# IMPLIES

Logical implication operator. "A IMPLIES B" is TRUE unless A is TRUE and B is FALSE.

## Syntax

```l4
expression1 IMPLIES expression2
expression1 => expression2
```

## Truth Table

| A     | B     | A IMPLIES B |
| ----- | ----- | ----------- |
| TRUE  | TRUE  | TRUE        |
| TRUE  | FALSE | FALSE       |
| FALSE | TRUE  | TRUE        |
| FALSE | FALSE | TRUE        |

## Semantics

`A IMPLIES B` is logically equivalent to `NOT A OR B`.

In legal terms: "If A, then B" - the rule is violated only when A is true but B is false.

## Examples

**Example file:** 

```l4-file
-- IMPLIES keyword examples

-- Basic IMPLIES Operations

#EVAL TRUE IMPLIES TRUE
#EVAL TRUE IMPLIES FALSE
#EVAL FALSE IMPLIES TRUE
#EVAL FALSE IMPLIES FALSE

-- Symbolic Alternative (=>)

#EVAL TRUE => TRUE
#EVAL TRUE => FALSE
#EVAL FALSE => TRUE

-- Equivalence to NOT A OR B

GIVEN a IS A BOOLEAN
      b IS A BOOLEAN

-- These are logically equivalent
impliesVersion a b MEANS a IMPLIES b
orVersion a b MEANS NOT a OR b

-- Test equivalence
#EVAL impliesVersion TRUE TRUE
#EVAL orVersion TRUE TRUE

#EVAL impliesVersion TRUE FALSE
#EVAL orVersion TRUE FALSE

#EVAL impliesVersion FALSE TRUE
#EVAL orVersion FALSE TRUE

-- Legal Rule Examples

-- "If you are an employee, you must have a badge"
ASSUME isEmployee IS A BOOLEAN
ASSUME hasBadge IS A BOOLEAN

DECIDE badgeRule IS isEmployee IMPLIES hasBadge

#CHECK badgeRule

-- "A minor must have parental consent"
ASSUME personAge IS A NUMBER
ASSUME hasParentalConsent IS A BOOLEAN

DECIDE consentRule IS (personAge < 18) IMPLIES hasParentalConsent

#CHECK consentRule

-- Chained Implications

-- Implication is right-associative: A => B => C means A => (B => C)
GIVEN x IS A BOOLEAN
      y IS A BOOLEAN
      z IS A BOOLEAN
chainedImpl x y z MEANS x => y => z

#EVAL chainedImpl TRUE TRUE TRUE
#EVAL chainedImpl TRUE TRUE FALSE
#EVAL chainedImpl FALSE FALSE FALSE

-- Combined with AND/OR

GIVEN p IS A BOOLEAN
      q IS A BOOLEAN
      r IS A BOOLEAN

-- "(P AND Q) implies R"
conjunctionImplies p q r MEANS (p AND q) IMPLIES r

-- "P implies (Q OR R)"
impliesDisjunction p q r MEANS p IMPLIES (q OR r)

#EVAL conjunctionImplies TRUE TRUE TRUE
#EVAL conjunctionImplies TRUE TRUE FALSE
#EVAL impliesDisjunction TRUE FALSE TRUE
#EVAL impliesDisjunction TRUE FALSE FALSE
```



### Basic Usage

```l4
#EVAL TRUE IMPLIES TRUE    -- TRUE
#EVAL TRUE IMPLIES FALSE   -- FALSE
#EVAL FALSE IMPLIES TRUE   -- TRUE
#EVAL FALSE IMPLIES FALSE  -- TRUE
```

### Symbolic Alternative

```l4
#EVAL TRUE => TRUE    -- TRUE
#EVAL TRUE => FALSE   -- FALSE
```

### Legal Rule Example

```l4
GIVEN isEmployee IS A BOOLEAN
GIVEN hasBadge IS A BOOLEAN
-- "If you are an employee, you must have a badge"
DECIDE badgeRule IS isEmployee IMPLIES hasBadge
```

### Chaining Implications

```l4
-- A => B => C means A => (B => C)
DECIDE chainedImpl IS TRUE => (TRUE => FALSE)
```

## Use in Legal Logic

Implication is common in legal rules:

```l4
-- "A minor must have parental consent"
GIVEN age IS A NUMBER
GIVEN hasParentalConsent IS A BOOLEAN
DECIDE consentRule IS (age < 18) IMPLIES hasParentalConsent
```

## Related Keywords

- **[AND](/l4/reference/operators/AND.md)** - Logical conjunction
- **[OR](/l4/reference/operators/OR.md)** - Logical disjunction
- **[NOT](/l4/reference/operators/NOT.md)** - Logical negation

## See Also

- **[Logical Operators](/l4/reference/operators.md#logical-operators)** - All logical operators