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:
To address this question, we need to analyze the logical structure of a schema constructed using only conjunction (AND) and disjunction (OR), and determine its properties: consistent, valid, and satisfiable.
Definitions:
- Consistent: A schema (or set of sentences) is consistent if there is at least one interpretation or model where all the sentences in the schema are true.
- Valid: A schema is valid if it is true under every possible interpretation or model.
- Satisfiable: A schema is satisfiable if there is at least one interpretation where the schema is true. This is slightly different from consistency, as it does not require the schema to be true in every model, only in some.
Key Points for Logical Connectives:
- Conjunction (AND): The result of a conjunction (e.g., ( A \land B )) is true only when both ( A ) and ( B ) are true. In other words, if any part of the conjunction is false, the entire expression becomes false.
- Disjunction (OR): The result of a disjunction (e.g., ( A \lor B )) is true if at least one of ( A ) or ( B ) is true. The disjunction only becomes false if both parts are false.
Construction of a Schema Using Only Conjunction and Disjunction:
- Schemas in Conjunction and Disjunction: A schema consisting solely of conjunctions and disjunctions, with no negations (¬) or implications (→), generally contains statements that combine sentences using these two logical operations. For example:
- ( A \land B )
- ( A \lor B )
- ( (A \land B) \lor (C \land D) )
Analysis:
- Consistency:
- Since the schema involves both conjunction and disjunction, it is generally possible to find an interpretation where the schema holds true. For example, in a disjunction, if any one of the components is true, the entire expression is true, and in a conjunction, all components can be true under a consistent interpretation. Thus, the schema is consistent.
- Validity:
- A schema is valid only if it holds true under every interpretation. However, since conjunction requires every component to be true and disjunction requires only one part to be true, the schema cannot be guaranteed to be true in all possible interpretations. For example, the schema ( A \land \neg A ) (if we allow negation) is inconsistent and invalid, but even simpler cases like ( A \lor B ) are not valid in all models (e.g., if both ( A ) and ( B ) are false, the schema is false). Thus, the schema is not valid.
- Satisfiability:
- A schema is satisfiable if there exists at least one interpretation where the schema holds true. Since disjunction allows for true results when any component is true, and conjunction can be satisfied if all components are true, there are cases where the schema can be satisfied. Thus, the schema is satisfiable.
Conclusion:
Given the analysis above, the correct answer is b) Satisfiable but not valid. A schema constructed using only conjunction and disjunction is satisfiable (since there are interpretations that can make it true), but it is not valid (since it is not true under all possible interpretations).