# Equivalent Translations

As we saw in section 2.4), two logical statements are said to be *logically equivalent* if and only if they have the same truth value for every truth assignment.

We can extend this idea to our propositional logic translations – two (English) statements are said to be *equivalent* iff they have the same underlying meaning, and iff their translations to propositional logic are logically equivalent.

## Common equivalences, revisited

We previously identified the following common logical equivalences:

- Double negative:
`¬ ¬ p`

and`p`

- Contrapositive:
`p → q`

and`¬ q → ¬ p`

- Expressing an implies using an OR:
`p → q`

and`¬ p ∨ q`

- One of DeMorgan’s laws:
`¬ (p ∧ q)`

and`( ¬ p ∨ ¬ q)`

- Another of DeMorgan’s laws:
`¬ (p ∨ q)`

and`( ¬ p ∧ ¬ q)`

## Equivalence example 1

Suppose we have the following propositional atoms:

```
p: I get cold
q: It is summer
```

Consider the following three statements:

*I get cold except possibly if it is summer.**If it’s not summer, then I get cold.**I get cold or it is summer.*

We translate each sentence to propositional logic:

*I get cold except possibly if it is summer.*`p → ¬q`

- Meaning: I promise that if I get cold, then it must not be summer…because I am always cold when it’s not summer.

*If it’s not summer, then I get cold.*`¬q → p`

- Meaning: I promise that anytime it isn’t summer, then I will get cold.

*I get cold or it is summer.*`p V q`

- Meaning: I’m either cold or it’s summer…because my being cold is true every time it isn’t summer.

As we can see, each of these statements is expressing the same idea.

## Equivalence example 2

Suppose we have the following propositional atoms:

```
p: I eat chips
q: I eat fries
```

Consider the following two statements:

*I don’t eat both chips and fries.**I don’t eat chips and/or I don’t eat fries.*

We translate each sentence to propositional logic:

*I don’t eat both chips and fries.*`¬(p ∧ q)`

*I don’t eat chips and/or I don’t eat fries.*`¬p V ¬q`

These statements are clearly expressing the same idea – if it’s not the case that I eat both, then it’s also true that there is at least one of the foods that I don’t eat. This is an application of one of DeMorgan’s laws: that `¬ (p ∧ q)`

is equivalent to `( ¬ p ∨ ¬ q)`

.

If we were to create truth tables for both `¬(p ∧ q)`

and `¬p V ¬q`

, we would see that they are *logically equivalent* (that the same truth assignments make each statement true).

## Equivalence example 3

Using the same propositional atoms as example 2, we consider two more statements:

*I don’t eat chips or fries.**I don’t eat chips and I don’t eat fries.*

We translate each sentence to propositional logic:

*I don’t eat chips or fries.*`¬(p V q)`

*I don’t eat chips and I don’t eat fries.*`¬p ∧ ¬q`

These propositions are clearly expressing the same idea – I have two foods (chips and fries), and I don’t eat either one. This demonstrates another of DeMorgan’s laws: that `¬ (p ∨ q)`

is equivalent to `( ¬ p ∧ ¬ q)`

. If we were to create truth tables for each proposition, we would see that they are logically equivalent as well.