Using truth tree decomposition, determine the consistency of the set of propositions. Include line numbers and justifications, and recover the truth values if consistent.
{A and B, ~B v C}
To determine the consistency of the set of propositions {A and B, ~B v C}, we can use truth tree decomposition. This method involves constructing a truth tree that exhaustively explores all possible truth value assignments to the propositions in order to check for contradictions.
Here is the step-by-step process for constructing the truth tree and determining the consistency:
1. Write down the given propositions:
Line 1: A and B -- Given
Line 2: ~B v C -- Given
2. Apply the decomposition rules for conjunction and disjunction:
For Line 1 (A and B):
Line 3: A -- Conjunction Elimination (Conj E)
Line 4: B -- Conjunction Elimination (Conj E)
For Line 2 (~B v C):
Line 5: ~B -- Disjunction Elimination (Disj E)
Line 6: C -- Disjunction Elimination (Disj E)
3. Apply the decomposition rules to further expand the tree:
Branch 1 (assuming A):
Line 7: A -- Assumption
Branch 2 (assuming ~B):
Line 8: ~B -- Assumption
Branch 3 (assuming C):
Line 9: C -- Assumption
4. Compare the propositions in the branches for any contradictions:
Branch 1:
Line 10: B -- Contradiction (Lines 4, 7)
Branch 2:
Line 11: -- Contradiction (Lines 8, 5)
Branch 3:
Line 12: -- No contradiction
5. Determine the truth values if all branches lead to contradictions:
Since Branches 1 and 2 lead to contradictions, the propositions are inconsistent.
Therefore, the set of propositions {A and B, ~B v C} is inconsistent. There are no truth values to recover in this case.