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## Symbolic Logic: The Language of Modern Logic

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**Symbolic Logic: The Language of Modern Logic**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. Avoid these difficulties to move into logical heart of argument: use symbolic language Now can formulate an argument with precision Symbols facilitate our thinking about an argument These are called “logical connectives”**Logical Connectives**• The relations between elements that every deductive argument must employ • Helps us focus on internal structure of propositions and arguments • We can translate arguments from sentences and propositions into symbolic logic form • “Simple statement”: does not contain any other statement as a component • “Charlie is neat” • “Compound statement”: does contain another statement as a component • “Charlie is neat and Charlie is sweet”**Conjunction**• Conjunction of two statements: “…and…” • Each statement is called a conjunct • “Charlie is neat” (conjunct 1) • “Charlie is sweet” (conjunct 2) • The symbol for conjunction is a dot • • (Can also be “&”) • p • q • P and q (2 conjuncts)**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 Values of Conjunction**• 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 • Therefore our symbol “•” (or “&”) is a truth-functional connective**Truth Table of Conjunction •**Given any two statements, p and q A conjunction is true if and only if both conjuncts are true**Abbreviation of Statements**• “Charlie’s neat and Charlie’s sweet.” • N • S • 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 • “Byron was a great poet and a great adventurer.” • P• A • “Lewis was a famous explorer and Clark was a famous explorer.” • L• C**“Jones entered the country at New York and went straight**to Chicago.” • “and” here does not signify a conjunction • Can’t say “Jones went straight to Chicago and entered the country at New York.” • Therefore cannot use the • here • Some other words that can signify conjunction: • But • Yet • Also • Still • However • Moreover • Nevertheless • (comma) • (semicolon)**Negation**• Negation: contradictory or denial of a statement • “not” • i.e. “It is not the case that…” • The symbol for negation is tilde ~ • If M=“All humans are mortal,” then • ~M=“It is not the case that all humans are mortal.” • ~M=“Some humans are not mortal.” • ~M=“Not all humans are mortal.” • ~M=“It is false that all humans are mortal.” • All these can be symbolized with ~M**Truth Table for Negation**Where p is any statement, its negation is ~p**Disjunction**• Disjunction of two statements: “…or…” • Symbol is “ v ” (wedge) (i.e. A v B = A or B) • Weak (inclusive) sense: can be either case, and possibly both • Ex. “Salad or dessert” (well, you can have both) • We will treat all disjunctions in this sense (unless a problem explicitly says otherwise) • Strong (exclusive) sense: one and only one • Ex. “A or B” (you can have A or B, at least onebut not both) • The two component statements so combined are called “disjuncts”**Disjunction Truth Table**A (weak) disjunction is false only in the case that both its disjuncts are false**Disjunction**• Translate:“You will do poorly on the exam unless you study.” • P=“You will do poorly on the exam.” • S=“You study.” • P v S • “Unless” = v**Punctuation**• As in mathematics, it is important to correctly punctuate logical parts of an argument • Ex. (2x3)+6 = 12 whereas 2x(3+6)= 18 • Ex. p • q v r (this is ambiguous) • To avoid ambiguity and make meaning clear • Make sure to order sets of parentheses when necessary: • Example: { A • [(B v C) • (C v D)] } • ~E • { [ ( ) ] }**Punctuation**• “Either Fillmore or Harding was the greatest American president.” • F v H • To say “Neither Fillmore nor Harding was the greatest American president.” (the negation of the first statement) • ~(F v H) OR (~F) • (~H)**Punctuation**• “Jamal and Derek will both not be elected.” • ~J • ~D • 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)]” • “Jamal and Derek both will not be elected.” • ~(J •D)**Example**• Rome is the capital of Italy or Rome is the capital of Spain. • I=“Rome is the capital of Italy” • S=“Rome is the capital of Spain” • 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 • When doing truth values, do the innermost conjunctions/disjunctions/negations first, working your way outwards**I v S**• We know that Rome is the capital of Italy and that Rome is not the capital of Spain. • So we know that “I” is True, and that “S” is False. We put these values directly under their corresponding letter • 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)**Note**• When doing truth values, do the innermost conjunctions/disjunctions/negations first, working your way outwards • Ex. Do ( ) first, then [ ], then finally { } • Homework: Page 309-310 Part I (try 5 of these)Page 310 Part II (try 10 of these)