6.1 Symbols And Translation

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Course lecture I developed over section 6.1 of Patrick Hurley\'s "A Concise Introduction to Logic".

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6.1 Symbols And Translation

  1. 1. 6.1 Symbols and translation
  2. 2. Formalizing logic <ul><li>We’re going to start diagramming statements now using logical operators (symbols). </li></ul><ul><li>We’ll be using these symbols together with different variables (S, P, etc.) to mean different things. </li></ul>
  3. 3. Different statements <ul><li>Different possible statements </li></ul><ul><ul><li>It is not the case that A. (A could mean “people are happy” or “dogs are animals”.) </li></ul></ul><ul><ul><li>D and C. (D could mean “people are happy” and “dogs are not people”, or any kind of combination.) </li></ul></ul><ul><ul><li>Either P or E. </li></ul></ul><ul><ul><li>If W then F. </li></ul></ul><ul><ul><li>B if and only if R. </li></ul></ul>
  4. 4. Five logical symbols <ul><li>~ (Tilde) </li></ul><ul><ul><li>Symbolizes negation. </li></ul></ul><ul><ul><li>Translates statements that say “not X”, or “it is not the case that X”. </li></ul></ul><ul><ul><li>Example: </li></ul></ul><ul><ul><ul><li>~A </li></ul></ul></ul><ul><ul><ul><ul><li>This means it is not the case that A. </li></ul></ul></ul></ul><ul><ul><ul><li>~S </li></ul></ul></ul><ul><ul><ul><ul><li>There is not S. </li></ul></ul></ul></ul>
  5. 5. Logical symbols, continued <ul><li>· (dot) </li></ul><ul><ul><li>Indicates a conjunction of two things. </li></ul></ul><ul><ul><li>And, but. </li></ul></ul><ul><ul><li>Example: </li></ul></ul><ul><ul><ul><li>S · P </li></ul></ul></ul><ul><ul><ul><ul><li>There is both S and P </li></ul></ul></ul></ul><ul><ul><ul><ul><ul><li>There are dogs and cats. </li></ul></ul></ul></ul></ul><ul><ul><ul><li>X · Y </li></ul></ul></ul><ul><ul><ul><ul><li>It is true that both X and Y. </li></ul></ul></ul></ul><ul><ul><ul><ul><ul><li>There are planets and moons. </li></ul></ul></ul></ul></ul>
  6. 6. Logical symbols, continued <ul><li>v (wedge) </li></ul><ul><ul><li>Disjunctive, it means either one thing or the other. </li></ul></ul><ul><ul><li>Or, unless. </li></ul></ul><ul><ul><li>P v Q </li></ul></ul><ul><ul><ul><li>Either we have P or we have Q. </li></ul></ul></ul><ul><ul><ul><ul><li>Either we’re having fish or chicken for dinner. </li></ul></ul></ul></ul><ul><ul><li>S v P </li></ul></ul><ul><ul><ul><li>There is either S or there is P. </li></ul></ul></ul><ul><ul><ul><ul><li>There are either happy students or sad students. </li></ul></ul></ul></ul>
  7. 7. Logical symbols, continued <ul><li>> (horseshoe) </li></ul><ul><ul><li>Implication (Conditional statement) </li></ul></ul><ul><ul><li>If/then, only if, implies. </li></ul></ul><ul><ul><ul><li>This symbol means “If one thing, then another”. </li></ul></ul></ul><ul><ul><ul><li>Example: </li></ul></ul></ul><ul><ul><ul><ul><li>P > Q </li></ul></ul></ul></ul><ul><ul><ul><ul><li>If P is true, then Q is true. </li></ul></ul></ul></ul><ul><ul><ul><ul><li>P entails Q. </li></ul></ul></ul></ul><ul><ul><ul><ul><ul><li>If there is no more liquor, then I’ll be very upset. </li></ul></ul></ul></ul></ul><ul><ul><ul><ul><li>S > P </li></ul></ul></ul></ul><ul><ul><ul><ul><li>If S is true, then P is true. </li></ul></ul></ul></ul><ul><ul><ul><ul><li>S entails P. </li></ul></ul></ul></ul><ul><ul><ul><ul><ul><li>Lots of friends entails a good social life. </li></ul></ul></ul></ul></ul>
  8. 8. Formulas not to be confused <ul><li>A if B </li></ul><ul><ul><li>B > A </li></ul></ul><ul><li>A only if B </li></ul><ul><ul><li>A > B </li></ul></ul><ul><li>A if and only if B </li></ul><ul><ul><li>A = B </li></ul></ul>
  9. 9. Logical symbols, continued <ul><li>= (Triple bar) </li></ul><ul><ul><li>Indicates equivalence (biconditional) </li></ul></ul><ul><ul><li>Translates the statement “if and only if” </li></ul></ul><ul><ul><li>Example: </li></ul></ul><ul><ul><ul><li>A = B </li></ul></ul></ul><ul><ul><ul><li>(A > B) · (B > A) </li></ul></ul></ul><ul><ul><ul><ul><li>If A then B, and if B then A. </li></ul></ul></ul></ul><ul><ul><ul><ul><li>A entails B and B entails A. </li></ul></ul></ul></ul><ul><ul><ul><ul><ul><li>If there is no poverty then people will be happy. And if people are happy, then it implies there is no poverty. </li></ul></ul></ul></ul></ul>
  10. 10. More complex formulas <ul><li>~M v P </li></ul><ul><ul><li>Either there is not M, or there is P. </li></ul></ul><ul><ul><ul><li>Either we have no food, or there are happy people. </li></ul></ul></ul><ul><li>(A v B) · (C v D) </li></ul><ul><ul><li>There is either A or B, and either C or D. </li></ul></ul><ul><ul><ul><li>People are either poor or wealthy, and people are either happy or sad. </li></ul></ul></ul><ul><li>(A > B) v (C > D) </li></ul><ul><ul><li>Either A entails B or C entails D. </li></ul></ul><ul><ul><ul><li>Either a good economy entails lots of jobs or a poor job market entails unhappy workers. </li></ul></ul></ul>
  11. 11. Complex formulas, continued <ul><li>Not both A and B. </li></ul><ul><ul><li>~ (A · B) </li></ul></ul><ul><ul><ul><li>Can also means ~A v ~B (Either there is not A, or there is not B) </li></ul></ul></ul><ul><li>Both not A and not B. </li></ul><ul><ul><li>~A · ~B </li></ul></ul><ul><ul><ul><li>There is not A and there is not B. </li></ul></ul></ul>

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