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Propositional Logic

Propositional Logic. 6.2 Truth Functions. Truth Functions. Truth functions for the tilde:. Plug in truth. Out comes falsehood. Out comes truth. Plug in falsehood. Truth Functions. Truth functions for the dot:. Let p = Jack went up the hill. Let q = Jill went up the hill.

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Propositional Logic

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  1. Propositional Logic 6.2 Truth Functions

  2. Truth Functions Truth functions for the tilde: Plug in truth Out comes falsehood Out comes truth Plug in falsehood

  3. Truth Functions Truth functions for the dot: Let p = Jack went up the hill. Let q = Jill went up the hill. If both of them actually went up the hill, what should we say about the sentence, p • q? T If Jack went but Jill didn’t, what should we say about the sentence, p • q? F If Jack didn’t go but Jill did, what should we say about the sentence, p • q? F If neither of them went, what should we say about the sentence, p • q? F

  4. Truth Functions Truth functions for the wedge: Let p = Jack went up the hill. Let q = Jill went up the hill. If both of them actually went up the hill, what should we say about the sentence, p v q? T If Jack went but Jill didn’t, what should we say about the sentence, p v q? T If Jack didn’t go but Jill did, what should we say about the sentence, p v q? T If neither of them went, what should we say about the sentence, p v q? F

  5. Truth Functions Truth functions for the horseshoe (arrow): Let p = Jack went up the hill. Let q = Jill went up the hill. If both of them actually went up the hill, what should we say about the sentence, p → q? T If Jack went but Jill didn’t, what should we say about the sentence, p → q? F If Jack didn’t go but Jill did, what should we say about the sentence, p → q? T If neither of them went, what should we say about the sentence, p → q? T

  6. Truth Functions Truth functions for the triple bar: Let p = Jack went up the hill. Let q = Jill went up the hill. If both of them actually went up the hill, what should we say about the sentence, p Ξ q? T If Jack went but Jill didn’t, what should we say about the sentence, p Ξ q? F If Jack didn’t go but Jill did, what should we say about the sentence, p Ξ q? F If neither of them went, what should we say about the sentence, p Ξ q? T

  7. Truth Functions If the truth table for the horseshoe bothers you, just translate it to this: ~p v q So, saying to a troublemaker in the bar: If you stay, I’ll flatten you (S  F) Is the same as saying Leave or I’ll flatten you (~S v F)

  8. Computing Truth Values of Big Propositions True: A, B, and C False: D, E, and F What’s the truth value of … (A v D)  E ?

  9. Computing Truth Values of Big Propositions True: A, B, and C False: D, E, and F (A v D)  E (T v F)  F (put in the truth values) T  F (simplify from truth table) F

  10. Computing Truth Values of Big Propositions True: A, B, and C False: D, E, and F (B • C)  (E  A) (T • T)  (F  T) (put in the truth values) T  T (simplify from truth table) T

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