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Going Formal

Going Formal. Meet the Connectives. The Language of Propositional Logic. Syntax (grammar, internal structure of the language) Vocabulary: grammatical categories Identifying Well-Formed Formulae ( “ WFFs ” ) Semantics (pertaining to meaning and truth value) Translation Truth functions

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Going Formal

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  1. Going Formal Meet the Connectives

  2. The Language of Propositional Logic • Syntax (grammar, internal structure of the language) • Vocabulary: grammatical categories • Identifying Well-Formed Formulae (“WFFs”) • Semantics (pertaining to meaning and truth value) • Translation • Truth functions • Truth tables for the connectives

  3. The Vocabulary of Propositional Logic • Sentence Letters: A, B, … Z • Connectives (“Sentence-Forming Operators”) ~ negation “not,”“it is not the case that” ⋅ conjunction “and” ∨ disjunction “or” (inclusive) ⊃ conditional “if – then,”“implies” ≣ biconditional “if and only if,”“iff” • “Parentheses”: (, ), [, ], {, and }

  4. Sentence Letters • Translate “atomic” sentences • Atomic sentences have no proper parts that are themselves sentences • Examples: • It is raining R • It is cold C

  5. Sentential Connectives • Connect to sentences to make new sentences • Negation attaches to one sentence • It is not raining ∼ R • Conjunction, disjunction, conditional and biconditional attach two sentences together • It is raining and it is cold R ∙ C • If it rains then it pours R ⊃ P

  6. Parentheses, brackets & braces • I’ll go to Amsterdam and Brussels or Calais • This is ambiguous and we can’t tolerate ambiguity! Brussels AND Amsterdam OR Calais Amsterdam Brussels OR AND Calais

  7. Parentheses, brackets & braces • Grouping devices avoid ambiguity (for “unique readability”): • I’ll go to Amsterdam, and then to either Brussels or CalaisA ∙ (B ∨ C) • I’ll either go to Amsterdam and Brussels, or else to Calais(A ∙ B) ∨ C Brussels AND Amsterdam OR Calais Amsterdam Brussels OR AND Calais

  8. Variables: p, q, … • Sometimes we want to talk about all sentences of a given form, e.g. A (BC) F (MX) (K  M)  [(N  O) P] • So we use variables as place-holders • Each of the above sentences is of the form: p (qr)

  9. Plugging into variables ModusPonens • Variables are like expandable boxes • To do proofs in logic you have to see how sentences plug into those boxes. Substitution Instance of Modus Ponens  p  q p q (D  (E  F)) ((A  B)  C) ((A  B)  C) (D  (E  F))

  10. Plugging into variables • Variables are like expandable boxes • To do proofs in logic you have to see how sentences plug into those boxes. ModusPonens Substitution Instance of Modus Ponens  pq p q ((A  B)  C) (D  (E  F)) ((A  B)  C) (D  (E  F))

  11. The Grammar of Propositional Logic • Constructing WFFs (Well-Formed Formulae) • Identifying WFFs • Identifying main connectives

  12. Rules for WFFs • A sentence letter by itself is a WFF A B Z • The result of putting  immediately in front of a WFF is a WFFA  B  B  (A  B)  ( C  D) • The result of putting  ,  ,  , or  between two WFFs and surrounding the whole thing with parentheses is a WFF (A  B) ( C  D) (( C  D)  (E  (F  G))) • Outside parentheses may be dropped A  B  C  D ( C  D)  (E  (F  G))

  13. WFFs • A sentence that can be constructed by applying the rules for constructing WFFs one at a time is a WFF • A sentence which can't be so constructed is not a WFF • No exceptions!!! woof

  14. Main Connective • In constructing a WFF, the connective that goes in last, which has the whole rest of the sentence in its scope, is the main connective. • This is the connective which is the “furthest out.” • Examples ( C  D) (E  (F  G)) ( C  D)

  15. Hints: When it’s not a WFF • You can't have two WFFs next to one another without a two-sided connective between them.BAD! AB C  D (E  F)G • Two-sided connectives have to have WFFs attached to both sides.BAD! A (B  C)  ( D  E) G  H • You can't have more than one two-sided connective at the same levelBAD! A  B  C ( C  D)  (E  F  G)

  16. Identifying WFFs & Main Connectives 1 (S  T)  ( U  W) 2  (K  L)  ( G  H) 3 (E  F)  (W  X) 4 (B  T)  ( C  U) 5 (F  Q)  (A  E  T) ∨ X X ≡ X

  17. Identifying WFFs & Main Connectives 1 (S  T)  ( U  W) X2  (K  L)  (G  H) X3 (E  F)  (W  X) 4 (B  T)  ( C  U) X5 (F  Q)  (A  E  T)

  18. Identifying WFFs & Main Connectives 6 D  [ ( P  Q)  (T  R) ] 7 [ (D  Q)  (P  E) ]  [A  (  H) ] 8 M (N  Q)  ( C  D) 9  (F  G)  [ (A  E)  H] 10 (R  S  T)  ( W  X) ∨ X X ⊃ X

  19. Identifying WFFs & Main Connectives 6  D  [ ( P  Q)  (T  R) ] X7 [ (D  Q)  (P  E) ]  [A  (  H) ] X8 M (N Q)  ( C  D) 9  (F  G)  [ (A  E)  H] X 10 (R  S  T)  ( W  X)

  20. Why should we care about this? • Because in formal logic we determine whether arguments are valid or not by reference to their form. • And that assumes we can identify the form of sentences, i.e. that we can identify main connectives. • In doing formal derivations in particular, we have be able to immediately see what the forms of sentences are in order to formulate strategies.

  21. Translation

  22. Conditionals & Biconditionals If P then Q P  Q P, if Q Q  P P only if Q P  Q P if and only if Q P  Q Note: A biconditional is a “conditional going both ways”: so P  Q is the conjunction of P  Q and Q  P

  23. Conditionals If P then Q P  Q P, if Q Q  P P only if Q P  Q 5 If Chanel has a rosewood fragrance then so does Lanvin. C  L 6 Chanel has a rosewood fragrance if Lanvin does. L  C 8 Reece Witherspoon wins best actress only if Martin Scorsese wins best director. W  S

  24. Biconditionals P if and only if Q P  Q 7 Maureen Dowd writes incisive editorials if and only if Paul Krugman does. D  K A biconditional is a “conditional going both ways”: so P  Q is the conjunction of P  Q and Q  P. “Only if” is only half of “if and only if.” Be careful!

  25. Not both and & neither/nor Not both P and Q ~ (P  Q) Neither P nor Q  (P  Q) You can’t both have your cake and eat it. ~ (H  E) She was neither young nor beautiful.  (Y  B)

  26. Not both and & neither/nor Not both P and Q ~ (P  Q) Neither P nor Q  (P  Q) 15 Not both Jaguar and Porsche make motorcycles. ~ (J  P) 16 Both Jaguar and Porsche do not make motorcycles. J  ~ P

  27. Not both and & neither/nor Not both P and Q ~ (P  Q) Neither P nor Q  (P  Q) 18 Not either Ferrari or Maserati makes economy cars.19 Neither Ferrari nor Maserati makes economy cars. (F  M) 20 Either Ferrari or Maserati does not make motorcycles. F  ~ M

  28. DeMorgan’s Laws ~ (P  Q) is equivalent to  P  Q  (P  Q) is equivalent to  P  Q “She was neither young nor beautiful” is equivalent to “She was old and ugly” - NOT“She was old or ugly.” “You can’t both have your cake and eat it” is equivalent to “You either don’t have your cake or you don’t eat your cake” - NOT“You don’t have your cake and you don’t eat your cake.”

  29. So, what do I need for the quiz? • Determining whether arguments are of the same form, identifying counterexamples, and understanding what that shows about validity and soundness • Identifying WFFs and main connectives • Translation: given an English sentence, which of the following symbolized sentences is the correct translation?

  30. The End WFF

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