LOGIC & CRITICAL THNKG
LOGIC & CRITICAL THNKG PHIL 1313
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Dr. Alisha Osinski
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This 0 page Class Notes was uploaded by Dr. Alisha Osinski on Sunday November 1, 2015. The Class Notes belongs to PHIL 1313 at Oklahoma State University taught by Lawrence Pasternack in Fall. Since its upload, it has received 26 views. For similar materials see /class/232939/phil-1313-oklahoma-state-university in PHIL-Philosophy at Oklahoma State University.
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Date Created: 11/01/15
Chapter 9 TruthFunctional Logic Overview Truth Functional logic also called propositional or sentential logic represents propositions and arguments through variables and operators For instance lfJohn is happy then he must have had a good day at work would be represented as H gt W H I John is happy W I good day at work gt I the if then relation lfJohn is happy then either he must have had a good day at work or he got back together with his girlfriend would be represented as H gt W V G H I John is happy W I good day at work G I got back together with his girlfriend gt I the if then relation v I the or relation As truth functional suggests variables can only have the values true or false And the operator are DEFINED according to truth functional rules I 39l hconlyvalidsymbolsue xe vmemmmam I wl chmemsnot I dw chMamsmi dxennh mwmqusegtmahmhoes nbob I vvhichmemsot I atme I ampchuckcu I Mummthcdi sioubemeenpwnisesmdcmdm I Herc smmpleofapmposi mnAvBampGCampDv m I chilllookatargumcntsncxtclass I A truth table depicts all possible truc false combinations of vatiables and the resultant values detem ned by each operator I Truth tables will be used to de ne the operators I I mmplen eformmucsk I mmmmmmmm rhuduprumdnmwn mumum num Mym nta pmumo nmdh The Whithenunhetofmiu2 whm s Mmhu mmdnhthenmhaofm BeghwithshngofTFon em mMMWW TP mumqknhnmm gmm mmHHE NH H39 H E wmwmaaq mmqqmmee mHmHmHmH De nitions of Operators P 13 P Q P850 Ple P gtQl T F T T T T T F T T F F T F F T F T T F F F F T Negation It changes the truth value of the claim variable to which it attaches Where the claim variable is T negation makes it F Where the variable is F negation makes it T It also operates over the contents of parentheses eg PampQ amp Conjunction The rule for this operator is as follows a conjunction is T if and only if both conjuncts are T Hence the conjunction is F if and only if either conjunct is V Disjunction The rule for this operator is as follows a disjunction is T if and only if at least one of the disjuncts is T Hence the disjunction is F if and only if both disjuncts are gt Conditional The rule for this operator is as follows a conditional is T mlm its antecedent is T and its consequent is false In other words a conditional is F if and only if its antecedent is T and its consequent is F The antecedent is the claim variable preceding the gt The consequent is the claim variable following the gt The Conditional gt i I The ONLY time a conditional relation is P Q Q FALSE is if the antecedent is TRUE and T the consequent is FALSE E I This is the hardest operator to understandgl A conditional relation is TRUE even if the antecedent is FALSE I Think of it this way The conditional is saying the antecedent is true the consequent 772ml e true The conditional claim is therefore incorrect When the antecedent is true and the consequent is false But it is making no claims about Whether or not the consequent is false When the antecedent is false So the conditional is not incorrect in these cases Given that logic is truth functional ie given that everything must have the status of either T or E if it is not incorrect P the value must be T H H H T T T F F T T T T F F F F F T F T F F T T T F F F F F F F T T T T F T F T F T T F T F F T T F F F F F F T F F F Notes 1 Place the values in columns under the operators 2 It is generally best to list the operatoxs From the most embedded in parentheses to the least embedded so farthest m the right should appear the main operator ddddddddllllillL ddddllllddddlllym ddllddllddll dl I Llllllllddddlllg aldl ldl ldl ld Translation Exercise Ben likes ice cream J Jerry likes ice cream M Marcia likes ice cream 1 Ben andJerry like ice cream or Marcia likes ice cream 2 It39s not the case that both Ben andJerry like ice cream 3 Either Ben doesn39t like ice cream orJerry doesn39t like ice cream lllUd A There are apples in the bag B There are bananas in the bag C There are cookies in the bag D There are doughnuts in the bag I 4 If there are bananas in the bag then there are cookies in the bag I 5 If there are apples and bananas in the bag then there are cookies and doughnuts in the bag I 6 If there are apples and bananas in the bag then there are either cookies or doughnuts in the bag P The plumber came C The carpenter came L The leak has been fixed G A good job was done I 7 Either the plumber or the carpenter came I 8 A good job was done only if the plumber and the carpenter came I 9 Either the plumber came and the leak was xed or the carpenter came and the leak was not xed A Alice went for a walk C Charles went for a walk B Betsy went for a walk D Doug drove I 10 Alice went for a walk if both Charles and Betsy did I 11 If either Doug drove or Charles went for a walk then neither Alice nor Betsy went for a walk I 12 If neither Alice nor Betsy went for a walk then Doug drove but if either Alice or Betsy went for a walk then Doug did not drive 1BampjvM aGPampC 2gtBampJ 9 PampLvltcacLgt u B v 10 C 8 B gtA 3 Bv1 11DvC gtAVB 4B C ag va bAampB 5 A ampB C ampD 12 AvB DampAvn D 6 AampB gt CvD 7PvC kwptnckofthcbmkm nmzyouaccdmmmmwn timoadetofapcmmmdwuidmhig ty dammloga ycq mm m t md myon n ow mmaym equivalent YOU WILL NEED TO DO A LOT OF PRACTICE TO MASTER WHAT FOLLOWS I There are two table methods for testing for validity the long and the short The long method can be very long given the rule for creating rows based upon the number of variables The short method requires a keencr grasp of logical operators The Long Method 1 Create a column for each claim variable that appears anywhere in the argument 2 Look at the rst premise create columns for each operation with its variables It is best to identify the main operator of the premise and place its column to the right of the set of operations contained Within it 3 Move to the next premise and follow 2 then move to the conclusion and follow 4 Alternate Ts and Fs in the right most claim variable column Alternate pairs of Ts and Fs in the next to right most claim variable column Alternate sets of four Ts and Fs in the second next to right most claim variable column Alternate sets of eight Ts and Fs in the third to next right most claim variable column and so forth 5 Fill in the operation columns by looking to the variable values in each row and according to the operator truth table rules 6 For the final step ONLY consider the columns that represent each premise as a Whole and the conclusion as a Whole That is ignore the columns for the variables and the operators that are not for each premise its main operator IF there is a ROW where the conclusion as a Whole is F and that same ROW has the value T for EVERY column that represents a premise as a whole the argument is INVALID OTHERWISE the argument is VALID See pages 304 305 as an example The Short Method The goal of the short method is to produce just one row the row whose values make the argument INVALID If this is possible the argument is invalid If not possible the argument is valid The main challenge here is only entering T s and F s when the logical structure of the argument requires them 1 Create a column for each claim variable that appears anywhere in the argument 2 Look over the argument and find those variables whose value is forced upon them as follows if the claim is a premise given the operators in the claim there is only one way that the values of the variables that compose the premise claim can make the premise TRUE if the claim is the conclusion given the operators there is only one way that the values of the variables that compose the conclusion claim can make the claim FALSE 3 For those variables whose values are forced upon them by 2 plug these values into the other occurrences of the same variables in the other claims and follow the logical structure of the claims to determine all other variable values forced upon them by the logical structure The goal remains to try to make all premises true and the conclusion false 4 If there is a set of values where all the premises are true and the conclusion false the argument is INVALID Otherwise it is valid GET VERY LOW GRADES ON THIS TEST TRUST ME I VE BEEN DOING THIS FOR YEARS YOU NEED TO PRACTICE TRANSLATING AND TESTING FOR VALIDITY WITH BOTH METHODS
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