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Unit 2: The Structure of Sentential Logic Friday, September 9, 2011 TA: Karin Howe Last week… • Unit 1 – Propositions • These are also known as "declarative sentences" (Klenk) or statements (all of these terms can be used more or less interchangeably in the context of this course) – Argument (and Statement) Forms, and Argument Instances – Inductive vs. Deductive Reasoning – Validity, Invalidity and Soundness – Counter-examples (for proving invalidity) This week … • Unit 2 – More on declarative sentences (statements/propositions) • Simple vs. compound sentences – Sentential operators – Finding the major operator (and why we care) Sentential Logic • • • This is also known as "propositional logic." In sentential logic, we will learn how to analyze arguments into simple sentences/propositions and operators. The two basic units: 1) The simple sentence 2) The sentential operator Simple and Compound Sentences • A compound sentence is a (declarative) sentence that contains another complete declarative sentence as a component. (Klenk, pp. 23, 30) • When is one sentence a component of another? – A sentence is a component of another if, whenever the first sentence is replaced by a third declarative sentence, the resulting whole is still a grammatical sentence. (Klenk, pp. 23, 31) • A simple sentence is a sentence that is not compound: it does not contain another complete declarative sentence as a component. (Klenk, pp. 23, 30) Examples: Simple Sentences 1. 2. 3. 4. 5. 6. The kitten frolicked in the teapot. The kitten knocked over the teapot. I bought a new teapot yesterday. Death likes cats. Cats are nice. Kangaroos can fly. Examples: Compound Sentences • Example: Either the kitten knocked over the teapot or it is raining tea in here. • Recall that: A sentence is a component of another if, whenever the first sentence is replaced by a third declarative sentence, the resulting whole is still a grammatical sentence. (Klenk, pp. 23, 31) • Try it out! – Either ____________ or it is raining tea in here. – Either the kitten knocked over the teapot or _________. • Note that it is not necessary for the new sentence to make sense. – Either the kitten knocked over the teapot or I like bananas. How many components do compound sentences have? • It varies -- some have two components, some have more than two, and some have only one component! • Examples of compound sentences with only one component: – I don't like bananas. – I believe that kangaroos can fly. – I am 86.0989% sure that kangaroos can fly. Compound or simple? 1. Jacques went to the store, and the store sells baguettes. 2. Libya is experiencing political instability. 3. Libya and Egypt are in Africa. 4. Obama loves peanut brittle and roasted puppy. 5. Rosa and Steve are hooking up. 6. Carlos and John are gay and married (to one another). 7. Canada is not defenseless. 8. Dinosaurs go roar. Sentential Operators • Recall that 'Obama loves peanut brittle and roasted puppy' has the structure: ___________ and ____________. • The 'and' here is playing the role of an operator. It operates on the simple sentences: here, it brings two of them together into a conjunction. (It conjoins them.) – To put it roughly, it asserts that both are the case. • Klenk defines an sentential operator as: "an expression containing blanks such that when the blanks are filled with complete sentences, the result is a sentence." (Klenk, pp. 26, 31) • Recall some earlier examples: – I don't like bananas. • It is false that I like bananas. – I believe that kangaroos can fly. • I believe that kangaroos can fly. – I am 86.0989% sure that kangaroos can fly. • I am 86.0989% sure that kangaroos can fly. • Claim: There are an infinite number of operators. • However, we will only be concerned with five of them in sentential logic. (phew!) Your Five New Best Friends • I like to eat apples and bananas. ______ and _______ (conjunction) • I like to eat apples or bananas. _________ or ________ (disjunction) • If I like to eat apples, then I like to eat bananas. If ________ then ________ (conditional) • I like to eat apples if and only if I like to eat bananas. ________ if and only if _________ (biconditional) • I don't like to eat bananas. It is false that _______ (negation) Symbolizing Statements • We will symbolize simple statements using capital letters (A, B, C, etc.) Example: A = I like aardvarks • We symbolize the sentential operators as follows: "____ and _____" using the dot (•): ____ • ____ "____ or ____" using the wedge (): ____ ____ "if ____ the ____" using the horseshoe (): ____ ____ "____ if and only if ____" using the triple bar (): ____ ____ "It is false that ____" using the tilde (~): ~ ____ Symbolizing Statements: Examples • Dictionary: – A = I like to eat apples – B = I like to eat bananas • • • • I like to eat apples and I like to eat bananas. A•B I like to eat apples or I like to eat bananas. AB If I like to eat apples, then I like to eat bananas. A B I like to eat apples if and only if I like to eat bananas. A B • It is false that I like to eat bananas. ~B A word about negations • The negation symbol (~) does not join two components, like our other operators do. • The negation symbol operates on only one component. (Recall the form of our negation operator: It is false that _____) • Thus, when we symbolize negations, we do not need extra punctuation: "It is false that p" will be symbolized as ~p. • Note: the component that is negated can be a compound component! – Example: It is false that I like apples and bananas. • Thus, the following two statements are different statements: ~(A • B) I don't like apples and bananas. (~A • B) I don't like apples but I do like bananas. Practice Makes Perfect: Simple or 1. Compound? Cats enjoy getting into teapots if and only if they have warm tea in them. 2. The gerbil likes to get into the teapot whether or not it has warm tea in it. 3. Kangaroos do not like to get into teapots. 4. The kitten's mother tries to keep the kitten out of the teapot. 5. Neither the cat nor the gerbil are fond of baths. 6. The cat and the gerbil will need a bath if they keep playing in teapots. 7. Life on Earth is doomed if gerbils do not stay out of our teapots. 8. The gerbil lay down in the teapot and took a nap. 9. The cat likes to play in the teapot with the gerbil. 10. The gerbil will have to either sink or swim if there is a lot of tea in the teapot. Finding the Major Operator • The major operator of a formula is the one that determines the overall form of the sentence. • Find the major operator in the following formulas: – (A B) – [(A • B) (C • D)] – {[A (B • C)] [A (C • D)]} The Two Chunk Rule • The Two Chunk Rule says: "Once more than one logical connective symbol is necessary to translate a statement, there must be punctuation that identifies the main connective of a symbolic statement. In addition, there cannot be any part of a statement in symbols that contains more than two statements, or chunks of statements, without punctuation." • Examples: – (A B C) – [(A B) C] – [A (B C)] (wrong!) (correct) (correct) WFFs (Well-Formed Formulas) • • WFF (pronounced "woof") stands for wellformed formulas A well-formed formula in sentential logic is defined recursively as follows: 1) (Base clause) Any statement constant is a WFF 2) (Recursion clause) If and are any WFFs, then all the following are also WFFs: (a) ( ), (b) ( ), (c) ( ), (d) ( ), and (e) ~. 3) (Closure clause) Nothing will count as a WFF unless it can be constructed according to clauses (1) and (2). Why do we care about the major operator?? • • We care about the main connective (major operator) because it tells us what kind of logical statement we are looking at. (e.g., if we see that the main connective is a "," then we know that the statement is a disjunction) This is important in two different contexts (preview of coming attractions): 1. Working With Truth Tables o It lets us know what the truth values of the statement are at each line of the truth table. We look at the main connective to tell us the truth value of that statement as a whole (on each line of the truth table). 2. Working With Derivation Rules o It tells us what derivation rules we can use on that statement - either to take it apart or to build it up (or to convert it into another logically equivalent statement). Practice Makes Perfect: Finding the major operator 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. A (B C) (A ~C) (~B A) ~(B A) ~(~B A) [A (B C)] [(A B) ~C] ~(~A ~C) (~~A ~C) 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. ~~(A ~C) [B ~(A C)] [A ~(C ~D)] [~(B C) ~D] [B ~(~B A)] ~[A ~(B ~D)] [(~C B) (A ~C)] [~(C B) (A ~C)] ~[(C B) (A ~C)] [~(C B) ~(A ~C)]