PHIL222 Lecture 6 - Logic Concepts

Uses of Trees

  • Trees can be used for various propositions and arguments, assisting in the logical analysis and evaluation of statements. They help determine:

    • Validity of arguments: By breaking down the premises and conclusions into their components, a tree can show whether a conclusion necessarily follows from the premises.

    • Satisfiability of propositions: Establishing whether there exists an interpretation that makes the proposition true.

    • Identification of contraries and contradictories: Distinguishing between propositions that can both be true but cannot both be false (contraries) and those that cannot both be true (contradictories).

    • Tautologies: Assessing whether a proposition is true in every possible interpretation, thus categorizing it as universally valid.

    • Equivalence of propositions: Determining if two different propositions express the same truth conditions.

Validity

  • A tree serves as a visual representation to determine if a set of propositions is satisfiable, which is crucial for assessing the validity of an argument. The process involves the following steps:

    • Construct a tree for the premises and the negation of the conclusion.

    • If an open branch remains, it suggests that there is a counterexample to the argument, making it invalid.

    • If all branches close, it asserts that the argument is valid, meaning that the conclusion necessarily follows from the premises.

Example 1

  • Given argument: A → B, B → C, therefore A → C

    • List the premises and the negation of the conclusion:

    • A → B

    • B → C

    • ¬(A → C)

    • Nonbranching Rules Applied:

    • A → B ✓

    • B → C ✓

    • ¬(A → C) ✓

    • Resulting in: A, ¬C

    • Branching Rules Applied:

    • A → B ✓

    • B → C ✓

    • ¬(A → C) ✓

    • This results in:

      • A

        • ¬C

        • ¬A × B

        • ¬B × C ×

    • Conclusion: Since all branches closed, the argument is confirmed valid.

Example 2

  • Argument: A → C, B → C, therefore B → A

    • List premises and negation:

    • A → C

    • B → C

    • ¬(B → A)

    • Nonbranching Rules: Based on the premises, resulting terms: B, ¬A

    • Branching Rules: Reveal that not all branches close, indicating the emergence of a counterexample.

    • Conclusion: The argument is deemed invalid.

Counterexamples

  • A tree can highlight counterexamples through open branches.

  • Interpretation: Letters represent true propositions, while their negated counterparts signify false propositions. This framework allows for effective exploration of logical structures.

Example 3

  • Argument: A therefore ¬(A ∧ B) ∧ ¬((A ∧ B) ∧ C)

    • Open branches lead to counterexamples based on potential truth value assignments to A, B, and C, showcasing possible logical inconsistencies.

Satisfiability

  • To ascertain if a proposition is satisfiable, the following steps are typically taken:

    • Write the proposition at the top of the tree and systematically explore branching paths. Closed paths indicate contradictions, while open paths reveal satisfiable propositions.

Example 4

  • Proposition: (A ∨ B) ∧ ¬(A ∨ B)

    • Results in all branches closing, confirming it as unsatisfiable, hence implying a contradiction in truth values.

Contraries and Contradictories

  • To effectively determine if two propositions are contraries or contradictories, the following methods are employed:

    1. Test for satisfiability of the original propositions to establish their truth conditions.

    2. If found unsatisfiable, further test the negations for satisfiability to conclude their relationship.

  • Case Study Example: Propositions A ∧ B and A ∧ ¬B exhibit contrary relationships by being true in mutually exclusive conditions.

Tautologies

  • The process to identify a tautology involves writing the negation of the proposition and evaluating it through a tree structure:

    • If all paths close, this confirms the proposition as a tautology, establishing its universal truth.

Example 6

  • Proposition: A → (A ∧ (B ∨ ¬B))

    • In this case, all branches close, thereby confirming the proposition as a tautology, representing a valid logical truth that holds under any circumstances.

Equivalence

  • To test for equivalence between two formulas α and β, the procedure involves:

    • Checking whether the biconditional statement α ↔ β is a tautology.

Example 7

  • Testing equivalence: A and A ∨ A.

    • Investigating the negation evidences closure of all branches, thus supporting their equivalence, meaning both expressions yield identical truth conditions in any interpretation.