Glossary

4.3 Decision Procedures and Truth Trees

3 min read•july 22, 2024

is a powerful tool for analyzing arguments and determining their . It uses truth tables and 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 by following steps:
    1. Identify propositional variables and list them in table header
    2. Determine number of rows needed (2n2^n rows for nn 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
  • has no rows where all premises are true and conclusion is false

Truth tree method for arguments

  • Test validity of propositional arguments by applying to break down formulas
    • Start with of conclusion and premises as initial branches
    • Apply decomposition rules to simplify complex formulas (, , )
    • 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:
    • ¬(ϕ∧ψ)\neg(\phi \land \psi) decomposes into ¬ϕ\neg\phi or ¬ψ\neg\psi (branching)
    • ¬(ϕ∨ψ)\neg(\phi \lor \psi) decomposes into ¬ϕ\neg\phi and ¬ψ\neg\psi (linear)
    • ϕ→ψ\phi \rightarrow \psi decomposes into ¬ϕ\neg\phi or ψ\psi (branching)
    • ϕ↔ψ\phi \leftrightarrow \psi decomposes into (ϕ∧ψ)∨(¬ϕ∧¬ψ)(\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

Key Terms to Review (14)

Conjunction: A conjunction is a logical operator that connects two statements to form a new statement that is true only if both of the original statements are true. This concept is crucial for understanding how complex logical expressions can be constructed and evaluated.
Decomposition Rules: Decomposition rules are a set of logical principles used to break down complex formulas into simpler components, making it easier to analyze their truth values. These rules play a crucial role in decision procedures and truth trees by allowing the systematic examination of logical expressions, ultimately aiding in determining their validity. By applying decomposition rules, one can transform intricate statements into more manageable parts, facilitating the construction of truth tables or the evaluation of propositional logic.
Disjunction: Disjunction is a logical connective that represents the 'or' relationship between two propositions, denoted by the symbol '∨'. It indicates that at least one of the propositions must be true for the disjunction itself to be true. Understanding disjunction helps in translating natural language statements into formal logic, constructing truth tables, applying rules of inference, and analyzing predicate logic.
Implication: Implication is a fundamental logical operation that expresses a relationship between two propositions, typically in the form 'if P, then Q', symbolically represented as $$P \rightarrow Q$$. This logical connective is essential for understanding how different statements interact, particularly in proofs, translations, and truth evaluations, linking antecedents and consequents across various forms of reasoning.
Modus ponens: Modus ponens is a fundamental rule of inference in propositional logic that states if a conditional statement is true and its antecedent is true, then the consequent must also be true. This logical form is vital for constructing valid arguments and making sound conclusions based on given premises.
Modus tollens: Modus tollens is a valid form of deductive reasoning that states if a conditional statement is accepted, and the consequent is false, then the antecedent must also be false. This logical structure is important for analyzing arguments and assessing their validity, especially when dealing with implications in various forms of reasoning.
Negation: Negation is a logical operation that takes a proposition and inverts its truth value, transforming a true statement into a false one, and vice versa. This fundamental concept is essential for understanding how statements relate to one another, particularly in logical reasoning and various forms of proof.
Proof by contradiction: Proof by contradiction is a logical method of proving that a statement is true by assuming the opposite of that statement is true, leading to a contradiction. This technique relies on the principle that if assuming the negation of a statement results in a contradiction, then the original statement must be true. It connects deeply with concepts like quantifiers, inference rules, and logical structures.
Propositional logic: Propositional logic is a branch of logic that deals with propositions and their relationships through logical connectives, focusing on the structure of arguments rather than the content. It simplifies complex statements into truth-functional components, allowing for the systematic evaluation of validity using rules and structures, such as truth trees and inference rules. This makes it essential for understanding logical reasoning in various fields, including mathematics and computer science.
Semantic Tableau: A semantic tableau is a decision procedure used in logic to determine the satisfiability of a set of propositions by systematically breaking them down into simpler components. This method involves constructing a tree-like structure that branches according to the logical relationships of the propositions, making it easier to visualize the relationships and contradictions between different statements.
Truth Table: A truth table is a mathematical table used to determine the truth value of logical expressions based on the truth values of their components. It systematically lists all possible combinations of truth values for the involved propositions, showing how these values combine to affect the overall truth of the statement. Truth tables are essential for evaluating logical relationships, assessing the validity of arguments, and understanding complex logical constructs.
Truth Trees: Truth trees are a visual method used in logic to determine the satisfiability of a set of propositions. They provide a systematic way of breaking down complex logical expressions into simpler components, allowing for the identification of contradictions and truths in formal reasoning. This technique is particularly useful in decision procedures, as it helps evaluate the validity of arguments by exploring all possible truth assignments.
Valid argument: A valid argument is a form of reasoning in which, if the premises are true, the conclusion must also be true. This concept is fundamental in evaluating the structure of arguments, ensuring that the logical flow from premises to conclusion maintains consistency. Validity does not concern itself with the actual truth of the premises, but rather with the relationship between them and the conclusion.
Validity: Validity refers to the property of an argument wherein if the premises are true, the conclusion must also be true. This concept is crucial in assessing the strength of arguments, as it determines whether an argument logically follows from its premises, linking directly to methods of analysis and various logical tools.
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