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Understanding Truth Functions in Propositional Logic: Tilde, Dot, Wedge, Arrow, and Triple Bar

This guide explores truth functions in propositional logic, focusing on the operations of conjunction (•), disjunction (v), implication (→), and biconditional (Ξ). Alongside examples involving Jack and Jill's adventures up the hill, this resource clarifies how different combinations produce truth values. The truth tables for each logical operation demonstrate when a proposition is true or false based on the truth of the individual statements. By understanding these principles, readers can enhance their comprehension of logical reasoning and implications in larger propositions.

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Understanding Truth Functions in Propositional Logic: Tilde, Dot, Wedge, Arrow, and Triple Bar

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Presentation Transcript

  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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