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.