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INTRO LOGIC

INTRO LOGIC

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INTRO LOGIC

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  1. INTRO LOGIC DAY 03

  2. Schedule for Unit 1 warm-up 40% of Exam 1 60% of Exam 1

  3. Chapter 2 Sentential Logic

  4. Review • An argument is valid or invalidpurely in virtue of its form. Form is a function of the arrangement of the terms in the argument, where theLOGICAL TERMS play a primary role.

  5. Classical Syllogistic Logic • Logical terms Example Arguments all some no are not all X are Yall Y are Z/ all X are Z all X are Yno Y are Z/ no X are Z all X are Ysome X arenot Z/ some Y arenot Z

  6. Sentential Logic • In sentential logic • the logical terms are statement connectives

  7. What is a Statement Connective? • A statement connective(or simply, a connective) • is an "incomplete" expression – • i.e., an expression with one or more blanks – • such that, • whenever the blanks are filled by statements, • the resulting expression is also a statement. statement1 statement2 statement3

  8. Example 1

  9. 1-Place, 2-Place, … • a 1-place connective has 1 blank • a 2-place connective has 2 blanks • a 3-place connective has 3 blanks • etc.

  10. Examples – 1-place

  11. Examples – 2-place

  12. Examples – 3-place

  13. Atoms and Molecules • A compound (molecular)statement isone that is constructed from one or more smaller statements by the application of a statement connective. A simple (atomic)statement isone that is not constructed out of smaller statements by the application of a statement connective.

  14. A Simplification • Intro Logic is not concerned • with all connectives, • but only special ones – namely… truth-functional connectives

  15. Truth-Values • the truth-value of a true statement is T the truth-value of a false statement is F

  16. Truth-Functional • To say that a connective istruth-functional is to say that • the truth-value of any compound statementproduced by that connective • is afunction of the truth-values • of its immediate parts. the whole is merely the sum of its parts

  17. Abbreviation Scheme • 1. atomic sentences are abbreviated by upper-case letters (of the Roman alphabet) 2. connectives are abbreviated by special symbols (logograms) 3. compound sentences are abbreviated by algebraic-combinations of 1 and 2

  18. Example 1 – Conjunction expression abbreviation it is raining R it is sleeting S and & it is raining and it is sleeting ( R & S )

  19. Terminology • The symbol ‘&’ is called ampersand, • which is a stylized way of writing • the Latin word ‘et’, • which means “and”. & &&&& &

  20. Terminology (cont) the word ‘ampersand’ is a children’s pronunciation of the original word and per se and R&S is called the conjunction of R and S. R and S are individually called conjuncts.

  21. Conjunction is truth-functional R S R&S case 1 T T T case 2 T F F case 3 F T F case 4 F F F

  22. Slogan • A conjunction & is trueif and only if both conjuncts  and  are true. A conjunction & is true if both conjuncts  and  are true; otherwise, it is false.

  23. Example 2 – Disjunction (‘or’) expression abbreviation it is raining R it is sleeting S or  it is raining or it is sleeting ( R  S )

  24. Terminology • The symbol ‘’ is called wedge, • which is a stylized way of writing the letter ‘v’, • which initializes the Latin word ‘vel’, • which means “or”. RS is called the disjunction of R and S. R and S are individually called disjuncts.

  25. Exclusive Sense vs. Inclusive Sense • would you like soup,OR salad? would you like a baked potato,OR French fries? would you like coffee or dessert? would you like cream or sugar?

  26. Exclusive ‘or’ vs. Inclusive ‘or’ • exclusive ‘or’ soup OR salad • inclusive ‘or’ cream or sugar • Latin has two words: • ‘aut’ is exclusive ‘or’ • ‘vel’ is inclusive ‘or’ Legalistic English has the word ‘and/or’ Logic concentrates on inclusive ‘or’.

  27. Disjunction is truth-functional R S RS case 1 T T T case 2 T F T case 3 F T T case 4 F F F inclusive ‘or’

  28. Slogan • A disjunction  is trueif and only if at least one disjunct  or  is true. A disjunction  is false if both disjuncts  and  are false; otherwise, it is true.

  29. a Connective that is not Truth-Functional R S R because S S because R T T ??? ??? T F F F F T F F F F F F merely knowing that R and S are both true tells us nothing about whether one is responsible for the other

  30. Example 3 – Negation (‘not’) expression abbreviation it is raining R not  it is notraining R

  31. Terminology • The symbol ‘’ is called “tilde” • (as in ‘matilda’); • which is a highly stylized way of writing the letter ‘N’, • which is short for ‘not’.

  32. Negation is truth-functional • if R is true, then R is false • if R is false, then R is true R and R have opposite truth-values

  33. Example 4 – ‘if...then...’ my car runs out of gas R my car stops S if… then…  ifmy car runs out of gas, then my car stops ( R  S ) ifmy car stops,then my car runs out of gas ( S  R ) RS is not equivalent to SR.

  34. Terminology AC is called a conditional(of A and C). A is called the antecedent. C is called the consequent. ifantecedent, thenconsequent

  35. Aside • the prefix ‘ante’ means ‘before’ other words that contain ‘ante’ ante antechamber antediluvian antebellum ante meridian (a.m.) antipasto (Italian form)

  36. Non-Truth-Functional ‘If-Then’

  37. NOT TRUTH-FUNCTIONAL!

  38. Truth-Functional ‘If-Then’

  39. Truth-Functional version of ‘if-then’ R S RS case 1 T T T case 2 T F F case 3 F T T case 4 F F T true by “default”

  40. The Oddness of Cases 3 and 4 • If you promise to shut the windowsIF it rains, then only one scenario (case) constitutes breaking your promise – • the scenario in which it rains but you don’t shut the windows. In case 3 and case 4, you keep your promise "by default".

  41. THE END