SDSD: Natural Deduction In : Natural Deduction In SS

Valid or Not?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

Deducing the ConclusionDeducing the Conclusion1. If Carol drives, Ann will go to the fair2. Carol will drive, if Bob goes and pays for gas3. Bob will pay for gas if Ann promises to pay him back4. If Bob lets Ann ride shotgun, she will promise to pay him back5. Bob goes to the fair and lets Ann ride shotgun6. Bob lets Ann ride shotgun from 57. Ann promises to pay Bob back from 4,68. Bob pays for gas from 3,79. Bob goes to the fair from 510. Bob goes and pays for gas from 8,911. Carol drives from 2,1012. Ann goes to the fair from 1,11

Symbolically Deriving the Symbolically Deriving the ConclusionConclusion

1. If Carol drives, Ann will go to the fair2. Carol will drive, if Bob goes and pays for

gas3. Bob will pay for gas if Ann promises to

pay him back4. If Bob lets Ann ride shotgun, she will

promise to pay him back5. Bob goes to the fair and lets Ann ride

shotgun6. Bob lets Ann ride shotgun from 57. Ann promises to pay Bob back from 4,68. Bob pays for gas from 3,79. Bob goes to the fair from 510. Bob goes and pays for gas from 8,911. Carol drives from 2,1012. Ann goes to the fair from 1,11

ReiterationReiteration

• 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

Wedge Intro & ElimWedge 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

Arrow Intro & ElimArrow 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

Valid?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

Deducing the ConclusionDeducing the Conclusion

1. If Bob goes to the fair, then he’ll eat lots of popcorn2. If Bob eats lots of popcorn, then if he rides the roller

coaster he’ll throw up3. Suppose Bob goes to the fair

and rides the roller coaster assumption4. Bob goes to the fair from 35. Bob eats lots of popcorn from 1,46. If Bob rides the roller coaster he’ll throw up from 2,57. Bob rides the roller coaster from 38. Bob throws up from 6,7

9. If Bob goes to the fair and rides the roller coaster, then he’ll throw up from 3-8

Symbolically Deriving the Symbolically Deriving the ConclusionConclusion

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 fair from 3

5. Bob eats lots of popcorn from 1,4

6. If Bob rides the roller coaster he’ll throw upfrom 2,5

7. Bob rides the roller coaster from 3

8. Bob throws up from 6,7

9. If Bob goes to the fair and rides the roller coaster, then he’ll throw up from 3-8

Anatomy of a DerivationAnatomy of a DerivationPrimary 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

Hook Intro & ElimHook 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

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?Valid?

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 does4. Suppose Carol does go

assumption

5. Ann goes to the fair from 3,4

6. Bob goes to the fair from 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 fair from 4-7

Deducing the ConclusionDeducing the Conclusion

Vee Intro & ElimVee 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

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

2. Ann goes

Carol goes to the fair

Valid?Valid?

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

2. Ann goes

3. Either Ann or Bob goes from 2

4. Carol goes to the fair from 1,3

Deducing the ConclusionDeducing the Conclusion

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?Valid?

1. If Carol goes to the fair, then Ann will drive2. If Bob goes to the fair, then he will pay for gas3. If Bob pays for gas, then Ann will drive4. Either Carol or Bob goes to the fair

5. Suppose Carol goes assn.6. Ann will drive from 1,5

7. Suppose Bob goes assn.8. Bob will pay for gas from 2,79. Ann will drive from 3,8

10. Ann will drive 4,5-6,7-9

Deducing the ConclusionDeducing the Conclusion

Double Arrow Intro & ElimDouble 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

Proof Theory in Proof Theory in SDSDDerivation 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.

P P is derivable from , syntactically entails P? P or P P is derivable from the empty set, P is a theorem

P P is not derivable from …

Validity, Theoremhood in Validity, Theoremhood in SDSD

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.

Equivalence and Inconsistency in Equivalence and Inconsistency in SDSD

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.

Proof Theory SummaryProof Theory SummaryDerivable 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).

SDESDE—An Extension to —An Extension to SDSD

The rules of SD, plus the following Inference Rules…

…plus the following Substitution Rules.

Inference Rules vs. Substitution Inference Rules vs. Substitution Rules: Two important DifferencesRules: 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.