Semantics
Introduction to Semantics in Propositional Logic
The semantics of propositional logic concerns the meaning of logical expressions—specifically, how their truth values are determined under different interpretations. This page introduces the foundational semantic concepts necessary to evaluate, categorize, and reason about logical formulas.You will begin by exploring the truth-functional nature of propositional logic: how the truth value of a compound proposition is derived from its components. From there, the page discusses how propositions can be classified as tautologies (always true), contradictions (always false), or contingent (true in some cases, false in others).
You will also examine logical equivalences, which identify when two expressions always yield the same truth value, and are fundamental to simplifying and transforming logical formulas.
A dedicated section on logical implication explains conditional statements like "" in both syntactic and semantic terms. Common misconceptions, properties of implication, and its use in proofs are addressed clearly.
Each section is supported by examples, truth tables, and conceptual explanations designed to connect formal definitions to intuitive reasoning. Together, these topics form a complete picture of how meaning is assigned to logical statements in propositional systems.
Logical Equivalences
Logical equivalences are statements in propositional logic that always have the same truth value regardless of the truth values of their individual components.
If two propositions and are logically equivalent, we write:.
This means that for every possible truth assignment, and yield the same truth value.
Logical equivalences form the foundation of propositional logic laws. Each logical law is a logical equivalence that helps us simplify, manipulate, or prove logical expressions.
However, not all logical equivalences are laws. Some are specific derivations that are still true but are not considered "fundamental" enough to be named as laws.
Logical equivalences are essential in mathematical logic, shaping both syntax and semantics. Syntactically, they allow transformations between logically identical formulas, crucial for normal forms like CNF and DNF. Semantically, they ensure truth preservation across interpretations, making them fundamental in truth tables and model theory. In proof techniques, equivalences justify valid transformations within formal systems, aiding in inferences and proof simplifications. By bridging truth conditions with formal derivations, logical equivalences unify reasoning in propositional logic, predicate logic, and beyond.
Check our equivalence validator tool to polish your practical understanding of logical equivalences.
Go to PageIf two propositions and are logically equivalent, we write:.
This means that for every possible truth assignment, and yield the same truth value.
Logical equivalences form the foundation of propositional logic laws. Each logical law is a logical equivalence that helps us simplify, manipulate, or prove logical expressions.
However, not all logical equivalences are laws. Some are specific derivations that are still true but are not considered "fundamental" enough to be named as laws.
Logical equivalences are essential in mathematical logic, shaping both syntax and semantics. Syntactically, they allow transformations between logically identical formulas, crucial for normal forms like CNF and DNF. Semantically, they ensure truth preservation across interpretations, making them fundamental in truth tables and model theory. In proof techniques, equivalences justify valid transformations within formal systems, aiding in inferences and proof simplifications. By bridging truth conditions with formal derivations, logical equivalences unify reasoning in propositional logic, predicate logic, and beyond.
Check our equivalence validator tool to polish your practical understanding of logical equivalences.
Contigency, Tautology or Contradiction
In classical propositional logic, every proposition falls into one of three categories based on its truth value across all possible interpretations:
Example:
("Either it will rain or it won’t"—this is always true.)
Validity and Tautologies
A proposition is valid if it is true in all possible interpretations—in other words, if it is a tautology.
If a proposition is valid, it means there is no possible scenario where it is false.
(Law of the Excluded Middle) is valid because it’s always true.
Example:
("It is raining and it is not raining"—this is never true.)
Unsatisfiability and Contradictions
A proposition is unsatisfiable if it is false in every possible interpretation—which means it is a contradiction.
is unsatisfiable because there is no way to make it true.
(If p is "It will rain" and q is "It will snow," this can be true or false depending on the weather.)
Satisfiability and Contingencies
A proposition is satisfiable if there is at least one interpretation where it is true.Otherwise it would be contradiction.
A contingent proposition (one that is sometimes true and sometimes false) is always satisfiable because it has at least one true case.
is satisfiable because it’s true in cases where either or is true.
These three categories exhaust all possibilities for a proposition in classical logic. Every proposition must be one of these.
1. Tautology
A proposition that is always true, no matter what.Example:
("Either it will rain or it won’t"—this is always true.)
Validity and Tautologies
A proposition is valid if it is true in all possible interpretations—in other words, if it is a tautology.
If a proposition is valid, it means there is no possible scenario where it is false.
(Law of the Excluded Middle) is valid because it’s always true.
2. Contradiction
A proposition that is always false, no matter what.Example:
("It is raining and it is not raining"—this is never true.)
Unsatisfiability and Contradictions
A proposition is unsatisfiable if it is false in every possible interpretation—which means it is a contradiction.
is unsatisfiable because there is no way to make it true.
3. Contingency
A proposition that is sometimes true and sometimes false, depending on the values of its variables. Example:(If p is "It will rain" and q is "It will snow," this can be true or false depending on the weather.)
Satisfiability and Contingencies
A proposition is satisfiable if there is at least one interpretation where it is true.Otherwise it would be contradiction.
A contingent proposition (one that is sometimes true and sometimes false) is always satisfiable because it has at least one true case.
is satisfiable because it’s true in cases where either or is true.
These three categories exhaust all possibilities for a proposition in classical logic. Every proposition must be one of these.
Logical Implication
Logical implication is a core semantic construct in propositional logic, expressing conditional relationships between propositions. On this page, we introduce the formal definition of implication (), along with common notation styles used in logic and mathematics.
A truth table illustrates how implication behaves under all combinations of truth values for its components. From there, several key properties are discussed, including reflexivity, transitivity, contraposition, and the equivalence to the disjunction form (). These properties are essential for both understanding logical behavior and applying simplification techniques in formal proofs.
To aid learners, the page also addresses common misconceptions—such as the counterintuitive truth of implications with false antecedents—and provides clarification on why implication is not a symmetric relationship.
Finally, the practical role of implication in proof strategies is outlined, emphasizing its use in direct reasoning, contrapositive arguments, and proofs by contradiction. Altogether, this resource provides a well-rounded foundation for understanding how conditional statements function within the semantic framework of propositional logic.
Go to PageA truth table illustrates how implication behaves under all combinations of truth values for its components. From there, several key properties are discussed, including reflexivity, transitivity, contraposition, and the equivalence to the disjunction form (). These properties are essential for both understanding logical behavior and applying simplification techniques in formal proofs.
To aid learners, the page also addresses common misconceptions—such as the counterintuitive truth of implications with false antecedents—and provides clarification on why implication is not a symmetric relationship.
Finally, the practical role of implication in proof strategies is outlined, emphasizing its use in direct reasoning, contrapositive arguments, and proofs by contradiction. Altogether, this resource provides a well-rounded foundation for understanding how conditional statements function within the semantic framework of propositional logic.