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Symbolic Logic: The Language of Modern Logic Technique for analysis of deductive arguments Technique for analysis of deductive arguments English (or any) language: can make any argument appear vague, ambiguous; especially with use of things like metaphors, idioms, emotional appeals, etc. English (or any) language: can make any argument appear vague, ambiguous; especially with use of things like metaphors, idioms, emotional appeals, etc. Avoid these difficulties to move into logical heart of argument: use symbolic language Avoid these difficulties to move into logical heart of argument: use symbolic language Now can formulate an argument with precision Now can formulate an argument with precision Symbols facilitate our thinking about an argument Symbols facilitate our thinking about an argument These are called “logical connectives” These are called “logical connectives”

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Logical Connectives The relations between elements that every deductive argument must employ The relations between elements that every deductive argument must employ Helps us focus on internal structure of propositions and arguments Helps us focus on internal structure of propositions and arguments We can translate arguments from sentences and propositions into symbolic logic form We can translate arguments from sentences and propositions into symbolic logic form “Simple statement”: does not contain any other statement as a component “Simple statement”: does not contain any other statement as a component “Charlie is neat” “Charlie is neat” “Compound statement”: does contain another statement as a component “Compound statement”: does contain another statement as a component “Charlie is neat and Charlie is sweet” “Charlie is neat and Charlie is sweet”

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Conjunction Conjunction of two statements: “…and…” Conjunction of two statements: “…and…” Each statement is called a conjunct Each statement is called a conjunct “Charlie is neat” (conjunct 1) “Charlie is neat” (conjunct 1) “Charlie is sweet” (conjunct 2) “Charlie is sweet” (conjunct 2) The symbol for conjunction is a dot The symbol for conjunction is a dot (Can also be “&”) (Can also be “&”) p q p q P and q (2 conjuncts) P and q (2 conjuncts)

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Truth Values Truth value: every statement is either T or F; the truth value of a true statement is true; the truth value of a false statement is false Truth value: every statement is either T or F; the truth value of a true statement is true; the truth value of a false statement is false

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Truth Values of Conjunction Truth value of conjunction of 2 statements is determined entirely by the truth values of its two conjuncts Truth value of conjunction of 2 statements is determined entirely by the truth values of its two conjuncts A conjunction statement is truth-functional compound statement A conjunction statement is truth-functional compound statement Therefore our symbol “” (or “&”) is a truth- functional connective Therefore our symbol “” (or “&”) is a truth- functional connective

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Truth Table of Conjunction Truth Table of Conjunction pq p q TTT TFF FTF FFF Given any two statements, p and q A conjunction is true if and only if both conjuncts are true

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Abbreviation of Statements “Charlie’s neat and Charlie’s sweet.” “Charlie’s neat and Charlie’s sweet.” N S N S Dictionary: N=“Charlie’s neat” S=“Charlie’s sweet” Dictionary: N=“Charlie’s neat” S=“Charlie’s sweet” Can choose any letter to symbolize each conjunct, but it is best to choose one relating to the content of that conjunct to make your job easier Can choose any letter to symbolize each conjunct, but it is best to choose one relating to the content of that conjunct to make your job easier “Byron was a great poet and a great adventurer.” “Byron was a great poet and a great adventurer.” P A P A “Lewis was a famous explorer and Clark was a famous explorer.” “Lewis was a famous explorer and Clark was a famous explorer.” L C L C

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“Jones entered the country at New York and went straight to Chicago.” “Jones entered the country at New York and went straight to Chicago.” “and” here does not signify a conjunction “and” here does not signify a conjunction Can’t say “Jones went straight to Chicago and entered the country at New York.” Can’t say “Jones went straight to Chicago and entered the country at New York.” Therefore cannot use the here Therefore cannot use the here Some other words that can signify conjunction: Some other words that can signify conjunction: But But Yet Yet Also Also Still Still However However Moreover Moreover Nevertheless Nevertheless (comma) (comma) (semicolon) (semicolon)

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Negation Negation: contradictory or denial of a statement Negation: contradictory or denial of a statement “not” “not” i.e. “It is not the case that…” i.e. “It is not the case that…” The symbol for negation is tilde ~ The symbol for negation is tilde ~ If M=“All humans are mortal,” then If M=“All humans are mortal,” then ~M=“It is not the case that all humans are mortal.” ~M=“It is not the case that all humans are mortal.” ~M=“Some humans are not mortal.” ~M=“Some humans are not mortal.” ~M=“Not all humans are mortal.” ~M=“Not all humans are mortal.” ~M=“It is false that all humans are mortal.” ~M=“It is false that all humans are mortal.” All these can be symbolized with ~M All these can be symbolized with ~M

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Truth Table for Negation p~p TF FT Where p is any statement, its negation is ~p

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Disjunction Disjunction of two statements: “…or…” Disjunction of two statements: “…or…” Symbol is “ v ” (wedge) (i.e. A v B = A or B) Symbol is “ v ” (wedge) (i.e. A v B = A or B) Weak (inclusive) sense: can be either case, and possibly both Weak (inclusive) sense: can be either case, and possibly both Ex. “Salad or dessert” (well, you can have both) Ex. “Salad or dessert” (well, you can have both) We will treat all disjunctions in this sense (unless a problem explicitly says otherwise) We will treat all disjunctions in this sense (unless a problem explicitly says otherwise) Strong (exclusive) sense: one and only one Strong (exclusive) sense: one and only one Ex. “A or B” (you can have A or B, at least one but not both) Ex. “A or B” (you can have A or B, at least one but not both) The two component statements so combined are called “disjuncts” The two component statements so combined are called “disjuncts”

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Disjunction Truth Table pq p v q TTT TFT FTT FFF A (weak) disjunction is false only in the case that both its disjuncts are false

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Disjunction Translate: “You will do poorly on the exam unless you study.” Translate: “You will do poorly on the exam unless you study.” P=“You will do poorly on the exam.” P=“You will do poorly on the exam.” S=“You study.” S=“You study.” P v S P v S “Unless” = v “Unless” = v

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Punctuation As in mathematics, it is important to correctly punctuate logical parts of an argument As in mathematics, it is important to correctly punctuate logical parts of an argument Ex. (2x3)+6 = 12 whereas 2x(3+6)= 18 Ex. (2x3)+6 = 12 whereas 2x(3+6)= 18 Ex. p q v r (this is ambiguous) Ex. p q v r (this is ambiguous) To avoid ambiguity and make meaning clear To avoid ambiguity and make meaning clear Make sure to order sets of parentheses when necessary: Make sure to order sets of parentheses when necessary: Example: { A [(B v C) (C v D)] } ~E Example: { A [(B v C) (C v D)] } ~E { [ ( ) ] } { [ ( ) ] }

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Punctuation “Either Fillmore or Harding was the greatest American president.” “Either Fillmore or Harding was the greatest American president.” F v H F v H To say “Neither Fillmore nor Harding was the greatest American president.” (the negation of the first statement) To say “Neither Fillmore nor Harding was the greatest American president.” (the negation of the first statement) ~(F v H) OR (~F) (~H) ~(F v H) OR (~F) (~H)

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Punctuation “Jamal and Derek will both not be elected.” “Jamal and Derek will both not be elected.” ~J ~D ~J ~D In any formula the negation symbol will be understood to apply to the smallest statement that the punctuation permits In any formula the negation symbol will be understood to apply to the smallest statement that the punctuation permits i.e. above is NOT taken to mean “~[J (~D)]” i.e. above is NOT taken to mean “~[J (~D)]” “Jamal and Derek both will not be elected.” “Jamal and Derek both will not be elected.” ~(J D) ~(J D)

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Example Rome is the capital of Italy or Rome is the capital of Spain. Rome is the capital of Italy or Rome is the capital of Spain. I=“Rome is the capital of Italy” I=“Rome is the capital of Italy” S=“Rome is the capital of Spain” S=“Rome is the capital of Spain” I v S I v S Now that we have the logical formula, we can use the truth tables to figure out the truth value of this statement Now that we have the logical formula, we can use the truth tables to figure out the truth value of this statement When doing truth values, do the innermost conjunctions/disjunctions/negations first, working your way outwards When doing truth values, do the innermost conjunctions/disjunctions/negations first, working your way outwards

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I v S 1.We know that Rome is the capital of Italy and that Rome is not the capital of Spain. 1.So we know that “I” is True, and that “S” is False. We put these values directly under their corresponding letter I v S T F We know that for a disjunction, if at least one of the disjuncts is T, this is enough to make the whole disjunction T We put this truth value (that of the whole disjunction) under the v (wedge) I v S T F T

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Note When doing truth values, do the innermost conjunctions/disjunctions/negations first, working your way outwards When doing truth values, do the innermost conjunctions/disjunctions/negations first, working your way outwards Ex. Do ( ) first, then [ ], then finally { } Ex. Do ( ) first, then [ ], then finally { } Homework: Page Part I (try 5 of these) Page 310 Part II (try 10 of these) Homework: Page Part I (try 5 of these) Page 310 Part II (try 10 of these)

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