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• Logic is a language allows to make association of the world in a precise way.
• it form the basis of planning

Why do we need formal logic?

Knowledge base: sentence in the language of logic; complete and correct Rules of Inference engine:

Inferences About Birds

Exercise: Inferences About Foos

The sentences are language of logic

Predicates

":" means indicates. If Feathers(animal): Then Bird(animal) reads if animal has feathers, then animal is bird.

Conjunctions and Disjunctions

Implies

Notation Equivalency

Exercise: Practicing Formal Logic

Truth Tables

Exercise: Truth Tables I

Exercise: Truth Tables II (more complicated sentence)

In theory, we can calculate True or False values for very complicated sentences.

Exercise: Commutative Property

A & B is the same as B & A

Exercise: Distributive Property

• A &(B or C) is equivalent to (A and B) or (A and C) Exercise: Associative Property

Exercise: Associative Property

• The associate property applies when there is only conjunction or disjunctions.

Exercise: De Morgan's Law

Truth of Implications

This is hard to understand: need to go back to the examples But remember A=> B is actually !A | B.

Implication Elimination

Rules of Inference

• The rules here can be used for reasoning.

Prove Harry is a bird

Although the rules exist for a long times, it's hard to use these rules in AI agent because the computation is not feasible.

Universal Quantifiers and Existential Quantifier

• Propositional logic (zero-order logic): no variables.
• Existential and Universal Quantifier can be used with variables.
• Universal Quantifier mean given all values for the variable
• Existential Quantifier means at least one value

A Simple Proof

1. convert every sentence into a conjunctive normal form.

   not finished for this part. did not understand
   

exercise

• A efficient way to proof sentences

Wrap Up

The Cognitive Connection

• Formal and precise way to reason
• Human uses logic sometimes.. but its not a fundamental method.
• Human uses induction and abduction more for reasoning. logic is deductive.