The Basic Logical Equivalence Laws (Statement Logic)

June 5, 2023

If two statements are logically equivalent, it means they have the same truth value in all possible scenarios. In other words, the two statements are equal and are basically saying the same thing.

Logical equivalence laws tell us which statements are logically equivalent to each other. They enable us to establish the equivalence between two statements, which means that if one statement is true, the other statement will also be true, and if one statement is false, the other statement will also be false.

By applying these laws, we can manipulate and transform statements to reveal their logical equivalence. This can be particularly useful when simplifying complex statements or when trying to determine whether two statements are logically equivalent.

In this post, we will look at the 10 basic logical equivalence laws (also referred to as laws of replacement):

1. Redundancy

P = (P and P)

P = (P or P)

The redundancy rule focuses on the presence of redundant components in a compound statement. It states that adding or removing a statement that has no impact on the truth value of the compound statement does not alter its logical equivalence.

For example:

  • Statement: Today is Monday and today is Monday
  • Redundancy: Today is Monday.

In this example, the redundancy rule allows us to eliminate the redundant component “and it is Monday” without changing the logical equivalence of the statement.

2. Double Negation

P = Not not P

The double negation rule states that a statement is logically equivalent to its double negation. In other words, negating a statement twice cancels out the negation.

For example:

  • Statement: It is raining.
  • Double Negation: It is not not raining.

In this example, the double negation rule tells us that “It is not not raining” is logically equivalent to “It is raining.”

3. Commutation

P or Q = Q or P

P and Q = Q and P

The commutation rule deals with the order of logical operators in a compound statement. It states that the order of conjunction (AND) and disjunction (OR) can be interchanged without changing the truth value of the statement.

For example:

  • Statement: I will have pizza and pasta.
  • Commutation: I will have pasta and pizza.

In this example, the commutation rule allows us to interchange the order of having pizza and pasta without altering the overall meaning of the statement.

4. Association

(P or (Q or R)) = ((P or Q) or R)

(P and (Q and R)) = ((P and Q) and R)

The association rule addresses the grouping of logical operators in a compound statement. It states that the grouping of conjunctions (AND) and disjunctions (OR) can be changed without affecting the truth value of the statement.

For example:

  • Statement: I will eat healthily or (I will eat pizza or ice cream).
  • Association: (I will eat healthily or I will eat pizza) or I will eat ice cream.

In this example, the association rule allows us to change the grouping of the logical operators (OR) while preserving the logical equivalence of the statement.

5. De Morgan’s Laws

De Morgan’s laws provide a relationship between negation and logical operators. There are two laws:

Note: When we use the word ‘or’ in logic it connects two statements and tells us that at least one of them needs to be true for the whole statement to be true. For example, if both P and Q are true, then the statement “P or Q” is also true. This is particularly important to make sense of De Morgan’s Laws.

Negation of Conjunction:

Not (P and Q) = (Not P) or (Not Q)

The law of negation of conjunction states that the negation of a conjunction is equivalent to the disjunction of the negations of its individual parts.

For example:

  • Statement: It is not the case that I am hungry and it is sunny outside
  • De Morgan’s Law: I am not hungry or it is not sunny outside

In this example, applying De Morgan’s law allows us to transform the negation of a conjunction into a disjunction of negations.

Negation of Disjunction:

Not (P or Q) = (Not P) and (Not Q)

The law of negation of disjunction states that the negation of a disjunction is equivalent to the conjunction of the negations of its individual parts.

Example:

  • Statement: It is not the case that it is raining or I am wearing a jacket
  • De Morgan’s Law: It is not raining and I am not wearing a jacket

In this example, De Morgan’s law enables us to rewrite the negation of a disjunction as a conjunction of negations.

6. Contraposition

If P then Q = If not Q then not P

The contraposition rule establishes the equivalence between a conditional statement and its contrapositive. A contrapositive switches the positions of the antecedent (the “if” part) and the consequent (the “then” part) while negating both.

Example:

  • Statement: If the phone is on, then the battery has power.
  • Contrapositive: If the battery does not have power, then the phone is not on.

In this example, the contraposition rule allows us to establish the logical equivalence between a conditional statement and its contrapositive.

7. Exportation

(If (P and Q) then R) = (If P then (If Q then R))

The exportation rule deals with the combination of conditional statements. It states that a conditional statement followed by another conditional statement can be simplified into a single conditional statement.

For example:

  • Statement: If it is raining outside and I have an umbrella, then I will stay dry.
  • Exportation: If it is raining outside, and if I have an umbrella, then I will stay dry.

In this example, the exportation rule allows us to merge the two conditional statements into a single statement, simplifying the expression.

8. Distribution

(P and (Q or R)) = ((P and Q) or (P and R))

(P or (Q and R)) = ((P or Q) and (P or R))

The distribution rule applies to conjunction and disjunction of statements. It states that a conjunction or disjunction applied to a group of statements can be distributed to each individual statement.

Example:

  • Statement: Spiders are animals and they are either insects or arachnids.
  • Distribution: Either spiders are animals and insects or spiders are animals and arachnids.

In this example, the distribution rule enables us to distribute the conjunction to each individual statement within the parentheses.

9. Material Equivalence

Biconditional statements are statements that connect two conditions using “if and only if.” And material equivalence gives us two ways to understand them.

The first version of material equivalence is:

P if and only if Q = ((If P then Q) and (if Q then P))

Here, material equivalence tells us that a biconditional statement is equivalent to a conjunction of two conditionals. In other words, if we have a statement of the form “P if and only if Q,” it can be expressed as the conjunction of two conditionals: “If P, then Q” and “If Q, then P.” This equivalence allows us to break down complex biconditional statements into simpler conditional statements.

For example:

  • Statement: You can attend the party if and only if you are invited.
  • Material Equivalence: If you are invited, then you can attend the party. And if you can attend the party, then you are invited.

The second version of material equivalence takes the form:

P if and only if Q = ((P and Q) or (not P and not Q))

Here, material equivalence shows us that a biconditional statement is equivalent to a disjunction and two conjunctions.

For example:

  • Statement: You can watch the movie if and only if you’ve done your homework.
  • Material Equivalence: Either you’ve done your homework and you can watch the movie, or you haven’t done your homework and you cannot watch the movie.

10. Material Implication

If P then Q = Not P or Q

The material implication rule establishes the logical relationship between a conditional statement and its truth values. It states that a conditional statement is true except when the antecedent is true and the consequent is false.

For example:

  • Statement: If it is sunny, then I will go to the beach.
  • Material Implication: Either it is not sunny or I will go to the beach.

Summary: Equivalence Laws

The 10 equivalence laws above enable us to replace statements with simpler statements without altering their truth values. By using these equivalence laws, we can simplify our arguments and make them easier to understand.

And being able to explain your arguments clearly is key to getting top marks in A level philosophy. I know philosophers often overcomplicate their arguments in order to make themselves sound smart, but this isn’t something you should seek to imitate! A clearly-explained argument is always better than a needlessly complicated one. Detail is good, but redundant complexity is not. So, if you can replace a complex statement with a simpler one – as long as it is logically equivalent – then do so!

The philosophy textbook written with the student in mind!