**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:

*may or may not give the same result as*

**x op y**

*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, á„€)

**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

## No comments:

## Post a Comment