Clausal Language (ver. 5.81.21, by P.J. Voda, J. Komara, J. Kluka)

Literature.

Chapter. Propositional Logic.

Section. Language.

Formulas.

[CL] Syntax. Examples:

Atomic formulas: $R$ , $Q$ .
Propositional connectives: $⊤$ , $⊥$ , $\neg R$ . $R \land Q$ , $R \lor Q$ , $R \to Q$ , $R \leftrightarrow Q$ .

Section. The Method of Propositional Tableaux.

[CL] Remark. The module Mlogicp in the file mlogicp.cl contains the syntax of propositional tableau rules.

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B

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Exercise. (1 point) Prove that logical disjunction is distributive over logical conjunction:

A \lor B \land C \to (A \lor B) \land (A \lor C)

A \lor B \land C \to (A \lor B) \land (A \lor C)

Exercise. Prove that logical implication is transitive:

(A \to B) \land (B \to C) \land A \to C

(A \to B) \land (B \to C) \land A \to C

Exercise. Prove that contrapositives of logical implication are logically equivalent:

(A \to B) \leftrightarrow \neg B \to \neg A

(A \to B) \leftrightarrow \neg B \to \neg A

[CL] Propositional logic with automatic rules.

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B

C

Exercise. Prove that logical disjunction is distributive over logical conjunction:

A \lor B \land C \to (A \lor B) \land (A \lor C)

A \lor B \land C \to (A \lor B) \land (A \lor C)

Exercise. Prove that logical implication is transitive:

(A \to B) \land (B \to C) \land A \to C

(A \to B) \land (B \to C) \land A \to C

Exercise. Prove that contrapositives of logical implication are logically equivalent:

(A \to B) \leftrightarrow \neg B \to \neg A

(A \to B) \leftrightarrow \neg B \to \neg A