1 / 27

INTRO LOGIC

INTRO LOGIC. Derivations in SL 2. DAY 10. Review. We demonstrate (show) that an argument is valid by deriving ( deducing ) its conclusion from its premises using a few fundamental modes of reasoning . Initial Modes of Reasoning.

shlomo
Download Presentation

INTRO LOGIC

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. INTRO LOGIC Derivations in SL2 DAY 10

  2. Review We demonstrate (show) that an argument is valid by deriving (deducing) its conclusion from its premises using a few fundamental modes of reasoning.

  3. Initial Modes of Reasoning Modus (Ponendo) Ponens(affirming by affirming) –––––– Modus (Tollendo) Tollens(denying by denying)–––––– Modus Tollendo Ponens (1)(affirming by denying)–––––– Modus Tollendo Ponens (2)(affirming by denying)–––––– aka disjunctive syllogism

  4. Example 1 (review) S ; RS ; RT ; PT ; PQ / Q 1. write all premises (1) S Pr (2) R  S Pr (3) R T Pr 2. write ‘show’ conclusion (4) P  T Pr (5) P Q Pr 3. apply rules to available lines (6) : Q DD (7) R 1,2, MT (8) T 3,7, MTP1 4. box and cancel(high-five!) (9) P 4,8, MTP2 (10) Q 5,9, MP

  5. other high-fives Tek gives Manny a high-five Tek gives A-Rod a high-five

  6. Rule Sheet provided on exams don’t makeupyourownrules available on courseweb page(textbook) keep this in front of you when doing homework

  7. Rules discussed Today

  8. Rules of Inference – Basic Idea • Every connective has • (1) a take-out rule • also called an: • elimination-rule • OUT-rule • (2) a put-in rule • also called an: • introduction-rule • IN-rule

  9. if you have a conjunction & then you are entitled to infer –––––– its first conjunct  Ampersand-Out (&O) if you have a conjunction & then you are entitled to infer –––––– its second conjunct  ‘have’ means ‘have as a whole line’ rules apply only to whole linesnot pieces of lines

  10. if you have a formula  and you have a formula  then you are entitled to infer –––––– their (first) conjunction& Ampersand-IN (&I) if you have a formula  and you have a formula  then you are entitled to infer ––––––their (second) conjunction &

  11. if you have a disjunction and you have the negation of its 1st disjunct  then you are entitled to infer –––––– its second disjunct  Wedge-Out (O) if you have a disjunction and you have the negation of its 2nd disjunct  then you are entitled to infer –––––– its first disjunct  what we earlier called ‘modus tollendo ponens’

  12. if you have a formula  then you are entitled to infer –––––– its disjunction with any formula to its right  Wedge-IN (I) if you have a formula  then you are entitled to infer –––––– its disjunction with any formula to its left  

  13. if you have a conditional and you have its antecedent  then you are entitled to infer –––––– its consequent  Arrow-Out (O) if you have a conditional and you have the negation of its consequent  then you are entitled to infer –––––– the negation of its antecedent  What we earlier called ‘modus ponens’ and ‘modus tollens’

  14. THERE IS NO SUCH THING ASARROW-IN (I)  Arrow-Introduction what we have instead is CONDITIONAL DERIVATION (CD)  which we examine later

  15. Double-Negation (DN) • if you have a formula  • then you are entitled to infer ––––– • its double-negative  if you have a double-negative  then you are entitled to infer ––––– the formula  rules apply only to whole linesnot pieces of lines

  16. Direct Derivation The Original and Fundamental -Rule • :  • ° • ° • ° •  DD In Direct Derivation (DD),one directly arrives at the very formula one is trying to show.

  17. Example 2 S;R  S;R T;P  T;P Q/Q (1) S Pr (2) R  S Pr (3) R T Pr (4) P  T Pr (5) P Q Pr (6) : Q DD (7) R 1,2, O (8) T 3,7, O (9) P 4,8, O (10) Q 5,9, O

  18. Arrow-Out Strategy • If you have a line of the form , • then try to apply arrow-out (O), • which requires a second formula as input, • in particular, either  or . have    find or deduce  

  19. Example 3 S;R  S;R T;P  T;P Q/Q (1) S Pr (2) R  S Pr (3) R T Pr (4) P  T Pr (5) P Q Pr (6) : Q DD (7) R 1,2, O (8) T 3,7, O (9) P 4,8, O (10) Q 5,9, O

  20. Wedge-Out Strategy • If you have a line of the form , • then try to apply wedge-out (O), • which requires a second formula as input, • in particular, either  or . have    find or deduce  

  21. Example 4 (P  Q)  R;[(P  Q)  R] R;(P  Q)  (Q  R)/Q (1) (P  Q)  R Pr (2) [(P  Q)  R] R Pr (3) (P  Q)  (Q  R) Pr (4) : Q DD (5) R 1,2, O (6) P  Q 1,5, O (7) Q  R 3,6, O (8) R 5,7, O

  22. Example 5 (P  Q) S ; (R  S) T) ; P /T (1) (P  Q) S Pr (2) (RS) T Pr (3) P Pr (4) : T DD (5) P  Q 3, I (6) S 1,5, O (7) R  S 6, I (8) T 2,7, O

  23. Wedge-In Strategy • If you need •  • then look for either disjunct: • find  • OR • find  • then • apply vI • to get 

  24. Example 6 P  Q ; R  (Q  S) ; R  T ;T & P / Q & S (1) P  Q Pr (2) R  (Q  S) Pr (3) R  T Pr (4) T & P Pr (5) Q & S DD (6) T 4,&O (7) P (8) Q 1,7, O (9) R 3,6, O (10) Q  S 2,9, O (11) S 8,10, O (12) Q & S 8,11, &I

  25. Ampersand-Out Strategy • If you have a line of the form • &, • then apply ampersand-out (&O), • which can be applied immediatelyto produce •  • and • 

  26. Ampersand-In Strategy • If you are trying to find or show • & • then look for both conjuncts: • find  • AND • find  • then • apply &I • to get &

  27. THE END

More Related