Definition and Notation

Definition

Two logical statements (or propositions) are logically equivalent if they always have the same truth value, regardless of the truth values of their individual components. This means that no matter what, both expressions will evaluate to either true or false together in every possible case.

Notation

Logical equivalence is usually written as:

P≡Q

Sometimes notation like this is used :

P⇔Q

Both of those forms are legitimate and acceptable to notate equivalence.Both notations indicate that

𝑃

and

𝑄

always produce the same truth values.
In boolean algebra also used this notation:

P=Q

The notations for logical equivalence and biconditional (

P↔Q

or

𝑃⟺𝑄

) can sometimes look similar, but they have different meanings, and they are not always interchangeable.

P≡Q

means

𝑃

and

𝑄

are logically equivalent in all possible cases.

𝑃↔𝑄

is just a statement that says

𝑃

and

𝑄

happen to be true or false together in a specific context - it can be true or false, depending on

𝑃

and

𝑄

.

How to Verify Logical Equivalence?

To verify that two logical statements

𝑃

and

𝑄

are logically equivalent (

𝑃≡𝑄

), we need to show that they always have the same truth value in all cases.
There are several methods to do this:

1.
Truth Tables (Brute-Force Method):
Construct a truth table for both expressions.
Check if the final column values are identical for all possible truth values of the variables.
If they match in every row, the expressions are logically equivalent.
✅ Best for: Propositional logic with a small number of variables.
❌ Not practical for more than 3-4 variables due to exponential growth in rows.
2.
Algebraic Manipulation Using Logical Laws:
Apply known logical equivalences (laws) (e.g., De Morgan’s laws, distributive, commutative, associative properties) to transform one statement into another.
If you can rewrite $𝑃$ into $𝑄$ (or vice versa), they are equivalent.
✅ Best for: Proofs and simplifying expressions without constructing tables.
❌ Requires familiarity with logical laws.
3.
Using Logical Implications:
Show that $𝑃→𝑄$ and $𝑄→𝑃$ are both true.
If both implications hold, then $𝑃≡𝑄$ .
✅ Best for: When equivalences involve implications.
❌ Requires proving two separate implications.
4.
Venn Diagrams (Set-Theoretic Approach):
Represent logical statements using sets and intersections/unions.
If two statements cover the same region, they are equivalent.
✅ Best for: Visualizing logical expressions.
❌ Not practical for complex expressions.

Fundamental Equivalences (Laws)

Some logical equivalences are considered laws of logic.A law is a fundamental equivalence—one that is taken as a basic principle, rather than something that needs to be proven from other rules.
On the other hand, many of logical equivalences, while being valid transformations, are not considered fundamental laws because they depend on definitions, are derived from the laws, or are specific to certain logical systems.

•
Identity Laws:
$P \lor \text{false} \equiv P$ ,
$P \land \text{true} \equiv P$
•
Domination Laws:
$P \lor \text{true} \equiv \text{true}$ ,
$P \land \text{false} \equiv \text{false}$
•
Idempotent Laws:
$P \lor P \equiv P$ ,
$P \land P \equiv P$
•
Double Negation Law:
$\neg (\neg P) \equiv P$
•
Commutative Laws:
$P \lor Q \equiv Q \lor P$ ,
$P \land Q \equiv Q \land P$
•
Associative Laws:
$P \lor (Q \lor R) \equiv (P \lor Q) \lor R$ ,
$P \land (Q \land R) \equiv (P \land Q) \land R$
•
Distributive Laws:
$P \land (Q \lor R) \equiv (P \land Q) \lor (P \land R)$ ,
$P \lor (Q \land R) \equiv (P \lor Q) \land (P \lor R)$
•
Absorption Laws:
$P \lor (P \land Q) \equiv P$ ,
$P \land (P \lor Q) \equiv P$
•
De Morgan’s Laws:
$\neg (P \lor Q) \equiv \neg P \land \neg Q$ ,
$\neg (P \land Q) \equiv \neg P \lor \neg Q$
•
Law of Excluded Middle:
$P \lor \neg P \equiv \text{true}$
•
Law of Non-Contradiction:
$P \land \neg P \equiv \text{false}$

For more information about laws of propositional logic, you can visit this page.

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Equivalences with Implications

In propositional logic, a conditional statement (implication) is written as:

𝑃→𝑄

which means "if

𝑃

, then

𝑄

".
However, implications can be rewritten using logical equivalences.
Here are the key ones:

name	equivalence	explanation
Material Implication	$p \rightarrow q \equiv \neg p \lor q$	"if $𝑝$ then $𝑞$ " is the same as saying "either $𝑝$ is false or $𝑞$ is true."
Contrapositive	$p \rightarrow q \equiv \neg q \rightarrow \neg p$	Reversing and negating an implication produces an equivalent statement.
Disjunction Form of Implication	$p \lor q \equiv \neg p \rightarrow q$	A disjunction can be rewritten as an implication.
Implication as a Conjunction	$p \land q \equiv \neg (p \rightarrow \neg q)$	A conjunction can be expressed using an implication and negation.
Negation of an Implication	$\neg (p \rightarrow q) \equiv p \land \neg q$	An implication is false only when the antecedent is true and the consequent is false.
Implication Distribution over Conjunction	$(p \rightarrow q) \land (p \rightarrow r) \equiv p \rightarrow (q \land r)$	If ( p ) implies both ( q ) and ( r ), then it implies their conjunction.
Nested Implications in a Disjunction	$(p \rightarrow r) \land (q \rightarrow r) \equiv (p \lor q) \rightarrow r$	If both ( p ) and ( q ) imply ( r ), then their disjunction also implies ( r ).
Implication Distribution over Disjunction	$(p \rightarrow q) \lor (p \rightarrow r) \equiv p \rightarrow (q \lor r)$	If ( p ) implies either ( q ) or ( r ), then ( p ) implies their disjunction.
Nested Implications in a Conjunction	$(p \rightarrow r) \lor (q \rightarrow r) \equiv (p \land q) \rightarrow r$	If either ( p ) or ( q ) implies ( r ), then their conjunction implies ( r ).

All logical equivalences involving implications provide ways to rewrite conditional statements in different but logically identical forms. A common theme among them is restructuring the relationship between the antecedent (

𝑝

) and the consequent (

𝑞

) using negation, disjunction, or conjunction while preserving truth values. These transformations help simplify logical expressions and proofs.

Equivalences with Biconditionals

Logical equivalences involving biconditional statements (

𝑝↔𝑞

) focus on expressing the mutual dependence of two propositions in different but logically equivalent ways. The biconditional means "p if and only if q," meaning both must have the same truth value (either both true or both false).
Here are some equivalences involving biconditional statements:

name	equivalence	explanation
Biconditional as Two Implications	$p \leftrightarrow q \equiv (p \rightarrow q) \land (q \rightarrow p)$	A biconditional means both directions of implication must be true.
Negation Preservation in Biconditional	$p \leftrightarrow q \equiv \neg p \leftrightarrow \neg q$	Negating both sides of a biconditional does not change its truth value.
Biconditional in Terms of AND and OR	$p \leftrightarrow q \equiv (p \land q) \lor (\neg p \land \neg q)$	Two statements are equivalent if both are true or both are false.
Negation of a Biconditional	$\neg (p \leftrightarrow q) \equiv p \leftrightarrow \neg q$	Negating a biconditional swaps one side, making it an XOR.

Conclusion on Biconditional Equivalences

Biconditional equivalences highlight the fundamental symmetry in logic: two statements are logically interchangeable if and only if they always have the same truth value. These equivalences allow us to express ( p leftrightarrow q ) in different but logically equivalent ways, making it easier to manipulate and analyze logical statements.
- They break down into implications, showing that mutual implication defines equivalence.
- They connect to conjunction and disjunction, emphasizing that two statements are equivalent when they share truth values.
- Negating a biconditional results in an XOR, reinforcing that logical opposition emerges when one statement is true and the other is false.

In formal logic, these properties make biconditionals a powerful tool for proofs, simplifications, and logical reasoning across mathematics, philosophy, and computing.

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