Logic of AI

What is logic ?

- We can also think of logic as an "algebra" for manipulating only two value : true(T) and false (F)
We will cover :
- Propositional logic -- the simplest kind

Propositional Logic

Propositional logic consists of:
- The logical values true and false (T and F)
- Propositions; "sentence", which

• Are atomic (that is, they must be treated as indivisible units, with no internal structure), and 
• Have a single logical value, either true and false 

 - Operators, both unary and binary; when applied to logic values, yield logical values

•  The usual operators are and, or not, and implies

Propositional logic: Syntax

- Propositional logic is the simplest logic -illustrates basic ideas
- The propositional symbols P1,P2 etc are sentences

• If S is sentence, -S is a sentence (negation ,not)
• If S1 and S2 are sentence S1^S2 is a sentence (conjunction, AND) 
• If S1 and S2 are sentence S1vS2 is a sentence (disjunction, OR)
• If S1 and S2 are sentence S1⇒S2 is a sentence (implication, IMPLIES)
• If S1 and S2 are sentence, S1⇔S2 is a sentence (biconditional)

Truth Table

Logic, like arithmetic, has operators, which apply to one, two, or more values (operands) 
A truth table lists the results for each possible arrangement of operands
  - Order is important: x op y may or may not give the same result as y op x
The rows in a truth table list all possible sequence of truth value for n operands, and specify a result for each sequence 
  - Hence, there are 2n rows in a truth table for n operands.

Unary Operators 

There are four possible unary operators:

Only the last of these (negation) is widely used  (and has a symbol- for the operation)

Useful binary operators

Here are the binary operators that are traditionally used:

Notice in particular that material implication (®️) only approximately means the same as the English word "implies"
Any other binary operators can be constructed from a combination of these (along with unary not, ᄀ)

Logical expressions

All logical expressions can be computed with some combination of and (∋), or ( ( ), and not (⇽) operators

For example, logical implication can be computed this way:

 Notice that ←X (Y is equivalent to X ®️Y