (機械学習) acquires those criteria from data automatically.
Key points
Predicate logic writes objects separately from their properties and relations, and its quantifiers can express general rules, which puts knowledge into a form that can be reasoned with.
Deductive reasoning applies a general rule to an individual fact, and as long as the premises are true, the conclusion is true.
Inductive reasoning builds a general rule out of what several cases have in common.
A rule-based approach makes decisions by following IF-THEN rules written by people, and it is how expert systems work.