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In this set of notes, we explore basic proof techniques, and how they can be understood by a grounding in propositional logic. We will show how to use these proof techniques with simple examples, and demonstrate that they work using truth tables and other logical tools.
Formal logic is used for specifying and verifying computer systems. The course should help you to understand Prolog and is a prerequisite for more advanced verification courses. It describes many techniques used in automated theorem provers.
Formal logic is used for specifying and verifying computer systems and (some-times) for representing knowledge in Artificial Intelligence programs. The course should help you with Prolog for AI and its treatment of logic should be helpful for understanding other theoretical courses.
Every theorem in mathematics must have a proof from a set of stated assump-tions. The proof demonstrates that the conclusion of the theorem follows from the assumptions by the laws of logic alone. Can the notions of “laws of logic” and “proof” be made precise? This text is devoted to establishing a positive answer to this question.
2 First-Order Logic 63 2.1 Languages 63 Predicates 63 Alphabets 67 Terms 70 Formulas 71 2.2 Substitution 75 Terms 75 Free Variables 76 Formulas 78 2.3 Syntactics 85 Quantifier Negation 85 Proofs with Universal Formulas 87 Proofs with Existential Formulas 90 2.4 Proof Methods 96 Universal Proofs 97 Existential Proofs 99 Multiple Quantifiers ...
A Proof in Propositional Logic To prove: :(p _(:p ^q)) :p ^:q:(p _(:p ^q)) :p ^:(:p ^q) by De Morgan’s 2nd law :p ^(:(:p)_:q) by De Morgan’s first law :p ^(p _:q) by the double negation law (:p ^p)_(:p ^:q) by the 2nd distributive law F_(:p ^:q) because :p ^p F (:p ^:q)_F by commutativity of disj. :p ^:q by the identity law for F
One of the principal aims of this course is to teach the student how to read and, to a lesser extent, write proofs. A proof is an argument intended to convince the reader that a general principle is true in all situations. The amount of detail that an author supplies in a proof should depend on the audience.
