1.3 Semantics and truth values

Logic's building blocks are semantics and syntax. Semantics assigns meaning and truth values to formulas, while syntax defines the rules for constructing them. Together, they form the foundation for evaluating logical statements and determining their validity.

Truth values in propositional logic are the cornerstone of logical reasoning. By assigning true or false to atomic propositions and using logical connectives, we can evaluate complex formulas. This process is crucial for mathematical proofs and understanding logical validity.

Semantics and Truth Values

Semantics and syntax in logic

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• Semantics deals with the meaning and interpretation of logical formulas focuses on assigning truth values (true or false) to well-formed formulas
• Syntax concerns the structure and rules for constructing well-formed formulas defines the formal language and its grammar
• Relationship between semantics and syntax
  • Syntax provides the rules for constructing valid formulas (well-formed formulas or wffs)
  • Semantics assigns meaning and truth values to syntactically correct formulas determines if a formula is true or false based on its interpretation

Truth values in propositional logic

• Propositional logic
  • Atomic propositions are simple declarative statements assigned truth values (true or false)
  • Compound propositions are built using logical connectives ($\neg$ for negation, $\wedge$ for conjunction, $\vee$ for disjunction, $\rightarrow$ for implication, $\leftrightarrow$ for biconditional)
• Examples of atomic propositions
  • "The sky is blue" (true)
  • "2 + 2 = 5" (false)
• Examples of compound propositions
  • "The sky is blue and the grass is green" ($P \wedge Q$)
  • "If it is raining, then the ground is wet" ($P \rightarrow Q$)

Evaluating complex logical formulas

• Truth tables exhaustively list all possible combinations of truth values for the atomic propositions determine the truth value of the compound proposition for each combination using the semantic rules of the logical connectives
• Semantic rules for logical connectives
  • Negation ($\neg$): $\neg P$ is true if and only if $P$ is false
  • Conjunction ($\wedge$): $P \wedge Q$ is true if and only if both $P$ and $Q$ are true
  • Disjunction ($\vee$): $P \vee Q$ is true if and only if at least one of $P$ or $Q$ is true
  • Implication ($\rightarrow$): $P \rightarrow Q$ is false if and only if $P$ is true and $Q$ is false otherwise, it is true
  • Biconditional ($\leftrightarrow$): $P \leftrightarrow Q$ is true if and only if $P$ and $Q$ have the same truth value
• Example: Evaluate the truth value of $(P \wedge Q) \rightarrow (\neg P \vee Q)$ using a truth table

Logical validity for mathematical reasoning

• Logical validity
  • A formula is logically valid (or a tautology) if it is true under all possible interpretations always true regardless of the truth values assigned to its atomic propositions
  • A formula is satisfiable if there exists at least one interpretation that makes it true
  • A formula is unsatisfiable (or a contradiction) if it is false under all possible interpretations always false regardless of the truth values assigned to its atomic propositions
• Significance in mathematical reasoning
  • Logically valid formulas represent universal truths can be used as axioms or theorems in proofs and derivations
  • Satisfiability helps determine the consistency of a set of formulas if a set of formulas is satisfiable, it is consistent (no contradictions)
  • Unsatisfiable formulas indicate contradictions can be used in proofs by contradiction to show that a statement leads to a logical inconsistency
• Examples of logically valid formulas
  • $P \vee \neg P$ (law of excluded middle)
  • $(P \wedge (P \rightarrow Q)) \rightarrow Q$ (modus ponens)