Copyright 2008, Scott Gray1 Propositional Logic 2) The Formal Language

Copyright 2008, Scott Gray 2 Some Examples of Last Week’s Topics □Some mammals are winged A bat is a mammal ∴ a bat is winged □Which is to say: Some ■ are ● ♦ is ■ ∴ ♦ is ● □ Invalid : substitute dog for bat – a winged bat?

Copyright 2008, Scott Gray 3 More Examples □No Supreme Court justice is a woman Ruth Ginsberg is a woman ∴ Ruth Ginsberg is not a Supreme Court justice □Some ■ are ● ♦ is ● ∴ ♦ is ■ □ Valid : We went from falsity (premise #1) to truth; however, T, T -> F would be invalid

Copyright 2008, Scott Gray 4 More Examples, cont. □Some NFL players are major league baseball players Some major league baseball players make millions ∴ some NFL players make millions □Some ■ are ● some ● is ♦ ∴ some ■ is ♦ □ Invalid : can you come up with a counter example?

Copyright 2008, Scott Gray 5 More Examples, cont. □Some cats are tailless animals Tailless animals are amphibians ∴ some cats are amphibians □It may help to think of this in terms of a Venn diagram: NFLMLB y = has millions $ y y y y y y x = doesn’t have millions $ x x x

Copyright 2008, Scott Gray 6 More Examples, cont. □Some transfinite sets are non-denumerable sets All non-denumerable sets are demonstrated by the diagonalization principle ∴ some transfinite sets are demonstrated by the diagonalization principle □Can you symbolize this? □Valid or invalid?

Copyright 2008, Scott Gray 7 More Examples, cont. □Some ■ are ● all ● is ♦ ∴ some ■ is ♦ T N D N T N D

Copyright 2008, Scott Gray 8 Elements of the Formal Language □Proposition letters: A, B, C,…Z □Connectives: ~ & v → ↔ □Brackets: () □You can say the connectives this way: ~ = tilde & = ampersand v = wedge → = arrow ↔ = double arrow □Your book uses additional brackets ([] & {}), but they are (IMO) unnecessary

Copyright 2008, Scott Gray 9 Dictionary □Sometime (usually?) called an interpretation □For instance, the dictionary will tell us that W = John won’t work □When first starting out, you should be explicit in your dictionary

Copyright 2008, Scott Gray 10 Notes on Connectives □ & – this approximates ( ≈ ) the English “and”; the letters can be flip-flopped (they are commutative) □ v – ≈ “or”; inclusive, meaning “possibly both” (exclusive means “not both”); we will treat this as commutative also □ ~ – ≈ “not” □ → – ≈ “if…then”; no sense of causality, nor A temporal proceeds B ; read “implies”

Copyright 2008, Scott Gray 11 Translating to Connectives □ A if & only if B == (B → A) & (A → B) ; this is the same as A ↔ B or B ↔ A □ A only if B == A → B □ A if B == B → A □The logical content of “but” is usually “and” □“Neither…nor” == ~(A v B) == (~A & ~B)

Copyright 2008, Scott Gray 12 Translating to Connectives, cont. □ ~A → ~B == A → B □necessary condition on the right of the → ; sufficient condition on the left □“only if” == necessary, not sufficient

Copyright 2008, Scott Gray 13 Translating to Connectives, cont. □“Unless” can be used in a weak or strong sense: A unless B == ~B → A A unless B == (A → ~B) & (B → ~A) We will use the weak sense □“Provided” == A provided B == B → A

Copyright 2008, Scott Gray 14 Additional Descriptions of Connectives □ ~ = negation □ & = conjunction □ v = disjunction □ → = conditional □ ↔ = bi-conditional conjuncts A & B disjuncts A v B antecedent A → B consequent

Copyright 2008, Scott Gray 15 Truth-Functional Sentence Connectives □Sentence connective = a symbol in a language such that by juxtaposing it in some appropriate way with some appropriate number of sentences of the language a new sentence of language is formed

Copyright 2008, Scott Gray 16 Truth-Functional Sentence Connectives, cont. □Truth-functional = a sentence connective is truth functional if and only if the value of the sentence it forms depends entirely upon the truth value of the component sentences

Copyright 2008, Scott Gray 17 Truth-Functional Sentence Connectives, cont. □AND: A & BA B T T F T T F F F T F F F □AND is truth-functional

Copyright 2008, Scott Gray 18 Truth-Functional Sentence Connectives, cont. □“Because” is not truth functional □Example: “Taft was fat and Taft didn’t exercise” □Both parts are true and the entire statement is true □But: “Taft was fat because Taft didn’t exercise” □Both parts are true, but “because” is not truth functional – we don’t cover this

Copyright 2008, Scott Gray 19 Assignments □Learn the connectives □Begin to collect translations (we have listed a few in this lesson) □Collect 10 truth-functional sentences and 10 non-truth-functional sentences from your reading □Be able to identify truth-functional and non-truth-functional sentences