1.1 a Introduction to Logic

Problem of the Day The weather at the Cape of Good Hope makes it a notorious location where many ships have been lost to the sea. Philip Wood, the famous diver, has discovered four ships that sank in the same spot, one on top of each other. With some skill, Philip was able to determine the name, captain, cargo, destination and year built for all of the ships. The question is, can you? 1. During his first exploration of the site, Philip determined there were four ships - the Red Rover, the ship built in 1743, the ship captained by Quigley and the ship carrying tea that was bound for North America. 2.The ship that carried a cargo of saffron was built after the ship that carried a cargo of tea. 3. An entry in one ship's log found at the site indicated that the ship built in 1522 was carrying a valuable cargo of gold. 4. One ship was carrying botanical specimens to France. Philip determined that this was not the Royal Bride. 5. The Scarlet Queen, a pirate ship, was certainly built before the nineteenth century but not as early as the sixteenth. The captain of this ship was the famous rogue Clubfoot. 6. The Wanderer was built after the Royal Bride. The Royal Bride's cargo was not tea and the Royal Bride was not travelling to the South Seas. 7. Captain Bolton's ship was built before Clubfoot's and before The Royal Bride.

Problem of the Day Ships: Red Rover, Royal Bride, Scarlet Queen, Wanderer Year Built: 1522, 1688, 1743, 1817 Captains: Bolton, Clubfoot, Quigley, Vickers Cargos: Gold, Saffron, Specimens, Tea Destinations: England, France, North America, South Seas Ship NameYear BuiltCaptainCargoDestination Ship NameYear BuiltCaptainCargoDestination Royal Bride1743VickersSaffronEngland Scarlet Queen1688ClubfootTeaNorth America Red Rover1522BoltonGoldSouth Seas Wanderer1817QuigleySpecimensFrance

Propositions and Truth Value Definition: Logic is the analysis, without regard to meaning or context, of the patterns of reasoning by which conclusions are validly derived from a set of premises. Consider the statement “Today is Monday.” Does it have a Truth Value? Yes, it has a truth value. It is either True or False. We can connect several statements together using logic connectives to form an argument. For Example: If today is Monday, then it will rain. It is raining, therefore it is Monday. Is this a valid argument? Why or Why not? Write out your response… Think-Pair-Share

Definition: A statement is a Declarative sentence, which has a truth value; that is, it is either True or False but not both true and false. Propositions and Truth Value

Example A: Determine whether the following sentences are statements. (a) 5 is a prime number. (b) = 8 (c) Is Indianapolis the capital of Indiana? (d) x < 12 (a) and (b) are statements. They can be assigned a truth value of True and False. (c) and (d) are not statements because they cannot be give a truth value. We could not say that they are true or false because (c) is not declaring anything, and (d) cannot be evaluated as true or false without knowing the value of x. Propositions and Truth Value

Example B: Part 1 - Which of the following sentences are statements? (a) Wyoming is a state in the United States (b) Today is not Monday. (c) Watch out for the guard dog! (d) Dalmatians are a breed of dogs. (e) A gambling die has eight sides. (f) 3 + x = 12 __________________________________________________ Part 2 – Determine the truth value of each sentence in Part 1 that is a statement. The Truth Value of the statements are… Propositions and Truth Value

Example C: Explain why each of the following is not a statement? (a) Water the garden this afternoon. ______________________________________________. (b) ______________________________________________. (c) Can you come to the party? ______________________________________________. (d) 6 + x = 36 ______________________________________________. Propositions and Truth Value

Because we want to study the structure and validity of the argument, apart from its context, we can represent statements with symbols. The symbolic form of the statement is made when symbols are substituted in. In the study of logic, we typically represent statements with lowercase letters such as p, q, and r. For example, assign p, q, and r … p: Today is Monday. q: It is raining. r: We will see a movie today. Propositions and Truth Value

Given a statement, we can change the statement’s truth value by negation. In language we can negate a statement by using the word “Not” in the sentence. Another way would be to start the sentence with “It is not the case that…”. “John is 21 years old.” Negate with “not” __________________________________________. Negate the same sentence with, “It is not the case that ___________________________.” You may remember negation taught in geometry used the symbol ~. We will follow the text’s notation and use ¬ for meaning negation. Propositions and Truth Value

Definition: The negation of a statement p, denoted ¬ p, and read not p, changes the truth value of p. If p is true, then ¬p is false; and if p is false, then ¬p is true. Propositions and Truth Value

We can also combine simple statements using connectives (such as and, or, if…then, etc.) to form a compound statement. Example D: Circle the connectives in each of the following compound statements. (a) Today is Monday and it is raining. (b) We will see a movie today or it is raining. (c) If today is Monday, then we will see a movie today. Write a compound statement and have your table partner find and circle the connective(s). Propositions and Truth Value

Definition: A statement is compound if it can be separated into two parts, each of which is a statement joined by a connective. A statement that cannot be separated into multiple parts is a called a simple statement. Propositions and Truth Value

Given two simple statements, say p and q, we can define various compound statements using different connectives. Let’s first define two connectives: Definition: The conjunction of two statements “p and q”, denoted p ∧ q, is true if both p and q are true; Otherwise, it is false. The disjunction of two statements “p or q”, denoted p ∨ q, is true if either p or q is true; Otherwise it is false. Propositions and Truth Value

We must understand that in mathematics the words and and or are very different and not interchangeable. Consider the statement “Enrique selected two of the red and blue marbles.” This statement does not necessarily mean that both a red and a blue marble were selected; in this case, both could be red or both could be blue. Also, the words and and or may not appear in the statement but be implied. For example, “Sebastian ate the banana, but not the apple.” The word but acts as a conjunction setting up the negation. Rewrite it as a compound statement with the same meaning. __________________________________________________________. In symbolic form, this statement would be p ∧ ¬q, where p: Sebastian ate the banana. q: Sebastian ate the apple. Propositions and Truth Value

Example B: Identify the simple statements of the compound statement. Express the statement in symbolic form. The time on the clock has expired and our team did not win the game. p: _________________________________________________________________ q: _________________________________________________________________ ¬q: _______________________________________________________________ Symbolic form __________________________________ Propositions and Truth Value

Open text and work #4 and #5 from page 5 Propositions and Truth Value

Write out complete sentences using p: Inga has two aces in her hand. q: Inga has a full house. (a) p ∧ q ________________________________________________________________________. (b) p ∨ q _________________________________________________________________________. (c) ¬q _________________________________________________________________________. (d) p ∧ ¬q _________________________________________________________________________. Propositions and Truth Value

Work #10 and #11 from pg 11 Exercises. IP – Textbook Pg. 11: problems Propositions and Truth Value