Week 8 Discussion: Ch 8, TA: Karin Howe PHIL 121 Spring 2012

Finding the Major Operator The major operator of a formula is the one that determines the overall form of the sentence (Klenk, Understanding Symbolic Logic, p. 30) Find the major operator in the following formulas: –A  B –(A B)  (C D) –[A  (B C)]  [A  (C D)]

WFFs (Well-Formed Formulas) WFF (pronounced "woof") stands for well-formed formula 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). (Formulas that are not constructed according to clauses (1) and (2) are called non-WFFs)

The Two Chunk Rule The Two Chunk Rule says: "Once more than one logical connective symbol (note: negations do not function as connectives!) 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." (Jim Schofield, Introduction to Logic, Onondaga Community College) Examples: –A  B  C(wrong!) –(A  B)  C(correct) –A  (B  C) (correct)

Why do we care about the major operator?? We care about the 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 (somewhat a preview of coming attractions): 1.Working With Truth Tables oIt 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 oIt 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: WFFs and Finding the Major Operator 1.A 2.~~A 3.A ~C 4.~B  A 5.~(B  A) 6.~(~B  A) 7.A  B  C 8.(A  B)  ~C 9.~(~A  ~C) 10.~~A  ~C 11.~~(A  ~C) 12.B  ~(A  C) 13.A  ~(C  ~D)] 14.~(B  C)  ~D 15.B  ~(~B  A) 16.~[A  ~(B  ~D)] 17.(~C  B)  A  ~C 18.~(C  B)  (A  ~C) 19.~[C  B  (A  ~C)] 20.~(~C  B)  ~(A  ~C)

Truth Tables This week in class we learned the basic truth tables for the four basic operators we have so far: conjunction (  ), disjunction (  ), negation (~), and conditionals (  ) Every full truth table will have 2 n rows, where n = the number of atomic statements in the formula or argument (more on truth tables and arguments next week) We can use truth tables to assess the truth or falsity of statements under all possible truth value assignments

Example: (p  q)  r pqr p  q(p  q)  r TTTTT TTFTF TFTFT TFFFT FTTTT FTFTF FFTTT FFFTF

Practice! Build truth tables for the following statements 1.~(P  Q) 2.~P  Q 3.~(P  Q) 4.~P  ~Q 5.(P  Q)  (R  S) 6.[P  (R  S)]  [Q  (R  S)]

Another Method for Finding the Truth or Falsity of a Formula Sometimes, rather than trying to find out all the possible truth values of a particular formula, what we want to do is find out the truth value of the formula under a particular assignment of truth values. We could do a truth table to find out, using only that particular truth value assignment However, there is another (easier?) way to find out the truth or falsity of a formula, given a particular assignment of truth values

Klenk's Tree Method for Computing Truth Values 1.Place the truth values of the simple sentences immediately above the atomic formulas. (A, B, C, etc.) 2.Check the statement for any negations attached to atomic formulas (~A, ~B, ~C, etc.) Write the truth value for these negations immediately below them. 3.Starting with the smallest subformulas and working your way out, determine the truth values of each subformula based on the truth values of its subformulas and the truth table for the major operator of the subformula. Write these truth value below the subformula, connecting it to the atomic formulas with arrows. "Bring down" the truth values for any atomic formulas that remain to be dealt with. Repeat Step 3 until you have worked your way out to the major operator of the formula as a whole.

Example 1 ~{[(A  B)  Z]  [(B  Z)  ~A]} Let A and B be true, and let Z be false TTTTFFStep 1 F Step 2 FT T Step 3a Step 3b F F F Step 3c Step 3d T Done!!

Example 2 Let A and B be true, and let X, Y and Z be false ({[(A  B)  X]  Z}  Y)  {[(X  Z)  A]  X} TTTFFFFFF Step 1 Step 2 (skip) T TT FFF F Step 3a Step 3b F FFF T Step 3c TF F F Step 3d Step 3e T Done!!

Practice! A, B, C are true; X, Y, Z are false 1.(A  X)  (B  Y) 2.~B  ~(X  Y) 3.[X  (Y  Z)]  [(A  B)  C] 4.~{A  [~A  (~B  X)]} 5.({[(A  B)  X]  Z}  Y)  [(X  Z)  A]

Practice with Unknown Truth Values A, B, C are true; X, Y, Z are false; G, H, I are unknown 1.(G  H)  ~A 2.~~A  [X  (G  ~A)] 3.H  (G  H) 4.H  (G  ~G) 5.(X  Y)  [(G  A)  ~(X  Z)]