A proof system for propositional and predicate logic is discussed. As a meta-language specifying the system, a logic programming language, namely, Prolog is adopted. All of proof rules, axioms, definitions, theorems and also proofs can be described as predicates of Prolog. So Prolog can be used to verify whether deductions are valid or not.

Logic, Proofs 1.1. Propositions A proposition is a declarative sentence that is either true or false (but not both). For instance, the following are propositions: “Paris is in France” (true), “London is in Denmark” (false), “2 < 4” (true), “4 = 7 (false)”. However the following are not propositions: “what

Proof. Given x, we need to nd ysuch that y2 >x. If x 1, then x 1 <232; so we can take y= 23. Otherwise x>1. Multiplying both sides of x>1 by the positive number x, we see that x2 >x; so we can take y= x. Alternatively, one could maybe make a case that the statement of Problem 1 is obvious. 2. Disprove 8x9y: y2 <x. Proof.

Propositional logic: proofs, semantics, normal forms, SAT solvers. Predicate logic: proofs, semantics. Proof calculi for program verification. Learning outcomes. By the end of the module, students...In logic, a disjunction is a compound sentence formed using the word or to join two simple sentences. The symbol for this is $$ ν $$ . (whenever you see $$ ν $$ read 'or') When two simple sentences, p and q, are joined in a disjunction statement, the disjunction is expressed symbolically as p $$ ν$$ q.