Theorems and Shortcuts Kareem Khalifa Department of Philosophy Middlebury College.

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Theorems and Shortcuts Kareem Khalifa Department of Philosophy Middlebury College

Overview Why this matters Theorems Shortcuts Sample Exercises

Why this matters Shortcuts are shortcuts. –They often decrease the length and difficulty of proofs. Shortcuts (derived rules) provide us with a number of natural patterns of reasoning, e.g., disjunctive syllogism, modus tollens, etc. Shortcuts give us more moves to use. –This means that it’s harder to get stuck, but easier to get lost.

Theorems A tautology –Any argument that can be proven valid without any premises. We represent this as “├ p” Essentially, this says, “regardless of what the premises are, p will be true”

Example ├ P  (P v Q) How do we prove this? 1. |PH for  I 2. | P v Q1 vI 3. P  (PvQ)1,2  I

Shortcuts/Derived Rules The 10 basic rules suffice to prove any valid inference. However, it’s often a pain to prove even very intuitive inferences. So, we use derived rules (Nolt, ) to make our lives easier. –For example…

4.3.7 ~P & ~Q ├ ~(P v Q) DM Without Shortcut 1.~P & ~QA 2.| P v QH for ~I 3.|| PH for→I 4.|| |~QH for ~I 5.||| ~P1 &E 6.||| P & ~P3,5 &I 7.|| Q4-6 ~I 8.| P →Q3-7 →I 9.|| QH for→I 10.| Q→Q9-9 →I 11.| Q 2,8,10 vE 12.| ~Q1 &E 13.| Q & ~Q11, 12 &I 14.~(P v Q)2-13 ~I

4.3.7 With Shortcut 1.~P & ~Q A 2.~(P v Q) 1, DM

Sample Exercise ├ P  ~~P 1. |PH for  I 2. | ~~P1, DN 3.P  ~~P1-2  I

~~P v ~~QA  P v Q 2.~(~P & ~Q)1, DM 3.P v Q2, DM

├ ~(P v Q)  ~P 1. |~(P v Q)H for  I 2. |~P & ~Q1 DM 3. | ~P2 &E 4.~(P v Q)  ~P1-3  I

From Copi & Cohen 1.(E&F) v (G  H)A 2.I  GA 3.~(E&F)A IHIH 4.G  H1,3 DS 5.I  H2,4 HS

More from C&C 1.Q  (RvS)A 2.(T&U)  RA 3.(RvS)  (T&U)A  Q  R 4.Q  (T&U)1,3 HS 5.Q  R2,4 HS

More C&C 1.A v (B  A)A 2.~A&CA  ~B 3.~A2 &E 4.B  A1,3 DS 5.~B3,4 MT