Propositional Logic 5) And Copyright 2008, Scott Gray

Conjunctions Statements joined by AND The component statements are called “conjuncts” AND is represented by the ampersand: & Copyright 2008, Scott Gray

English Equivalents Bear is black and Coda is yellow. Bear is black, but Coda is yellow. Bear is black, however Coda is yellow. Bear is black, moreover Coda is yellow. Bear is black although Coda is yellow. Bear is black yet Coda is yellow. Bear is black even though Coda is yellow. All of these may be symbolized: B & Y Copyright 2008, Scott Gray

AND Reminders We view the conjuncts as being commutative There are occasions where you will need parentheses: A & B → C does this mean (A & B) → C or A & (B → C) Copyright 2008, Scott Gray

AND Gotchas Not every English “and” is a logical AND Are the following two sentences equivalent? Marvin and Nancy are cousins Marvin is a cousin and Nancy is a cousin Sometimes an English and should be represented as an arrow: Give us the tools of war and we shall finish the job You must think, not blindly see a word in English and use a logic operator which we have given the same name Copyright 2008, Scott Gray

More for Your Notebook Common introduction to conclusions: therefore so hence thus consequently it follows that proves that shows that Copyright 2008, Scott Gray

More for Your Notebook, cont. Common introduction to premises: since because for These follow conclusions and introduce premises – which means that sometimes the conclusion is stated before the premises Copyright 2008, Scott Gray

Ampersand In You can form a conjunction statement from two statements If you have a sentence Q and a sentence B, then you may write down Q & B The justification column has the two conjunct lines and “&I”; i.e.: 2,6 &I Copyright 2008, Scott Gray

Ampersand Out From a conjunction derive either conjunct If you have a sentence Q & P, you may write down Q (you may also write down P) The justification has the conjunction line and “&O”; i.e.: 3 &O Copyright 2008, Scott Gray

Ampersand Notes A & (B & C) is equivalent to (A & B) & C Can you prove this? 1 A & (B & C) A 2 A 1 &O 3 B & C 1 &O 4 B 3 &O 5 C 3 &O 6 A & B 2,4 &I 7 (A & B) & C 6,5 &I Copyright 2008, Scott Gray

Sample Proofs P → Q, P → R, P ∴ R & Q 1 P → Q A 2 P → R A 3 P A 4 Q 1,3 →O 5 R 2,3 →O 6 R & Q 5,4 &I Copyright 2008, Scott Gray

Sample Proofs, cont. (P & R) → (Q → T), Q → P, Q & (P → R) ∴ T 1 (P & R) → (Q → T) A 2 Q → P A 3 Q & (P → R) A 4 Q 3 &O 5 P → R 3 &O 6 P 2,4 →O 7 R 5,6 →O 8 P & R 6,7 &I 9 Q → T 1,8 →O 10 T 9,4 →O Copyright 2008, Scott Gray

Sample Proofs, cont. P & Q, P → R, Q → T ∴ R & T 1 P & Q A 2 P → R A 3 Q → T A 4 P 1 &O 5 Q 1 &O 6 R 2,4 →O 7 T 3,5 →O 8 R & T 6,7 &I Copyright 2008, Scott Gray

Sample Proofs, cont. (P & (P → R)) & (R → T) ∴ T 1 (P & (P → R)) & (R → T) A 2 P & (P → R) 1 &O 3 P 2 &O 4 P → R 2 &O 5 R 4,3 →O 6 R → T 1 &O 7 T 6,5 →O Copyright 2008, Scott Gray

Assignments Read Chapter 3 Do all of the exercises Be sure to ask me questions if you don’t understand something or can’t solve a problem Copyright 2008, Scott Gray