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# Chapter 6 Notes PHIL-1210-05

Tulane

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This 7 page Class Notes was uploaded by Lisa Chupp on Sunday September 27, 2015. The Class Notes belongs to PHIL-1210-05 at Tulane University taught by Franklin (Frankie) Worrell in Fall 2015. Since its upload, it has received 91 views. For similar materials see Symbolic Logic 1210-05 in PHIL-Philosophy at Tulane University.

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Date Created: 09/27/15

61 Symbols and Translations operatorsconnectives are the symbols that convert arguments into formulas to convert an argument into a formula simple statements must first be recognized 0 Simple statements a statement that does not contain other statements Example 0 The house is red 0 A dolphin is a mammal Dogs are fantastic 0 Simple statements are symbolized by upper case letters Example 0 The house is red R A dolphin is a mammal D 0 Dogs are fantastic X Note that any uppercase letter can symbolize any simple statement but making the symbol similar to What a simple statement is saying may be helpful 0 A compound statement contains at least on simple statement Example 0 My house is not blue 0 A dolphin is a mammal and lives in the water 0 If dogs are fantastic then they are lovable 0 In compound statements containing one ore more simple statement make sure that the symbolic let ters are different Example 0 My house is not blue It is not the case that B Note that the simple statement here is My house is blue 0 A dolphin is a mammal and lives in the water D and W The two simple statements in this compound statement are a dolphin is a mammal and A dolphin lives in the water 0 If dogs are fantastic then they are lovable If X then L The expressions surrounding these representative letters are replaced by operators Each operator has a logical function that translates into words or phrases Operator Name Logical function Translation tilde negation notit is not the case that clot conjunction andalsomoreover V wedge disjunction orunless 3 horseshoe implication if then only if E triple bar equivalence if and only if Note that tilde is the only single operator because it negates a simple statement While the other operators link statements The tilde negates the preposition it precedes When using operators to translate compound statements a formulaic phrase is made My house is not blue It is not the case that B B A dolphin is a mammal and lives in the water D and W D 0 W Jane reads books or plays on the computer B or C B v C If dogs are fantastic then they are lovable If X then L X D L Katie studies if and only if there is a test K if and only if T K g T To be a proper statement no more than two prepositions can be used Parentheses are used in compound statements where there are more than two simple statements A statement can have numerous parentheses From inside out you use the following l and so on depending on the complexity of the statement for example 0 Example Cats dogs and rabbits require maintenance C and D and R C 0 D 0 R or C 0 D 0 R 0 Theses mean the same thing but the parentheses are nonetheless necessary If I feed my dog and take him on walks then he will be healthy If F and W then H F 0 W D H 0 The order here matters because or the main operator horseshoe Every statement has a main operator 0 Main operator the operator that has as its scope ever tying else in the statement The main operator determines what logic function a statement is 0 If the main operator is a tilde the logical function is a negation and so on Examples Negations main operator is a tilde P P Q P QVA Bl Conjunctions main operator is a dot P Q P Q P QVA Bl Q3H Disjunctions main operator is a wedge B v C B v C B A C v CDB Conditionals main operator is a horseshoe A 3 B A B D C a A Cv ABDC EA Biconditionalsmaterial equivalences main operator is a triple bar L E I L I E B L 13A Cl C D B Placement of parentheses is important To know where the parentheses goes look for commas which of ten indicate the main operator I suggest reading a formulaic statement out to insure that the parentheses are placed properly For example If I run and hide then the murder will not nd and eat me R H D F E This is correct because if you read it back the concepts that are linked together in the sentence are linked together by parentheses as well However R H D F E does not make sense If you were to convert this statement into a sentence it would be I run and if I hide then the murder will find me and the murder will eat me The goal here is to create wellformed formulas or WFF S To form WFF s simply follow the rules of this chapter 62 Truth Functions Statement variables lowercase letters that can stand for any statement 0 Example p can stand for H D F F and so on 0 Statement variables are used to create statement forms Statement form an arrangement of statement variables and operators that farther simplify a statement 0 Example p 3 q p and so on Truth Tables Truth tables show the possible truth values of a statement depending on the truth and falsity of the statement variables The potential amount of truth values depends on the amount of operators in a statement 0 This can be shown by 2An This is because with more statements there are more combinations of truth and false values Negations P P T F F T If P is true then tilde makes it false negation If P is false then tilde makes it true Conjunction P Q P Q T T T T F F F T F F F F This truth table shows that the only way for a conjunction to be true is if both statements are true This is because the dot indicates and Disjunction P Q PVQ T T T T F T F T T F F F The only way for a disjunction to be false is if both statements are false This is because the wedge indi cates or so a compound statement can be true if one simple statement or the other is true Conditional P Q P 3 Q T T T T F F F T T F F T This is the trickiest truth table Because horseshoe means if then you must keep in mind the order of the statements Horseshoe is all about cause and effect so think of a statement like this If you run a mile every day then you Will succeed in math If you actually do run a mile everyday and do not succeed in math then this statement is not true or Who ever told you this statement is a liar This corresponds With the only false truth value of horseshoe If this does not make sense to you I suggest simply memorizing that With horseshoe if the first statement is true and the second is false than the truth value is false Biconditional P Q P g Q T T T T F F F T F F F T Because the triple bar relies on equivalence then the truth value of each individual statement must match up for the truth value to be true Computing the truth value of longer propositions I nd it easiest to simply write below a compound proposition to nd the truth value and work from the inside out There are a few ways to nd the truth value you will have to gure out what works best for you For this section I will demonstrate what the book uses however the book does alter it s method as the proposition get more complicated Let us evaluate B R D C given that B and C are true and R is false First input the truth value using T for true and F for false B R D C T F D T Next evaluate the statements separately B R D C T F D T F D T Lastly use the last line to nd the nal truth value of the compound statement B R D C T F D T F D T T So the nal answer is True Order of operations here is key 63 Truth tables for Propositions The number of lines in a truth table depends on the number of different simple propositions The relationship can be expressed by 2An where n is the amount of different simple propositions Example B R D B Because there are 2 simple propositions use 22 to nd that there are 4 lines in the truth table Next write all the possible truth and false values beneath each letter Make sure that if the same letter ap pears twice the truth values line up BRDB T T T T F T F T F F F F Now work from the inside out This means that you rst must nd the truth value of what is in the inner most parentheses in this case that is the dot BRDB TTT T TFF T FFT F FFF F Using the concepts from the truth tables you can nd the truth values under the wedge Make sure that you use the truth yalue s under the dot to B R D B T T T T T T F F T T F F T T F F F F T F The column under the main operator represents the entire compound propositions Again there are other ways to set up truth tables this is they method that I prefer Classifying statements Compound statements can be one of three things 0 Tautologous logically true all true B R D B T T T T T T F F T T F F T T F F F F T F 0 Selfcontradictory logically false all false B R 3 B F T T T T T F T F F T T F F F T T F F F F F T F 0 Contingent at least one true at least one false B v R FT T T FT F F TF T T TF T F Comparing Statements Truth tables aid in comparing two propositions to each other There are four possible relationships between propositions Logically equivalent same truth value on each line BDR BVR TTT FTTT TFF FTFF FTT TFTT FTF TFTF Contradictory opposite truth value on each line B v R B DR TTT FFTTT TTF FFTTF FTT FTFTT FFF TTFFF Cmmmwm wmhwnbmhmcmwonwhmhmeumhwmwsmeM Hnm BVR lm TTT TTT TTF TFF FTT FTT FFF FTF imam am mmemn0mmonwmmhmeuthmMwebthm BI QJ33R TTT FFTTT TFF FFTTF FFT FTFTT FFF TTFFF

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