1. SD: Natural Deduction In S

2. Valid or Not? 1.If Carol drives, Ann will go to the fair 2.Carol will drive, if Bob goes and pays for gas 3.Bob will pay for gas, if Ann promises to pay him back 4.If Bob lets Ann ride shotgun, she will promise to pay him back 5.Bob goes to the fair and lets Ann ride shotgun So, Ann goes to the fair

3. Deducing the Conclusion 1.If Carol drives, Ann will go to the fair 2.Carol will drive, if Bob goes and pays for gas 3.Bob will pay for gas if Ann promises to pay him back 4.If Bob lets Ann ride shotgun, she will promise to pay him back 5.Bob goes to the fair and lets Ann ride shotgun 6.Bob lets Ann ride shotgunfrom 5 7.Ann promises to pay Bob backfrom 4,6 8.Bob pays for gasfrom 3,7 9.Bob goes to the fairfrom 5 10.Bob goes and pays for gasfrom 8,9 11.Carol drivesfrom 2,10 12.Ann goes to the fairfrom 1,11

4. Symbolically Deriving the Conclusion 1.If Carol drives, Ann will go to the fair 2.Carol will drive, if Bob goes and pays for gas 3.Bob will pay for gas if Ann promises to pay him back 4.If Bob lets Ann ride shotgun, she will promise to pay him back 5.Bob goes to the fair and lets Ann ride shotgun 6.Bob lets Ann ride shotgunfrom 5 7.Ann promises to pay Bob back from 4,6 8.Bob pays for gasfrom 3,7 9.Bob goes to the fairfrom 5 10.Bob goes and pays for gasfrom 8,9 11.Carol drivesfrom 2,10 12.Ann goes to the fairfrom 1,11

5. Reiteration You may reiterate any wff on some later line The line number of the reiterated wff appears in the justification Remember that the metavariables represent wffs of arbitrary complexity—the reiterated wff need not be atomic Usually only useful in certain special circumstances The same input lines may be used repeatedly

6. Wedge Intro & Elim You may form the conjunction of any two previous wffs Order of input wffs is irrelevant The line numbers of the two input wffs are cited in the justification Remember that the metavariables represent wffs of arbitrary complexity—the input wffs need not be atomic The same input lines may be used repeatedly You may derive either conjunct of an exiting conjunction The line number of the conjunction is cited in the justification Again, remember that the metavariables indicate wffs of arbitrary complexity The same input lines may be used repeatedly

7. Arrow Intro & Elim From a conditional and a wff exactly matching the antecedent, you may derive the consequent Order of input wffs is irrelevant The line numbers of the two input wffs are cited in the justification Having started a subderivation with the auxiliary assumption P, and derived Q, you may end the subderivation and derive the conditional P ! Q Once the subderivation is closed, wffs inside it are inaccessible The whole subderivation is cited in the justification

8. Valid? 1.If Bob goes to the fair, then he’ll eat lots of popcorn 2.If Bob eats lots of popcorn, then if he rides the roller coaster he’ll throw up If Bob goes to the fair and rides the roller coaster, then he’ll throw up

9. Deducing the Conclusion 1.If Bob goes to the fair, then he’ll eat lots of popcorn 2.If Bob eats lots of popcorn, then if he rides the roller coaster he’ll throw up 3.Suppose Bob goes to the fair and rides the roller coasterassumption 4.Bob goes to the fairfrom 3 5.Bob eats lots of popcornfrom 1,4 6.If Bob rides the roller coaster he’ll throw upfrom 2,5 7.Bob rides the roller coasterfrom 3 8.Bob throws upfrom 6,7 9.If Bob goes to the fair and rides the roller coaster, then he’ll throw upfrom 3-8

10. Symbolically Deriving the Conclusion 1.If Bob goes to the fair, then he’ll eat lots of popcorn 2.If Bob eats lots of popcorn, then if he rides the roller coaster he’ll throw up 3.Suppose Bob goes to the fair and rides the roller coasterassumption 4.Bob goes to the fairfrom 3 5.Bob eats lots of popcornfrom 1,4 6.If Bob rides the roller coaster he’ll throw up from 2,5 7.Bob rides the roller coasterfrom 3 8.Bob throws upfrom 6,7 9.If Bob goes to the fair and rides the roller coaster, then he’ll throw upfrom 3-8

11. Anatomy of a Derivation Primary Assumptions indicated with a ‘P’ Goal wff indicated by a ‘ ` ’ Every derivation has a Main Scope Line Most derivations we’ll do will have primary assumptions attached to the main scope line by a Horizontal Justification for each step after the primary assumptions is listed to the right, with the proper citations of input line(s) and rule abbreviation Note that input lines may be used repeatedly The fact that the goal sentence is on the main scope line shows that it has been derived from the primary assumptions If a subderivation appears, it will have its own scope line and horizontal, any wff right of this line depends on the aux assumption—a subderivation only ever has one assumption Auxiliary Assumptions indicated with an ‘A’ This derivation is not as short as it could be—though not required, shorter derivations are better

12. Hook Intro & Elim Having started a subderivation with the auxiliary assumption P ( : P) and derived both Q and : Q (a contradiction), you may end the subderivation and derive : P (P) Note that neither the P nor the Q need be atomic In some cases P may be the same wff as Q

13. 1.If Bob goes to the fair, then Carol won’t 2.If Ann goes to the fair then Bob will 3.Carol will go to the fair only if Ann does Carol does not go to the fair Valid?

14. 1.If Bob goes to the fair, then Carol won’t 2.If Ann goes to the fair then Bob will 3.Carol will go to the fair only if Ann does 4.Suppose Carol does goassumption 5.Ann goes to the fairfrom 3,4 6.Bob goes to the fairfrom 2,5 7.Carol won’t go from 1,6 (Carol both goes and doesn’t go from 4,7) 8.So Carol doesn’t go to the fairfrom 4-7 Deducing the Conclusion

15. Vee Intro & Elim Any wff may be attached to an existing one with the vee This simple rule is often overlooked Having shown that each side of a disjunction leads to the same result, you may derive that result You must cite 3 things: the disjunction and two subderivations

16. 1.If either Ann or Bob goes to the fair, then Carol will go 2.Ann goes Carol goes to the fair Valid?

17. 1.If either Ann or Bob goes to the fair, then Carol will go 2.Ann goes 3.Either Ann or Bob goesfrom 2 4.Carol goes to the fairfrom 1,3 Deducing the Conclusion

18. 1.If Carol goes to the fair, then Ann will drive 2.If Bob goes to the fair, then he will pay for gas 3.If Bob pays for gas, then Ann will drive 4.Either Carol or Bob goes to the fair Ann will drive Valid?

19. 1.If Carol goes to the fair, then Ann will drive 2.If Bob goes to the fair, then he will pay for gas 3.If Bob pays for gas, then Ann will drive 4.Either Carol or Bob goes to the fair 5.Suppose Carol goesassn. 6.Ann will drivefrom 1,5 7.Suppose Bob goesassn. 8.Bob will pay for gasfrom 2,7 9.Ann will drivefrom 3,8 10.Ann will drive4,5-6,7-9 Deducing the Conclusion

20. Double Arrow Intro & Elim Essentially you are just doing two ! I Both Subderivations must be cited A biconditional wff and one side of it allow you to derive the other side This is much like ! E, but here you may go in either direction

21. Proof Theory in SD Derivation in SD: A derivation in SD is a finite sequence of wffs of S such that each wff is either an assumption with scope indicated or justified by one of the rules of SD. Derivable in SD,  ` P: A wff P of S is derivable in SD from a set  of wffs of S iff there is a derivation in SD the primary assumptions of which are members of  and P depends on only those assumptions.   PP is derivable from ,  syntactically entails P ?  P or  P P is derivable from the empty set, P is a theorem   PP is not derivable from  …

22. Validity, Theoremhood in SD Valid in SD: An argument of S is valid in SD iff the conclusion is derivable in SD from the set consisting of only the premises, otherwise it is invalid in SD. Theorem of SD: A wff P is a theorem of SD iff P is derivable in SD from the empty set; i.e., iff  P.

23. Equivalence and Inconsistency in SD Equivalent in SD: Two wffs P and Q are equivalent in SD iff they are interderivable in SD; i.e., iff both P  Q and Q  P. Inconsistent in SD: A set  of wffs is inconsistent in SD iff, for some wff P, both   P and   : P.

25. Proof Theory Summary Derivable in SD,   P: A wff P of S is derivable in SD from a set  of wffs of S iff there is a derivation in SD the primary assumptions of which are members of  and P depends on only those assumptions. Valid in SD: An argument of S is valid in SD iff the conclusion is derivable in SD from the set consisting of only the premises, otherwise it is invalid in SD. Theorem of SD: A wff P is a theorem of SD iff P is derivable in SD from the empty set; i.e., iff  P. Equivalent in SD: Two wffs P and Q are equivalent in SD iff they are interderivable in SD; i.e., iff both P  Q and Q  P. Inconsistent in SD: A set  of wffs is inconsistent in SD iff, for some wff P, both   P and   : P (this can be shown using a single derivation).

26. SDE—An Extension to SD The rules of SD, plus the following Inference Rules…

27. …plus the following Substitution Rules.

28. Inference Rules vs. Substitution Rules: Two important Differences Substitution rules are bidirectional, whereas inference rules are unidirectional. Whereas inference rules require that the wff on a given line (i.e., the whole wff) match the form in the rule, substitution rules require only that some well- formed component of a wff on a given line match the form in the rule. That is, whereas inference rules work only on whole lines, substitution rules work on whole lines or well-formed subcomponents.