4.3 Decision Procedures and Truth Trees

Propositional logic is a powerful tool for analyzing arguments and determining their validity. It uses truth tables and truth trees to systematically evaluate the relationship between premises and conclusions, providing a rigorous framework for logical reasoning.

These methods allow us to break down complex arguments into their component parts and test their logical structure. By mastering these techniques, we can strengthen our critical thinking skills and improve our ability to construct and evaluate arguments in various contexts.

Propositional Logic and Validity

Truth tables for propositional validity

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• Determine validity of propositional arguments by evaluating truth values of premises and conclusion
  • List all possible combinations of truth values for propositional variables (P, Q, R)
  • Evaluate truth value of each premise and conclusion for each combination
  • Argument is valid if conclusion is true whenever all premises are true
• Construct truth table by following steps:
  1. Identify propositional variables and list them in table header
  2. Determine number of rows needed ($2^n$ rows for $n$ variables)
  3. Fill in truth values for each variable in all possible combinations
  4. Write premises and conclusion as propositional formulas
  5. Evaluate truth value of each premise and conclusion for each row
• Valid argument has no rows where all premises are true and conclusion is false

Truth tree method for arguments

• Test validity of propositional arguments by applying decomposition rules to break down formulas
  • Start with negation of conclusion and premises as initial branches
  • Apply decomposition rules to simplify complex formulas (conjunction, disjunction, implication)
  • Close branches when contradictions are found (formula and its negation appear)
  • Argument is valid if all branches close, invalid if any branch remains open
• Decomposition rules for common logical connectives:
  • $\neg(\phi \land \psi)$ decomposes into $\neg\phi$ or $\neg\psi$ (branching)
  • $\neg(\phi \lor \psi)$ decomposes into $\neg\phi$ and $\neg\psi$ (linear)
  • $\phi \rightarrow \psi$ decomposes into $\neg\phi$ or $\psi$ (branching)
  • $\phi \leftrightarrow \psi$ decomposes into $(\phi \land \psi) \lor (\neg\phi \land \neg\psi)$ (branching)
• Strategies for applying decomposition rules efficiently:
  • Prioritize linear rules over branching rules to minimize number of branches
  • Decompose formulas containing negations before those without negations
  • Avoid decomposing the same formula multiple times on the same branch

Truth tables vs truth trees

• Both methods determine validity of propositional arguments based on semantics and definition of validity
  • Truth tables list all possible truth value combinations, truth trees focus on finding contradictions
  • Truth tables evaluate truth values of individual formulas, truth trees primarily test validity
  • Truth tables grow exponentially with number of variables, truth trees can be more efficient
• Advantages of truth tables:
  • Provide complete overview of truth values for all formulas in argument
  • Can evaluate truth value of individual formulas and test for logical equivalence (tautology, contradiction)
• Advantages of truth trees:
  • More efficient than truth tables for arguments with many variables or complex formulas
  • Provide visual representation of decomposition process and search for contradictions

Decision procedures in propositional logic

• Algorithms that systematically apply rules to determine validity of propositional arguments
  • Truth tables and truth trees are examples of decision procedures
• Steps for using decision procedures:
  1. Translate argument into propositional logic notation
  2. Choose appropriate decision procedure based on complexity and personal preference
  3. Apply rules of chosen procedure systematically and accurately
  4. Interpret results to determine validity of argument
• Strategies for solving problems involving validity:
  • Break down complex arguments into smaller, manageable components
  • Use truth tables for arguments with few variables or to evaluate individual formulas
  • Use truth trees for arguments with many variables or complex formulas
  • Double-check work to ensure accuracy and avoid errors in applying rules