MATH 211 Lecture 3, Spring 2012

1 Summer 2007 S. Abu Diab 1 Ch1- 1.5: Rules of Inference Summer 2007 S. Abu Diab 2 Ch1- 1.5: Rules of Inference Definition: An argument in propositional logic is a sequence of propositions. All but the final proposition in the argument are called premises and the final proposition is called the conclusion. An argument is valid if the truth of all its premises implies that the conclusion is true. Summer 2007 S. Abu Diab 3 Ch1- 1.5: Rules of Inference Note: A sequence of compound arguments is called an argument form. From the definition, an argument with prem ises and conclusion is valid if is a tautology. np p p ,..., , 2 1 q ( ) q p p p n → ∧ ∧ ... 2 1 Summer 2007 S. Abu Diab 4 Ch1- 1.5: Rules of Inference Sometimes we may have many propositional variables so we can not draw a truth table to determine the validity of an argument since it will need rows. We will now prove the validity of some arguments, call them particular names, and use them to prove the validity of an argument form! We will call them Rules of Inference. 1024 210 = Summer 2007 S. Abu Diab 5 Ch1- 1.5: Rules of Inference •• An An Inference Rule Inference Rule is is –– A pattern establishing that if we A pattern establishing that if we know that a set of know that a set of premises premises are all are all true, then we can deduce that a true, then we can deduce that a certain certain conclusion conclusion statement is true. statement is true. •• premise 1 premise 1 premise 2 premise 2 …… ∴ ∴ conclusion conclusion ““∴ ∴”” means means ““therefore therefore”” * Summer 2007 S. Abu Diab 6 Ch1- 1.5: Rules of Inference Some Rules of Inference: • p Rule of Addition ∴ p∨q • p∧q Rule of Simplification ∴ p • p Rule of Conjunction q ∴ p ∧q 2 Summer 2007 S. Abu Diab 7 Ch1- 1.5: Rules of Inference • p Rule of modus ponens p →q (law of detachment) ∴q mode of affirming • ¬q p →q Rule of modus tollens ∴¬p mode of denying Summer 2007 S. Abu Diab 8 Ch1- 1.5: Rules of Inference • p →q Rule of hypothetical q →r syllogism (transitivity) ∴p →r • p ∨ q Rule of disjunctive ¬p syllogism ∴ q Aristotle (384-322 B.C.) Summer 2007 S. Abu Diab 9 Ch1- 1.5: Rules of Inference • p ∨q ¬ p ∨r ∴q ∨r Resolution Summer 2007 S. Abu Diab 10 Ch1- 1.5: Rules of Inference • For each of these rules, we can easily prove their validity: If their premises are true than their conclusion must be true. • Example: Rule of disjunctive syllogism: “If p ∨ q and ¬p then q” Use truth tables to show this rule is valid. * Summer 2007 S. Abu Diab 11 Ch1- 1.5: Rules of Inference Refer to worksheet 1 Summer 2007 S. Abu Diab 12 Ch1- 1.5: Rules of Inference • Each valid logical inference rule corresponds to an implication that is a tautology. • premise 1 Inference rule premise 2 … ∴ conclusion • Corresponding tautology: ((premise 1) ∧ (premise 2) ∧ …) → conclusion * 3 Summer 2007 S. Abu Diab 13 Ch1- 1.5: Rules of Inference More generally, if there is a valid argument premise 1,.., premise n ∴ conclusion then the implication ((premise 1) ∧ .. ∧ (premise n)) → conclusion is a tautology. Summer 2007 S. Abu Diab 14 Ch1- 1.5: Rules of Inference Incorrect arguments involve several fallacies. Some common ones are: 1) Fallacy of affirming the conclusion: 1) Fallacy of denying the hypothesis: ( ) [ ] p q q p → ∧ → ( ) [ ] q p q p → ¬ ∧ ¬ → Summer 2007 S. Abu Diab 15 Ch1- 1.5: Rules of Inference • ∀x P (x) ∴P (a) (substitute any object a) Summer 2007 S. Abu Diab 16 Ch1- 1.5: Rules of Inference • P (g) ∴∀x P (x)? Universal Generalization!!! • This is not a valid inference of course. But suppose you can prove P (g) without using any information about g ... • ... then the inference to ∀x P(x) is valid! • In other words ... Summer 2007 S. Abu Diab 17 Ch1- 1.5: Rules of Inference Probably the trickiest of the four rules. • What do you think of this inference: • ∃x P (x) ∴P (c) ? Summer 2007 S. Abu Diab 18 Ch1- 1.5: Rules of Inference ••∀∀xx P P ((xx)) ∴ ∴P P ((aa)) universal instantiation universal instantiation ••P P ((gg) for ) for gg an an arbitrary arbitrary element element ∴∀ ∴∀xx P P ((xx) ) universal generalization universal generalization ••∃∃xx P P ((xx) ) existential instantiation existential instantiation ∴ ∴P P ((bb) for some element ) for some element bb ••P P ((cc) for some object ) for some object cc ∴∃ ∴∃xx P P ((xx) ) existential generalization existential generalization 4 Summer 2007 S. Abu Diab 19 Ch1- 1.5: Rules of Inference • Unproblematic rules: – Univ. instantiation – Existential generalisation • Be more careful with: – Universal generalisation (“any object” in premise) – Existential instantiation (“new object” in conclusion) Summer 2007 S. Abu Diab 20 Next Ch1- 1.6: Introduction to Proofs Summer 2007 S. Abu Diab 21 Ch1- 1.6: Introduction to Proofs Definition: A proof is a valid argument that establishes the truth of a mathematical statement Summer 2007 S. Abu Diab 22 Ch1- 1.6: Introduction to Proofs Terminology: • A theorem is a statement that can be shown to be true. In mathematical writing, the term theorem is usually reserved for a statement that is considered somewhat important. • Less important theorems are sometimes called propositions. (both can be referred to as facts or results) Summer 2007 S. Abu Diab 23 Ch1- 1.6: Introduction to Proofs • We demonstrate that a theorem is true using a proof. • A less important statement that is helpful in the proof of other results is called a lemma. • A corollary is a theorem that can be established directly from a theorem that has been proved. Summer 2007 S. Abu Diab 24 Ch1- 1.6: Introduction to Proofs • An axiom (or postulate) is a statement that is assumed true without proof. • A conjecture is a statement that is proposed to be a true statement usually on the basis of some partial evidence. • A theory is the set of all theorems that can be proven from a given set of axioms. 5 Summer 2007 S. Abu Diab 25 Ch1- 1.6: Introduction to Proofs • Methods for Proofs: - Direct Proof - Proof by contraposition - Proof by contradiction - Exhaustive proof or proof by cases (section 1.7) Refer to worksheet 1 Summer 2007 S. Abu Diab 26 Ch1- 1.7: Proof methods and Strategy You may also want to know terms like: -begging the question or circular reasoning (one or more step of the proof are based on the truth of the statement being proved. In other words, a statement is proved using itself or a statement equivalent to it) -without loss of generality or WLOG (after proving one case of a theorem, wlog means that no additional argument is needed for the other cases. I.e, other cases follow in a similar way) wrong Use carefully Summer 2007 S. Abu Diab 27 Ch1- 1.7: Proof methods and Strategy -existence proof (can be constructive or non-constructive) (show that an element x with a desired property exists i.e., ∃x P (x) ) -uniqueness proof (shows that if y x then y does not have the desired property) ≠ Summer 2007 S. Abu Diab 28 Ch1- 1.7: Proof methods and Strategy -forward reasoning (start from the premises use certain steps to reach the conclusion) -backward reasoning (find a statement that will lead to the conclusion and move backward in the same way to reach the premises) WARNING: You must never start from the conclusion or assume that it is true. Summer 2007 S. Abu Diab 29 Ch1- 1.7: Proof methods and Strategy • Exercise: Use backward reasoning to show that the arithmetic mean of two positive real numbers x and y, is greater than their geometric mean, that is xy y x ≥ + 2 Summer 2007 S. Abu Diab 30 Ch1- 1.7: Proof methods and Strategy • Other proof methods and strategies will be seen later in this course!