Show that if a schema is constructed from sentence letters using only conjunction and disjunction, then it is:
a) Consistent and valid
b) Satisfiable but not valid
c) Valid but not satisfiable
d) Inconsistent and not valid
The Correct Answer and Explanation is:
The correct answer is: a) Consistent and valid
Explanation:
A schema constructed from sentence letters using only conjunction (AND, ∧) and disjunction (OR, ∨) is consistent and valid under certain conditions. Here’s a breakdown of why this is true:
- Consistency:
A logical schema is consistent if it does not lead to a contradiction, i.e., it can be satisfied by at least one interpretation or truth assignment. In the case of conjunction and disjunction, the schema can be interpreted in such a way that there is no inherent contradiction in the structure.
- For conjunction (A ∧ B), both A and B must be true for the schema to hold true.
- For disjunction (A ∨ B), the schema holds true if at least one of A or B is true. In other words, there is always a way to assign truth values to the components of the schema such that the schema doesn’t lead to a contradiction. This makes the schema consistent.
- Validity:
A logical schema is valid if it is true in every possible interpretation or truth assignment. For a schema constructed using only conjunction and disjunction, validity holds in certain cases:
- If the schema is a tautology (such as (A \vee \neg A)), then it is valid because it holds true in every interpretation.
- For example, the disjunction (A \vee \neg A) is valid because in every possible case (whether A is true or false), the schema is true. In this case, the schema constructed from conjunctions and disjunctions can be valid if it is structured to always evaluate as true (for instance, using a form like (A \vee (B \vee \neg B)), which is valid since it covers all cases).
Conclusion:
Because schemas built with conjunctions and disjunctions can be structured to be free of contradictions (consistent) and hold true in all cases (valid), they are both consistent and valid. Therefore, the correct answer is (a).