Semantics: Truth Tables and Logical Equivalence

In the previous post of this thread, we examined the syntax of propositional logic, focusing on how logical statements are constructed using propositions and logical connectives. Now, we turn to the semantics of propositional logic, which determines how the truth values of logical expressions are evaluated. This is achieved using truth tables, a fundamental tool for analyzing logical statements.

Truth Tables for Basic Connectives

A truth table is a systematic way to display the truth values of a logical expression based on all possible truth values of its atomic propositions. Each row of a truth table corresponds to a possible assignment of truth values to the atomic propositions, and the columns show how the logical connectives operate on these values.

It is important to emphasize that the truth tables for the basic logical connectives should be understood as their definitions. In the previous post, we introduced these connectives in natural language, but their precise meaning is formally established by these truth tables.

Below are the truth tables that define the basic logical connectives:

1. Negation (NOT, \(\neg P\)):

   \( P \)	\( \neg P \)
   T	F
   F	T

2. Conjunction (AND, \(P \land Q\)):

   \( P \)	\( Q \)	\( P \land Q \)
   T	T	T
   T	F	F
   F	T	F
   F	F	F

3. Disjunction (OR, \(P \lor Q\)):

   \( P \)	\( Q \)	\( P \lor Q \)
   T	T	T
   T	F	T
   F	T	T
   F	F	F

4. Implication (IMPLIES, \(P \rightarrow Q\)): Note: Implication is often misunderstood because it is considered true when the antecedent (P) is false, regardless of Q. This is due to its interpretation in classical logic as asserting that “if P is true, then Q must also be true.”

   \( P \)	\( Q \)	\( P \rightarrow Q \)
   T	T	T
   T	F	F
   F	T	T
   F	F	T

5. Biconditional (IF AND ONLY IF, \(P \leftrightarrow Q\)): The biconditional is true only when PP and QQ have the same truth value.

   \( P \)	\( Q \)	\( P \leftrightarrow Q \)
   T	T	T
   T	F	F
   F	T	F
   F	F	T

Tautologies, Contradictions, and Contingencies

Using truth tables, we can classify logical statements based on their truth values under all possible circumstances:

1. Tautology: A statement that is always true, regardless of the truth values of its components.
   • Example: \(P \lor \neg P\) (The law of the excluded middle)
2. Contradiction: A statement that is always false, no matter what values its components take.
   • Example: \(P \land \neg P\) (A proposition and its negation cannot both be true)
3. Contingency: A statement that is neither always true nor always false; its truth value depends on the values of its components.
   • Example: \(P \rightarrow Q\)

Logical Equivalence and Important Identities

Two statements A and B are logically equivalent if they always have the same truth values under all possible truth assignments. We write this as \(A \equiv B\).

Many logical identities can be proven using truth tables. As an example, let us prove De Morgan’s first law:

• Statement: \(\neg (P \land Q) \equiv \neg P \lor \neg Q\)

\( P \)	\( Q \)	\( P \land Q \)	\( \neg (P \land Q) \)	\( \neg P \)	\( \neg Q \)	\( \neg P \lor \neg Q \)
T	T	T	F	F	F	F
T	F	F	T	F	T	T
F	T	F	T	T	F	T
F	F	F	T	T	T	T

Since the columns for \(\neg (P \land Q)\) and \(\neg P \lor \neg Q \) are identical, the equivalence is proven.

Other important logical identities include:

1. Double Negation: \(\neg (\neg P) \equiv P\)
2. Implication as Disjunction: \(P \rightarrow Q \equiv \neg P \lor Q\)
3. Commutative Laws: \(P \lor Q \equiv Q \lor P\), \(P \land Q \equiv Q \land P\)
4. Associative Laws: \((P \lor Q) \lor R \equiv P \lor (Q \lor R)\)
5. Distributive Laws: \(P \land (Q \lor R) \equiv (P \land Q) \lor (P \land R)\)

The remaining identities can be verified using truth tables as an exercise.

Exercises

1. Construct the truth table for \(P \rightarrow Q \equiv \neg P \lor Q\) to prove their equivalence.
2. Use truth tables to verify De Morgan’s second law: \(\neg (P \lor Q) \equiv \neg P \land \neg Q\).
3. Prove the associative law for disjunction using truth tables: \((P \lor Q) \lor R \equiv P \lor (Q \lor R)\).

Next Steps

Now that we understand the semantics of propositional logic through truth tables and logical equivalence, the next step is to explore proof techniques in propositional logic, where we formalize reasoning through structured argumentation and derivations.