Lecture 1-2: Propositional Logic

Propositional Logic Deriving a logical conclusion by combining many propositions and using formal logic: hence, determining the truth of arguments. Definition of Argument: –An argument is a sequence of statements in which the conjunction of the initial statements (called the premises/hypotheses) is said to imply the final statement (called the conclusion). An argument can be presented symbolically as (P 1 Λ P 2 Λ... Λ P n )  Q where P 1, P 2,..., P n represent the hypotheses and Q represents the conclusion.

Valid Argument What is a valid argument?  When does Q logically follow from P 1, P 2,..., P n. Informal answer: Whenever the truth of hypotheses leads to the conclusion Note: We need to focus on the relationship of the conclusion to the hypotheses and not just any knowledge we might have about the conclusion Q. Example: –P 1 : Neil Armstrong was the first human to step on the moon. –P 2 : Mars is a red planet And the conclusion –Q: No human has ever been to Mars. –Thus, P 1 Λ P 2  Q is not a valid argument

More Examples Example 1 –If George Washington was the first president of the United States, then John Adams was the first vice president. George Washington was the first president of the United States. Therefore John Adams was the first vice president. (A -> B)  A -> B (Is this a valid argument? A: Yes, B: No) Example 2 –George Washington was the first president of the United States. Thomas Jefferson wrote the Declaration of Independence. Therefore, Thomas Jefferson was the first vice president. Is this a valid argument? A: Yes, B: No

Valid Argument Definition of valid argument: –An argument is valid if whenever the hypotheses are all true, the conclusion must also be true. –i.e. (P 1 Λ P 2 Λ... Λ P n )  Q is Tautology How to arrive at a valid argument? –You prove it to be true using either equivalence rules or inference rules

Equivalence Rules These rules state that certain pairs of wffs are equivalent, hence one can be substituted for the other with no change to truth values. The set of equivalence rules are summarized here: ExpressionEquivalent toAbbreviation for rule R V SS V RCommutative - comm (R V S) V Q (R Λ S) Λ Q R V (S V Q) R Λ (S Λ Q) Associative- asso (R V S) (R Λ S) R Λ S R V S De-Morgan’s Laws De-Morgan R  SR V S implication - imp R (R) Double Negation- dn PQPQ(P  Q) Λ (Q  P) Equivalence - equ

Inference Rules Inference rules allow us to add a wff to the last part of the proof sequence, if one or more wffs that match the first part already exist in the proof sequence. Note: Inference rules do not work in both directions, unlike equivalence rules. FromCan DeriveAbbreviation for rule R, R  S SModus Ponens- mp R  S, SR Modus Tollens- mt R, SR Λ SConjunction-con R Λ SR, SSimplification- sim RR V SAddition- add

Examples Example for using Equivalence rule in a proof sequence: –Simplify (A V B) V C 1. (A V B) V C 2. (A Λ B) V C1, De Morgan 3. (A Λ B)  C2, imp Example of using Inference Rule –If it is bright and sunny today, then I will wear my sunglasses. Modus Ponens It is bright and sunny today. Therefore, I will wear my sunglasses. Modus Tollens I will not wear my sunglasses. Therefore, it is not (bright and sunny) today.

Quick Practice What inference rule is illustrated by the argument give? Argument: If Martina is the author, then the book is fiction. But the book is nonfiction. Therefore, Martina is not the author. A.Simplification B.Conjunction C.Modus ponens (From P,P->Q derive Q) D.Modus tollens (From P->Q,Q’ derive P’)

Practice Prove that (P  Q)  (Q  P) is a valid argument – This means, if P  Q is true, then Q’  P’ is also true.

More Inference Rules These rules can be derived using the previous rules. They provide a faster way of proving arguments. FromCan DeriveName / Abbreviation P  Q, Q  RP  R Hypothetical syllogism- hs P V Q, PQDisjunctive syllogism- ds P  QQ  P Contraposition- cont Q  PP  Q Contraposition- cont PP Λ PSelf-reference - self P V PPSelf-reference - self (P Λ Q)  RP  (Q  R) Exportation - exp P, PQInconsistency - inc P Λ (Q V R)(P Λ Q) V (P Λ R)Distributive - dist P V (Q Λ R)(P V Q) Λ (P V R)Distributive - dist

Group Exercise Write down the propositional form of the following argument: –If my client is guilty, then the knife was in the drawer. Either the knife was not in the drawer or Jason Pritchard saw the knife. If the knife was not there on October 10, it follows that Jason Pritchard didn’t see the knife. Furthermore, if the knife was there on October 10, then the knife was in the drawer and also the hammer was in the barn. But we all know that the hammer was not in the barn. Therefore, ladies and gentlemen of the jury, my client is innocent.